Design of viscoelastic damping for vibration and noise control ...

Université Libre de Bruxelles
Faculté des Sciences Appliquées
Université Libre de Bruxelles
Structural and Material Computational Mechanics department
Structural and Material Computational
Mechanics department
Design of viscoelastic damping
for vibration and noise control:
modelling, experiments
and optimisation
Laurent Hazard
Committee in charge:
Laurent Hazard
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Design of viscoelastic damping for
vibration
Reporters
and noise control: modellin
experiments and optimisation
Thèse originale présentée en vue de l’obtention du grade de
Docteur Chairman en Sciences Appliquées
Examiners
Année académique 2006-2007
Advisor
- Almost the last version ...
Promoteur:
Prof. Philippe BOUILLARD
Co-promoteur: Prof. Jean-Louis MIGEOT
December 9, 2006

Laurent Hazard 21/12/2006
Laurent Hazard
Design of viscoelastic damping for
vibration and noise control: modelling,
experiments and optimisation
December 21, 2006
Université Libre de Bruxelles

Laurent Hazard 21/12/2006

Laurent Hazard 21/12/2006
Acknowledgements
A PhD thesis is a very personal work but, at the same time, involves so many
people that I find of capital importance to acknowledge their contributions.
First of all, this research work would not have been possible without
the support, motivation and strong faith in the project of a few people: Dr.
Jean-Yves Sener (ex. Arcelor), Prof. Jean-Louis Migeot (ULB, FFT) and
Prof. Philippe Bouillard (ULB). Thank you all for making it happen...
I am grateful to both the Arcelor group and the Région Wallonne for
sharing the financial support of this thesis. I would particularly like to acknowledge
Mr Raymond Montfort (DGTRE), for showing a real interest in
the project and always asking pertinent questions during meetings.
I would like to thank all the people at Arcelor Liège who where involved
in this research: Dr. Jean-Yves Sener, Emmanuel Bortolloti, Muriel Chaidron
and Jacques Mignon.
I am deeply grateful to the Structural and Material Computational Mechanics
department of the Université Libre de Bruxelles, for allowing me to
make my comeback to the academic world after many years of industrial wanderings.
I especially thank Prof. Guy Warzée, head of the department, for
his kindness and patience. I would like to thank Prof. Philippe Bouillard for
his interest in the project from the start and for his personal involvement as
thesis advisor. I would also like to express my gratitude to Prof. Jean-Louis
Migeot for his support and for accepting the role of co-advisor.
I thank Guy-Michel Hustinx and Philippe Lemaire from OPTRION SA,
for the help they gave me during the setup of the experimental measurements.
I am also grateful to Dr. Ezio Gandin from the Solvay Central Laboratory, for
his valuable recommendations concerning the characterisation of viscoelastic
materials.
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Acknowledgements
I am, of course, indebted to all my collegues and friends, past and present,
at the ULB for making this short moment in my life a great and unforgettable
journey. I would like to thank Tanguy, Geneviève, Katy, Yannick, Kfir,
Benoit,Louise,Berta,Adama,Peter,Vincent, Erik, Nathalie, Guy and others
that I apologise for not naming.
A special “Thank you !” to Guy Paulus for his great support in both hard
skills (such as L A TEX or UNIX) and soft skills (such as english grammar).
He contributed immensely to the readability of this dissertation and it was
always a pleasure to share long discussions with him about physics, politics
and other “-ics” .
I would also like to thank all my friends for their support and encouragements.
Laurent, Carine, Philippe, Cécile, Eric, Vincent: I am lucky to know
you all and that you still like to spend time with me, even if I am obsessed
with physics, mathematics and punk-rock music...
Finally, I would like to express a very warm thanks to my family. My
parents have always supported me in all my adventures, even when I decided
to start this PhD research. I know that I can always rely on them and I am
proud of the education and values they gave me.
Last but not least, I want to thank Julie for her constant support and
Nathanäel, simply for being the most wonderful little boy I know. We shared
together good and hard times during these last years. Too often, I had so
many things to do and so little time that it all perturbed our life as a family.
Julie handled these moments with care and patience and I am immensely
grateful to her for all she has done and continues to do...
Laurent Hazard,
Brussels, December 2006.
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Contents
1 INTRODUCTION ........................................ 1
2 VISCOELASTIC MATERIALS ........................... 5
2.1 Viscoelasticmaterials.................................... 8
2.2 Linearviscoelasticity andcomplexmodulus................. 10
2.2.1 Dynamicloading................................... 11
2.3 Selectionofthematerial ................................. 12
2.3.1 Themethodofreducedvariables(RVM) .............. 12
2.3.2 Experimental determination of material data: the
ISD112case....................................... 14
2.3.3 RepresentationoftheComplex Modulus .............. 15
2.3.4 Manufacturer’sdata:whatarenomograms........... 19
2.4 Summary .............................................. 20
3 INTRODUCTION TO THE PARTITION OF UNITY
METHOD ................................................ 21
3.1 Modelproblem ......................................... 22
3.1.1 Variationalformulation............................. 23
3.2 Galerkinmethod........................................ 23
3.3 Finiteelementmethod................................... 24
3.4 Partitionofunityfiniteelementmethod.................... 26
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3.5 Backtotheonedimensional modelproblem................. 30
3.5.1 Lineardependencies................................ 32
3.6 Twodimensionalapplication .............................. 37
3.7 Summary .............................................. 42
4 PUFEM MINDLIN PLATE ELEMENT .................. 43
4.1 Strain-displacement and
stress-strainrelations.................................... 43
4.2 Resultingeffort-strainrelations ........................... 44
4.3 Variationalmodel....................................... 45
4.4 PUFEMapproximationfields............................. 46
4.5 Stiffnessandmassmatrices............................... 47
4.6 Treatment of essential boundary conditions . ................ 48
4.7 Choiceofenrichment functions ........................... 48
4.8 Summary .............................................. 50
5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT 51
5.1 Statictests............................................. 52
5.1.1 Shearlockinganalysis .............................. 52
5.1.2 Convergence analysis............................... 59
5.2 Dynamictests.......................................... 63
5.2.1 Closed-formsolution ............................... 63
5.2.2 Dynamicplateconvergence test...................... 72
5.3 Summary .............................................. 80
6 INTERFACE ELEMENT FORMULATION .............. 81
6.1 Displacementfieldintheviscoelastic core .................. 81
6.2 Strainsintheviscoelastic core ............................ 83
6.3 Stresses-strains relationsintheviscoelasticcore............. 83
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6.4 Variationalexpressions fortheviscoelastic core ............. 84
6.5 Discretisedvariationalformfortheviscoelasticcore ......... 85
6.6 Summary .............................................. 87
7 DYNAMIC & ACOUSTIC ANALYSIS OF STRUCTURES 89
7.1 Dynamicanalysisofelasticandviscoelastic structures ....... 89
7.1.1 Modalanalysis .................................... 89
7.1.2 Dynamicanalysisbydirectfrequency response......... 94
7.2 Acousticanalysis ....................................... 96
7.2.1 Soundintensity andsoundpower .................... 98
7.2.2 Rayleighintegralmethod ........................... 99
7.2.3 Structural-acousticcoupling......................... 101
7.2.4 Radiationefficiencies ............................... 101
7.2.5 Soundpowerexpressed intermsofradiationmodes .... 107
7.3 Summary .............................................. 108
8 APPLICATIONS ......................................... 109
8.1 Twobondedplateswithstructuraladhesive ................ 109
8.1.1 Description ....................................... 109
8.1.2 Models ........................................... 111
8.1.3 Resultsandconclusions............................. 112
8.2 Moreira,RodriguesandFerreiravalidation ................. 115
8.2.1 Descriptionofthetestbenchandspecimens........... 115
8.2.2 Experimental measurements......................... 116
8.2.3 Results........................................... 117
8.2.4 Conclusions....................................... 117
8.3 WangandWereleyexperiment............................ 120
8.3.1 Description of the experimental setup and specimens . . . 120
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8.3.2 Modeldescription.................................. 121
8.3.3 Resultsandconclusions............................. 121
8.4 Calculationofmodallossfactors.......................... 124
8.4.1 Description ....................................... 124
8.4.2 Results........................................... 125
8.5 Summary .............................................. 126
9 EXPERIMENTAL APPLICATION: PLATE TESTS AT
OPTRION S.A. .......................................... 127
9.1 Experimentalsetup ..................................... 128
9.1.1 Descriptionofthetestedspecimens .................. 129
9.1.2 The OPTRION holographic interferometry camera . . . . . 131
9.1.3 Practicalconsiderations............................. 133
9.1.4 Free-Free setup (FFFF) . ............................ 135
9.1.5 Clampedsetup(CFCF)............................. 138
9.2 Correlation between PUFEM and ACTRAN software results . . 139
9.2.1 Free-Free setup (FFFF) . ............................ 140
9.2.2 Clampedsetup(CFCF)............................. 148
9.3 Correlationwithexperimentalmeasurements ............... 150
9.3.1 Free-Free setup (FFFF) . ............................ 150
9.3.2 Clampedsetup(CFCF)............................. 157
9.4 Summary .............................................. 159
10 DAMPING SANDWICH DESIGN RULES FROM
PARAMETRIC STUDIES ............................... 161
10.1SandwichOberstbeam .................................. 161
10.2Partialsandwich Oberstbeam ............................ 164
10.3Summary .............................................. 167
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11 OPTIMISATION ......................................... 169
11.1Numericaloptimisationinacoustics ....................... 169
11.1.1Literaturereview .................................. 170
11.2Anoptimisationapplication .............................. 173
11.2.1Preliminary undamped model study . ................. 173
11.2.2Theparametricmodelofthepatchedstructure ........ 174
11.2.3Thechoiceoftheobjectivefunction .................. 176
11.2.4Parametricstudies ................................. 178
11.2.5Partialconclusions................................. 184
11.2.6Optimisationstrategy .............................. 185
11.2.7Application of MFHO strategy to the first optimisation
problem .......................................... 191
11.2.8Application of MFHO strategy to a second optimisation
problem .......................................... 195
11.3Summary .............................................. 200
12 CONCLUSIONS ......................................... 203
12.1Conclusions&discussions................................ 204
12.2Perspectives............................................ 208
A ISD112 data .............................................. 211
B
Bending relationships for simply supported rectangular
plates .................................................... 213
C Bending relationships for circular plates .................. 217
D Solver choice and computational complexity .............. 219
E Sound power levels L W ................................... 221
References ................................................... 223
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1
INTRODUCTION
Nowadays, the acoustic and vibrational properties are increasingly considered
as key constraints during the design process of new products. Many factors
explain this evolution:
• public realisation of the importance of noise as a major pollution and
annoyance source;
• the publication of extremely rigorous noise emission norms, specifically
at the european level;
• the emergence of the sound quality notion among the consumer’s choice
criteria.
Consequently, market pressure and regulations dictate quiet products. Steelmakers,
especially, are experiencing hard times: the competition with other
materials, like aluminium and polymers, is tough and they must continuously
struggle to innovate. While steel has many intrinsic qualities that make it
the most widely used material for many industrial applications, ranging from
transportation to civil engineering or home appliances, it also has a major
drawback in an NVH (noise, vibration and harshness) context: flat steel products
are naturally noisy and this is essentially due to the very low natural
damping of thin steel sheets.
Many steelmakers are now proposing damped sandwich sheets, composed
of two steel layers separated by a dissipative material. These products are
based on the concept of passive damping (as opposed to active damping)
where, usually viscoelastic, dissipative material is added to the steel to complete
the properties of the designed product with noise and vibration performance.
These sandwich products have already found applications in the
transportation industries, such as railway or automotive transport, but are
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1 INTRODUCTION
actually too expensive to really be competitive in the building or civil engineering
markets.
However, an alternative design is possible: recent developments in manufacturing
tools allow the economic production of flat steel products with
localised bonded reinforcements. Initially developed by the steel-makers for
the production of light car body members (B-pillar, body sides, etc.) with
variable thicknesses (the patchwork technique), this methodology has already
been adopted by many automotive manufacturers for recent car models
[Arc05].
The adaptation of the concept to noise and vibration constraints leads to
a variant of the viscoelastic sandwich that we call viscoelastic patches (see
figure 1).
Fig. 1.1. Full and partial sandwich plate. The left image illustrates the classical constrained
layer sandwich concept where the whole structure contains a dissipative core. The right image
illustrates the concept of patch constrained layer damping where only a small portion of the
structure is locally covered with a dissipative material and a constraining plate.
The physical damping principle behind the patches is the so-called Constrained
Layer Damping (CLD) technique [Ker59]. The energy loss comes
from the shear deformation energy of the viscoelastic material layer which is
partially dissipated in the form of heat.
Since it is generally impossible to foresee the effects of alterations of
designs by experiments, designers require predictive numerical simulation
tools. Such tools should help to position the damping patches and optimise
its use.
The first aim of the thesis is to develop original and efficient modelling
techniques to simulate the dynamic and acoustic behaviour of structures with
viscoelastic damping devices such as patches. The finite element method offers
a solution that is now widely available in the literature and in commercial
codes such as SYSNOISE or ACTRAN. The usual FEM approach involves
the use of different layers of elements, with shared nodes or nodal constraints,
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1 INTRODUCTION
to simulate the behaviour of each material layers. It is possible to describe
accurately the shear deformation and the damping of the viscoelastic layer
by using tridimensional elements [BB02] for the core. One drawback of this
technique is that any modifications of the thickness of the polymer layer
involves a re-meshing step, which is sometimes expensive. Another drawback
that affects all FEM models is the following: as the frequency increases,
the use of low-order finite elements leads to prohibitive mesh refinement,
very large matrices and huge amounts of computational resources, limiting
their potential application to low frequencies. The main issue with low-order
Galerkin discretizations is the dispersion error [IB95] [BI99]. For high wave
numbers, Ainsworth [Ain03] proved improved efficiency by using higher polynomial
degrees in the approximation. This idea is revisited in this thesis in
a partition of unity framework [BM97]. Other alternatives include the use of
wave-based methods (WBM [Des98], developed by Desmet and his team), the
variational theory of complex rays (VTCR, [LARB01]) or the developments
of the discontinuous Galerkin method (DGM, [FHF01][FHH03]).
The manuscript is organised as follows: the first chapter introduces the
use of viscoelastic materials. We work under the assumptions of linear viscoelasticity,
applied to homogeneous and isotropic materials, and introduce
the concept of complex modulus. We present both tabulated and parametric
models and discuss their use. The processing of experimental data is also
illustrated on the 3M ISD112 material, that we tested in laboratory.
In the second chapter, we introduce the partition of unity finite element
method as a generalisation of the finite element method and discuss advantages
and drawbacks on uni- and bidimensional problems.
Chapter 3 focuses on the formulation of an original PUFEM Mindlin plate
element that answers our needs for an efficient element for vibration analysis.
This element benefits from polynomial enrichment and is more efficient than
classical, low-order finite elements. Our first contribution consists in the
application of the partition of unity finite element method (PUFEM), based
on the work of Babuška et al [BM97], to the development of efficient Mindlin
plate elements. These elements perform significantly better than the elements
available in the commercial software ACTRAN [Fre05]. Convergence studies
on both static and dynamic tests are performed in Chapter 4. This part also
demonstrates the performance and cost of the polynomial PUFEM approach,
as opposed to standard elements available in a commercial software. This
work has been already presented in [HB06].
Chapter 5 introduces an interface element technique for the modelling of
the viscoelastic core layer, in full and partial sandwich configurations. This
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1 INTRODUCTION
interface technique was developed initially for the modelling of bond assemblies
by FEM, but its application to PUFEM elements and to viscoelastic
layers is the second original contribution of this thesis.
Chapter 6 offers a review of classical analysis schemes for the dynamics
of structures, such as modal extraction and direct frequency response.
Some peculiarities linked to the analysis of structures containing viscoelastic
materials are developed and implementation tricks are also discussed. Most
authors addressing the design of passive damping devices in the literature
only focus on the dynamic behaviour of structures. We present also a simple
acoustic propagation model, based on the Rayleigh integral, that allows
the acoustic design of such devices. This approach, the third contribution
of this thesis, is lacking in most passive damping works though common in
publications on active control of structures.
Chapter 7 covers a handful of applications and focus on the validation
of our approach (PUFEM + interface element + modal or direct frequency
response). Our results are compared to published numerical or experimental
data. The methodology and some applications were already presented in
[HBS06].
We also developed our own experimental test bench for the medium frequency
validation, in collaboration with OPTRION SA. This validation test
is based on the measurement of frequency response curves for different configurations
of sample plates (naked or patched). This work is an important
part of the thesis and is covered in Chapter 8. The quality of the measured
data and the broad range of frequencies covered make it a major achievement
and the fourth contribution of this dissertation.
Finally, we tackle the design optimisation of viscoelastic patches. We
give some simple design rules based on bidimensional models. This work
is presented in chapter 9 and was already the subject of conference articles
[HDBB + 04],[HBS05] and [BHB06]. An original optimisation technique is proposed
for the optimal positioning of patches on structures in chapter 10. This
strategy takes advantage of the flexibility brought by the polynomial enrichment
of the Mindlin element. A variable fidelity optimisation framework is
monitored by a hybrid optimisation sequence. This strategy does perform
well on our application and succesfully reaches optimal design points, while
coping with the multimodal character of the response functions. In this chapter,
we also compare optimisation based on damping response functions with
optimisation based on the acoustic sound power level. The latter strategy is
an original development in the field and is the fifth contribution of this
thesis.
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2
VISCOELASTIC MATERIALS
Vibrations represent a major engineering issue in many areas of civil, mechanical,
or aeronautic design: wind excitation of long bridges and skyscraper
buildings, earthquake-proof foundations, sea waves actions on offshore platforms,
takeoff efforts on satellites transported rocket launchers, turbofaninduced
noise in airplanes - to name a few hot topics of active research.
Quite often, vibrations are undesirable and engineers struggle to reduce
them or to damp the structure. The term damping refers to the removal of
some part of the vibration energy from a vibrating structure. This suppression
may result from transferring energy to other structural components or fluids
which are not of concern or from converting this energy into other forms like
heat or radiation, for instance.
Since a long time, characterisation of damping forces of vibrating structures
has been an area of active research in the broad field of structural dynamics.
Since the publication of Lord Rayleigh’s book “Theory of Sound” at
the end of the 19th century [Ray45], a vast body of literature can be found
on this subject. Nevertheless, even if this topic is an old one, the demands
of modern engineering have led to a continuous increase of research in the
recent years.
At this point of the discussion, the term damping is still purely theoretical;
in fact, it covers a large number of physical mechanisms involving the
dissipation of vibration energy in a system. Some of these processes can be
linked to the intrinsic properties of the structural material, like the friction
of macromolecular chains in polymers, for instance. Others can be generated
by various types of boundary effects: friction at the contact of two bodies,
fluid-structure interaction - a moving body in a fluid or a moving fluid in
a hollow body -, by radiative energy loss, etc. In all these cases, a fraction
of the vibration energy is transformed into another, generally unrecoverable,
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2 VISCOELASTIC MATERIALS
form of energy (often, heat) or transported to another part of the system
where it is dissipated.
In this work, we are interested in the damping brought about by the use of
viscoelastic materials (VEM) in structures. The concept seems to date back
to a few years after the end of the First World War (WWI), when the Lord
Corporation company was founded by H.C. Lord in 1919 with the objective
of “exploring the potential of bonding rubber to metal to isolate and control
shock, noise and vibration.” 1
The first publications appeared in the 1950’s, by Liénard in France and
Oberst in Germany (see [Aus98]). The work of Oberst was focused on the application
of rubber layers on automotive panels to reduce acoustic radiation.
Most of his articles dealt with experimental methods of testing the performance
of various combinations of materials. At that time, the viscoelastic
layer was considered alone, with free-layer damping as the only loss mechanism.
Later, Kerwin ([Ker59], 1959), introduced the concept of constrained layer
damping (CLD), in which the VEM is sandwiched between two layers of solid
material. In that case, the loss mechanism is primarily due to shear in the
VEM core. Kerwin was also the first to introduce mathematical modelling
of this dissipative sandwich configuration. Together with Ross and Ungar,
Kerwin compared the effectiveness of free- and constrained-layer damping
on large plates and concluded that the constrained-layer treatments were the
most weight efficient in most cases. Since the 1950’s, hundreds of papers have
been published on the theory and application of constrained layer damping.
The relationship between damping and energy had already been noted
by Ross, Ungar and Kerwin in 1962 (see [BV92], [Cro98]). In 1981, Johnson,
Kienholz and Rogers presented an important work, introducing the Modal
Strain Energy (MSE) technique for the specific design of viscoelastic damping
structures by the finite element method [JK82]. With this technique, the
analysis of damping treatments by numerical simulation became much easier
than before, and analytic formulations lost some appeal, essentially because
of their limitation to very simple geometries. The rise of the finite element
method, together with the MSE technique, opened the way for the industrial
damping applications, primarily in aerospace engineering.
1 http://www.lord.com
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2 VISCOELASTIC MATERIALS
The design of a viscoelastic damping treatment involves five main design
options (see [Aus98]):
1. the thickness of the VEM layer,
2. the modulus of the VEM (which is both temperature- and frequencydependent),
3. the location of the VEM,
4. the thickness of the constraining layer
5. and the modulus of the constraining layer.
The design process requires finding the right combination of all these options
which will result in the maximum damping of the vibration modes of interest.
An efficient damping treatment configuration is the one that focus the strain
energy into the VEM, and the ideal VEM is the one that dissipates this
energy the most.
We treat this matter extensively in the chapter devoted to the optimisation
of damping patches (see chapter 11). Specifically, we study the optimal
thickness ratio of damping material and constraining material for viscoelastic
sandwiches and partial damping treatments. We also address the problem of
optimal placement of damping patches on product surface.
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2.1 Viscoelastic materials
Materials for which the relationship between stress and strain depends on
time are called viscoelastic.
Some phenomena are typical of such behaviour: creep or relaxation,
strain-rate dependent apparent stiffness, hysteresis in the stress-strain curves
(dissipation of mechanical energy), etc.
Most polymers are viscoelastic materials made up of long chains of molecules
or macromolecules. When submitted to an applied stress, polymers can
deform by either one or both two fundamentally different atomistic mechanisms
[Roy01]:
1. The lengths and angles of the chemical bonds connecting the atoms may
distort, moving atoms to new positions of greater internal energy; this
small motion can occur quickly.
2. If the polymer has sufficient molecular mobility, larger-scale rearrangements
of atoms are possible. This mobility is influenced by various physical
and chemical factors such as molecular architecture, temperature or
presence of fluid phases in the polymer.
The degree of mobility is determined by the rate of conformational
change, which is often described by an Arrhenius expression [MBB97] of
the form
rate ∝ e −E act
RT (2.1)
where E act is the activation energy of the process, R =8.314J/molK is the
Gas constant and T is the temperature, expressed in Kelvin.
The glass transition temperature T g is an important property of polymer
materials. At temperatures much below T g , the rates are so slow as to be
negligible. The mobility is, sort of, frozen and the polymer can only adapt to
applied stress by bond stretching. It responds in a glassy manner, incapable of
being strained beyond some few percent before brittle fracture. The material
exhibit a high modulus and low loss factor (“glassy” region (a) in figure 2.1).
In the neighbourhood of T g , the material is in a transition phase. Its response
is a combination of viscous fluidity and elastic solidity. This change
of phase corresponds to a fast variation of the modulus and to the highest
value of the loss factor, obtained at the glass transition temperature T g
(“transition” region (b) in figure 2.1). Dissipative materials used for damping
applications are those for which the transition zone occur at ambient
temperatures.
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2.1 Viscoelastic materials
At temperatures much above T g , the rates are so fast that the changes are
almost instantaneous, and the polymer acts in a rubbery manner in which
it exhibits large, instantaneous and fully reversible strains in response to
applied stress. It is characterised by low storage modulus and loss factor,
both varying weakly with temperature (“rubbery” region (c) in figure 2.1).
At even higher temperatures, far above the T g , the material exhibits the
properties of a fluid and is therefore instable (“flow” region (d) in figure 2.1).
Fig. 2.1. Variation of storage modulus (continuous line) and loss factor (dashed line) of a viscoelastic
polymer with temperature. We can distinguish four regions of different viscoelastic behaviour,
namely: (a) the glassy region, (b) the transition region, (c) the rubbery region and (d)
the flow region. Existing materials can be found in each regions at ambient temperature: grease,
rubbers, elastomers, epoxies or others polymers are all viscoelastic materials. [Roy01]
The effect of frequency on the dynamic properties of viscoelastic materials
is similar (but inverse) to the effect of temperature. The increase of excitation
frequency has the same effect on materials than a decreasing temperature.
This is called the temperature-frequency equivalence and this property is
used in most experimental measurement techniques for the characterisation
of the dynamic properties of materials.
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2.2 Linear viscoelasticity and complex modulus
In linear viscoelasticity (see [Oya04]), the stress σ is supposed to be a linear
function of the strain history. For a onedimensional tensile test, there exists a
relaxation function h such that the response σ (t) toastrainε (t) is obtained
by
∫ t
σ (t) = h (t − τ) dε (τ) dτ. (2.2)
dτ
−∞
This assumption is strictly equivalent to the existence of a complex modulus,
noted Λ. First let us operate a change of variable in relation 2.2: s = t−τ,
then
∫ +∞
σ (t) =− h (s) dε (t − s) ds. (2.3)
dτ
0
In a second step, we suppose an harmonic excitation (ie. strain) of the form 2
ε = ε 0 e jωt ; the response (stress) is then σ = σ ∗ 0 ejωt , for which the amplitude
is a complex variable (the stress is out-of-phase with the strain).
Substituting these expressions into 2.3 leads to the next equation
∫ +∞
σ0 ∗ = jωε 0 h (s)e −jωs ds. (2.4)
Previous expression can be rewritten in the form :
0
σ ∗ 0
ε 0
= Λ ′ (ω)+jΛ ′′ (ω) (2.5)
where 3 ∫ +∞
Λ ′ (ω) =ωε 0 h (s)sin(ωs) ds
0
∫ +∞
Λ ′′ (ω) =ωε 0 h (s)cos(ωs) ds. (2.6)
0
The terms Λ ′ and Λ ′′ are the real and imaginary parts of the complex modulus,
respectively. The complex modulus approach treats the viscoelastic materials
like frequency dependent elastic materials with complex properties.
2 Here j = √ −1andtheasterisk() ∗ denotes a complex quantity.
3 Using the Euler relation e jθ = cosθ + jsinθ.
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2.2 Linear viscoelasticity and complex modulus
2.2.1 Dynamic loading
For isotropic and homogeneous materials, the complex modulus is described
by a complex Young modulus E ∗ and a complex Poisson ratio ν ∗ (see
[Ker04]). The experimental measurement of these two quantities is however
very tricky, such that the Poisson ratio is usually considered real and constant
(not frequency-dependent) and that the experiment aims at the determination
of the complex Young modulus E ∗ or complex shear modulus G ∗ alone.
In the frequency domain, if we consider a simple dynamic, onedimensional,
tensile test, the following relations hold
σ (ω)=E ∗ (ω) ε (ω)
=[E ′ (ω)+jE ′′ (ω)] ε (ω)
= E ′ (ω)[1+jη (ω)] ε (ω) (2.7)
where we introduce the storage modulus E ′ (ω), the loss modulus E ′′ (ω) and
the loss factor η (ω) =E ′′ (ω) /E ′ (ω).
Fig. 2.2. Stress-strain cycle for a linear viscoelastic material [Ker04].
For each frequency, the complex modulus describes an elliptical trajectory
in the stress/strain plane (see figure 2.2):
σ =Re ( E ∗ ε 0 e iωt) )
=Re
(E ′ (1 + iη) ε 0 e iωt = E ′ ε 0 (cos ωt − η sin ωt) .
(2.8)
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2 VISCOELASTIC MATERIALS
The shape of the ellipse varies with the loss factor η; it describes an hysteresis
trajectory and the surface of this hysteresis is proportional to the dissipated
energy.
2.3 Selection of the material
The choice of a viscoelastic material will be based on its properties (loss factor
and Young modulus), in the frequency and temperature range of interest for
the application. But these properties are also affected by the stress state of
the material and environmental factors such as humidity.
The temperature dependance of viscoelastic materials has already been
covered in a previous section, and is intrinsequely linked to the nature of the
polymeric material. Temperature is the factor which has the most influence
on the VEM properties [Fer80]. Referring to figure 2.1, we can say that
the material of choice for damping should have its transition temperature
right in the range of functionning temperature of the application of concern.
Specifically, the loss factor peek should ideally corresponds to the standard
temperature for the system in function. The ideal material should also has a
broad loss factor peek, to exhibit good damping properties for a wide range
of temperatures.
The frequency dependence of the material is taken into account by the
complex modulus approach, with tabulated frequency properties. A methodology,
called reduced variables methods (RVM [Fer80]) can be used to modify
the parameters of this frequency-dependent model to reflect the temperature
dependence. The use of RVM leads to a unique set of material parameters
that can be handled at any predefined temperature, with the help of a single
scalar variable.
2.3.1 The method of reduced variables (RVM)
The effect of temperature and frequency on viscoelastic materials can be
combined into a single parameter dependence. This can be achieved through
a temperature function variable, called the shift factor, which means that the
viscoelastic behaviour at different temperatures can be related to each other
by a change (or shifting) in the time-scale only.
Figure 2.3, taken from [dS03], illustrates the methodology: the top plots
represent the frequency dependence of the Young modulus and loss factor for
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2.3 Selection of the material
Fig. 2.3. Representation of the time-temperature superposition principle (from [dS03])
an arbitrary viscoelastic material, at three different temperatures T1, T2 and
T3. When the method of reduced variables is applied to the curves, one of the
curve is chosen as reference and the others are shifted along the frequency axis
to overlap the reference curve. This horizontal shift, extrapolated from the
raw material data, is log(a T ), where a T is the shift factor. This characteristic
behaviour is referred to as the time- or frequency-temperature superposition
principle, which implies the existence of a reduced frequency, related to the
actual one through the shift factor:
ω r = a T ω. (2.9)
The single reference curve that comes up from the method is called master
curve.
The materials for which this superposition principle applies well are numerous
and are generally called thermorheologically simple (TS) materials.
Some complex materials do not obey the rule but they will not be considered
in the scope of this research: they are essentially non-homogeneous materials,
like copolymers (obtained by the chemical assembly of two different
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2 VISCOELASTIC MATERIALS
polymers) or polymers including added inorganic or organic material in the
melted phase (like fiber-reinforced or chalk-reinforced polymers).
The reduced variable method is often combined with an analytical expression
for the shift factor temperature dependence. A simple data fit of
the experimental shift values is sufficient to obtain a convenient analytical
represention of the shift factor temperature dependence. In 1955, Williams,
Landel and Ferry [WLF55] proposed an empirical relation, referred since as
the WLF equation.
To quote the original article: “In an amorphous polymer above its glass
transition temperature, a single empirical function can describe the temperature
dependence of all mechanical and electrical relaxation processes. The
ratio A(T) of any mechanical relaxation time at temperature T to its value
at a reference temperature, T(0), derived from transient or dynamic viscoelastic
measurements or from steady flow viscosity, and the corresponding ratio
b(T) of the values of any electrical relaxation time, appear to be identical over
wide ranges of time scale.”
The WLF equation is written :
with T 0 , the reference temperature.
log 10 (a T )= −C 1 (T − T 0 )
C 2 +(T − T 0 ) , (2.10)
For a given material, the constants C 1 and C 2 are obtained from a plot
of (T − T 0 )/log 10 (a T )versus(T − T 0 ).
2.3.2 Experimental determination of material data: the ISD112
case
Our objective is not to detail every steps of the procedure: the technique is
developed in many documents, refer for instance to [dS03] for a comprehensive
overview.
To obtain the shift factor and master curve for a given material, we need to
carry on a few sets of experiments at different temperatures. For the ISD112
material, manufactured by 3M Company in the form of tapes, the tests were
performed at the Central Laboratory of Solvay S.A., under supervision of E.
Gandin (see report [Gan04]). Dynamic shear tests on samples were made at
temperatures ranging from −30 o C to +80 o C (with steps of 4 o C). At each
temperature, a frequency sweep from 1 Hz to 100 Hz (7 intermediate steps)
was done.
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Laurent Hazard 21/12/2006
2.3 Selection of the material
To validate the superposition principle for this material, we can manually
translate the curves for the modulus, in order to ensure the continuity of
the storage module. The resulting modulus curve is the master curve of the
ISD112 material (see figure 2.4). The reference temperature was chosen equal
to 20 o C. One can see that, by doing so, the loss factor segments also form a
continuous curve (2.5). The behaviour of the ISD112 perfectly matches the
superposition assumption. The shift factor, necessary to align each modulus
segment, is also plotted against the temperature (expressed in Kelvin) (2.6).
Using the material shift factor curve and the two master curves, we can
generate modulus and loss factor frequency dependent data at all temperatures
covered by the shift factor plot.
Fig. 2.4. Master curve of the shear modulus of the ISD112 material, as a function of the reduced
frequency. This curve was build from experimental measurements carried out at Solvay S.A.
Central Laboratory.
2.3.3 Representation of the Complex Modulus
In this section, we briefly present different numerical representations of the
complex modulus of a material. We present the tabulated data model and
parametric models.
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2 VISCOELASTIC MATERIALS
Fig. 2.5. Master curve of the loss factor of the ISD112 material, as a function of the reduced
frequency.
5
4
Shift factor log(a T )
3
2
1
0
-1
240 250 260 270 280 290 300 310
Temperature [Kelvin]
Fig. 2.6. Shift factor, for the ISD112 material, as a function of temperature.
Tabulated data
After the experimental measurements and the data manipulation developed
in 2.3.2, we get a representation of the complex modulus as a function of the
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2.3 Selection of the material
reduced frequency and the shift factor temperature dependence. For each
temperature, we can generate tabulated data containing, for example, the
storage and loss modulus as functions of the frequency, using simple interpolation
procedures.
The advantage of this tabulated approach is that the material behaviour
law is very general and that complex frequency and temperature dependence
can be captured, even for wide ranges of frequencies and/or temperatures.
The direct use of data also avoid the process of model parameters fitting that
can be a source of discrepancy between the virtual and the real behaviour.
To conclude, the tabulated model is the most general form of complex
modulus representation and is also convenient for most numerical simulations.
Parametric models
With a parametric model, we attempt to approach the material behaviour
with analytical expressions. The first type of models are the rheological models,
which are build by combining springs and dashpots: such simple models
like Maxwell, Kelvin-Voigt or Zener models are present in the literature
for the representation of linear viscoelastic material behaviour. These simple
models are however limited, for real materials, to narrow range of frequencies.
More complex frequency dependence can be taken into account by fractional
derivatives models, such as those presented in [Ker04].
To be complete, recent work involving damping materials often mention
the Golla-Hughes-McTavish parametric model developed by Golla et
al. (GHM, see [GH85], [MH93]) and the Anelastic Displacement Field model
(ADF, see [LB95]) proposed by Lesieutre and Bianchini. Both models are
compatible with frequency domain and transient simulations.
For the ISD112, da Silva present the following analytical expressions,
involving fractional derivatives, for the complex (shear) modulus and the
shift factor:
G = G real + iG imag = B 1 + ( ) −B6 ( ) −B4
, (2.11)
if
1+B r 5 B 3
+
if r
B 3
where f r is the reduced frequency (in Hz). The material loss factor is obtained
as η = G imag
G real
.
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2 VISCOELASTIC MATERIALS
The shift factor can be found from the expression:
( 1
log (a T )=a
T − 1 ) ( ) ( )
2a
T
+2.303 − b log
T 0 T 0 T
( 0
b
+ − a )
− S AZ (T − T 0 ) . (2.12)
T 0
The parameters B i ,i=1, ..., 6anda, b, S AZ have no physical meaning and
are just chosen for the best data fit. Their values, for the ISD112 material,
are given in table A.1, in appendix. Figures 2.3.3 and 2.3.3 show respectively
the master curves and the shift factor curve corresponding to this parametric
model.
T 2 0
10 3 Storage modulus [MPa]
Loss modulus [MPa]
Loss factor
Reduced frequency f r
[Hz]
10 2
10 1
10 0
10 −1
10 −2
10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8
Fig. 2.7. ISD112 material : Master curves for the storage modulus, the loss modulus and the loss
factor (reference temperature = 290 K)
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2.3 Selection of the material
10
8
6
log 10
(a T
)
4
2
0
−2
−4
200 220 240 260 280 300 320 340 360
T [K]
Fig. 2.8. ISD112 material : Shift factor curve.
2.3.4 Manufacturer’s data: what are nomograms
Often, performing experimental measurements for the determination of material
properties is just too expensive or too time-consuming to be afforded
in a short project, and there are no analytical models for the chosen material
in the literature. If that is the case, the only source of information is the
material’s brochure delivered by the manufacturer.
Manufacturer data are almost always found in the form of nomograms,
giving the essential material properties as function of both the temperature
and the frequency. A typical example is presented in figure 2.3.4, extracted
from the ISD112 material brochure. To each temperature corresponds a shift
factor a T that defines an isothermal line in a plate (a T f,f), where f is the
frequency. The frequency values are then read on the vertical right axis.
For a given frequency f i , and a temperature T j , the nomogram can be
read as follows:
1. The intersection point (P) between the horizontal at frequency f i and the
isothermal (oblique) line T j is found;
2. drawing a vertical line through point P, the intersection with the modulus
or loss factor curves give the corresponding values, respectively.
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2 VISCOELASTIC MATERIALS
Fig. 2.9. Nomogram for 3M’s ISD112 material.
2.4 Summary
In this chapter, we gave a short history of the use of viscoelastic materials
for the damping of structural vibrations. The characteristics of viscoelastic
materials was developed and their temperature dependence explained.
In the context of the linear viscoelasticity theory, we introduced the notion
of Complex Modulus. The dependence with frequency was explained
and the superposition principle was defined. Application of this frequencytemperature
superposition, through the reduced variables methodology, allows
to express simply the temperature and frequency dependence of materials,
through the introduction of the shift factor. This assumption leads to
nomograms, giving modulus and loss properties as function of the reduced
frequency. We illustrated the technique on experimental data that we collected
for the ISD112 material.
To take into account the complex modulus data of viscoelastic materials
in computer simulations, differents representations are possible. We presented
both tabulated and parametric models principles. In our own calculations,
we choose to use tabulated data, defined from experimental measurements.
The implementation of such tables of frequency dependence in our numerical
code is straightforward and does not need to be detailed here.
The general formalism used for the numerical calculation of damped structures,
including viscoelastic materials, will be covered in chapter 7.
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3
INTRODUCTION TO THE PARTITION
OF UNITY METHOD
The fundamental concepts of the partition of unity (PU) approximation were
established initially by Babuška and Melenk, with the partition of unity
method [BM97] and the partition of unity finite element method (or PUFEM)
[MB96]. Early work was also found in the doctoral thesis of J.M. Melenk,
concerning generalized finite element methods [Mel95], in 1995, or in the
thesis of C.A. Duarte, which concerned the hp-cloud method [Dua96], in
1996.
To summarise, the partition of unity approximation was applied to develop
different kinds of generalised finite element methods (GFEM, or PUbased
GFEM). The PU-based GFEM family includes the hp−cloud method
of Duarte and Oden [ODZ98], the generalised finite element method of
Strouboulis et al. [SCB00] [SBC00], the generalised finite element method
of Duarte et al. [DBO00], and the partition of unity-based hierarchical finite
element method of Taylor et al. [TZO98].
A well-known problem of the PU-based GFEM is the arising of linear
dependencies in the system matrices. Even after correct specification of the
Dirichlet boundary conditions, the number of unknowns is generally larger
than the number of generalised shape functions generated by the PU approximation.
These unknowns must therefore be linearly dependent which leads
to the rank deficiency of the stiffness matrix.
In this chapter, we further develop the partition of unity finite element
method (PUFEM) on a onedimensional model problem and illustrate the
linear dependencies phenomenon. We start by presenting the model problem
and the corresponding differential equation. We then develop its variational
form and introduce the Galerkin method to solve the problem. The classical
finite element method is covered and the PUFEM technique is presented as
a generalisation of the finite element concept.
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3 INTRODUCTION TO THE PARTITION OF UNITY METHOD
3.1 Model problem
Consider the problem in onedimensional linear elasticity of a rod of length L,
subjected to a distributed traction load f(x) along its length, and a localised
traction T at position x = L. The rod is constrained at the left end (x =0).
The location of each point is given by a coordinate x and the displacement of
each point from its original position is denoted u(x). To simplify the developments
in this section, we assume a constant section area and Young modulus
along the rod (no material changes, no section changes).
Fig. 3.1. Onedimensional model problem: a rod of length L is subjected to a distributed traction
load f(x) and a traction force T at location x = L.
Under these assumptions, the boundary value problem (BVP) which describes
the displacement u(x) as a function of the position x along the rod is
given by the differential equation (strong form):
AE d2 u
= f (x) , ∀x ∈ Ω =[0,L]
dx2 u (0) = 0
AE du
dx (L) =T. (3.1)
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3.2 Galerkin method
3.1.1 Variational formulation
Let V be a linear space. The variational boundary value problem equivalent
to 3.1 is stated as follows:
Find the function u ∈V, that satisfies
a (u, v) =b (v) ∀v ∈V, (3.2)
where a : V×V→R is the bilinear form
a (u, v) =
∫ L
0
and b : V→R is the linear functional
AE du dv
dx (3.3)
dx dx
b (v) =
∫ L
0
f (x) vdx + Tv(L). (3.4)
3.2 Galerkin method
The Galerkin method builds an approximate solution to the variational
boundary value problem from a finite dimensional subspace V h of V spanned
by N linear independent functions in V:
V h ⊂V, span {Φ i } N i=1 = Vh (3.5)
where the Φ i are the basis functions (or approximation functions). The parameter
h characterises the dimension of the subspace and decreases with an
increasing number of basis functions N. We pose the variational form of the
BVP in V h as follows:
Find the function u h ∈V h , that satisfies
and
a ( u h ,v h) = b ( v h) ∀v h ∈V h (3.6)
b ( v h) =
∫ L
0
f (x) v h dx + Tv h (L). (3.7)
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3 INTRODUCTION TO THE PARTITION OF UNITY METHOD
To solve for the approximation u h , we write :
u h (x) =
N∑
Φ i (x)u i , v h (x) =
i=1
N∑
Φ i (x)v i (3.8)
where u i and v i are the unknown coefficients of the approximation.
We now substitute the approximations of equations 3.8 in relations 3.6
and 3.7, and use the arbitrariness of the values v i to obtain the following
linear system of equation
i=1
Ku = F (3.9)
where
K ij = a (Φ i ,Φ j ) (3.10)
F j = b (Φ j ) . (3.11)
3.3 Finite element method
The finite element method (FEM) is based on two principles: first, the original
problem is approximated following the application of the Galerkin method;
secondly, the constuction of an approximation space of finite dimension based
on finite partitioning of the domain and the definition of approximation function
on each subdomains. The Galerkin method has been intoduced previously.
We focus here on the definition of the approximation functions Φ i in
thecaseoftheFEM.
We start by partitioning the domain Ω into a set of M subdomains (or
elements). Nodes are located at the vertices of each elements. A total of N
nodes is distributed through the domain. The coordinates of the nodes are
labelled x 1 ,x 2 , ..., x N and the elements are labelled Ω 1 ,Ω 2 , ..., Ω M .
We associate to each node x i a shape function Φ i with a compact support
ω i . The support of the nodal shape functions is defined as the union of the
element subdomains sharing the node x i .
The finite element basis functions are the nodal shape functions spanning
a space of piecewise polynomials of at least order one. Figure 3.2 shows a
nodal shape function and the corresponding support on a one-dimensional
mesh, corresponding to the one-dimensional model problem.
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3.3 Finite element method
Fig. 3.2. Mesh for the one-dimensional problem. The linear shape function Φ i used at node x i,
defined on the support ω i is illustrated.
The finite element approximation is written as follows
u h (x) =
N∑
Φ i (x)u i . (3.12)
i=1
The nodal coefficients u i have a physical meaning: they are the values
of the approximated field (lateral displacements, for our 1D model problem)
at the nodes x i . The linear precision property follows from the fact that the
shape functions satisfy
∑
∑
Φ i (x) =1, Φ i (x)x i = x. (3.13)
i
Hence, if the nodal values are prescribed according to an arbitrary linear field,
the FE approximation will reproduce this field exactly. The equations 3.13
are therefore often called reproducing conditions. The first expression implies
that the shape functions form a partition of unity [BM97]. This property
relates to the ability of the FE model to represent rigid body modes and is
closely linked to the convergence properties of the approximation.
For the construction of the stiffness matrix K and force vector F, theintegrals
over Ω are replaced by a sum of elemental integrals of the subdomains
ω i . One important thing to note is that the the shape functions restricted to
the subdomains are polynomial functions. These functions are often numerically
integrated with a Gauss quadrature, since the order of the scheme can
be chosen such as to integrate the bilinear form 3.10 exactly.
i
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3 INTRODUCTION TO THE PARTITION OF UNITY METHOD
3.4 Partition of unity finite element method
The Partition of unity finite element method (PUFEM) was first proposed
by Melenk and Babuška [BM97] and is also known today as the Generalized
Finite Element Method (GFEM) [Mel95] [SCB00]. It is a particular case
of the general class formed by the Partition of Unity Methods, thatcovers
numerous different meshless approaches such as the Diffuse element method
(DEM)[NTV92], the Element Free Galerkin method (EFGM)[BLG94] or the
Reproducing Kernel Particle method (RKPM)[LJZ95].
The basic idea consists in the use of partition of unity functions, a set of
functions whose sum equals the unity on the whole domain. Let the functions
ϕ α , α =1, ..., N, denoteapartition of unity subordinate to the open covering
domain T N = {ω α } N α=1
of the domain Ω , such that
∑
α ϕ α(x) =1, ∀x ∈ Ω. (3.14)
This set of functions ϕ α composes the partition of unity attached to the
support ω α . In the case of the partition of unity finite element method (or
generalized FEM), ω α is the union of the finite element sharing the vertex
node x α .
We now introduce the local spaces χ α defined on ω α , α =1, ..., N
χ α (ω α )=span{L iα } i∈I(α)
, (3.15)
where I (α) are index sets and L iα denotes the local approximation functions
or local enrichment functions.
The family of generalized finite element shape functions of order p, F p N ,
is constructed by multiplying each partition of unity function ϕ α by the local
approximation functions
F p N = {Φα i = ϕ α L iα , α =1, ..., N, i ∈I(α)} . (3.16)
An obvious choice to form a basis for χ α are polynomial functions, which
can approximate well smooth functions. We use the following notation:
L iα (x) =ˆL i ◦ F −1
α
(x) (3.17)
where ˆL i is a polynomial of order i in R, chosen inside the space of polynomials
of degree less or equal to p.
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3.4 Partition of unity finite element method
The operator F −1
α
operates a change of coordinate system such that
F −1
α : ω α → ˆω, F −1
α (x) :=x − x α
(3.18)
h α
where x α is the coordinate of the node α, h α is the diameter of the largest
element sharing the node α.
The PUFEM approximation for u(x) can be written in the following form:
N
u h GF EM (x) = ∑ ∑
ϕ α (x) L iα (x) a α i = 〈Φ〉{a} . (3.19)
α=1
i∈I
Figures 3.3 and 3.4 illustrate the principle of the PUFEM, in the case
of the one dimension model problem presented earlier (for nonshifted and
shifted nodal enrichment functions). The problem is discretised by a two
elements mesh. The central node as coordinate x = 5, for a rod of length
L = 10. For this central node, the support is the union of both elements and
the PU functions are simply the linear hat functions. The nodal enrichment
functions, defined on this support, take different forms in the non-shifted case
andtheshiftedcase.
The PUFEM technique is also illustrated on a two-dimensional support
mesh in figure 3.5.
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3 INTRODUCTION TO THE PARTITION OF UNITY METHOD
1
PU function
Nodal enrichment functions
2
Generalized shape functions
1
0.5
PU
1.5
1
{1}
0.5
PU X {1}
0.5
0
0 2 4 6 8
x
0
0 5 10
x
10
5
0
100
{x}
0 5 10
x
0
0 2 4 6 8
x
5
4
3
2
1
PU X {x}
0
0 2 4 6 8
x
50
{x 2 }
20
10
PU X {x 2 }
0
0 5 10
x
0
0 2 4 6 8
x
Fig. 3.3. Illustration of the principle of the PUFEM, for a 2 elements mesh. The generalized
shape functions are obtained as the product of the PU functions, defined over the support around
node 2, with the unshifted nodal enrichment functions (monomials of x). The last column shows
each generalized shape function generated in the process (up to cubic functions).
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3.4 Partition of unity finite element method
1
PU function
Nodal enrichment functions
2
Generalized shape functions
1
PU
1.5
{1}
PU X {1}
0.5
1
0.5
0.5
0
0 2 4 6 8
x
0
0 2 4 6 8
x
4
2
0
−2
−4
0 2 4 6 8
x
25
20
15
10
5
{x−x α
}
{(x−x α
) 2 }
0
0 2 4 6 8
x
0
0 2 4 6 8
x
1
0.5
0
−0.5
−1
3
2
1
PU X {x−x α
}
0 2 4 6 8
x
PU X {(x−x α
) 2 }
0
0 2 4 6 8
x
Fig. 3.4. Illustration of the principle of the PUFEM, for a 2 elements mesh. The generalized
shape functions are obtained as the product of the PU functions, defined over the support around
node 2, with the shifted nodal enrichment functions (monomials of (x − x α)). The last column
shows each generalized shape function generated in the process (up to cubic functions).
Partition of unity function
Enrichment function
Shape function
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
0.5
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0.5
0.04
0.03
0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
0.5
1
1
1
0
0.5
0
0.5
0
0.5
−0.5
0
−0.5
0
−0.5
0
Fig. 3.5. Partition of unity finite element principle, for a problem in two dimensions. The support
mesh is made of four elements. The PU functions at the central node are plotted is the first window,
the enrichment function in the second window and the corresponding generalised shape function
in the last window.
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3 INTRODUCTION TO THE PARTITION OF UNITY METHOD
3.5 Back to the onedimensional model problem
We consider a particular case of the rod model problem introduced previously
(see 3.1). The rod, of length L=10, is fixed at both ends and supports a
distributed traction load f(x) =−2x. We consider both section area and
material Young modulus equal to unity (AE =1).
The differential equation to solve is written:
d 2 u
= −2x, ∀x ∈ Ω =[0, 10],
dx2 u (0) = 0,
u (10) = 0.
(3.20)
It is possible to find the exact solution to the differential equation by integration.
The exact solution obtained, taking into account the two boundary
conditions, is:
u EX =
3( 1 x 3 − 100x ) . (3.21)
We now apply the PUFEM technique to solve the model problem. To
further illustrate how the PUFEM technique works, we propose to decompose
analytically the different steps that bring to the final matrix form. We do this
for a single element mesh, containing two nodes (1 and 2).
The generalised shape functions are built from the product of the partition
of unity functions ϕ α and linear monomials of the coordinate x.Sinceweknow
that the exact solution contains terms up to x 3 , we choose the terms of the
enrichment basis up to x 2 . In order to simplify the notations and clarify the
development, we consider the case in which the basis functions L iα of χ α are
given by {1,x,x 2 }.
In this case, the PUFEM shape functions are given by
S τ = ϕ α × { 1,x,x 2} , α =1, 2. (3.22)
30

Laurent Hazard 21/12/2006
3.5 Back to the onedimensional model problem
Therefore, switching to matrix notations, we get the expressions for the
generalised shape functions, where L e stands for the element length:
{
Φ 1 Le − x
= ; L e − x
x; L }
e − x
x 2 (3.23)
L e L e L e
{ } x
Φ 2 = ; x2
; x3
(3.24)
L e L e L e
and their derivatives with respect to x:
{
dΦ 1
dx = − 1 ; L }
e − 2x
;2x − 3 x2
L e L e
L e
(3.25)
dΦ 2
dx = { 1
L e
; 2x
L e
;3 x2
L e
}
. (3.26)
Using the definition of the stiffness matrix, we get, in global coordinates:
k (e)
12 = ∫ Le
0
{ } dΦ
1 T { dΦ
2
dx
dx
}
dx. (3.27)
Then:
⎡
k (e) ⎢
12 = ⎣
k (e)
− 1 L e
⎤
−1 −L e
0 − Le − L2 e
3 2
0 − L2 e
6
− 3L3 e
10
⎡
1
L
11 = ⎢
e
0 0
⎣
k (e)
22 =
0 Le
3
L 2 e
6
0 L2 e 2L 3 e
6 15
⎤
⎡ ⎤
1
L
⎢
e
1 L e
3L
⎣
2 e
1 4Le
3
L e
3L 2 e
2
2
9L 3 e
5
⎥
⎦ (3.28)
⎥
⎦ (3.29)
⎥
⎦ . (3.30)
The force vector, for a variable transverse distributed load f(x) =−2x,
can be calculated as follows:
⎧ ⎫
∫ Le
f (e)
1 = (−2x) { ⎪⎨
Φ 1} − L2 e
3
⎪⎬
dx = − L3 e
6
0
⎪⎩ ⎪⎭ . (3.31)
− L4 e
10
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Laurent Hazard 21/12/2006
3 INTRODUCTION TO THE PARTITION OF UNITY METHOD
Identically,
f (e)
2 =
⎧
⎪⎨
⎪⎩
− 2L2 e
3
− L3 e
2
− 2L4 e
5
⎫
⎪⎬
⎪⎭ . (3.32)
The system obtained after assembly is a 6 × 6 matrix with 6 unknowns.
The approximated displacement field can be expressed as follows:
u h GF EM (x) =Φ 1 1a 11 + Φ 1 2a 12 + Φ 1 3a 13 + Φ 2 1a 21 + Φ 2 2a 22 + Φ 2
( ) ( ) ( 3a
) 23
Le − x Le − x Le − x
= a 11 + xa 12 + x 2 a 13
L e L e L e
+ x L e
a 21 + x2
L e
a 22 + x3
L e
a 23 .
(3.33)
Solving the linear system of equation, after application of the boundary
conditions, the solution found is {a} = { 0., − 100,
− 10
3 3 ;0.;0.;0.}T (for L e =
L = 10). In this case, the approximation is able to reproduce exactly the
exact solution and
u h GF EM = 1 3 (x3 − 100x). (3.34)
3.5.1 Linear dependencies
Unfortunately, the use of polynomial enrichment functions introduces linear
dependencies in the system matrix and therefore deteriorates the matrix
spectral conditioning. We illustrate this phenomenon on the model problem;
for different local enrichment basis, three discretisations are tested, containing
1, 2 and 10 elements. The boundary conditions are initially prescribed by
application of a penalty method. This technique will be detailed later (see
chapter 4).
The observed properties, for the different approximations, are:
• A measure of the spectral condition number of the stiffness matrix K,
as estimated by the Matlab function rcond. Thecalltorcond(K) sends
back the LAPACK condition estimator (reciprocal of the condition of the
matrix, in 1-norm). If K is well-conditioned, rcond(K) iscloseto1.IfK
is badly conditioned, rcond(K) is near the floating point relative accuracy
of the machine;
32

Laurent Hazard 21/12/2006
3.5 Back to the onedimensional model problem
• the error estimate, in L 2 -norm, calculated by :
⎛
⎞
ε L 2 = ∥ u EX − u h∥ ∫
(
∥
L 2
= ⎝ u EX − u h) 2
dx ⎠
• the size of matrix K, or number of unknowns;
Ω
1
2
; (3.35)
• the rank of matrix K, estimated by the Matlab function rank(K). It
provides an estimate of the number of linearly independent columns or
rows in K.
In table 3.1, we summarise results and properties obtained with different
approximations for the one dimensional problem.
Table 3.1. Test of different local enrichment basis functions for the one dimensional model
problem. Boundary conditions prescribed by penalty method.
Local functions Number of elements cond(K) ε L 2 size(K) rank(K)
1 0.9999 4.11 × 10 2 2 2
{1} 2 3.99 × 10 −7 1.44 × 10 2 3 3
10 7.99 × 10 −8 12.91 11 11
1 0. 51.43 4 3
{1, x} 2 0. 9.09 6 5
10 0. 0.16 22 21
¨1, © x,x
2
1 1.03 × 10 −20 2.28 × 10 −4 6 4
2 8.69 × 10 −28 3.18 × 10 −4 9 7
10 2.54 × 10 −28 7.07 × 10 −4 33 27
¨1, © x
2
1 1.19 × 10 −12 2.27 × 10 −4 4 4
2 2.12 × 10 −13 3.20 × 10 −4 6 6
10 2.23 × 10 −15 7.10 × 10 −4 22 21
The linear finite element approximation (single enrichment function {1})
does not capture well the displacement of the rope, even with the finer mesh
(10 elements). The second configuration tested includes the set {1, x} as
nodal enrichment functions. The approximation performs better than FEM,
thanks to the quadratic shape functions obtained, but exhibits a very bad
conditioning. The UMFPACK semi-definite direct solver is used to overcome
this difficulty [Dav06].
The third configuration includes the nodal enrichment set {1, x,x 2 }.As
illustrated above, in that case, the generalised shape functions are capable
of reproducing the exact solution and that is true whatever the number of
elements. It can be observed that the conditioning of the system matrix is
still bad.
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3 INTRODUCTION TO THE PARTITION OF UNITY METHOD
Duarte et al. [ODZ98] suggested to remove the identical elements existing
in the PU function and in the enrichment set. This leads to the last
configuration test, where the set contains only {1, x 2 }. The accuracy of the
approximation is identical as for the full set {1, x,x 2 }, but with better conditioning.
In table 3.2, we present results obtained for the same 1D model problem,
using bounded enrichment functions in the form of monomials of x − x α ,
where x α is the current node coordinate. This is called a shifted enrichment
basis, since the enrichment coordinate system is shifted to the local node.
Note that, for better conditioning, it is recommended to shift the basis, but
also to normalize it with respect to the local element length h α ,suchasin
expression 3.18. This was not done for the current application.
Table 3.2. Test of different local enrichment basis functions for the one dimensional model
problem. Bounded enrichment functions, as functions of x − x α. Boundary conditions prescribed
by penalty method.
Local functions Number of elements cond(K) ε L 2 size(K) rank(K)
1 2.22 × 10 −22 51.43 4 3
{1, x− x α} 2 7.40 × 10 −23 9.09 6 5
10 5.05 × 10 −24 0.16 22 21
¨1, x− xα, (x − x α) 2© 1 1.23 × 10 −22 2.28 × 10 −4 6 4
2 7.40 × 10 −23 3.18 × 10 −4 9 7
10 7.08 × 10 −24 7.07 × 10 −4 33 31
¨1, (x − xα) 2© 1 9.99 × 10 −5 2.27 × 10 −4 4 4
2 3.99 × 10 −7 42.64 6 6
10 7.99 × 10 −8 0.66 22 21
With the shifted basis, the set {1, x− x α } lead to the same accuracy
than its unshifted version, but with better conditioning. The same behaviour
is observed for the set {1, x− x α , (x − x α ) 2 }. Surprisingly, the incomplete
set, {1, (x − x α ) 2 }, does not perform well for more than one single element.
It can be shown, by a simple analytical development similar to that of section
3.5, that for a mesh containing more than one element, the generalised shape
functions generated by the shifted basis are no more sufficient to reproduce
the exact solution.
We now try to propose a technique to suppress the linear dependencies.
When performing some analytical developments, we noticed that at the
nodes supporting essential boundary conditions, often the nodal enrichment
generates unnecessary unknowns. These unknowns contribute to the rank deficiency
of the system matrix and therefore to the bad conditioning. One way
to solve this problem is to avoid the nodal enrichment of the nodes with pre-
34

Laurent Hazard 21/12/2006
3.5 Back to the onedimensional model problem
scribed boundary conditions, by using only a constant function (L iα = {1})
at those nodes. This technique also simplify the prescription of the boundary
conditions since a direct method can be efficient. Unfortunately, by doing
so, a poor approximation is obtained in the neighbourhood of the boundary
nodes. The accuracy of the procedure can be regained by using a small layer
of elements at the boundaries. Typically, for the current application of global
length L = 10, an additional layer of elements of size L e =0.1 was added
at the two extremities. This means that, to compare with previous results,
the configuration with one element becomes a configuration withonelarge
element and two boundary elements.
For the sake of comparison, the results in table 3.3 are obtained using
the penalty method for the application of the boundary conditions, while the
results of table 3.4 are obtained with a direct method.
Table 3.3. Test of different local enrichment basis functions for the one dimensional model
problem. Shifted enrichment functions, as functions of x−x α. The nodes with prescribed essential
BCs do not benefit enrichment from local functions. Boundary conditions prescribed by penalty
method.
Local functions Number of elements cond(K) ε L 2 size(K) rank(K)
1 (+2 boundary elements) 3.33 × 10 −11 51.09 6 6
{1, x− x α} 2(+2 boundary elements) 2.22 × 10 −11 9.05 8 8
10 (+2 boundary elements) 5.92 × 10 −12 0.71 24 24
1 (+2 boundary elements) 2.77 × 10 −11 7.05 × 10 −6 8 8
¨1, x− xα, (x − x α) 2© 2 (+2 boundary elements) 1.85 × 10 −11 7.90 × 10 −6 11 11
10 (+2 boundary elements) 1.09 × 10 −12 1.17 × 10 −5 35 35
1 (+2 boundary elements) 9.94 × 10 −7 3.30 × 10 −5 6 6
¨1, (x − xα) 2© 2 (+2 boundary elements) 3.98 × 10 −9 42.35 8 8
10 (+2 boundary elements) 7.05 × 10 −10 0.53 24 24
Table 3.4. Test of different local enrichment basis functions for the one dimensional model
problem. Shifted enrichment functions, as functions of x−x α. The nodes with prescribed essential
BCs do not benefit enrichment from local functions. Boundary conditions prescribed by direct
method.
Local functions Number of elements cond(K) ε L 2 size(K) rank(K)
1 (+2 boundary elements) 3.32 × 10 −5 51.09 6 6
{1, x− x α} 2 (+2 boundary elements) 2.21 × 10 −5 9.05 8 8
10 (+2 boundary elements) 5.85 × 10 −6 0.71 24 24
1 (+2 boundary elements) 1.39 × 10 −5 5.01 × 10 −6 8 8
¨1, x− xα, (x − x α) 2© 2 (+2 boundary elements) 1.84 × 10 −5 5.09 × 10 −6 11 11
10 (+2 boundary elements) 1.08 × 10 −6 6.18 × 10 −6 35 35
¨1, (x − xα) 2© 1 (+2 boundary elements) 0.006 3.11 × 10 −5 6 6
2 (+2 boundary elements) 0.004 42.35 8 8
10 (+2 boundary elements) 6.97 × 10 −4 0.53 24 24
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3 INTRODUCTION TO THE PARTITION OF UNITY METHOD
The accuracy obtained with both the penalty and direct method are similar.
The technique proposed is efficient and suppress the rank deficiency of
the system matrix. However, the direct method leads to a better matrix conditioning
than the penalty method. This is due to the use of a penalty term
(which is a very large number, of the order 1. × 10 6 for our application), that
contributes to the conditioning of the scheme.
We can write the following remarks:
• In the general case, both the enrichment by the PUFEM and the BCs
penalty method contribute to the degradation of system matrix conditioning.
• With the shifted nodal enrichment, the suppression of the redundant term
(x − x α ) from the basis is not efficient; the resulting approximation performs
poorly and there is serious loss of accuracy when compared with
the full enrichment set, for more than one element.
• With the basis {1, x− x α , (x − x α ) 2 }, the use of degenerated basis at
boundary nodes (constant function {1}, together with a small layer of
boundary elements is efficient; the matrix conditioning is good and the
level of accuracy reached is excellent.
The origin of the linear dependencies problem is well understood and
we proposed a solution. However, this solution implies an adaptation of the
support mesh close to the edges supporting boundary conditions. We have
access to an efficient and fast direct solver that handles well bad conditioned
matrices (UMFPACK, see [Dav06]). Therefore, we did not apply the propose
solution in the latter developments and used the PUFEM technique
with standard support meshes and the penalty method for prescribing the
boundary conditions.
36

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3.6 Twodimensional application
3.6 Twodimensional application
The PUFEM can be applied to any finite element topology. We introduce here
the Q4-PUM element, which is based on the Q4 isoparametric finite element.
This element is adapted to the modelling of 2D plane stress problems. In the
PUFEM version, the linear Lagrange functions are used as PU functions.
The stiffness K and mass matrix M are constructed as follows:
K = ∑ e
K e (3.36)
M = ∑ M e (3.37)
e
∫
K e = B T HBdΩ (3.38)
Ω
M e =
∫
M u =
[ ]
Mu 0
0 M v
(3.39)
ρΦ T u Φ udΩ (3.40)
Ω
∫
M v =
ρΦ T v Φ vdΩ (3.41)
Ω
where the vectors Φ u and Φ v contain the generalized shape functions
respectively for the horizontal displacement u and vertical displacement v (in
local element axis). In this work, we choose to work with the same enrichment
basis for all displacement unknowns.
The B matrix contains the derivatives of the generalized shape functions:
⎡
⎤
〈Φ u,x 〉 〈0〉
B = ⎣ 〈0〉 〈Φ v,y 〉 ⎦ . (3.42)
〈Φ u,y 〉〈Φ v,x 〉
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Laurent Hazard 21/12/2006
3 INTRODUCTION TO THE PARTITION OF UNITY METHOD
The Hooke matrix H (plane stress) is expressed as follows, under the
plane stress assumptions, where E is the Young modulus and ν the Poisson
ratio of the material :
H =
⎡
E
1 − ν 2
1 ν 0
⎣
ν 1 0
00 1−ν
2
⎤
⎦ . (3.43)
The element matrices described in expressions (3.38) and (3.39) are evaluated
by a numerical integration procedure, based on a high order Gauss-
Legendre integration scheme. The order of this scheme is adapted to the
highest degree of the polynomial terms involved in the expression to integrate
i.e. in the expression of M e (exact integration).
We illustrate the application of the PUFEM to a simple static cantilever
beam application. The beam has a length L =48m and an height D =12m
(unit thickness). The Young modulus E is taken equal to 3.107 N/m2 and
the Poisson’s ratio is 0.3. A parabolic traction force is applied to the free
end : the upper and lower edges are free from load, and shearing forces,
having a resultant P , are distributed along the end x = L. The total load
P is chosen equal to -1000 N. Timoshenko and Goodier [TG82] developed
analytical expressions for the displacements and the stresses in the beam:
u x = − Py
6EI
u y = − Py
6EI
[
(
(6L − 3x) x +(2+ν)
[
3νy 2 (L − x)+(4+5ν) D2 x
y 2 − D2
4
)]
4
+(3L − x) x 2 ] (3.44)
where I is the moment of inertia of the beam (I = D3 ). The expressions for
12
the normal stresses and the shear stress are :
σ xx = − P (L−x) y
I
σ yy =0
[ ] (3.45)
τ xy = P D 2
− 2I 4 y2
Figure 3.7 illustrates the convergence behaviour of the Q4-PUM element
with different enrichment basis. The model is composed of a single element
in the thickness of beam and is progressively refined along the length. The
relative vertical displacement at the tip is plotted against the number of
elements in the length. As foreseen, the linear FEM solution (enrichment
by {1}) shows a very bad behaviour. The other solutions are build with
shifted enrichment polynomials. It is possible to obtain excellent convergence
behaviour by a judicious choice of enrichment functions.
38

Laurent Hazard 21/12/2006
3.6 Twodimensional application
Fig. 3.6. Cantilever beam problem (see Timoshenko and Goodier [TG82]).
Fig. 3.7. We plot the relative vertical tip displacement (v/v ref ) as a function of the number of
elements in the mesh. The mesh contains only one element in the thickness and is progressively
refined in the length. Shifted enrichment polynomials are used.
In figures 3.8, 3.9 and 3.10, we plot the strain energy error in L 2 norm,
as a function of the size of element h (in the beam length) for different beam
thickness ratio h . Different shifted polynomial basis are tested as enrichment.
L
Very good error levels are obtained with the enrichment basis
{1,x− x α ,y − y α , (x − x α ) 2 , (y − y α ) 2 }. If we suppress the terms (x − x α )and
(y − y α ), to improve the conditionning of the matrices, the error level is good
for a single element mesh but deteriorates when more elements are used.
This confirms the behaviour that we already observed on the one dimension
problem (see tables 3.3 and 3.3).
39

Laurent Hazard 21/12/2006
3 INTRODUCTION TO THE PARTITION OF UNITY METHOD
Fig. 3.8. The error in strain energy (L 2 norm) is plotted against the element size h, for a cantilever
beam with an aspect ratio h =0.25. The rightmost point of each curves corresponds to a single
L
element mesh. Shifted enrichment polynomials are used.
Fig. 3.9. The error in strain energy (L 2 norm) is plotted against the element size h, for a cantilever
beam with an aspect ratio h =0.041. The rightmost point of each curves corresponds to a single
L
element mesh. Shifted enrichment polynomials are used.
40

Laurent Hazard 21/12/2006
3.6 Twodimensional application
Fig. 3.10. The error in strain energy (L 2 norm) is plotted against the element size h, fora
cantilever beam with an aspect ratio h =0.0104. The rightmost point of each curves corresponds
L
to a single element mesh. Shifted enrichment polynomials are used.
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3 INTRODUCTION TO THE PARTITION OF UNITY METHOD
3.7 Summary
In this chapter, we introduced the partition of unity finite element method
(PUFEM) as a generalisation of the finite element method (FEM). A model
problem was first presented in strong form and the variational formulation
was derived. The Galerkin method was then presented, together with the
principles of FEM.
The concepts of PUFEM were introduced and illustrated on both one- and
bidimensional models. The advantages and drawbacks were covered, and the
problem of linear dependencies was deeply developed in the onedimensional
case. The source of these linear dependencies was explained and a solution
was proposed.
The PUFEM technique is then applied to a 2D problem and lead to
the development of Q4-PUM elements for the study of 2D beams problems.
The convergence properties of this element are studied on simple theoretical
cantilever beam configuration, using both relative displacement and error in
strain energy as quality indicators. These elements will also be used for the
study of different composite configurations in chapter 10.
In the next chapter, we apply the PUFEM technique to the development
of Mindlin plate elements.
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Laurent Hazard 21/12/2006
4
PUFEM MINDLIN PLATE ELEMENT
The first-order shear deformation theory (Mindlin theory, or FSDT) is used to
develop a Mindlin plate element. We will first review the strain-displacement
and stress-strain relations. We will then introduce the variational form and
derive the discretised form with PUFEM approximations of the displacement
fields.
4.1 Strain-displacement and
stress-strain relations
As pointed out by H.C. Huang [Hua89], a good starting point for the analysis
of both thin and moderately thick plates is a theory in which the classical
Kirchhoff assumption of zero transverse shear strain is relaxed. Reissner
[Rei45] proposed first to introduce the rotations of the normal to the plate
midsurface as independent variables in the plate theory. Mindlin [Min51] proposed
the simplified assumption that “normals to the plate midsurface before
deformation remain straight but not necessarily normal to the plate after deformation”
and the stress normal to the plate midsurface is disregarded as
in the Kirchhoff theory.
We adopt Mindlin’s theory for the further developments. The displacements
of all points of a plate are therefore expressed by:
⎧
⎨
U (x, y, z) =u (x, y)+zβ x (x, y)
V (x, y, z) =v (x, y)+zβ y (x, y)
⎩
W (x, y, z) =w (x, y)
(4.1)
where u(x, y), v(x, y) andw(x, y) are the translational displacements of a
point of the midsurface along the x, y and z axis respectively. β x (x, y) and
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Laurent Hazard 21/12/2006
4 PUFEM MINDLIN PLATE ELEMENT
β y (x, y) are the rotation angles of a normal to the midsurface around the
x and y axis. The linear bending and shear strains are expressed by the
relations (where the superscrip () t denotes the transposed of the matrix of
vector):
{ε} = {e} + z {χ} {γ} t = 〈 β x + ∂w β 〉
∂x y + ∂w
∂y
(4.2)
where
{ε} t = 〈 ε x ε y 2ε xy
〉
{χ} t =
〈
∂βx
∂x
∂β y
∂y
{e} t = 〈 ∂u
∂β x
∂y
+ ∂βy
∂x
∂x
∂v
∂y
∂u
+ 〉
∂v
∂y ∂x
(4.3)
〉
. (4.4)
4.2 Resulting effort-strain relations
We make the assumption that the midsurface is the xy plane placed at z =0
(neutral plane) and that there is a material symmetry with respect to the
midsurface. In this case, the membrane and bending effects are decoupled.
This means that in-plane membrane forces acting in the neutral plane can
not generate bending strains (no curvature change) and that bending couples
can not generate membrane strains (no in-plane displacements). For a
homogeneous and isotropic plate, the efforts are related to the strains by the
relations:
{N} =[H m ] {e} = t [H] {e}
{M} =[H f ] {χ} = t3
[H] {χ}
12
{T } =[H c ] {γ} = κt [H τ ] {γ}
(4.5)
where
⎡
[H] =
E
1 − ν 2
1 ν 0
⎣
ν 1 0
00 1−ν
2
⎤
⎦ , [H τ ]=
E
2(1+ν)
[ ] 10
, (4.6)
01
with E the Young modulus, ν the Poisson coefficient, t the thickness of the
plate and κ the shear correction factor.
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Laurent Hazard 21/12/2006
4.3 Variational model
4.3 Variational model
The plate vibration problem consists in finding the resulting efforts {N},
{M}, {T } and the kinematic variables {u} and {β}, that satisfy the virtual
work principle. The virtual work principle is stated as follow:
W = W int − W ext =0 ∀{δu} , {δβ} (4.7)
where {δu} and {δβ} are virtual displacements and rotations. The term W int
represents the virtual work done by the internal stress state in the body;
W ext is the virtual work done by the external (body forces and traction on
the boundary) and the inertial forces. The application of the virtual work
principle to the Mindlin plate leads to the weak variational form, with these
expressions for the virtual works :
∫
(
W int = {δe} t [H m ] {e} + {δχ} t [H f ] {χ} + {δγ} t [H c ] {γ} ) dΩ, (4.8)
Ω
∫
W ext =
Ω
∫
+
Γ t
∫
−
Ω
({δu} t {b} + {δβ} t {m})dΩ
({δu} t {¯t} + {δβ} t {m s })dΓ
[ ( { }) ( { })]
{δu} t ρ m {ü} + ρ mf ¨β + {δβ} t ρ mf {ü} + ρ f ¨β dΩ,
(4.9)
with {u} t = 〈u(x, y) v(x, y) w(x, y)〉 and {β} t = 〈β x (x, y) β y (x, y)〉. The
vectors {b} and {m} contain loadings (forces and couples) acting on the
surface of the plate Ω (by unit of surface). The vectors {¯t} and {m s } contain
loadings (traction forces and couples) acting along the border of the plate Γ t
(by unit of contour). The density terms ρ m , ρ mf and ρ f are obtained from
an integration in the thickness of the plate :
ρ m =
∫ h/2
−h/2
ρ(z)dz ρ mf =
∫ h/2
−h/2
ρ(z)zdz ρ f =
∫ h/2
−h/2
ρ(z)z 2 dz. (4.10)
For an isotropic, homogeneous and symmetric plate material, ρ mf =0.
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Laurent Hazard 21/12/2006
4 PUFEM MINDLIN PLATE ELEMENT
4.4 PUFEM approximation fields
We choose the approximation fields according to the PUFEM principles.
The in-plane displacements, out-of-plane displacement and rotations are expressed
as follow:
⎧
⎪⎨
⎪⎩
u h = 〈Φ u 〉{u m n }
v h = 〈Φ v 〉{v m n }
w h = 〈Φ w 〉{w m n }
β h x = 〈 Φ βx 〉
{β
m
xn }
β h y = 〈 Φ βy 〉{
β
m
yn
}
{u m n } t = 〈u 1 1 ···u m n 〉
{v m n }t = 〈v 1 1 ···vm n 〉
{w m n }t = 〈w 1 1 ···wm n 〉
{β m xn} t = 〈β 1 x1 ···β m xn〉
{
β
m
yn
} t
= 〈 β 1 y1 ···βm yn〉
(4.11)
In the most general case, different shape functions are used for the approximation
of the translations u, v, w and the rotations β x and β y , depending on
the set of enrichment functions used for each field:
〈Φ〉 = 〈···Φ α ···〉 for α =1...n {Φ α } t = 〈ϕ α L 1α ···ϕ α L mα 〉 , (4.12)
where n is the number of nodes per element and m, the number of enrichment
functions per approximation field.
The nodal variables obtained from these approximations are stored in one
large vector {U n } = 〈 u 1 1 ... um n v1 1 ... vm n ... ... βm yn〉
.
The curvatures are linked to the nodal variables through the next relation
(involving rotations only) :
{χ} =[B f ] {U n } . (4.13)
The shear strains are linked to the nodal variables (involving the out-of-plane
displacement and the rotations only) by :
{γ} =[B c ] {U n } . (4.14)
In these expressions, [B f ]and[B c ] are the matrices of the shape functions
derivatives :
⎡
〈0〉 〈0〉〈0〉 〈 ⎤
∂Φ
∂x〉
〈 〈0〉
〉
∂Φ
[B f ]= ⎢ 〈0〉〈0〉〈0〉〈0〉
⎣
∂y
⎥
〈 〉
∂Φ 〈
〈0〉〈0〉〈0〉
∂Φ
〉 ⎦ (4.15)
∂y ∂x
[ 〈
〈0〉〈0〉
∂Φ
]
∂x〉
〈0〉〈Φ〉 〈0〉
[B c ]=
〈 〉
. (4.16)
〈0〉〈0〉〈0〉 〈0〉〈Φ〉
46
∂Φ
∂y

Laurent Hazard 21/12/2006
4.5 Stiffness and mass matrices
The expression of the in-plane strains {e} is written as follow:
{e} t =[B m ] {U n } (4.17)
where :
〈 ∂Φ
⎡
⎤
∂x〉
〈 〈0〉〈0〉〈0〉〈0〉
〉
∂Φ
[B m ]= ⎢ 〈0〉 〈0〉〈0〉〈0〉
⎣
∂y
⎥
〈 〉 〈 ∂Φ
〉 ⎦ . (4.18)
〈0〉 〈0〉〈0〉
∂Φ
∂y
∂x
4.5 Stiffness and mass matrices
The development of the stiffness matrix is based on the previous expression
of the internal virtual work (see 4.8). The membrane behaviour is uncoupled
from the bending and shear strains (midplane assumption). Each element
is characterised by the matrices of strain derivatives [B m ],[B c ]and[B f ], accounting
respectively for the membrane, shear and bending behaviours.
⎡ [ ] ⎤ ⎡
⎤
Km 0 00 00 000
0 00
[K] =
⎢ 00 0
[
00
] ⎥
⎣ 0 0 0 Kf ⎦ + 00 000
⎡ ⎤
⎢ 0 0
⎥
⎣ 0 0 ⎣ K c
⎦ ⎦ , (4.19)
0 0 0
0 0
[K c ]= ∫ Ω
[B c ] t [H c ][B c ] dΩ,
[K f ]= ∫ [B f ] t [H f ][B f ] dΩ,
Ω
[K m ]= ∫ [B m ] t [H m ][B m ] dΩ.
Ω
(4.20)
The mass matrix expressions is obtained from the expression of the inertial
terms in the external virtual work expression. The element mass matrix is
made of two different contributions:
⎡
[M w ] 0 0 0 0
0 [M w ] 0 0 0
[M] =
⎢ 0 0 [M w ] 0 0
⎥
⎣ 0 0 0 [M β ] 0 ⎦ , (4.21)
0 0 0 0 [M β ]
[M w ]= ∫ Ω {Φ}ρ m 〈Φ〉 dΩ,
[M β ]= ∫ Ω {Φ}ρ f 〈Φ〉 dΩ.
⎤
(4.22)
The element stiffness and mass matrices are obtained by numerical integration
with a Gauss-Legendre scheme.
47

Laurent Hazard 21/12/2006
4 PUFEM MINDLIN PLATE ELEMENT
4.6 Treatment of essential boundary conditions
With the PUFEM, the degrees of freedom associated to each node are the
unknowns of the approximation (see equation 3.19). The prescription of a
linear constraint on an edge, between two nodes, is not straightforward. With
linear FEM, it is sufficient to prescribe nodal values since the shape functions
can represent exactly a linear variation, but this technique is no more suitable
with PUFEM (for general enrichment). It is however possible to degenerate
the enrichment function set at nodes where linear edge constraints are needed.
The terms of the basis are simply multiplied by a function which vanishes
on the prescribed edge (see [DBVB05]). This technique works well for simple
linear boundary conditions. We adopt in this paper a weak penalty method
[Liu03] in order to prescribe the essential boundary conditions. This method
is very efficient and is suitable for very general boundary condition types.
The weak form is modified by adding the penalty term to the left side of
equation 4.9:
∫
∫
{δu} t [α β ]({u}−{ū}) dΓ + {δβ} t [α β ] ( {β}− { ¯β}) dΓ (4.23)
Γ u Γ β
where {ū} contains prescribed displacements on boundary Γ u , { ¯β} contains
imposed rotations on boundary Γ β .[α u ]and[α β ] are diagonal penalty matrices.
The diagonal terms of the penalty matrices are the penalty factors
associated to each displacement and rotation fields.
4.7 Choice of enrichment functions
The introduction of polynomials inside the enrichment set leads to the generation
of high order polynomial shape functions. In this form, the (polynomial)
PUFEM behaves very much like a p-FE method. For a Q4 element, quadratic
shape functions are built from the product of the linear Lagrange shape functions
(partition of unity functions) and linear monomials (see section 3.4 and
[DBO00]), such as:
ϕ α ×
{
1, x − x α
h α
,
}
y − y α
, α =1, ..., N. (4.24)
h α
Different enrichment basis can be chosen for each approximation field.
In particular, in-plane displacements, deflection and rotations need different
polynomial degrees.
48

Laurent Hazard 21/12/2006
4.7 Choice of enrichment functions
In this paper, we adopt two types of polynomial enrichments: the full
polynomial enrichment and the reduced polynomial enrichment.
The full polynomial enrichment technique is based on an identical choice
of local spaces for all approximation fields of the Mindlin element. The p =2
full polynomial enrichment corresponds to
where ξ = x−xα
h α
and η = y−yα
h α
.
χ α =span { 1; ξ; η; ξ 2 ; η 2 ; ξη } (4.25)
In a more general form, the local space is build from the minimal set of
complete polynomials of order less or equal to p, suchas
χ α =span{L iα } (4.26)
with
L iα = ξ a η b , 0 ≤ a, b ≤ p, a + b ≤ p. (4.27)
In the reduced polynomial enrichment technique, we adopt different local
enrichment spaces for each approximation field. The reduced p =2enrichment
corresponds to a choice where:
• the out-of-plane displacement, w, benefits from a p =2enrichment:
L iα = {1; ξ; η; ξ 2 ; η 2 ; ξη};
• and the rotations β x and β y benefit from a p =1enrichment:
= {1,ξ,η}.
V j
i
In the reduced polynomial enrichment, the enrichment of the rotations unknowns
are chosen one order less than the enrichment of the out-of-plane
displacements.
The in-plane displacement field, when necessary, benefits always from a
first-order enrichment (p =1).
The efficiency and convergence properties of each type of enrichment
strategy will be discussed in chapter 5.
49

Laurent Hazard 21/12/2006
4 PUFEM MINDLIN PLATE ELEMENT
4.8 Summary
In this chapter, we detailed the formulation of the PUFEM Mindlin element.
The expressions for the stiffness and mass matrices were obtained, starting
from the variational form. We also covered the treatment of the essential
boundary conditions and the choice of enrichment for each approximation
fields.
We now need to validate and assess the convergence properties of this
element. This is the aim of chapter 5.
50

Laurent Hazard 21/12/2006
5
CONVERGENCE OF THE PUFEM
MINDLIN ELEMENT
In this chapter, we present numerical examples to demonstrate the advantages
and performances of the PUFEM approach applied to Mindlin plates
elements for both static and dynamic applications.
The first static tests aim at the illustration of the performance of the element
in situations where shear locking could occur, with regular and irregular
meshes.
The dynamic tests assert the convergence of the element on dynamic
problems.
We use the two different types of enrichment introduced in chapter 4:
• full polynomial enrichment basis, where all displacement and rotation
fields receive the same enrichment polynomials;
• reduced polynomial enrichment basis, where the rotations benefits from
an enrichment basis which is one order less than that of the out-of-plane
displacement.
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Laurent Hazard 21/12/2006
5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
5.1 Static tests
5.1.1 Shear locking analysis
It is necessary to verify that the Mindlin plate elements are able to capture
the correct physical behaviour of extreme cases of thin plates. The discretised
form of the static problem can be written in the following form (see [BD90]
and [War94]), with adimensional stiffness matrices:
( [ ]
0 0
Et 3 0 K b
[
+ κGtL 2 Kw K wβ
Kwβ
T K β
]){ w
}
L
= β
{ }
Fw
0
(5.1)
where E is a characteristic bending modulus, G is a characteristic shear
modulus, t is the plate thickness, L is a characteristic length of the plate and
k is the shear correction factor.
We introduce the following parameter:
Φ = E κG ( t L )2 (5.2)
which is a measure of the contribution of the shear terms to the total stiffness
matrix.
The equation 5.1 can be rewritten in the form (with the subscripts () b for
bending and () s for shear):
Et
([K 3 b ]+ 1 )
Φ [K s] {U} = {F } . (5.3)
The thin or thick character of the problem can be strictly defined by the
evaluation of the parameter Φ.
“Thin”plates are characterised by Φ

Laurent Hazard 21/12/2006
5.1 Static tests
Techniques to prevent shear locking, for standard engineering applications,
have been developed for a long time. The developed techniques can be
ranged in one of these three categories:
1. Use of high order shape functions: plate elements based on the p-FEM are
known to be considerably less susceptible to shear locking than standard
lower order finite element methods (see [Sch96], [SBS05] and, recently,
the work of Rank et al. [RDN + 95]).
2. Build shape functions for the rotations β x and β y based on the firstorder
derivative of the shape functions that are used for the transverse
displacement w and therefore satisfy naturally the Kirchhoff assumptions;
the approach has, for instance, been applied with success to EFGM beams
and plates in [KNBSYB01].
3. Evaluate the shear strain energy in a particular way: in FEM, we may cite
the Selective Reduced Integration or the Assumed Shear Strain techniques
(see [Hug87], [Hua89])
In our work, we choose to work with polynomial enrichment of the shape
functions and therefore we put in practice the first option (full polynomial
enrichment). The development of reduced enrichment basis where different
basis are used for the transverse displacement and the rotations should be
classified in the second category. The results obtained with both approaches
will be presented in the next sections.
Simply supported square plate
This example consists in a simply supported square plate (hard support) with
a uniform distributed load (see figure 5.1). The material properties are listed
in table 5.1. Only one quarter of the plate is meshed and symmetry conditions
are prescribed on the corresponding edges. The boundary conditions
are described in table 5.2.
Table 5.1. Square plate properties.
Symbol Plate property Value
E Young modulus 1.E9 Pa
ν Poisson ratio 0.3
L edge length 10 m
q distributed load 1. N/m 2
t plate thickness variable
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5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
6
5
4
3
2
1
0
−1
−1 0 1 2 3 4 5 6
(b)
6
5
4
3
2
1
0
−1
−1 0 1 2 3 4 5 6
(a)
(c)
Fig. 5.1. (a) Simply supported (hard) square plate. Only one quarter of the plate is modelled
and symmetry conditions are applied on the unsupported edges. (b) Regular plate mesh (2 × 2
elements) (c) Irregular plate mesh (2 × 2 elements).
Table 5.2. Square plate boundary conditions (w is the transverse displacement, β t is the rotation
around a direction tangent to the contour and β n is the rotation around a direction normal to
the contour.
Type
Conditions
Simply supported (soft) w =0
Simply supported (hard) w =0β t =0
Clamped w =0β t =0β n =0
Symmetry β n =0
The evolution of the maximum deflection (obtained at the centre of the
plate) is calculated for different values of the ratio L/t. Thenumericalvalues
are normalised by the analytical solution obtained by Timoshenko and
Woinowsky-Krieger [TWK95], under Kirchhoff assumptions (Classical Plate
Theory, or thin plate theory). The results are produced with both regular
and irregular meshes to illustrate the influence of mesh distortion on the
behaviour of the element. For both cases, a 4 element mesh (2 × 2) is used.
The evolution of the maximum displacement in the plate for an increasing
aspect ratio L/t is illustrated on figure 5.2. For an enrichment order p =2,
the behaviour of the element appear too stiff for L/t ≥ 10 4 .
Higher polynomial enrichment improves the element quality, even for extreme
aspect ratio (L/t ≥ 10 6 ). Identical conclusions are obtained for an
54

Laurent Hazard 21/12/2006
5.1 Static tests
irregular mesh (see figure 5.3): mesh distortion does not affect too much the
quality of the element.
Center displacement normalized with respect to thin plate theory
1.15
1.1
1.05
1
0.95
0.9
0.85
p=2
p=3
p=4
Classical Plate Theory (Timoshenko et al)
0.8
10 1 10 2 10 3 10 4 10 5 10 6 10 7
L/t
Fig. 5.2. Aspect ratio study for square plate (regular mesh). The influence of the order of the
polynomial enrichment basis is illustrated for homogeneous enrichment with p =2,p =3and
p =4.
Center displacement normalized with respect to thin plate theory
1.2
1.1
1
0.9
0.8
0.7
0.6
p=2
p=3
p=4
Classical Plate Theory (Timoshenko et al)
0.5
10 1 10 2 10 3 10 4 10 5 10 6 10 7
L/t
Fig. 5.3. Aspect ratio study for square plate (irregular mesh). The influence of the order of the
polynomial enrichment basis is illustrated for homogeneous enrichment with p =2,p =3and
p =4.
55

Laurent Hazard 21/12/2006
5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
The same test is also applied with a reduced basis of polynomial enrichment
(see figure 5.4). The “reduced” p = 2 basis is not as efficient as the
“full” p = 2 basis, but is still very satisfactory. Large deviations only occur
for L/t ≥ 10 4 , just like for the “full” p = 2 basis. The “reduced” p=3 basis
is comparable to the “full” p=3 basis. In this case, the response starts to
deviate from the analytical prediction for ratio L/t ≥ 10 6 .
Center displacement normalized with respect to thin plate theory
1.2
1.1
1
0.9
0.8
0.7
0.6
Reduced p=2
Reduced p=3
Classical Plate Theory (Timoshenko et al)
0.5
10 1 10 2 10 3 10 4 10 5 10 6 10 7
L/t
Fig. 5.4. Aspect ratio study for square plate (regular mesh). Polynomial enrichment is different
for both w, β x and β y (reduced p order).
56

Laurent Hazard 21/12/2006
5.1 Static tests
Clamped circular plate
This example consists in a circular plate (or disk) clamped at the border and
supporting a uniform distributed load. The radius of the disk is 5 m. We
exploit the symmetry of the problem and mesh only a quarter of the domain,
as illustrated in figure 5.5. The mesh is made of 12 elements.
We use both full and reduced enrichment basis with different maximum
polynomial order p =2,p =3andp =4.
The results are presented in figures 5.6 and 5.7. The reference maximum
displacement, obtained at the centre of the plate, was calculated analytically
by Timoshenko and Woinowsky-Krieger [TWK95], following the classical thin
plate theory. The numerical results are normalised with respect to the classical
plate theory prediction.
For full polynomial enrichment, the locking manifests itself for ratio
2R/t ≥ 10 2 when p = 2. With higher order polynomial enrichment, the locking
effect becomes important at greater ratio: approximately for 2R/t ≥ 10 3
for p = 3 and even further for p =4.
Good results are also obtained with the reduced polynomial basis, as
illustrated on figure 5.7.
6
5
4
3
2
1
0
0 1 2 3 4 5 6
(a)
(b)
Fig. 5.5. (a) Simply supported (hard) circular plate. Only one quarter of the disk is modelled
and symmetry conditions are applied on the unsupported edges. (b) Circular plate mesh (12
elements).
57

Laurent Hazard 21/12/2006
5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
Center displacement normalized with respect to thin plate theory
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
p=2
p=3
p=4
Classical Plate Theory (Timoshenko et al)
10 2 10 3 10 4 10 5
2R/t
Fig. 5.6. Aspect ratio study for circular plate. The influence of the order of the polynomial
enrichment basis is illustrated for homogeneous enrichment with p =2,p =3andp =4.
Center displacement normalized with respect to thin plate theory
1.4
1.2
1
0.8
0.6
0.4
0.2
Reduced p=2
Reduced p=3
Classical Plate Theory (Timoshenko et al)
0
10 1 10 2 10 3 10 4 10 5
2R/t
Fig. 5.7. Aspect ratio study for circular plate. Polynomial enrichment is different for both w, β x
and β y (reduced p order).
58

Laurent Hazard 21/12/2006
5.1 Static tests
5.1.2 Convergence analysis
In this section, we analyse the performance of the PUFEM Mindlin plate
on standard plate problems. The numerical solution obtained for the whole
plate is compared to analytical expressions, for different enrichment basis.
Simply supported square plate
In this example, we use the same geometry than in the previous section (see
figure 5.1). The edge length stays equal to 10 m, but the thickness is fixed to
0.1 m (aspect ratio L/t =10 2 ). A uniform distributed load on the domain is
equal to 1 MPa.
TherelativeerrorinL 2 norm of the transverse displacement is calculated
for the whole plate. An analytic expression is derived for the expression of the
reference solution (transverse displacement for all the plate); this expression
isbasedontheLévy methodology applied to Mindlin plates and developed
in the work of Wang, Reddy and Lee (see appendix B and [WRL00]). The
relative error in L 2 norm is expressed by
‖e w ‖ L2 (Ω) = ‖w − w h‖ L2 (Ω)
‖w h ‖ L2 (Ω)
=
√ ∫
(w − w h ) 2 dΩ
Ω
√ ∫ . (5.4)
wh 2dΩ Ω
Full and reduced p-enrichment approximation spaces were used. Two
types of convergence curves are plotted :
• Relative error in L 2 norm of the transverse displacement with respect to
the size of the system matrix;
• transverse displacement of the centre point of the plate, with respect to
the size of the system matrix.
Both convergence curves are presented in figure 5.8, for the square plate
problem. For both convergence plots, the increase of system size (or number
of unknowns) is obtained by increasing the degree of the enrichment basis,
for a fixed, constant, mesh.
In the case of a reduced enrichment, the reduced basis of enrichment
is specified by two numbers : p w and p β ; for instance, the point p w = 3
59

Laurent Hazard 21/12/2006
5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
and p β = 2 represent an approximation build with an enrichment basis up
to degree 3 for the out-of-plane displacement and up to degree 2 for the
rotations.
The two plots show a very fast convergence for both type of enrichment.
The reduced enrichment technique is less expensive but a bit less accurate.
The convergence rates are similar.
Clamped circular plate
In this example, we use the same geometry as in previous section (figure 5.5).
The radius of the disk is 5 m and the thickness is fixed to 0.01 m (aspect
ratio L/t =10 3 ). The uniform distributed load on the disk is equal to 1 MPa.
TherelativeerrorinL 2 norm of the transverse displacement is calculated
for the whole plate. An analytic expression is obtained for the reference solution
(transverse displacement for all the disk); this expression is derived
from the work of Wang, Reddy and Lee (see appendix C and [WRL00]).
Both full and reduced p-enrichment approximation spaces were used.
Again, convergence curves in terms of relative error in L 2 norm of the transverse
displacement and transverse displacement of the centre point of the
disk are plotted in figure 5.9. The same notation is adopted.
The convergence is slower than for the plate problem. The different enrichments
studied hardly reach relative error levels of 10 −2 , whereas in the
plate problem, error level close to 10 −4 where obtained with system sizes well
below 10 3 unknowns. The rate of convergence with reduced and homogeneous
basis are similar.
The convergence is here handicapped by the poor fidelity of the support
mesh to the original problem: the contour of the disk is only poorly described
by the straight border of the linear element used for the partition of unity
definition. This fact explains that the obtained relative errors are higher than
for the square plate example.
60

Laurent Hazard 21/12/2006
5.1 Static tests
10 0
10 −1
||e w
||
L2 (Ω)
10 −2
p=1
10 1 Number of unknowns
p w
=1, p β
=0
Homogeneous p−enrichment
Reduced p−enrichment
p w
=2, p β
=1
10 −3
p=2
p w
=3, p β
=2
10 −4
p=3
p w
=4, p β
=3
p=4
10 2 10 3
(a)
Center displacement normalized with respect to reference solution
1
0.8
0.6
0.4
0.2
0
Homogeneous p−enrichment
Reduced p−enrichment
Reference solution
10 2 10 3
Number of unknowns
(b)
Fig. 5.8. Simply supported square plate convergence study for both full and reduced p-enrichment
sets. (a) Relative error in L 2 norm of the transverse displacement against model size (total number
of unknowns). (b) Normalised transverse displacement of the plate centre against the model size
(total number of unknowns).
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5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
10 0 Number of unknowns
p w
=1, p β
=0
p=1
p w
=2, p β
=1
p=2
||e w
||
L2 (Ω)
10 −1
p w
=3, p β
=2
p=3
10 −2
p=4
p w
=4, p β
=3
Homogeneous p−enrichment
Reduced p−enrichment
10 2 10 3
(a)
Center displacement normalized with respect to reference solution
1
0.8
0.6
0.4
0.2
0
Homogeneous p−enrichment
Reduced p−enrichment
Reference solution
10 2 10 3
Number of unknowns
(b)
Fig. 5.9. Clamped disk convergence study for both full and reduced p-enrichment sets. (a)
Relative error in L 2 norm of the transverse displacement against model size (total number of
unknowns). (b) Normalised transverse displacement of the disk centre against the model size
(total number of unknowns).
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Laurent Hazard 21/12/2006
5.2 Dynamic tests
5.2 Dynamic tests
It is now necessary to assert the performance and the convergence properties
of our elements on dynamic problems. Typically, we want to use these
elements for medium-frequency computations of plate vibrations.
5.2.1 Closed-form solution
The convergence of the PUFEM Mindlin element is studied by comparison
with a closed-form solution developed by G.B. Warburton for particular rectangular
plates with special boundary conditions (see [War54]).
Warburton consider plates assumed to be isotropic, elastic, and of uniform
thickness; the analysis is based on the ordinary theory of thin plates.
The thickness is also assumed to be small in comparison with the wavelength.
He used the Rayleigh method, assuming waveforms identical to those
of onedimensional beams, to derive expressions for all modes of vibrations.
By prescribing a particular set of boundary conditions on all four edges of
rectangular plates, derived from the characteristic beam functions, we obtain
a simple solution for the transverse (or out-of-plane) displacement that is
exact and limited to the first term of the Rayleigh development.
Specifically, we consider an aluminium plate Ω= [0, x max
]×[0,y max ]where
and y max = nπ
k y
. The material properties are given in table 5.3. The
x max = mπ
k x
terms m and n define the number of waves used along each edges, respectively
the x- and y-axis. The terms k x and k y are the projections of the wave number
k b in each axis directions, expressed for one direction of propagation θ by
{
kx = k b cos (θ)
k y = k b sin (θ) , (5.5)
k b = 4 √
ρhω
2
D , D = Eh 3
12(1 − υ 2 ) . (5.6)
When the following boundary conditions are applied on the edges
⎧
for x =0, w =0,β x = k x sin (k y y) ,β y =0
⎪⎨
for x = x max , w =0,β x = k x cos (k x x max )sin(k y y) ,β y =0
for y =0, w =0,β x =0,β y = k y sin (k x x)
⎪⎩
for y = y max , w =0,β x =0,β y = k y sin (k x x)cos(k y y max )
(5.7)
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Laurent Hazard 21/12/2006
5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
the rectangular plate vibration solution takes the simple form
⎧
⎨ w ref (x, y) =sin(k x x)sin(k y y)
β x,ref (x, y) =k x cos (k x x)sin(k y y) . (5.8)
⎩
β y,ref (x, y) =k y cos (k x x)cos(k y y)
Table 5.3. Aluminium properties.
Symbol Plate property Value
E Young modulus 68 × 10 9 Pa
ν Poisson ratio 0.3
ρ Density 2470 kg/m 3
t plate thickness 1 × 10 −3 m
The reference solution obtained for the three different configurations used
in the convergence study are illustrated in figure 5.10.
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Laurent Hazard 21/12/2006
5.2 Dynamic tests
0.14
u REF
0.8
u REF
0.8
0.12
0.6
0.25
0.6
0.1
0.4
0.2
0.2
0.4
0.2
0.08
0
0.15
0
0.06
−0.2
0.1
−0.2
0.04
−0.4
−0.4
0.02
−0.6
0.05
−0.6
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
−0.8
−1
0
0 0.05 0.1 0.15 0.2 0.25 0.3
−0.8
(a)
(b)
u REF
0.8
0.5
0.6
0.4
0.4
0.2
0.3
0
0.2
−0.2
−0.4
0.1
−0.6
0
0 0.1 0.2 0.3 0.4 0.5 0.6
−0.8
(c)
Fig. 5.10. Closed form Warburton solutions, used as references in the convergence studies. Contour
plots of out-of-plane displacement for (a) m=n=2, θ =45 ◦ .(b) m=n=4, θ =45 ◦ (c)m=n=8,
θ =45 ◦ .
65

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5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
Full polynomial enrichment
The convergence curves, for each configuration m = n =2,m = n =4and
m = n = 8, are presented in the figures 5.11, 5.12 and 5.13. These results are
obtained by increasing regularly the mesh size. The relative error in L 2 norm
of the transverse displacement (calculated by relation 5.4) is plotted against
model size (total number of unknowns). Each curve in the plots is obtained
for one particular enrichment basis.
In the captions of these figures, the term “FEM” stands for a standard
linear Mindlin finite element (no enrichment functions) for which we implemented
a selective reduced integration (SRI) technique; the shear terms of
the stiffness matrix are integrated by a Gauss-Legendre scheme with a single
point per element, while the other stiffness contributions are based on a 2×2
scheme (bilinear, 4-nodes Lagrange plate bending element with SRI, called
S1 in [Hug87]).
10 0
10 −1
Number of unknowns
FEM
p=2
p=3
p=4
p=5
||e w
||
L2 (Ω)
10 −2
10 −3
10 1 10 2 10 3 10 4
Fig. 5.11. Convergence curves for m=n=2, θ =45 ◦ for full polynomial enrichment basis.
The first case, n=m=2, θ =45 ◦ , illustrated in figure 5.11, corresponds to
a simple “modal” pattern and very low error levels are easily achieved at low
expense. One can see that even the linear FEM solution does perform well on
this case. Relative errors below 10 −2 are reached with low polynomial order,
for models of less than 10 3 unknowns.
66

Laurent Hazard 21/12/2006
5.2 Dynamic tests
10 0
10 −1
Number of unknowns
FEM
p=2
p=3
p=4
p=5
||e w
||
L2 (Ω)
10 −2
10 −3
10 1 10 2 10 3 10 4
Fig. 5.12. Convergence curves for m=n=4, θ =45 ◦ for full polynomial enrichment basis.
10 1
10 0
Number of unknowns
FEM
p=2
p=3
p=4
p=5
||e w
||
L2 (Ω)
10 −1
10 −2
10 3 10 4
Fig. 5.13. Convergence curves for m=n=8, θ =45 ◦ for full polynomial enrichment basis.
The configurations n=m=4, and n=m=8, allwithθ =45 ◦ , correspond
to more complex patterns of out-of-plane displacements. The FEM does perform
poorly on these two problems, as can be seen in figures 5.12 and 5.13.
Basically, we can see that for higher polynomial enrichment order, we get
better and faster convergence: to achieve a certain accuracy, the models get
67

Laurent Hazard 21/12/2006
5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
smaller and smaller, while the convergence rates increase generally with the
order of enrichment.
It is also interesting to study the evolution of the relative error for constant
mesh discretizations. The results are plotted in figure 5.14. For each
problems, we use the same support meshes and increase progressively the
degree of polynomial enrichment. Each curve in this plot corresponds to one
configuration (m = n = 2, etc). The first point of each curve is the FEM
solution, and so on up to the highest polynomial degree tested in the configuration.
The advantage of the enrichment appears clearly on this plot: considering
the m=n=8 problem, we see that the error drops almost vertically
(low increase of the number of unknowns) when we increase the enrichment
order. The error decreases faster than the growth of the model size.
10 0 Number of unknowns
FEM
FEM
FEM
10 −1
||e w
||
L2 (Ω)
p=5
10 −2
10 −3
p=5
n=m=2,θ=45 deg.
p=4
n=m=4,θ=45 deg.
n=m=8,θ=45 deg.
10 1 10 2 10 3 10 4
Fig. 5.14. Convergence curves for m=n=2, m=n=4, m=n=8 and θ =45 ◦ . The mesh size is kept
constant for each curve. For case m=n=2, we used a 4 × 4 mesh; for case m=n=4, a 5 × 5mesh
and for m=n=8, a 8 × 8mesh.
68

Laurent Hazard 21/12/2006
5.2 Dynamic tests
Reduced polynomial enrichment
The convergence curves, for each configuration m = n =2,m = n =4and
m = n = 8, are presented in the figures 5.15, 5.16 and 5.17. These results are
obtained by increasing regularly the mesh size. The relative error in L 2 norm
of the transverse displacement is plotted against model size (total number of
unknowns). Each curve in the plots is obtained for one particular choice of
enrichment basis.
10 0 Number of unknowns
FEM
p=2 reduced
p=3 reduced
p=4 reduced
10 −1
||e w
||
L2 (Ω)
10 −2
10 −3
10 1 10 2 10 3 10 4
Fig. 5.15. Convergence curves for m=n=2, θ =45 ◦ for reduced polynomial enrichment basis.
The same behaviour is observed for the reduced enrichment than for the
full, homogeneous enrichment. For the configuration m = n = 2, the displacement
pattern is not complex and easy to capture even with low order
enrichment. Again, the FEM solution performs well and it is hard to distinct
the performance of p = 4 (reduced) with p = 3 (reduced).
In the configurations m = n =4andm = n = 8, the enriched approximations
clearly outperform the FEM approximation and allow higher accuracy
at lower cost. The rate of convergence is also improved by the enrichment
basis.
The convergence curves showed in figure 5.18 are obtained by increasing
the order of the enrichment polynomials, with constant mesh size. Like before,
each curve of the plot is drawn for one configuration (m = n = 2, etc). The
first point of each curve is the FEM solution, the second is the reduced
69

Laurent Hazard 21/12/2006
5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
10 0 Number of unknowns
10 −1
||e w
||
L2 (Ω)
10 −2
10 −3
FEM
p=2 reduced
p=3 reduced
p=4 reduced
10 1 10 2 10 3 10 4
Fig. 5.16. Convergence curves for m=n=4, θ =45 ◦ for reduced polynomial enrichment basis.
10 1 Number of unknowns
FEM
p=2 reduced
p=3 reduced
p=4 reduced
10 0
||e w
||
L2 (Ω)
10 −1
10 −2
10 2 10 3 10 4
Fig. 5.17. Convergence curves for m=n=8, θ =45 ◦ for reduced polynomial enrichment basis.
enrichment solution with p = 2 for the out-of-plane displacement and p =1
for the rotations (noted p w = 2, p β = 1), and so on, up to the highest
enrichment tested (p w =4,p β =3).
70

Laurent Hazard 21/12/2006
5.2 Dynamic tests
10 0 Number of unknowns
FEM
p w
=2,p β
=1
10 −1
||e w
||
L2 (Ω)
p w
=4,p β
=3
p w
=4,p β
=3
10 −2
p w
=3,p β
=2
10 −3
n=m=2,θ=45 deg.
p w
=4,p β
=3
n=m=4,θ=45 deg.
n=m=8,θ=45 deg.
10 1 10 2 10 3 10 4
Fig. 5.18. Convergence curves for m=n=2, m=n=4, m=n=8 and θ =45 ◦ . The mesh size is kept
constant for each curve. For case m=n=2, we used a 4 × 4 mesh; for case m=n=4, a 5 × 5mesh
and for m=n=8, a 10 × 10 mesh.
71

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5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
5.2.2 Dynamic plate convergence test
The model problem consists of a rectangular vibrating steel plate in free-free
boundary conditions (plate properties are given in table 5.4); the plate is
excited by a unit point force (see figure caption for locations). The reference
solution used is an overkilled model obtained with a 10 × 10 support mesh
and a polynomial enrichment basis p = 5 (7623 unknowns). The PUFEM solutions
are compared to three different FEM implementations: the first one
is directly derived from our Mindlin plate development and is referenced as
‘FEM’ in the figures; it is obtained by using selective reduced integration
in the PUFEM Mindlin plate element, with no enrichment functions. Convergence
of two other FEM solutions (obtained with the commercial FEM
software ACTRAN [Fre05]) are also presented, respectively for linear and
quadratic ACTRAN brick plate elements.
Table 5.4. Steel plate properties.
Symbol Plate property Value
E Young modulus 210.10 9 [Pa]
ν Poisson ratio 0.3 [/]
ρ Density 7825 [kg/m 3 ]
t Thickness 2.10 −3 [m]
u REFERENCE
x 10 −7
u REFERENCE
x 10 −5
0.2
6
0.2
1.5
0.18
0.16
0.14
0.12
4
2
0
0.18
0.16
0.14
0.12
1
0.5
0.1
−2
0.1
0
0.08
0.06
−4
0.08
0.06
−0.5
0.04
−6
0.04
−1
0.02
−8
0.02
0
0 0.05 0.1 0.15 0.2 0.25
−10
0
0 0.05 0.1 0.15 0.2 0.25
−1.5
(a)
(b)
Fig. 5.19. Optrion plate contour plot (a) at 1000 Hz, for a plate excited at (0.05,0.05). (b) at
2000 Hz, for a plate excited at (0.,0.).
72

Laurent Hazard 21/12/2006
5.2 Dynamic tests
Full polynomial enrichment
10 0 Number of unknowns
10 −1
||e w
||
L2 (Ω)
10 −2
10 −3
10 −4
FEM
p=2
p=3
p=4
ACTRAN − LIN. FEM
ACTRAN − QUAD. FEM
10 2 10 3 10 4
Fig. 5.20. Convergence curves at 1000 Hz, for a plate excited at (0.05,0.05). We show the
performance for different FEM implementations, and PUFEM Mindlin approximation with full
polynomial basis. We added also ACTRAN linear and quadratic elements.
Both linear FEM elements (in-house development and ACTRAN linear
brick) exhibit extremely low convergence rates. The quadratic implementation
of the ACTRAN FEM brick is more efficient but, as foreseen, tremendous
numerical efforts are necessary to reach acceptable error levels.
These convergence curves (5.20 for 1000 Hz and 5.21 for 2000 Hz) exhibit
an asymptotic behaviour. An oscillatory convergence domain is clearly
observed at all enrichment degrees for small model sizes (below 1.10 3 unknowns,
for the 2000 Hz curve): the dynamic behaviour of the plate is not
accurately captured by coarse meshes and the increase of enrichment degree
is not sufficient to improve this fact.
For finer meshes, the behaviour is regular and an improvement of the rate
of convergence with higher polynomial enrichment is observed.
For the highest enrichment degrees, the convergence curves ends with an
increased convergence rate.
73

Laurent Hazard 21/12/2006
5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
10 0 Number of unknowns
10 −1
||e w
||
L2 (Ω)
10 −2
10 −3
FEM
p=2
p=3
p=4
ACTRAN − LIN. FEM
ACTRAN − QUAD. FEM
10 2 10 3 10 4
Fig. 5.21. Convergence curves at 2000 Hz, for a plate excited at (0.,0.). We show the performance
for different FEM implementations, and PUFEM Mindlin approximation with full polynomial
basis. We added also ACTRAN linear and quadratic elements.
74

Laurent Hazard 21/12/2006
5.2 Dynamic tests
Reduced polynomial enrichment
10 0 Number of unknowns
10 −1
FEM
p=2 reduced
p=3 reduced
p=4 reduced
ACTRAN − LIN. FEM
ACTRAN − QUAD. FEM
||e w
||
L2 (Ω)
10 −2
10 −3
10 2 10 3 10 4 10 5
Fig. 5.22. Convergence curves at 1000 Hz, for a plate excited at (0.05,0.05). We show the performance
for different FEM implementations, and PUFEM Mindlin approximation with reduced
polynomial basis. We added also ACTRAN linear and quadratic elements.
The same behaviour is observed with reduced enrichment than with full
enrichment. However, the full enrichment technique allows lower error levels
to be reached than the reduced enrichment technique.
75

Laurent Hazard 21/12/2006
5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
10 0 Number of unknowns
||e w
||
L2 (Ω)
10 −1
FEM
p=2 reduced
p=3 reduced
p=4 reduced
ACTRAN − LIN. FEM
ACTRAN − QUAD. FEM
10 −2
10 2 10 3 10 4
Fig. 5.23. Convergence curves at 2000 Hz, for a plate excited at (0.,0.). We show the performance
for different FEM implementations, and PUFEM Mindlin approximation with reduced polynomial
basis. We added also ACTRAN linear and quadratic elements.
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Laurent Hazard 21/12/2006
5.2 Dynamic tests
Number of Flops
The application of the PUFEM leads to sparse, semi-positive definite linear
systems of equations with real variables. The semi-positive character comes
from the polynomial enrichment procedure which often brings linear dependencies
between equations (see chapter 3). These properties of the system
matrix push us to choose a robust direct solver adapted to these type of matrix.
After a review of some existing codes (Matlab solver, MA47 from HSL
library, etc) and some tests, we choose the UMFPACK software, designed by
Tim Davis (see chapter 7.1.2 and reference [Dav06]).
The number of Flops (floating point operations) for the solution of a
single linear symmetric system is ≈ 1 2 N (β +1)2 +2Nβ,forN →∞(with
N the matrix size and β the matrix bandwidth, see appendix D). We should
however add two remarks concerning this topic :
• The resolution of indefinite system requires some more steps, linked to
the fact that the implementation of some black-magic pivoting algorithms
are needed. These steps are however marginal in the operations count and
we choose to ignore them in the total complexity.
• We only consider in this section the computational effort devoted to the
solution of the system matrix, and ignore the resources needed for the creation
and assembly of the matrices. Intrinsically, the polynomial PUFEM
technique involves the numerical integration of high-order polynomials
and is therefore more expensive than classical low-order FEM at this
step. However, we are interested, in our applications, by the calculation
of the response at a large number of frequencies and the additional cost of
matrix creation is small compared to the gain in the ratio cost/accuracy.
We now use the free plate vibration problem of the previous subsection
and compare the computational cost of the different solution procedures.
We present, for two meshes (6 × 4and12× 8 elements), the system size,
system bandwidth, number of flops and relative errors obtained with each
methods (see table 5.5). Three classical finite element approaches are included
for comparison purposes: both linear and quadratic ACTRAN brick-shaped
shell elements and our linear FEM plate element. The results obtained for
PUFEM Mindlin elements (full polynomial enrichment, p =2top =5)are
also included.
It is interesting to note that the bandwidth value progress regularly with
the system size when the PUFEM is used with increasing enrichment polyno-
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Laurent Hazard 21/12/2006
5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
mial order. The fill-in of the matrix does not explode and the estimated computational
cost (flops) are kept reasonable, compared to the different FEM
estimations. With the coarse mesh, a similar relative error level is obtained
with the ACTRAN quadratic elements and PUFEM p = 2, at approximately
the same cost. For a finer mesh (12×8 elements), the cost is smaller with the
PUFEM p = 2 procedure, while the error level achieved is largely favourable
to the PUFEM implementation.
Table 5.5. We present, for two meshes (6 × 4 and 12 × 8 elements), the system size, system
bandwidth, number of flops and relative errors obtained with different solution procedures.
Three classical finite element approaches are included for comparison purposes : both linear and
quadratic ACTRAN brick-shaped shell elements and our linear FEM plate element. The results
obtained for PUFEM Mindlin elements (full polynomial enrichment, p =2top =5)arealso
included.
Support mesh Solution System System Flops ‖e w‖ L2
‖e w‖ L2
type procedure size N bandwidth β ≈ at 1000 Hz at 2000 Hz
6 × 4 elements ACTRAN (linear) 210 66 91.47 × 10 4 3.9342 0.9684
ACTRAN (quad.) 663 186 22.94 × 10 6 4.8688 0.8793
FEM (linear) 105 33 11.40 × 10 4 3.7837 0.9693
PUFEM p =2 630 198 24.69 × 10 6 0.0919 0.8777
PUFEM p =3 1050 300 94.50 × 10 6 0.0226 0.7370
PUFEM p =4 1575 450 31.89 × 10 7 0.0128 0.2909
PUFEM p =5 2205 693 10.59 × 10 8 0.0094 0.2041
12 × 8 elements ACTRAN (linear) 702 108 81.88 × 10 5 1.7768 1.6253
ACTRAN (quad.) 2325 375 32.69 × 10 7 0.3074 0.8095
FEM (linear) 351 54 10.23 × 10 5 1.2697 0.7694
PUFEM p =2 2106 324 22.10 × 10 7 0.0181 0.5947
PUFEM p =3 3510 540 10.23 × 10 8 0.0093 0.1998
PUFEM p =4 5265 810 34.54 × 10 8 0.0043 0.0899
PUFEM p =5 7371 1134 94.79 × 10 8 0.0013 0.0018
78

Laurent Hazard 21/12/2006
5.2 Dynamic tests
In table 5.6, obtained for the same problem at 1000 Hz, we estimated
the computational cost of different solution procedure in order to achieve
a relative error smaller than 7%. The advantage of the polynomial PUFEM
implementation in term of computational resources is obvious when compared
to the classical FEM solution:
1. The PUFEM p = 2 configuration allows us to reach approximately the
same accuracy than the FEM approximation, at half the computational
cost.
2. With the PUFEM p = 4 configuration, we get half the error than with
FEM (lin.), at a similar computational cost.
Table 5.6. Free plate vibration problem (1000 Hz): Flops evolution for different configuration
leading to a relative error ≤ 7%.
Support mesh Solution System System Flops ‖e w‖ L2
type procedure size N bandwidth β ≈ at 1000 Hz
30 × 30 FEM (lin.) 2883 186 51.48 × 10 6 0.0641
6 × 6 PUFEM p =2 882 252 28.67 × 10 6 0.0515
4 × 4 PUFEM p =3 750 300 34.42 × 10 6 0.0322
3 × 3 PUFEM p =4 720 360 47.43 × 10 6 0.0292
79

Laurent Hazard 21/12/2006
5 CONVERGENCE OF THE PUFEM MINDLIN ELEMENT
5.3 Summary
In this chapter, we perform convergence studies of the PUFEM Mindlin plate
element and compare his performances with classical, low order, finite elements.
Two types of enrichment strategies were tested: full and reduced.
First of all, we validated the static behaviour of the PUFEM element. On
these tests, both full and reduced enrichments sets performed well, with an
advantage in accuracy for the full enrichment.
In the dynamic tests, the full enrichment clearly outperformed the reduced
enrichment technique proposed, allowing, in the different dynamic tests, better
error levels to be reached for similar system sizes.
We also evaluated the computational cost of the PUFEM technique, based
on an estimation of the number of Flops required to solve the final system of
equations. The PUFEM technique was shown to be economical, even if larger
system bandwidth are generated as opposed to FEM. Finally, we should add
that this solution procedure is a critical step for what we intend to do in the
following: we will see in the next chapters that the dynamic study of structures
involves modal extractions or direct frequency responses calculations.
Both solution procedures involves a large number of matrix manipulations
and it is therefore capital to keep the computational cost of these operations
as low as possible.
80

Laurent Hazard 21/12/2006
6
INTERFACE ELEMENT FORMULATION
At this point, we have introduced the PUFEM technique and we have developed
a Mindlin plate element based on this technique. It was demonstrated
in the previous chapter that this element could adequately be used for both
static and dynamic simulation and that it outperformed the FEM in computational
cost and accuracy. However, this element has one major shortcoming
for the applications of interest in this work: we assumed the plate to be made
of one isotropic and homogeneous material. As such, the formulation developed
for the Mindlin PUFEM element is not adapted to the modelling of
viscoelastic sandwich or patched structures.
In the most recent works, such structures are modelled by using two plate
or shell elements and bonding them with a linear brick element, that accounts
for the viscoelastic layer properties (see for instance, [PB99] and [BB02]).
In this chapter, we propose an original approach, where an interface element
is substituted to the linear brick element. The core polymer layer
between the host plate and the constraining patch plate is modelled as an
interface element, while each plate uses a PUFEM Mindlin discretisation.
The technique adopted is inspired by early work by Lin and Ko (see
[LK97], [KLC95]); the interface element philosophy has also been applied to
the modelling of active shape control of composite plates by Sun and Tong
[ST04].
6.1 Displacement field in the viscoelastic core
The polymer layer is assumed to be thin, such that we can make the assumption
that the shear and peel stresses are constant in the thickness layer.
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Laurent Hazard 21/12/2006
6 INTERFACE ELEMENT FORMULATION
The displacement of point A, placed at z = hv (on the interface between the
2
patch and the polymer) is written
⎧
⎨ u A = u p − hp
β 2 xp
v
⎩ A = v p − hp
β 2 yp
(6.1)
w A = w p
The displacement of point B, placed at z = − hv (on the interface between
2
the host plate and the viscoelastic polymer) is written
⎧
⎨ u B = u h + h h
2
β xh
v
⎩ B = v h + h h
2
β yh
(6.2)
w B = w h
In theses relations, h v , h h and h p are respectively the thickness of the viscoelastic
polymer layer, the host plate and the patch plate; u p , v p , w p , β xp and
β yp are the mid-plane displacement and rotation component for the patch.
u h , v h , w h , β xh and β yh represent the same components for the host plate.
The expression of the displacements in the polymer layer is obtained (on the
whole thickness) by
⎧
⎪⎨
⎪⎩
u v =
v v =
w v =
(
1
(
1
2 + z
h v
)
u A +
) + z
2 h v
v A +
( )
1
+ z
2 h v
w A +
(
1
(
1
2 − z
2 − z
h v
)
u B
(
1
2 − z
h v
)
h v
)
v B
w B
(6.3)
Condensing the previous relations in matrix form, we obtain the following
expression where the displacements in the polymer are expressed in terms of
displacements and rotations of the patch and host plates
{ū v } =[Q h ] {ū h } +[Q p ] {ū p } (6.4)
where {ū v } t = 〈u v v v w v 〉, {ū h } t = 〈u h v h w h β xh β yh 〉,
{ū p } t = 〈u p v p w p β xp β yp 〉 and
⎡
( )
1
− z
1
2 h v
0 0 h h − z
4 2h v
0
(
[Q h (z)] = ⎢ 1
⎣ 0 − z
1
2 h v
0 0 h h
1
0 0 − z
2 h v
0 0
⎡
( )
1
+ z
1
2 h v
0 0 −h p + z
4 2h v
0
(
[Q p (z)] = ⎢ 1
⎣ 0 + z
1
2 h v
0 0 −h p
1
0 0 + z
2 h v
0 0
82
)
− z
4 2h v
⎤
)
+ z
4 2h v
⎥
⎦ , (6.5)
⎤
⎥
⎦ . (6.6)

Laurent Hazard 21/12/2006
6.3 Stresses-strains relations in the viscoelastic core
6.2 Strains in the viscoelastic core
From our assumptions, we consider only three strain components in the viscoelastic
polymer layer. These components are expressed as follows:
ε v z = ∂w v
∂z = w p − w h
,
h v
(6.7)
γxz v = ∂u v
∂z + ∂w v
∂x = 1 (
u p − u h − h p
h v 2 β xp − h )
h
2 β xh , (6.8)
γ v yz = ∂v v
∂z + ∂w v
∂y = 1 h v
(
v p − v h − h p
2 β yp − h h
2 β yh
)
. (6.9)
Again, we condense theses relations in the following matrix expression
where {¯ε v } t = 〈 ε v z γ v xz γ v yz〉
and
{¯ε v } =[L vh ] {ū h } +[L vp ] {ū p } (6.10)
⎡
⎤
[L vh ]= 1 0 0 −1 0 0
⎣ −1 0 0 − h h
h 2
0 ⎦ ,
v
0 −1 0 0 − h h
2
(6.11)
⎡
⎤
[L vp ]= 1 001 0 0
⎣ 100− hp
0 ⎦ .
2
h v
010 0 − hp
2
(6.12)
6.3 Stresses-strains relations in the viscoelastic core
The stresses-strains relations are written in matrix form
{¯σ v } =[H v ] {¯ε v } (6.13)
with {¯σ v } t = 〈 σz v τ xz v τ yz〉 v and
⎡ ⎤
E v 0 0
[H v ]= ⎣ 0 G v 0 ⎦ . (6.14)
0 0 G v
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Laurent Hazard 21/12/2006
6 INTERFACE ELEMENT FORMULATION
6.4 Variational expressions for the viscoelastic core
The internal virtual work developed in the core layer is expressed as follows
∫
Wint core = {δ¯ε v } t [ ]
Hv
PS {¯εv }dΩ (6.15)
Ω
with [ ]
Hv
PS = hv [H v ] (PS for peel and shear). Since no forces or couples
will be directly applied to the core material, the only contribution to the
external virtual work will be brought by the inertial effects. These inertial
contributions to the external virtual work are given by
W inertial,core
ext
∫
= −
V
}
{δU v } t [ρ v ]
{Üv dΩ (6.16)
} t
with
{Üv
(3 × 3)).
= 〈uv v v w v 〉 and [ρ v ]=ρ v I 3 (I 3 is the identity matrix of size
84

Laurent Hazard 21/12/2006
6.5 Discretised variational form for the viscoelastic core
6.5 Discretised variational form for the viscoelastic
core
We use partition of unity finite element approximations for the displacements
fields in the host and in the patch. These approximations are stored in vector
form as illustrated below for the patch displacement fields
⎧ ⎧ ⎫
u p 〈Φ〉{u pn }
⎪⎨ v p
⎫⎪ ⎬ ⎪⎨ 〈Φ〉{v pn } ⎪⎬
{U p } = w p = 〈Φ〉{w pn }
β xp ⎪⎩ ⎪ 〈Φ〉{β xpn }
⎭ ⎪⎩ ⎪⎭
β yp 〈Φ〉{β ypn }
⎡
⎤ ⎧
〈Φ〉 〈0〉 〈0〉 〈0〉 〈0〉 u p1
〈0〉 〈Φ〉 〈0〉 〈0〉 〈0〉
⎪⎨ ...
⎫⎪ ⎬
=
⎢ 〈0〉 〈0〉 〈Φ〉 〈0〉 〈0〉
⎥ u pN (6.17)
⎣ 〈0〉 〈0〉 〈0〉 〈Φ〉 〈0〉 ⎦ ... ⎪⎩ ⎪ ⎭
〈0〉 〈0〉 〈0〉 〈0〉 〈Φ〉 β ypN
or
{U p } =[Φ] {U pn } (6.18)
where {U p } are the solution functions and {U pn } are the nodal variables
used for the approximation of the solutions functions. Note that the previous
expressions are written here with the same PUFEM approximations for each
displacements fields; this was only done to simplify the notations: in the
case of the Mindlin plates, the correct enrichment fields must be used for
each fields (membrane displacements, out-of-plane displacements, rotations).
By application of the virtual work principle, we obtain the general dynamic
equilibrium system
[ ] ⎧ } ⎫
[Mh + M vh ] [M vv ] t ⎨
{Ühn ⎬
=
{
{Fsh } uvw
+ {F th } uvw
{0}
{Üpn
}
[M vv ] [M p + M vp ] ⎩ ⎭
[ ]{ }
[Kh + K
+
vh ] [K vv ] t {Uhn }
[K vv ] [K p + K vp ] {U pn }
}
+
{
{Msh } βxβ y
+ {M th } βxβ y
{0}
}
. (6.19)
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6 INTERFACE ELEMENT FORMULATION
The stiffness contributions for the host, the patch and the viscoelastic
material are given by
∫
[K h ]= [B m ] t [ ∫
]
Hm
h [Bm ] dΩ + [B f ] t [ ]
Hf
h [Bf ] dΩ
Ω h
∫
+
[B c ] t [ ]
Hc
h [Bc ] dΩ
Ω h
Ω h
∫
∫
[K p ]= [B m ] t [Hm p ][B m] dΩ +
Ω p Ω p
∫
+ [B c ] t [Hc p ][B c] dΩ
[B f ] t [ H p f
]
[Bf ] dΩ
∫
[K vh ]=
Ω p
[Φ] t [L vh ] t [ H PS
v
]
[Lvh ][Φ] dΩ
Ω p
∫
[K vp ]=
Ω p
∫
[K vv ]=
[Φ] t [L vp ] t [ ]
Hv
PS [Lvp ][Φ] dΩ
[Φ] t [L vp ] t [ ]
Hv
PS [Lvh ][Φ] dΩ (6.20)
Ω p
and the mass contributions for the host, the patch and the viscoelastic material
are
∫
[M h ]= [Φ] t [ρ h ][Φ] dΩ
Ω h
∫
[M p ]=
Ω p
∫
[M vh ]=
V v
∫
[M vp ]=
[Φ] t [ρ p ][Φ] dΩ
[Φ] t [Q h (z)] t [ρ v ][Q h (z)] [Φ] dV
[Φ] t [Q p (z)] t [ρ v ][Q p (z)] [Φ] dV
V v
∫
[M vv ]=
V v
[Φ] t [Q p (z)] t [ρ v ][Q h (z)] [Φ] dV (6.21)
where [ρ h ] is the host plate density matrix, [ρ p ] the patch density matrix and
[ρ v ] the viscoelastic core density matrix (see 6.16). The second member of
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Laurent Hazard 21/12/2006
6.6 Summary
equation 6.19 contains the contributions of the external forces and couples
acting on the host plate
[F sh ]= ∫ [ ] t
Φ(uvw) {b} dΩ [Fth ]= ∫ [ ] t
Φ(uvw) {¯t} dΓ
Ω
Γ t
[ ] t
Φ(βxβy)
{m} dΩ [M th ]= ∫ [ ] t
Φ(βxβy)
{m s } dΓ
Γ t
(6.22)
[M sh ]= ∫ Ω
where [ Φ (uvw)
]
and
[
Φ(βxβ y)]
are submatrices from [Φ], related to, respectively,
the displacements (u, v, w) and the rotations (β x ,β y ).
6.6 Summary
In this chapter, we developed the interface element formulation that was
necessary to be able to benefit from the PUFEM Mindlin plate element performance
in the modelling of viscoelastic sandwich structures. This interface
element benefit from the enrichment of shape functions developed in the host
and patch plates.
The use of the interface element simply adds contributions to the stiffness
and mass matrices, without additional degrees of freedom. As such, this
technique is very economical. We will demonstrate in the next chapters the
validity of this modelling choice.
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Laurent Hazard 21/12/2006
7
DYNAMIC & ACOUSTIC ANALYSIS OF
STRUCTURES
In this chapter, we cover both the dynamic behaviour of structures and their
acoustic performances. The first section introduces techniques to analyse the
dynamic behaviour of damped structures, including viscoelastic materials.
The modal approach is presented and leads to the introduction of the modal
strain energy (MSE) technique. The direct frequency analysis is also presented.
In the second section, we present the Rayleigh integral method to account
for the acoustic propagation around the vibrating structure: we show how to
calculate the sound pressure at some points in the surrounding medium and
how to estimate quantities such as the radiated sound power level. This is
one originality of this work, since the quality of the developed products will
not only be evaluated based on damping but also on acoustic performance.
7.1 Dynamic analysis of elastic and viscoelastic
structures
This section presents the motion equations defining the modal and dynamic
behaviour of elastic plates with or without viscoelastic patches. The modal
strain energy (MSE) [JK82] method will be introduced as a tool to estimate
the equivalent modal damping. The mean square velocity will be presented
as a characteristic of the dynamic behaviour of the structure.
7.1.1 Modal analysis
The modal analysis characterises the free vibrations of a structure. It gives
the natural vibration frequencies (eigenfrequencies) and modal shapes (eigenmodes).
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7 DYNAMIC & ACOUSTIC ANALYSIS OF STRUCTURES
Equation of motion
Free vibrations of a system are obtained when no external forces are applied
to the system. The discretised form of the differential equation which
describes the free vibration of a plate (in the time domain) is
[M g ] {ü ∗ (t)} + [ K ∗ g
]
{u ∗ (t)} = {0} (7.1)
where the stiffness matrix [ Kg] ∗ is complex when the structure contains viscoelastic
materials, represented by the complex modulus approach (see section
2.2.1). The quantity {u ∗ (t)} is the displacement vector. We assume that
each non trivial solution of the previous equation can be expressed in the
harmonic form
{u ∗ (t)} = { Ψ ∗(r)} e jω∗(r) t
(7.2)
where { Ψ ∗(r)} and ω ∗(r) represent the complex eigenmode and corresponding
rth complex eigenvalue. By substituting 7.2 into movement equation 7.1, we
obtain the generalised eigenproblem
([
K
∗
g
]
− ω ∗(r)2 [M g ] ){ Ψ ∗(r)} = {0} . (7.3)
This eigenvalue problem has N solutions where N is the size of the matrices
[ ]
Kg ∗ and [Mg ]. Each solution gives both the complex eigenvalues ω ∗(r)2
and the corresponding eigenmodes { Ψ ∗(r)} of the structure. The complex
eigenvectors can be written as follow
{ } { } { }
Ψ
∗(r)
= Ψ ′ (r)
+ j Ψ ′′ (r)
(7.4)
while the complex eigenvalue can be used to calcute the complex pulsation
ω ∗(r) = ω ′ (r) + jω ′′ (r)
= ω ′ (r) √ 1+jη (r) . (7.5)
The vibration eigenfrequency of the structure is obtained as
f (r) = ω′ (r)
2π
and the modal damping factor is calculated by
(7.6)
( ) ω ′′
η (r) (r) 2
= . (7.7)
ω ′ (r)
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7.1 Dynamic analysis of elastic and viscoelastic structures
The “exact” numerical resolution of such a complex eigenvalue problem
has the important advantage to give access, in a single analysis, to both the
vibration modes and the modal damping factor of the damped structure.
However, this approach has a high computational cost, which is sometimes
up to 3 times the cost of the same structure without damping. The modal
strain energy method is presented as an alternative to this approach.
Modal strain energy method (MSE)
The modal strain energy method was developed by Johnson and Kienholz
[JK82] for the design of viscoelastic damping treatment by mean of the finite
element method, in the early 80’s. First, by extension of the Rayleigh factor
definition to the complex domain, we can express the complex eigenvalues as
follow
{ } Ψ
∗(r) t [ ]{ }
K
∗
ω ∗(r)2 g Ψ
∗(r)
=
{Ψ ∗(r) } t [ ]
M g {Ψ
∗(r)
} . (7.8)
As a first step, we neglect the loss effect of the viscoelastic material (imaginary
part of the modulus) and we only take into account its stiffness contribution
to the global stiffness matrix. This means that we suppress the
imaginary part of the stiffness matrix
[ ] [ ] [
K
∗
g = K ′ g + j
K ′′
g
]
. (7.9)
The generalised, complex eigenvalue problem 7.3 turns into a real eigenvalue
problem ([ ]
{Ψ
K ′ g − ω ′(r)2 (r)
[M g ]) } = {0} (7.10)
with real eigenvalues and real eigenvectors. An approximation of the modal
damping factor η (r) can be calculated by substituting the complex eigenmode
{
Ψ
∗(r) } by the real one { Ψ (r)} . With these assumptions, the Rayleigh factor
in relation 7.8 becomes
ω ′ (r)2 =
{ } Ψ
(r) t [
′
]{ }
K g Ψ
(r)
{Ψ (r) } t [ ]
M g {Ψ
(r)
} . (7.11)
From equation 7.5, we derive under theses assumptions that
ω ∗(r)2 = ω ( { }
′ (r)2
1+jη (r)) Ψ
(r) t [
′
]{ } { }
K g Ψ
(r) Ψ
(r) t [
′′]{ }
K
=
{Ψ (r) } t [ ]
M g {Ψ
(r)
} + j g Ψ
(r)
{Ψ (r) } t [ ]
M g {Ψ
(r)
} .
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7 DYNAMIC & ACOUSTIC ANALYSIS OF STRUCTURES
Separating the real and imaginary part, and dividing them, we find the new
expression of the modal damping factor (under MSE assumptions)
{ } Ψ
η (r)
(r) t [
′′]{ }
K
MSE = g Ψ
(r)
{Ψ (r) } t [ ]
K ′ g {Ψ
(r)
} . (7.13)
We will now explain the physical meaning for the expression of η (r)
MSE .Weconsider
the general case of a viscoelastic patched plate: the structure contains
N layers of different materials, both elastic and viscoelastic. The stiffness of
each layer is expressed as follow
] [ ]
[Kn [K ∗ ]= ′ n + j K ′′
n
[ ]
= K ′ n (1 + jη n ) (7.14)
with n the indice referring to the layer, and η n , the material loss factor of
the layer n. The global stiffness matrix is simply the sum of the stiffness
contribution of each layer
[
K
∗
g
]
=
N
∑
n=1
([ ] )
K ′ n (1 + jη n ) . (7.15)
The strain energy of layer n, associated to the eigenmode r is expressed by
V (r)
n = { Ψ (r)} t [ K ′ n] {Ψ
(r) } . (7.16)
The total strain energy of the patched plate associated with an eigenmode r
is therefore calculated as follow
V (r)
g =
N∑
n=1
V (r)
n
= { Ψ (r)} t [ K ′ g] {Ψ
(r) } . (7.17)
By combining the last equations, we can express the modal damping factor
as a function of the strain energy of each layers
η (r)
MSE = N
∑
n=1
V n
(r)
η n
V (r)
g
. (7.18)
The modal damping factor depends directly on the material loss factors and
the fraction of modal strain energy of the layers. To obtain high damping
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7.1 Dynamic analysis of elastic and viscoelastic structures
factors, it is therefore necessary to use materials with high loss factors for
the layers which undergo the largest strain energies.
In the particular case of a simple three layer sandwich, the intermediate
layer (called “core”) is made of viscoelastic material and the other layers are
considered perfectly elastic (i.e. null material loss factors). The expression
7.18 therefore becomes
η (r)
MSE = η c
V (r)
c
V (r)
g
with the core layer strain energy V (r)
c and material loss factor η c .
(7.19)
Remark
Since most viscoelastic materials are frequency dependent, the stiffness matrix
contributions related to the viscoelastic layers exhibit the same dependency.
Particular resolution procedures are therefore often needed to solve
the resulting nonlinear eigenvalue problem. The simplest one consists in an
iterative resolution of successive eigenproblems with different material input
data; starting with an estimated first eigenfrequency, we build and assemble
all problem matrices and solve the problem for the first frequency. We
then update the system data at the calculated eigenfrequency and iterate
until convergence. The same procedure has to be repeated for each modal
frequency of interest.
More advanced techniques for the resolution of the nonlinear complex
eigenvalue problem have been proposed recently by Daya et al. [DPF01][DDPF03].
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7 DYNAMIC & ACOUSTIC ANALYSIS OF STRUCTURES
7.1.2 Dynamic analysis by direct frequency response
The dynamic analysis gives the response of a structure excited by an external
force. It concerns the forced response of the structure.
Equation of motion
To determine the forced response of a plate, we must solve the following
equation of motion (in the time domain):
[M g ] {ü ∗ (t)} + [ K ∗ g
]
{u ∗ (t)} = {F ∗ (t)} (7.20)
where the stiffness matrix [ K ∗ g]
is complex when the structure contains viscoelastic
materials, represented by the complex modulus approach (see section
2.2.1). The quantity {u ∗ (t)} is the displacement vector; {F ∗ (t)} is the
external load vector. Assuming harmonic excitation and response of the form
{F ∗ (t)} = {F } e jωt ,
{u ∗ (t)} = {U ∗ } e jωt , (7.21)
we obtain the equation of motion, expressed in the frequency domain:
[Z(E ∗ i ,ω)] = [ −ω 2 [M g ]+ [ K ∗ g (ω) ]] {U ∗ } = {F } . (7.22)
The dynamic system matrix [Z(Ei ∗ ,ω)] in 7.22 exhibits two types of frequency
dependence: first, the circular frequency ω appears explicitly in the
expression [ ] and secondly, it is also present in the complex stiffness matrix
K
∗
g , through the viscoelastic material complex modulus E
∗
i . The different
contributions of the complex modulus presented in 2.2.1 allow effective representation
of the viscoelastic material behaviour law. The frequency and
temperature dependence is taken into account through the frequency shift
function influence on the master curve, denoted Ê:
E ∗ (ω, T) =Ê (a T ω) . (7.23)
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7.1 Dynamic analysis of elastic and viscoelastic structures
A convenient methodology to handle the frequency dependence in the
stiffness matrix is to consider a linear combination of constant matrices,
such as in the following relation:
[
[Z(Ei ∗ ,ω)] = −ω 2 [M g ]+[K e ]+ ∑ ]
Ê i (a T ω) [K vi(E i0 )]
(7.24)
E
i
i0
where [K e ] is the elastic part of the stiffness matrix, [K vi ] is the part associated
to the viscoelastic material with complex modulus Ei ∗ (ω, T) andE i0
is a reference value of the modulus for which the [K vi ] matrix is assembled
once. An usual choice for E i0 is the real part of Ei
∗ at a very low frequency
value (quasi-static response).
The system matrix [Z(Ei ∗ ,ω)] is symmetric and definite-positive in the
case of the finite element method. When developed with a PUFEM method,
like in our applications, the matrix is only symmetric and semi-definite positive,
since the generalised finite element technique can bring singularities
in the matrix. To solve the resulting system, we use a direct solver based
on a multi-frontal technique and that was developed to handle well semidefinite
matrices by using a sophisticated pivot selection. This solver, called
UMFPACK, was created and is maintained by Prof. T. Davis [Dav06].
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7 DYNAMIC & ACOUSTIC ANALYSIS OF STRUCTURES
7.2 Acoustic analysis
A vibrating structure, surrounded by an acoustic medium, causes perturbations
of the pressure field that can be perceived as sounds. Reciprocally, these
pressures perturbations act as a load on the structure.
In our field of applications, the medium is the air and we assume that the
influence of the medium on the structure can be neglected. We are therefore
allowed to perform an uncoupled analysis of the structural-acoustic response.
In the first part of this analysis scheme, the structural vibration response under
one or more loadings is calculated. In our case, we developed the PUFEM
elements for that specific purpose. In the second part of analysis, the free field
sound radiation associated with the structural vibration is calculated. The
normal surface velocity field from the structural model is used as input for
this second analysis. The procedure is illustrated in figure 7.1.
Fig. 7.1. Relevant properties for passive noise and vibration control.
In the present work, the analysis is restricted to baffled plates (deformable
plates inserted into infinite hard walls), for which the Rayleigh integral method
can be applied. This method is valid for planar or nearly-planar structures.
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7.2 Acoustic analysis
The acoustic wave propagation inside an homogeneous elastic medium like
air or water is defined by the classical wave equation. In the case of harmonic
time dependence, the equation reduces to the Helmholtz differential equation,
given by the following expression:
∇ 2 p (r)+k 2 p (r) =−iωρ 0 q (r) (7.25)
where p (r) is the complex pressure amplitude at position r, k is the acoustic
wave number, ρ 0 is the density of the medium and q (r) issomeexternal
volume source. The Helmholtz equation can be rewritten in an integral form
[Mig03]; the normal surface velocity field v n (r S ), defined on the surface of
the vibrating object, is related to the radiated pressure field p (r) by:
∮
α (r) p (r) =
S
{
p (r S ) ∂G (r, r }
S)
+ iωρ 0 v n (r S ) G (r, r S ) dS (7.26)
∂n
where r S is a point on the boundary S of the vibrating structure, r is a
field point in the surrounding medium and G (r, r S ) represents the Green’s
function. The angle function α (r) depends on the field point position where
the pressure is calculated (α=1 if the point lies outside the closed boundary,
and 1 if the point is on the boundary S).
2
For an object vibrating in free field condition (Sommerfeld radiation condition),
the Green function is the solution of the Helmholtz differential equation
excited by a Dirac impulse:
G (r, r S )= e−ik|r−r S|
(7.27)
4π |r − r S |
where |r − r S | is the distance between a surface point and a field point.
Once the normal velocity distribution v n (r S ) has been calculated with the
structural model, the associated pressure field can be obtained from equation
7.26.
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7 DYNAMIC & ACOUSTIC ANALYSIS OF STRUCTURES
7.2.1 Sound intensity and sound power
The sound pressure depends on the position of the receiver with respect to
the sound source. A more convenient measure to characterise the source is
the time-averaged sound power. We first establish the definition of the timeaveraged
sound intensity, in the case of harmonic time dependence :
Ī(r) = 1 2 Re (p (r) v∗ (r)) (7.28)
where v (r) is the acoustic particle velocity and the superscript ∗ denotes
the complex conjugate. The sound intensity vector Ī gives the amount and
direction of the flow of acoustic energy per unit area at a given position. The
sound power generated within a given volume of medium equals the surface
integral of the normal component of the sound intensity and is expressed as
follows :
∮
¯W =
where n (r S ) is the surface normal.
S
Ī (r S ).n (r S ) dS (7.29)
When the surface of the vibrating body is chosen for the integral evaluation,
the sound power can be evaluated by the following expression :
¯W = 1 2 Re (∮
S
)
p (r S )vn ∗ (r S ) dS . (7.30)
The sound power is often expressed in a logarithmic scale that is better
suited to the human ear sensibility. The sound power level is then defined as:
L W =10log 10
(
) ¯W
¯W ref
(7.31)
where ¯W ref is a reference sound power (standardised at 10 −12 W). The sound
power level is expressed in decibels (dB Re 10 −12 W).
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7.2 Acoustic analysis
7.2.2 Rayleigh integral method
Weessentiallyfocusinthisworkonpractically flat or plate-like structures.
When such structures are baffled, the Helmholtz integral equation 7.26 reduces
to the Rayleigh integral, given by:
p (r) = iωρ 0
2π
∫
S
v n (r S ) e−ik|r−r S|
dS. (7.32)
|r − r S |
The figure 7.2 illustrates the terms involved in the Rayleigh integral. The
baffle is an infinitely extended rigid surface around the vibrating plate. The
sound fields on both sides of the plate are equal in magnitude, with opposite
phase.
Fig. 7.2. Illustration of the Rayleigh integral terms.
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7 DYNAMIC & ACOUSTIC ANALYSIS OF STRUCTURES
The Rayleigh integral can be solved easily by considering a discretisation
of the plate into N rectangular elements, which should be small compared
to the acoustic wavelength. It is then assumed that each element support
a constant normal velocity. This is equivalent as considering that the plate
is composed of N rigid vibrating pistons that each moves with constant
harmonic velocity. The equation 7.32 can be written in the linear matrix
form:
p f = Z f v n (7.33)
where p f is a vector containing the pressure at a set of field points, v n is the
vector of normal surface velocities of the N pistons and Z f is a frequencydependent
matrix with elements of the form:
(Z f ) ij
= iωρ 0S e
2π
e −ikr ij
r ij
(7.34)
where S e is the area of an elemental piston, r ij = |r i − r j | is the distance
between a field point i and a surface point j (at the centre of the piston).
Under these assumptions, the expression of the sound power reduces to
the simple matricial expression:
¯W = S e
2 Re ( vn H p ) (7.35)
where p is the vector containing surface pressures, evaluated at the same
points of the surface as v n , and the superscript ( ) H denotes the Hermitian
transpose operator. By substitution of p = Zv n in the last expression, where
Z is the impedance matrix evaluated on the vibrating surface, the sound
power can be written as:
¯W = S e
2 Re ( v H n Zv n
)
= v
H
n Rv n . (7.36)
In equation 7.36, the matrix R = Se Re (Z) is called radiation resistance
2
matrix and can be written, after development, as follows:
⎡
sin(kr
1
12
⎤
)
kr 12
··· sin(kr 1N )
kr 1N
R = ω2 ρ 0 Se
2 sin(kr 21 )
kr 21
1 .
4πc 0
⎢
.
⎣ .
.. ⎥ . ⎦ . (7.37)
sin(kr N1 )
kr N1
··· ··· 1
The elements of the radiation resistance matrix R depend on the properties
of the acoustic medium, the frequency and the size of the plate (or
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7.2 Acoustic analysis
size of elemental pistons). In the case developed here, the radiated sound
power is evaluated on the surface of the plate. It may be calculated on any
other surface enveloping the radiating object (i.e. an hemisphere or an ellipse
enclosing the baffled plate).
From the expression of the impedance matrix Z f (see 7.34), it can be
seen that a problem may arise when the pressure is evaluated directly at the
surface of the plate: the surface impedance matrix Z has singular diagonal
elements (r ii = 0). The singularity can be avoided by replacing the diagonal
elements of the impedance matrix by the terms :
Z ii = ρ 0 c 0
(
1 − e −ik √ Se
π
)
. (7.38)
This expression comes from the impedance seen by a baffled circular piston of
area S e , moving with uniform velocity. In this case, an analytical evaluation
of the Rayleigh integral is available. Note that the singularity problem is not
encountered through the evaluation of the radiation resistance, since R is
defined only by the real part of the surface impedance matrix.
7.2.3 Structural-acoustic coupling
The normal surface velocities should be exported from the structural model
to the acoustical model. These velocities are generated at the centres of the
elemental radiators used in the discretisation of the Rayleigh integral.
An important advantage of this modelling technique is that the radiation
matrix does only depend on the geometry of the radiating object and properties
of the fluid medium. The radiation matrix therefore remains unchanged
when different patch locations or sizes are tested, as long as the radiating
surface size stays the same.
7.2.4 Radiation efficiencies
The radiation efficiency is a measure of how well the vibration object radiates
the sound energy. It is defined by the ratio of the radiated sound power per
unit area of the object to the radiated sound power per unit area of a reference
source. This reference source is a circular baffled piston, vibrating at a high
frequency (kR >> 1, where R is the effective piston radius) and with a
velocity equal to the space-averaged mean-square normal velocity 〈¯v n 2 〉 of the
object.
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The radiation efficiency can be written as follows:
σ =
¯W
ρ 0 c 0 S 〈¯v 2 n 〉 (7.39)
where S is the total area of the vibrating object. The space-averaged meansquare
normal velocity is given by the following expression:
∫
〉
2 1 〈¯v n = |v n (r S )| 2 dS = vn H 2S
Nv n. (7.40)
S
The matrix N is a real, symmetric positive definite matrix, which in our
case is simply N = 1
2N I.
By inserting equations 7.36 and 7.40 into 7.39, we get the following expression
for the radiation efficiency:
σ =
vn HRv n
ρ 0 c 0 Svn HNv . (7.41)
n
Cunefare and Currey [CC94] introduced the radiation modes and the correponding
radiation mode efficiencies as solutions of the generalised eigenvalue
problem, involving the matrices R and N, at each frequency:
λNv n = Rv n . (7.42)
The eigenvalue solution of 7.42 yields a set of real, positive eigenvalues
λ i and the corresponding eigenvectors γ i . Each eigenvector is called a radiation
mode and the corresponding eigenvalue is directly proportional to the
radiation efficiency of that mode, through the relation:
σ i =
λ i
ρ 0 c 0 S
(7.43)
where S is the total radiating surface of the plate. The eigenvalues and radiation
modes are stored, respectively, in two matrices Λ and Γ:
⎡
⎤
λ 1 0 ··· 0
0 λ 2 ··· 0
Λ = ⎢
⎣ . .
..
⎥ . . ⎦ , Γ = [ ]
γ 1 γ 2 ··· γ N (7.44)
0 0 ··· λ N
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7.2 Acoustic analysis
with normalisation such that:
Γ t NΓ = I, (7.45)
Γ t RΓ = Λ. (7.46)
It should be noted that, because the radiation resistance matrix R depends
on the frequency, this eigenvalue decomposition must be performed for each
frequency step. It results from this remark that the radiation modes and
efficiencies are also frequency-dependent.
The evolution of the radiation efficiencies of the first six radiation modes
for a baffled rectangular plate is illustrated in figure 7.3, for increasing frequencies.
The radiation efficiencies of each radiation mode is plotted against
kl y ,wherel y is the length of the plate (l x =0.2 m and l y =0.28 m).
10 0 kl y
[/]
10 −2
(1)
Radiation efficiency σ [/]
10 −4
10 −6
10 −8
10 −10
(2)
(3)
(4)
(5)
(6)
10 −12
10 −1 10 0 10 1
Fig. 7.3. Radiation efficiencies of the first six radiation modes of a rectangular baffled plate, as
a function of kl y. The first curve above, tagged (1), is the first radiation mode, and so on.
These results were obtained with a discretisation of 20 × 20 elemental
radiators.
It can be seen that, at low values of kl y (low frequencies), the radiation
efficiencies decreases quickly with increasing radiation mode number. Clearly,
the first radiation modes are dominant and the higher could be neglected
since they do not contribute significally to the radiated sound power. The six
most efficient radiation mode shapes are shown in figure 7.4, at low frequency
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7 DYNAMIC & ACOUSTIC ANALYSIS OF STRUCTURES
(kl y =0.05). In this frequency region, the shape of the most efficient radiation
mode is a piston-like mode, i.e. the surface moves with uniform velocity. The
radiation mode shape become more and more complex as the mode number
increase, showing more and more oscillations.
At higher frequencies, typically for kl y > 2π, when the plate length exceeds
the acoustic wavelength, the radiation efficiencies of all radiation modes
become significant (see figure 7.3). It is no more possible to distinguish strong
and weak radiation modes. The radiation mode shapes change from their low
frequency shapes, specially for the first mode that is no more perfectly uniform
(see figure 7.5, the six most efficient radiation mode shapes for kl y =4).
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7.2 Acoustic analysis
Fig. 7.4. The first six radiation modes of a rectangular baffled plate, for kl y =0.05.
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7 DYNAMIC & ACOUSTIC ANALYSIS OF STRUCTURES
Fig. 7.5. The first six radiation modes of a rectangular baffled plate, for kl y =4.
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7.2 Acoustic analysis
7.2.5 Sound power expressed in terms of radiation modes
We introduce the radiation modes participation factors, which are a measure
of the amplitude of each radiation modes excited by the normal velocity
vector v n (i.e. the product of the radiation mode shape vectors with the
velocity vector). These radiation modes participation factors are stored in
vector form:
a = Γ T Nv n (7.47)
where the normalisation by the operator N follows from 7.46.
Alternatively, using this definition, the velocity response of the structure
can be developed in terms of the radiation modes:
v n = Γa (7.48)
The sound power can now be expressed in terms of the radiation modes
participation factors, as follows:
¯W = vn H Rv n
= a H Γ T RΓa
= a H Λa
N∑
= λ i |a i | 2 . (7.49)
i=1
With the last expression, we can see that the radiation modes contribute
independently to the sound power.
In the low frequency range (for kl y

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7 DYNAMIC & ACOUSTIC ANALYSIS OF STRUCTURES
7.3 Summary
In this chapter, we presented the different approaches to study the dynamic
behaviour of elastic structures equipped with viscoelastic patches. From a
vibration point of view, the modal approach was introduced together with
the Modal Strain Energy technique to enable the estimation of the modal
loss factor of structures. The direct frequency response approach was also
presented. Both can be used to gain information on the patched structure
behaviour and are complementary.
The modal approach gives the modal frequencies, mode shapes and the
modal loss factor for the modes inside a certain frequency range. This information
is useful because it immediatly gives the whole picture of what is
going on and how the structures behaves. The modal analysis is also independant
of the sollicitation type or location.
The direct frequency response approach, when applied to a set of frequencies,
gives the response of the patched structure to a certain sollicitation, in
the form of frequency response functions (FRF’s). It is evident that this
response is the greatest at the modal frequencies.
In the applications treated in following chapters, both techniques will be
used, depending on the type of indicator that is selected for the validation.
We presented also a technique to calculate the acoustic propagation
around a vibrating object. This technique, based on the Rayleigh integral,
is limited to the simulation of nearly-planar, baffled structures, radiating in
a semi-infinite, light medium (ie. air). This acoustic propagation method allows
the calculation of both pressure at any point in the fluid surrounding the
structure and total radiated sound power, for instance. This last indicator
will be used for optimisation purposes in chapter 11.
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8
APPLICATIONS
This chapter covers both numerical and experimental validation of the
proposed formulations. These applications illustrate the advantages of the
PUFEM Mindlin approach and validate the interface element technique for
the behaviour of the viscoelastic core. In all the examples presented in this
chapter, the material behaviour of this viscoelastic material is modelled by
a complex modulus using tabulated frequency dependent storage and loss
module, obtained from the polymer manufacturers.
Although the direct use of the complex modulus requires expensive direct
frequency analysis, it seems to be the correct procedure to validate our
numerical models. Using directly the measured complex modulus of the viscoelastic
material and the direct frequency analysis to generate frequency
response functions, we avoid any approximation errors inherent to other representation
methods of the viscoelastic material properties (like fractional
derivatives models, see 2.3.3) or to approximate eigenmodes extraction techniques
(see remark in section 7.1.1).
8.1 Two bonded plates with structural adhesive
To validate the interface element technique developed in chapter 6, we consider
different configurations where two plates are bonded together with a
structural adhesive. The case was studied by Lin and Ko in [LK97].
8.1.1 Description
A square aluminium plate is locally reinforced by the bonding of a central
aluminium patch. Different patch sizes are considered. The host plate itself
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8 APPLICATIONS
is clamped at all edges. The geometric data of the problem are presented in
figure 8.1; the material properties for the aluminium and the adhesive are
given in tables 8.1 and 8.2.
Fig. 8.1. Geometric data for the Lin and Ko bonded plates problem : the squared patch plate is
centred relatively to the host plate, with a = 508 mm, b = 3a mm. The thicknesses are given by
7
h h = h p =5.08 mm and h a =0.1016 mm.
Table 8.1. Aluminium properties.
Symbol Property Value
E Young modulus 6.895 × 10 10 Pa
ν Poisson ratio 0.3
ρ Density 2770 kg/m 3
Table 8.2. Structural adhesive properties.
Symbol Property Value
E Young modulus 3.068 × 10 9 Pa
ν Poisson ratio 0.3485
ρ Density 332.4 kg/m 3
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8.1 Two bonded plates with structural adhesive
8.1.2 Models
Lin and Ko published numerical results obtained with an FE-based technique
in the form of non-dimensional natural frequencies for different configurations
of plates. They developed an eight-node isoparametric plate element based
on the first-order shear deformation theory. Their element was used to model
the adherends, and a 16-node interface element is used to account for the
adhesive bond.
We introduce first some quantities that are used in the following result
presentation. The number Γ is the surface ratio between the patch surface
and the host plate surface; with A p , the patch area and A h , the host plate
area, we introduce the definition
The normalised natural frequency ¯ω is obtained by
Γ = A p
A h
. (8.1)
¯ω = ω ω 0
(8.2)
where ω is the non-dimensional natural frequency of a patched plate configuration,
corresponding to a pulsation Φ (expressed in s −1 ); ω 0 is the
non-dimensional natural frequency for the equivalent unpatched plate, corresponding
to the pulsation Φ 0 (expressed in s −1 ). The non-dimensional quantities
are obtained with the expressions :
ω 0 = Φ 0 h h
√ ρ
G ,
ω = Φh h
√ ρ
G
(8.3)
where ρ, G and h h are, respectively, the density, shear modulus and thickness
of the host plate, in all configurations.
Our PUFEM model for the host plate is a square of 7 × 7elements(see
figure 8.2). The patch itself is modelled by a mesh of 3 × 3elements(forthe
Γ = 9 configuration) for a total of 58 elements. A full enrichment basis p =2
49
(see chapter 5) is used for this example. A total of 80 nodes are involved in the
model, leading to a final system size of 1600 unknowns (which corresponds
to a computational complexity of 3.5911 × 10 8 flops).
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Fig. 8.2. Mesh for the Lin and Ko bonded plates problem : the squared patch plate is centred
relatively to the host plate, with a = 508 mm, b = 3a mm. 7
8.1.3 Results and conclusions
The results obtained with our PUFEM approach are compared with those
of Lin and Ko in table 8.3 . The agreement showed is very satisfying. It
is worthwhile to give some comments about the physical meaning of this
numerical experiment. The numerical results are consistent with the physics
of the bonded assembly: larger patch sizes make the assembly stiffer, therefore
leading to an increase of the natural frequencies. It is also worth to note that
the frequencies obtained for Γ = 1 are close to those obtained for a single
unpatched plate of thickness h h =10.16 mm.
The effect of the modulus of the adhesive was also studied by Lin and
Ko. Table 8.4 shows the evolution of the natural frequencies ¯ω for different
adhesive stiffness, in the configuration Γ = 9 . Once again, the agreement
49
between the published results of Lin and Ko and our PUFEM model is good.
Obviously, the natural frequencies of the patched plate decrease with the
decrease in stiffness of the adhesive.
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8.1 Two bonded plates with structural adhesive
Table 8.3. Natural frequencies of different patch configurations for clamped boundary conditions.
The results in the first column are those obtained and published by Lin and Ko. The second column
(italic) contains results obtained with our PUFEM model.
Patched plates
Modes
Γ 1 2and3 4
9/49 1.1334 1.1213 2.2361 2.1917 3.8636 3.7813
25/49 1.3904 1.4106 2.8764 2.9186 4.2028 4.2525
49/49 1.9851 1.9915 4.0497 4.0563 6.0125 6.0276
Unpatched plates
h h =5.08 mm 1.0000 1.0000 2.0488 2.0459 3.0749 3.0283
h h =10.16 mm 1.9898 1.9931 4.0641 4.0646 6.0392 5.9833
Table 8.4. Natural frequencies for one patched configuration (Γ = 9 ), under clamped boundary
49
conditions and for different adhesive modulus (Ea
new = M × E a,whereM is a parameter used
to scale the adhesive modulus, and E a is the normal value of the adhesive modulus from table
8.2). The results in the first column are those obtained and published by Lin and Ko. The second
column (italic) contains results obtained with our PUFEM model (p = 2 full).
Modulus of adhesive
Modes
M 1 2and3 4
1 1.1334 1.1213 2.2361 2.1917 3.8636 3.7813
0.1 1.1188 1.1068 2.2057 2.1603 3.8138 3.7317
0.01 1.0552 1.0477 2.0939 2.0562 3.6036 3.5306
0.001 0.9301 0.9271 1.9439 1.9192 3.2467 3.1567
0.0001 0.8599 0.8575 1.8522 1.8369 3.0607 2.9744
To complete this part of the validation, a simulation of the Γ = 9 49 configuration
was performed with the commercial FE code ACTRAN [Fre05].
The figure 8.3 shows the frequency response functions obtained with both
our PUFEM model and the ACTRAN model (21213 unknowns, corresponding
to a computational complexity of 2.3390 × 10 10 flops). The agreement
between both methodologies is satisfying, with a lower computational cost
and increased accuracy for the PUFEM model.
In figure 8.4, we illustrate the contours of out-of-plane displacement and
in-plane displacement vectors for both the host plate and the patch, in the
case of the Γ = 9 configuration and for the fourth mode of vibration of the
49
bonded plate. It can be seen that the patch plate follows the displacement
of the host plate and that in-plane displacements are generated inside both
adherends due to the transmission of efforts through the bonding interface.
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10 −3 Frequency [Hz]
10 −4
Actran model
PUFEM model
10 −5
Displacement [m]
10 −6
10 −7
10 −8
10 −9
10 −10
0 100 200 300 400 500 600 700 800 900 1000 1100
Fig. 8.3. Reinforced plate 09/49 (Ko and Lin) : frequency response functions obtained with the
polynomial PUFEM technique and ACTRAN FEM model. The excitation point is the same as
the measurement point and is located at (0.1088;0.1088).
Out−of−plane w − host (3D)
x 10 −4
x 10 −4
4
0.5
Out−of−plane w − host (2D)
4
3
0.5
In−plane displacements in host plate
x 10 −4 0 0.1 0.2 0.3 0.4 0.5
5
0
−5
−10
1
0.5
0 0
0.5
1
3
2
1
0
−1
−2
−3
−4
−5
0.4
0.3
0.2
0.1
0
0 0.2 0.4
2
1
0
−1
−2
−3
−4
−5
0.4
0.3
0.2
0.1
0
Out−of−plane w − patch (3D)
x 10 −4
6
4
2
0
−2
−4
0.4
0.4
0.2
0.2
0 0
x 10 −4
4
3
2
1
0
−1
−2
−3
Out−of−plane w − patch (2D)
0.5
0.4
0.3
0.2
0.1
0
0 0.2 0.4
4
3
2
1
0
−1
−2
−3
0.5
0.4
0.3
0.2
0.1
0
In−plane displacements in patch plate
x 10 −4 0 0.1 0.2 0.3 0.4 0.5
Fig. 8.4. Reinforced plate Γ = 9 (Lin and Ko) : out-of-plane and in-plane displacements for the
49
host plate (top line) and patch plate (bottom line). For a structural adhesive, the patch follow
the displacements of the host plate.
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8.2 Moreira, Rodrigues and Ferreira validation
8.2 Moreira, Rodrigues and Ferreira validation
Before conducting our own experiments, we wish to assess the validity of our
numerical technique on published experimental data. This section is based
on two articles by Moreira and Rodrigues [MR04] [MRF06], in which they
test different approaches for the modelling of Constrained Layer Damping
treatments. The dissipative material used is the 3M Scotchdamp viscoelastic
tape. This product consists of a pressure-sensitive viscoelastic polymer (3M
ISD112 formulation) and an aluminum foil constraining layer. In our simulation,
the VEM material is modelled by its complex modulus, as described
by the parametric model given in section 2.3.3. In this application, we only
consider full viscoelastic sandwich plates.
8.2.1 Description of the test bench and specimens
The specimens studied are aluminium sandwich plates with different layer
thicknesses. All plates have the same size : 298 ×198 mm 2 . All geometric and
material data for the plates are summarised in tables 8.5 and 8.6.
The respective layer thicknesses are illustrated in figure 8.5. The first
two specimens share the same host and patch plates (1 mm thick) but have
different core thicknesses. The last specimen (number 3) is based on a thicker
host plate (2 mm) and a thinner patch plate.
These specimens were supported by rubber bands to simulate the free-free
boundary conditions.
1 mm
0.125 mm 1 mm
1 mm 0.250 mm
1 mm
1 mm 0.125mm 0.250mm
Fig. 8.5. Illustrations of the layer thicknesses for each specimens. The core layer is represented
in black color.
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Table 8.5. Geometric data for Moreira and Rodrigues specimens.
Geometry Specimen 1 Specimen 2 Specimen 3
Length a 298 mm 298 mm 298 mm
Width b 198 mm 198 mm 198 mm
Thicknesses H 1 1mm 1mm 2mm
H 2 0.125 mm 0.250 mm 0.125 mm
H 3 1mm 1mm 0.250 mm
Table 8.6. Aluminium properties (Moreira and Rodrigues specimens).
Symbol Property Value
E Young modulus 6.95 × 10 10 Pa
ν Poisson ratio 0.32
ρ Density 2710 kg/m 3
8.2.2 Experimental measurements
For the experimental test bench, Moreira et al. used a measuring mesh of
25 points, defined on each tested specimen. The samples were excited at one
point (located at x=146.5 mm, y=94 mm, with the origin of the coordinate
system chosen at the lower left corner of the plate) by a magnetic shaker,
driven by random signal in the frequency range [1-400 Hz]. The velocity at
each point of the measurement mesh was acquired by using a laser Doppler
vibrometer. A dynamic signal analyser was used to record both excitation
and response signals. The velocity response signal was differentiated and the
accelerance frequency response functions were evaluated for different mesh
points.
All the experimental results presented here were obtained at room temperature,
close to 17.5 0 C.
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8.2 Moreira, Rodrigues and Ferreira validation
8.2.3 Results
We present here the experimental FRFs measured at point 17 (same as excitation
point), along with numerical simulations curves (see figures 8.6, 8.7
and 8.8).
Moreira and Rodrigues used these experimental data to validate two different
numerical approaches. The numerical results obtained are superposed
to the experimental curves in the figures 8.6 to 8.8.
For specimen 1 and 2, the numerical simulation used by Moreira et al. is
based on a generalised layerwise finite element that is detailed in [MRF06].
According to the layerwise theory (see [Red04]), the sandwich structure is
divided into layers, each one being treated individually as a thick plate, under
Mindlin’s assumption. Continuity assumptions are imposed at interfaces for
the respective displacement fields.
For specimen 3, their numerical solution was obtained with a mixed model
of plate and solid brick elements. Plates elements are used for the aluminium
host plate and patch plates, taking into account the offset of the thickness
to the contact distance with the solid brick elements, used to model the
viscoelastic core behaviour. This last model results in coincident nodes and
translational degrees of freedom for the plates and the adjacent face of the
solid element.
Our PUFEM model (p = 2, full) is based on a mesh of 8 × 6elements
for each aluminium layers, for a total of 96 elements. The system matrix size
is 4032 and the system bandwidth is 997, which leads to a estimate of the
computational complexity of 2.160 × 10 9 flops.
For all simulations, the viscoelastic material is considered as an elastic
material with a complex modulus of elasticity (complex modulus approach).
8.2.4 Conclusions
The correlation between experimental and numerical solutions is excellent,
both for the frequency response function (acceleration) and for the signal
phase (figures 8.6, 8.7 and 8.8). The weaker correlation in the very low frequency
range is due to irregularities in the experimental data which are
brought by rigid body modes at low excitation frequencies. These are only
present in the experimental data and do not affect the quality of the validation.
Some discrepancies between our numerical solutions and the results
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100
Acceleration [m/s²]
10
1
Experimental data [Moreira et al., 2004]
FE model [Moreira et al., 2004]
PUFEM model
0.1
0 100 200 300 400
π
Phase
0
-π
0 100 200 300 400
Frequency [Hz]
Fig. 8.6. Frequency response functions for specimen 1: experimental data obtained by Moreira
et al., FE simulation by Moreira et al. and our PUFEM results.
obtained by Moreira et al. are imputable to the difference in material properties
used for the viscoelastic core. No exact data was given in the papers
by Moreira et al. and the complex modulus tables where generated by the
parametric model given in section 2.3.3.
Nevertheless, the correlation is excellent and our approach is shown to
compete well with other recent modelling techniques.
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8.2 Moreira, Rodrigues and Ferreira validation
100
Acceleration [m/s²]
10
1
Experimental data [Moreira et al., 2004]
FE model [Moreira et al., 2004]
PUFEM model
0.1
0 100 200 300 400
π
Phase
0
-π
0 100 200 300 400
Frequency [Hz]
Fig. 8.7. Frequency response functions for specimen 2: experimental data obtained by Moreira
et al., FE simulation by Moreira et al. and our PUFEM results.
100
Acceleration [m/s²]
10
1
Experimental data [Moreira et al., 2004]
FE model [Moreira et al., 2004]
PUFEM model
0.1
0 100 200 300 400
π
Phase
0
-π
0 100 200 300 400
Frequency [Hz]
Fig. 8.8. Frequency response functions for specimen 3: experimental data obtained by Moreira
et al., FE simulation by Moreira et al. and our PUFEM results.
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8.3 Wang and Wereley experiment
The original experimental test was performed by Wang and Wereley (see
[WVW00][Wan01][WWC02]) for the validation of their numerical approach
(based on the assumed modes method) on clamped sandwich plates specimens.
8.3.1 Description of the experimental setup and specimens
In the experimental setup developed by Wang and Wereley, the specimens
are clamped at two parallel edges along the y direction, while edges parallel
to the x direction are left free. These specimens are placed horizontally, with
a suspended shaker for the excitation of the plate. This configuration was
chosen to minimise the influence of the shaker on plate vibrations: it should
excite dynamically the sample at the required frequency without restraining
its motions. The output force of the shaker is transmitted to the plate
via a stiff steel rod and through a load cell which provides the magnitude
of the excitation input force. The rod is positioned exactly normal to the
plate. A non-contact laser vibrometer is used to measure the vertical plate
displacements.
The sample host plate is 304.8 mm (12 in.) long and 254 mm(10 in.) wide
with a thickness is 1.51765 mm (0.05975 in.). This host plate is partly covered
by a viscoelastic patch. The passive treatment is centred on the plate, from
x = 101.6 mm(4in.)tox = 203.2 mm (8 in.). The constraining layer is
101.6 mm (4 in.) long, 254 mm (10 in.) wide and 0.381 mm (0.015 in.) thick.
The viscoelastic core is made of 3M ISD112 polymer with a thickness of
0.0508 mm (0.002 in.). The base plate and the constraining layer are made
of aluminium 6061T6 (properties are detailed in table 8.7, since they differ
slightly from the one used previously). The VEM complex modulus data is
obtained from the parametric model described in section 2.3.3.
Table 8.7. Aluminium 6061T6 properties (Wang and Wereley specimens).
Symbol Property Value
E Young modulus 6.8 × 10 10 Pa
ν Poisson ratio 0.3
ρ Density 2470 kg/m 3
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8.3 Wang and Wereley experiment
8.3.2 Model description
The complex modulus data for the ISD112 VEM is generated at ambient temperature,
to be as close as possible to the experimental conditions described
in the article [WVW00].
The PUFEM mesh for the host plate contains 12 × 8elements.Thesame
mesh is used for the naked plate calculation and for the patched plate calculation.
The patch itself is modelled by 4 × 8 elements for a total of 128
elements. The support mesh is illustrated in figure 8.9. For this analysis, we
choose an enrichment of the type p = 2 reduced, which means that different
enrichment basis were used for the transverse displacement and for the rotations.
A total of 162 nodes are involved in the model, for a final system size
of 2916 unknowns.
0.25
0.2
0.15
0.1
0.05
0
0 0.1 0.2 0.3
Fig. 8.9. Support mesh used for PUFEM simulation of the Wang and Wereley test. The central
area represents the constraining layer mesh.
8.3.3 Results and conclusions
The frequency response functions for the naked host plate and the damped
specimen are showed in figures 8.10 and 8.11. The numerical results obtained
by Wang and Wereley [WWC02] with their assumed modes approach are
shown for completeness. The contours plots of out-of-plane displacement, for
the patched specimen, are illustrated in figure 8.12. The modes shape are
well captured by our technique and correlate well with both experimental
and numerical data from Wang and Wereley.
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10 −2 Receptance FRF Wang−Wereley naked plate
10 −3
PUFEM Model
Experimental (Wang and Wereley 2001)
Assumed modes (Wang and Wereley 2001)
Amplitude [m/N]
10 −4
10 −5
10 −6
10 −7
10 −8
50 100 150 200 250 300 350 400 450
Frequency [Hz]
Fig. 8.10. Naked aluminium plate. Frequency response function for the displacement of a
point located on the plate at. The excitation point is located at (x, y) = (269.810 −3 , 50.810 −3 )
mm=(7.5, 1 32 ) inches.
10 −4 Receptance FRF Wang−Wereley plate with PCLD
PUFEM Model
Experimental (Wang and Wereley 2001)
Assumed modes (Wang and Wereley 2001)
Amplitude [m/N]
10 −5
10 −6
80 100 120 140 160 180 200 220 240 260 280 300
Frequency [Hz]
Fig. 8.11. Aluminium plate with partial constrained layer damping treatment. Frequency response
function for the displacement of a point located on the host plate at. The excitation point
is located at (x, y) = (269.810 −3 , 50.810 −3 ) mm=(7 9 , 9 29 ) inches.
16 32
A good correlation is found for the frequency response functions of the
naked host plate. The correlation of the frequency response for the damped
panel relies on the viscoelastic material data that we got from the manufacturer.
It is understood that this can be a source of discrepancy since the
exact material properties for the tested samples are not available. Nevertheless,
based on these assumptions, the trend of the damped response is well
captured. We can conclude that the PUFEM approach is well adapted to the
modelling of damped sandwich plates.
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0.5
0.5
0.5
0.5
0.5
0.25
0.5
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0.5
0.5
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8.3 Wang and Wereley experiment
0.25
0.75
0.75
0.5
0.25
0.2
0.15
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Mode 1 : (a) EXP. 83.1 Hz; (b) AMM 87.8 Hz; (c) PUFEM 87.1 Hz
0.25
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Mode 2 : (a) EXP. 104.9 Hz;(b) AMM 108. Hz; (c) PUFEM 108.2 Hz
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Mode 3 : (a) EXP. 218.9 Hz; (b) AMM 220.7 Hz; (c) PUFEM 217.2 Hz
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Mode 4 :(a) EXP. 234.2 Hz; (b) AMM 241.2 Hz; (c) PUFEM 239.3 Hz
0.25
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Mode 5 : (a) EXP. 277.8 Hz; (b) AMM 285.6 Hz; (c) PUFEM 278.7 Hz
Fig. 8.12. Contour plots from experimental measurements (EXP., left column), numerical solution
obtained with the assumed modes method by Wang and Wereley (AMM, middle column)
and PUFEM model (PUFEM, right column).
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8 APPLICATIONS
8.4 Calculation of modal loss factors
The previous applications showed the ability of the PUFEM approach to
reproduce the vibration behaviour of patched structures. For some analysis,
modal loss factors are evaluated. It is therefore interesting to validate our approach
on the prediction of modal loss factors. The demonstration is inspired
by the work from Johnson and Kienholz [JK82] and Soni [Son81].
8.4.1 Description
The closed-form solution of Soni for natural frequencies and modal loss factors
of sandwich beams has been employed by Johnson and Kienholz as a
test of their finite element/MSE methodology (where MSE stands for Modal
Strain Energy). The analytical solution was first developed by Soni [Son81],
leading to a sixth order differential equation.
The problem studied consists of a cantilever sandwich beam, made of two
aluminium layers separated by one viscoelastic core. The material data and
beam dimensions are listed in tables 8.8. Constant values were used for the
core shear modulus and loss factor in order to simplify the comparisons with
a single simulation run. Two different material loss factor values (η c )areused
for the analysis.
Table 8.8. Material properties and beam dimensions.
Symbol Property Value
Elastic layers
E Young modulus 6.9 × 10 10 Pa
ν Poisson ratio 0.3
ρ Density 2766 kg/m 3
t h = t p Host and patch thickness 1.524 mm
Viscoelastic layers
E c Young modulus 1794 × 10 3 Pa
ν c Poisson ratio 0.3
η c Material loss factor 0.1 or1.
ρ c Density 968.1 kg/m 3
h c Core thickness 0.127 mm
Whole beam
L Length 177.8 mm
l Width 12.7 mm
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8.4 Calculation of modal loss factors
8.4.2 Results
In table 8.9, the natural frequencies and corresponding modal loss factors
η m are computed with one PUFEM model and compared to the analytical
results obtained by Soni. We adopt the same notation as Soni and present
modal loss factors normalised with respect to the material loss factors. The
loss factors predicted by the PUFEM approach correlate perfectly with the
analytical calculations.
Table 8.9. Natural frequencies and loss factors for cantilever sandwich beam with two different
core material loss factors.
Material loss factor Mode Analytical PUFEM
η c number model
f[Hz]η m/η c f[Hz]η m/η c
1 64.1 0.282 64.2 0.281
2 296.4 0.242 297.3 0.242
0.1 3 743.7 0.154 745.8 0.154
4 1393.9 0.089 1397. 0.088
5 2261.1 0.057 2264.4 0.057
6 3343.6 0.039 3345.4 0.039
1 67.4 0.202 67.6 0.202
2 302.8 0.218 303.7 0.217
1. 3 748.6 0.150 750.7 0.150
4 1396.6 0.088 1399.7 0.088
5 2262.9 0.057 2266.2 0.057
6 3345. 0.039 3346.6 0.039
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8 APPLICATIONS
8.5 Summary
In this chapter, we presented different validations based on published results.
The first example treated, developed by Lin and Ko, was chosen to validate
the interface element model proposed in chapter 6. The interface element was
proven reliable in a vibration plate problem.
The second problem covered in this chapter consists in the virtual study of
experimental measurements performed by Moreira et al. on sandwich plates
with viscoelastic cores. The PUFEM simulated frequency response correlated
well with the experimental data (both amplitude and phase), in a frequency
range from 0 to 400 Hz. The agreement is fine for all sandwich configurations
tested.
The third application consisted in the reproduction of an experimental
test developed by Wang and Wereley. An aluminium plate, clamped at two
opposite sides, is covered by a central patch. Wang et al. used this test to
validate their own numerical simulation tool, based on the assumed modes
method. We were able to reproduce the results measured by Wang et al. with
our PUFEM technique.
Finally, the last application completed the validation procedure by demonstrating
the ability of our technique to predict efficiently the modal loss factors
on a simple application. The validation was performed by comparison
with analytical predictions.
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9
EXPERIMENTAL APPLICATION: PLATE
TESTS AT OPTRION S.A.
The experimental measurements of the vibration response of structures
equipped with viscoelastic patches was one of the three axis of research foreseen
for this thesis work. Although the validations in the previous chapter
already demonstrated the good behaviour and performances of the approach
adoptedinthisthesiswork,weperformedourownmeasurementsinorder
to fully master all the details of the specimen preparation and test setup.
The previous validations, based on numerical tests and experiments from
the literature, are also limited to relatively low frequencies. We want to address
higher frequencies, covering a range between 1 Hz to more than 2000
Hz.
Another objective of these tests was to elaborate an experimental strategy
for new products, that could be combined with the virtual prototyping
approach for the development of patched products.
These experimental tests were performed in the laboratory of OPTRION
S.A. [OPT06], a Belgian spin-off of the Space Centre of Liège. OPTRION
offers testing capabilities and expertise in area such as vibration analysis,
metrology and material characterisation, non destructive testing, etc. They
developed holographic interferometry equipments based on a high resolution
camera and photo-refractive crystals.
One of the objective is to couple the use of a laser vibrometer with the
holographic interferometry camera. The holographic interferometry equipment
developed by OPTRION is very useful to capture modal shapes and
can therefore help to identify potential damping patch locations. On the
other hand, this modal snapshot is taken at one particular frequency and it
is difficult to sweep a full frequency range without information of the modal
landscape of the studied system. It is therefore interesting to try to first use
the laser vibrometer technique to get frequency response functions (FRFs)
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9 EXPERIMENTAL APPLICATION: PLATE TESTS AT OPTRION S.A.
for few representative points of the product. Using this technique, the whole
range of frequencies is swept and modal frequencies appear clearly on the
plots. It is then possible to focus on these eigenfrequencies with the camera
to grab a snapshot of the mode itself.
Some practical questions should however be answered before using the
proposed methodology, like:
• does the laser light from the vibrometer interfere with the camera
• do we need reflective powder everywhere on the specimen for the holographic
recording
• is the holographic technique usable on free-free specimens
In the first rounds of the tests, we tried to answer theses questions, in order
to define an experimental strategy for the study of viscoelastic damped
products.
In this chapter, we first present the experimental setup that was used
to carry on our own frequency response measurements. Secondly, we reproduce
numerically the different configurations of the tests with our PUFEM
approach and start by a correlation with results obtained with the commercial
software ACTRAN. We end this chapter with the correlation with the
experimental data and conclusions.
9.1 Experimental setup
Two types of setup have been adopted for the testing of the samples. The first
one was developed to simulate free-free conditions (or FFFF). This setup allows
the simulation of the vibration behaviour of the specimens without influence
of any boundary conditions: we try as much as possible to let the sample
vibrate just like it would if it was floating in the air without constraints. The
second setup reproduces partially clamped boundary conditions (or CFCF):
the two opposite short edges are clamped in a specially designed tool, while
the two other edges are left free. The two setups are defined in more details
in the following subsections.
The experimental vibrations (in the form of normal velocities) were measured
using a laser Doppler vibrometer and classical transfer frequency response
measurement methodology. A dynamic signal analyser is used to
record both input and output channels (excitation of the shaker and velocity
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9.1 Experimental setup
response from the vibrometer). The measured vibrations are characterised by
FRFs, in the form of point receptances R ik = R ik (x, x e ,s)(withs = iω), defined
as the ratios between displacement field component spectra u i (x,s), at
response point x, and corresponding point force component spectra F k (x e ,s),
at excitation point x e .
9.1.1 Description of the tested specimens
Four different specimens were tested on our test benches. They are all based
on the same host plate which is a 280 mm × 200 mm steel plate, 2 mm
thick. The patches that are bonded on the host plate are all obtained from
a 3M product called Scotch-Damp (ref. SJ2052X T1005) which is sold in the
form of tape rolls. The tape itself is made of one layer of viscoelastic material
(3M ISD112), completed with a constraining aluminium layer. The tape roll
can be bought in different widths. Our sample is 5 cm wide. The thicknesses
of the layers and extended product references are given in table 9.1. The
specimens are described below (see also figure 9.1):
1. The naked host plate alone, called NAKED configuration, in the text;
2. the host plate with an horizontal central patch (in the length of the plate),
called PA1 configuration, in the text;
3. the host plate with a vertical central patch (in the width of the plate),
called PA2 configuration, in the text;
4. the host plate with three vertical patches, called PA3 configuration, in
the text.
Table 9.1. 3M Scotch-Damp tape details.
3M Scotch-Damp SJ2052X T1005
Viscoelastic layer thickness 0.127 mm
Aluminium thickness 0.254 mm
Run batch 4028
Date of manufacturing 04/04
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9 EXPERIMENTAL APPLICATION: PLATE TESTS AT OPTRION S.A.
50 mm 00000 11111
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280 mm
(c)
Fig. 9.1. Different configurations chosen for the patched plates: (a) a single, centred, vertical
patch, called PA1; (b) a single, centred, horizontal patch, called PA2; (c) three vertical patches,
called PA3.
01
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200 mm
The locations of the excitation and measurement points are detailed in
the figure 9.2. The same locations are used for all the plates. These points
were chosen based on preliminary numerical simulations, in order to avoid
nodal vibration positions.
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9.1 Experimental setup
Point 1
00 11
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Point 2
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Point 3
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Fig. 9.2. Locations of excitation and measurement points in OPTRION tests. The excitation
point (dark circle on the plot) is located at position (35;25), the measurement points 1 to 3
are respectively located at coordinates (35;50), (70;175) and (140;150) (in a Cartesian system of
coordinate with origin placed at the lower left corner of the plate).
9.1.2 The OPTRION holographic interferometry camera
The holographic interferometry (HI) is a well known technique for measuring
the displacements of diffusive objects. It has already found applications in
many areas such as damage detection in composite assemblies, the observation
of convection in fluids or material law characterization (see [OPT06]).
The major advantages of the principle is that it is a non-contact measurement
technique and that it allow a whole field capture of the object.
The specific instrumentation developed by OPTRION is adapted to the
measurement of relatively large objects (approximately 50 x 50 cm 2 , for static
tests), as opposed to most HI equipments in the field. It also allows the
observation of the vibration modes of structures, by application of the stroboscopic
real-time holographic interferometry technique (see [GL98], [GSL01]).
The term “real-time”refers to the superposition of a hologram of the object
over the object itself during the time it is subjected to excitation.
In the basic real-time HI technique, the hologram of the object at rest is
recorded and the readout is performed when it is vibrated. This readout phase
is done in stroboscopic mode, which means that the laser beam is activated at
atimet = 0 when the object is at a maximal modal displacement, i.e. when
the speed of each point at the surface of the object is zero. An interferogram is
then formed in a photorefractive crystal, where the phase difference between
the object at rest and the highest modal deformation can be observed.
The phase-shift (PS) technique is also applied, in order to quantify the
information (displacements) from the interferometric hologram. The PS technique
consists in adding constant phase shifts in the laser beam signal and
successive acquisition of four “shifted” interferograms. The phase can then
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9 EXPERIMENTAL APPLICATION: PLATE TESTS AT OPTRION S.A.
be extrapolated from the four phase-shifted interference images recorded by
the CCD camera.
The final results, for each observed modes, consists in a raw interferogram
of the mode and the corresponding phase map (obtained after phase-shifting
calculation). The phase map can later be converted in displacement map.
The final displacement maps have a high accuracy, of approximately λ/40,
where λ is the wavelength, after correction of some errors introduced during
the stroboscopic process.
A continuous observation of the vibratory deformation patterns of the
object, during the frequency sweep, is possible; this method can be used for
the determination of the resonant frequencies but, practically, is not feasible
in most real-life situations, where high modal densities are observed. In such
cases, it is much more simple and time-effective to find the resonance frequencies
with a laser vibrometer and afterwards to record the modal patterns with
the HI camera. This is the technique that we adopted in our experimental
strategy.
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9.1 Experimental setup
9.1.3 Practical considerations
During our experimental measurement campaign, we tested some combined
measurements on plate samples (with viscoelastic patches) using both a vibrometer
equipment (Polytec) and an holographic camera (holographic interferometry).
These tests where realised to assess the compatibility of both
equipments and to develop a future combined strategy for new products evaluation.
Preparation of the specimens
The evaluated specimens are all made of steel and are lightly polished (light
mirror finish). To be effective, the holographic interferometry procedure needs
the measured samples not to be reflective. To eliminate any light reflection,
we usually cover the specimens with a white powder, conditioned in spray
(final texture close to magnesite or chalk, used in athletism).
However, this white powder coating does not reflect light enough for the
laser vibrometer. When wiped locally, to regain the normal, polished, aspect
of the steel, the improvement is small. It is therefore needed to glue a
small sample (approx. 5 × 5mm 2 ) of reflective tape on the plate, at each
measurement points.
Data acquisition system
The SigLab software guides the input to the shaker and monitor both the output
signal and the input from the Polytec tool. From these data, it generates
transfer functions in the form of frequency response functions (amplitude vs.
frequency). A periodic signal is obtained from the vibrometer, in the form
of both velocity and displacement output. The product documentation advises
us to choose the velocity, since the signal is usually cleaner than the
displacement. The recorded velocity frequency response functions are later
transformed to displacement response functions for the correlation with numerical
predictions.
When doing a frequency sweep measurement, an FFT analysis of the
signal is necessary to trace the FRF curves (velocity or displacement of one
point with frequency). These type of curves are needed for the comparison
with the simulation results as well as for the determination of the damping
of each modes.
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9 EXPERIMENTAL APPLICATION: PLATE TESTS AT OPTRION S.A.
The recording of FRFs curves is fast; it is however necessary to use different
point locations to get a picture of all the modes appearing in the
product. This procedure is still fast because the Polytec laser can simply be
re-oriented to focus at each point one after the other; the laser does not need
to be perfectly normal to the specimen plane since reflective material is used
at each measurement points.
General comments
The reflective tape (needed for the Polytec measure) does not appear on the
deformation contour from the Optrion equipment. The fringes are clean in
the area of the reflective spot. As a result, it should be fine to glue many
spots of this type on the samples.
However, when the two laser equipments were activated simultaneously,
the holographic measurement was locally perturbed by the laser spot light of
the Polytec system; this only affects a small portion of the plot. It is therefore
suggested to seal the Polytec laser during the snapshot of the Optrion system.
On the other side, it is interesting to note that the Polytec vibrometer
measurement is not affected at all by the Optrion equipment.
We have checked the coherence of both measurement techniques by evaluating
the displacement of one particular point of the plate with both methods.
The amplitude of the displacement of the plate measured by holography (estimated
by fringe counting) is of the same order as the Polytec measurement.
Finally, the holographic measurement is easily perturbed by the constant
vibration of the plate: excitation by the acoustic environment and building
activity in the neighbourhood. This can become a source of problem with
large panels.
Suggested methodology for future experimental studies
From these preliminary measurements on plate specimen, we propose to
adopt the following methodology for future panels:
• We propose to use an horizontal bench for the study of larger panels;
the panel should be positioned in such a way that there is enough free
space between it and the table for the installation of Polytec tools. The
holographic camera could then look at one side of the product from the
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9.1 Experimental setup
top while the Polytec laser look at the same product from the other side
(lower side). Doing so, both systems can be used simultaneously without
perturbing one another. Please note that it is otherwise always possible
to seal the Polytec laser for the holographic acquisition.
• The studied product should first be equipped with reflective spots, placed
at interesting measurement points (FRFs). These locations can be given
by preliminary numerical simulations. The goal is to be able to capture
the frequencies of appearance of all the modes.
• The vibrometer is then used and FRFs are obtained for all the spots.
A simple superposition of these curves can help to locate the modal frequencies
of the product. These curves can also be used for validation and
updating of the numerical models.
• Once the modal frequencies are obtained, the camera is used to get the
modal patterns at these frequencies and understand what occurs.
9.1.4 Free-Free setup (FFFF)
In this setup, a stiff square aluminium bench was build on the breadboard
of the Newport table, using aluminium structural beams and accessories; the
specimens are suspended with their face oriented vertically.
Rubber bands were attached through holes in the corners of the samples
and on the other side to the bench surrounding the studied plate, as illustrated
in figures 9.3. This setup was inspired by the articles of Moreira et al.
[MR04], Dalenbring [Dal02] and Read and Dean [RD78].
Read and Dean detail the flexural resonance technique, which is an experimental
method used to measure dynamic Young’s modulus and loss factors
of relatively stiff, low-loss materials. The technique involves the application of
a harmonic force at some point along a strip of material and the monitoring
of the vibration amplitude as a function of the forcing frequency. At so-called
resonance frequencies, maxima are observed in the vibration amplitude, corresponding
to the appearance of standing waves in the sample. The storage
modulus can be obtained from the values of the resonance frequencies and
sample dimensions. It is of particular interest to note that this technique is
often applied in free-free conditions, in order to avoid clamping errors or misalignments.
Read and Dean suggest the use of fine nylon filaments to suspend
the specimens in order to simulate free-free conditions. The attachments of
the filaments on the samples should be made at points located on nodal lines
for the different vibrational modes considered.
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9 EXPERIMENTAL APPLICATION: PLATE TESTS AT OPTRION S.A.
Fig. 9.3. Experimental setup for free-free boundary conditions.
In our setup, we choose to use rubber bands to suspend the samples
because it present different advantages:
• The elongational stiffness of rubber bands is a lot smaller than nylon or
steel strings;
• low traction forces are sufficient to stabilise the sample and avoid annoying
rigid body modes at low frequencies;
• specimens can easily be installed and removed from the bench, while good
result reproduction is achieved.
One problem that could appear with rubber bands is that some damping
can be brought to the system by the use of rubber material (Dalenbring
suspension damping [Dal02]). In our case, this damping is small compared to
the intrinseque damping of our patched samples. It will be shown that the
untreated samples do not exhibit damping in their response functions.
The table 9.2 summarises the equipment that was used for the measurement
of the FRFs.
Our test bench is installed on an optical table, including vibration isolation
systems (Newport Corporation, RS1000). The isolation system floats the
table and very low frequency disturbances from the ground floor are filtered.
Ashakerisusedastheexcitationsource. Since the sample plates are
suspended vertically in our test bench, the shaker rod is horizontal and is
positioned such as to be normal to the plate surface. The shaker itself is
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9.1 Experimental setup
Table 9.2. Experimental equipment used for the FRFs measurements at OPTRION S.A.
Excitation Equipment Settings
Response
Dynamic
signal analyser
Electro-magnetic shaker
Signal amplification for shaker: TIRA E60 0-60W
Force transducer Bruel&Kjaer type 8200
Signal amplification for force signal: B&K type 2635
Polytec vibrometer OFV502 (fiber interferometer)
Vibrometer controller OFV3001S
Siglab DSP Model 20-42,
coupled to a notebook computer
running a data acquisition software under Matlab
Range: 10mV/unit out
unit out: acc (1 m/s 2 )
or 100 N/V
Velocity output selected
Range: 25 mm/s/V
V-filter: 5kHz
Displacement range 40 µm/V
suspended in an elastic bed, made of rubber bands (see figure 9.4). This is
done to avoid as much as possible an additional stiffness contribution to the
specimens.
The rigid rod tip is glued to the surface of the plate using cyanoacrylate
glue (Cyanofix, manufacturered by Sondal). This solution was adopted after
performing some tests with a magnetised head fixed on the tip of the shaker
rod. Unfortunately, the diameter of the magnet contact area was too wide
to still be considered as a point force. With a diameter of approximately
15 mm, some vibration modes of the plates were constrained by the magnet
contact force and this phenomenon affected the quality of the recorded FRFs.
Much better results were obtained with the bonded connection: it allows some
flexibility in the assembly and does less restrain the studied samples.
On the other hand, the perpendicularity of the rod with respect to the
plate was naturally obtained with the magnetised head. Special care should
be taken when installing the shaker with the glued connection: we let the rod
just touch the surface of the plate by adjusting the position of the shaker
bed, ensuring the perpendicularity also. The glue is applied afterwards to fill
the gap between the rod tip and the surface of the sample.
The force output from the shaker is transmitted through a load cell and
a rigid rod to the specimen. The load cell provides the magnitude of force
input to the measured samples.
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9 EXPERIMENTAL APPLICATION: PLATE TESTS AT OPTRION S.A.
Fig. 9.4. Elastic rubber bed for the shaker.
9.1.5 Clamped setup (CFCF)
In this setup, illustrated in figure 9.5 and 9.6, the samples are clamped by
fixtures at two opposite sides and left free on the others. The fixtures are
designed to provide clamped boundary conditions and are made of top and
bottom steel parts, of the same size (30 cm × 8cm× 3cmthick).
A massive steel plate of large dimension (5 cm thick) is positioned vertically
on the Newport isolation table and fixed with nuts and bolts. Aluminium
bars were also added to stiffen the assembly. The two pairs of fixtures
arethenboltedontheverticalplateandservetoclamptheplatespecimens.
The methodology employed for the measurements is the same as for the
free-free setup.
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9.2 Correlation between PUFEM and ACTRAN software results
Fig. 9.5. Clamped (CFCF) bench test setup.
Fig. 9.6. Clamped (CFCF) bench test setup, when holographic measurements are going-on.
9.2 Correlation between PUFEM and ACTRAN
software results
The PUFEM technique is applied to the different plates configurations tested
at the OPTRION lab and results are compared with ACTRAN models. The
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9 EXPERIMENTAL APPLICATION: PLATE TESTS AT OPTRION S.A.
frequency response functions at the three measurements points are used for
correlation. The chosen comparison data is the normal displacement.
For each configuration, different PUFEM and ACTRAN models are built,
due to the different problem geometries. The same enrichment basis is chosen
(p = 2, full).
The FRF curves are compared with ACTRAN FRFs obtained with an
overkilled model (a converged model with a high number of degrees of freedom).
Details for each models are given next to the result figures, in the
following subsections.
For all these simulations (PUFEM and ACTRAN), we used a tabulated
representation of the VEM material, based on the 3M Scotchdamp ISD112
material evaluated at the Solvay Central Laboratory (see section 2.3.2), for
an average temperature of 20 0 C.
9.2.1 Free-Free setup (FFFF)
Naked host plate
The naked host plate serves as base plate for all configuration; it is therefore
of capital importance to be able to validate the capability of our PUFEM
approach to capture the dynamical content of this base plate. Our element
has already proven to be efficient on many examples from the literature, but
these tests were limited to relatively low frequencies. This application extend
to higher frequencies, covering a range from 1 Hz to 2000 Hz.
The ACTRAN mesh resolution is valid up to approximately 4000 Hz
(based on the estimation of the flexural wave length in the plate and application
of a conservative rule of the thumb). It is therefore an overkilled model
and is used here as a solution reference.
Table 9.3. ACTRAN and PUFEM models characteristics for OPTRION naked configuration.
Models Host plate Matrix problem Bandwidth Estimated
mesh size (N) flops
PUFEM model (p =2full) 80 elements 1980 324 1.06 × 10 8
ACTRANmodel(quad.FE)1120 elements 24549 1215 1.82 × 10 10
As can be seen on the figures, the correlation between the PUFEM and
ACTRAN naked plate simulation is nearly perfect. The modes are well captured
and appear at nearly the same frequencies.
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9.2 Correlation between PUFEM and ACTRAN software results
10 0 Point 1
10 −2
10 −4
Amplitude [m]
10 −6
10 −8
10 −10
10 −12
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(a)Point1atlocationx =35.10 −3 m, y =50.10 −3 m.
10 0 Point 2
10 −2
10 −4
Amplitude [m]
10 −6
10 −8
10 −10
10 −12
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(b) Point 2 at location x =70.10 −3 m, y = 175.10 −3 m.
10 −2 Point 3
10 −3
10 −4
Amplitude [m]
10 −5
10 −6
10 −7
10 −8
10 −9
10 −10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(c)Point3atlocationx = 140.10 −3 m, y = 150.10 −3 m.
Fig. 9.7. Comparison between ACTRAN (dashed line) and PUFEM (p=2 full, continuous line)
simulation for the OPTRION naked plate configuration (all three measurement points).
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9 EXPERIMENTAL APPLICATION: PLATE TESTS AT OPTRION S.A.
Patched plate PA1
For this PA1 configuration, the base plate is the same as the one used for the
naked plate simulation and counts 80 elements. The patch itself is modelled
by 16 elements and interface elements. In this configuration, the material
properties for the interface elements introduce damping in the model.
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0 0.05 0.1 0.15 0.2 0.25
Fig. 9.8. Mesh used as support for the PUFEM model (PA1 configuration). Host plate is modelled
by 10 × 8elements.
Table 9.4. ACTRAN and PUFEM models characteristics for PA1 OPTRION configuration.
Models Mesh Matrix problem Bandwidth Estimated
size (N)
flops
PUFEM model (p =2full) 96 elements 2520 756 7.258 × 10 8
ACTRAN model (quad. FE) 1440 elements 29193 1374 2.767 × 10 10
10 0 Point 1
10 −1
10 −2
10 −3
Amplitude [m]
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(a)Point1atlocationx =35.10 −3 m, y =50.10 −3 m.
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9.2 Correlation between PUFEM and ACTRAN software results
10 −1 Point 2
10 −2
10 −3
Amplitude [m]
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(b) Point 2 at location x =70.10 −3 m, y = 175.10 −3 m.
10 −2 Point 3
10 −3
10 −4
Amplitude [m]
10 −5
10 −6
10 −7
10 −8
10 −9
10 −10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(c)Point3atlocationx = 140.10 −3 m, y = 150.10 −3 m.
Fig. 9.9. Comparison between ACTRAN (dashed line) and PUFEM (p=2 full, continuous line)
simulation for the PA1 OPTRION plate configuration (all three measurement points).
The correlation of results for both simulation models is again perfect. The
frequency response curves are close to each other and the damping is well
captured by the interface element procedure.
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9 EXPERIMENTAL APPLICATION: PLATE TESTS AT OPTRION S.A.
Patched plate PA2
For this PA2 configuration, the base plate is the same as the one used for
the naked plate simulation and counts 80 elements. The patch itself is modelled
by 20 elements and interface elements. Again, in this configuration,
some damping is brought to the model by the specifications of the material
properties for the interface elements.
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0 0.05 0.1 0.15 0.2 0.25
Fig. 9.10. Mesh used as support for the PUFEM model (PA2 configuration). Host plate is
modelled by 10 × 8elements.
Table 9.5. ACTRAN and PUFEM models characteristics for PA2 OPTRION configuration.
Models Mesh Matrix problem Bandwidth Estimated
size (N)
flops
PUFEM model (p =2full) 100 elements 2640 590 4.641 × 10 8
ACTRAN model (quad. FE) 1680 elements 31995 2007 6.457 × 10 10
10 0 Point 1
10 −1
PUFEM Mindlin p=2 full (2640 dof’s)
ACTRAN model (31995 dof’s)
10 −2
Amplitude [m]
10 −3
10 −4
10 −5
10 −6
10 −7
10 −8
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(a)Point1atlocationx =35.10 −3 m, y =50.10 −3 m.
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9.2 Correlation between PUFEM and ACTRAN software results
10 −1 Point 2
10 −2
10 −3
Amplitude [m]
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(b) Point 2 at location x =70.10 −3 m, y = 175.10 −3 m.
10 −2 Point 3
10 −3
10 −4
Amplitude [m]
10 −5
10 −6
10 −7
10 −8
10 −9
10 −10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(c)Point3atlocationx = 140.10 −3 m, y = 150.10 −3 m.
Fig. 9.11. Comparison between ACTRAN (dashed line) and PUFEM (p=2 full, continuous line)
simulation for the PA2 OPTRION plate configuration (all three measurement points).
The correlation of results for both simulation models is again fine. The
frequency response curves are close to each other and the damping is well
captured by the interface element procedure.
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9 EXPERIMENTAL APPLICATION: PLATE TESTS AT OPTRION S.A.
Patched plate PA3
For this PA3 configuration, the base plate is finer than the one used for
the naked plate simulation and counts 96 elements alone. The patches are
modelled by 3 × 16 elements and interface elements, for a grand total of 144
elements in the PUFEM model. Again, in this configuration, the damping is
brought to the model by the specifications of the material properties for the
interface elements.
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0 0.05 0.1 0.15 0.2 0.25
Fig. 9.12. Mesh used as support for the PUFEM model (PA3 configuration). Host plate is
modelled by 12 × 8elements.
Table 9.6. ACTRAN and PUFEM models characteristics for PA3 OPTRION configuration.
Models Mesh Matrix problem Bandwidth Estimated
size (N)
flops
PUFEM model (p =2full) 144 elements 3960 752 1.128 × 10 9
ACTRAN model (quad. FE) 2240 elements 40437 2235 1.012 × 10 11
The correlation of results for both simulation models is fine. The frequency
response curves are close to each other and the damping is well captured
by the interface element procedure, even in this configuration where
the damping is high.
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9.2 Correlation between PUFEM and ACTRAN software results
10 0 Point 1
10 −1
10 −2
Amplitude [m]
10 −3
10 −4
10 −5
10 −6
10 −7
10 −8
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(a)Point1atlocationx =35.10 −3 m, y =50.10 −3 m.
10 −1 Point 2
10 −2
10 −3
Amplitude [m]
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(b) Point 2 at location x =70.10 −3 m, y = 175.10 −3 m.
10 −2 Point 3
10 −3
10 −4
Amplitude [m]
10 −5
10 −6
10 −7
10 −8
10 −9
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(c)Point3atlocationx = 140.10 −3 m, y = 150.10 −3 m.
Fig. 9.13. Comparison between ACTRAN (dashed line) and PUFEM (p=2 full, continuous line)
simulation for the PA3 OPTRION plate configuration (all three measurement points).
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9 EXPERIMENTAL APPLICATION: PLATE TESTS AT OPTRION S.A.
9.2.2 Clamped setup (CFCF)
Only the naked and PA1 plates were evaluated experimentally with the
clamped setup (CFCF). Only these two configurations were also used for
correlation with the ACTRAN software. The correlation is done in the frequency
range from 1 to 1000 Hz, because of the lack of experimental data
beyond that point.
Naked host plate
The PUFEM mesh is the same as the one used for the naked plate simulation
in free-free conditions and counts 80 elements.
Point 2
10 −3 Frequency [Hz]
ACTRAN model
PUFEM Mindlin p=2 full
10 −4
Amplitude [m/N]
10 −5
10 −6
10 −7
10 −8
0 100 200 300 400 500 600 700 800 900 1000
(a)Point2atlocationx =70.10 −3 m, y = 175.10 −3 m.
Point 3
10 −3 Frequency [Hz]
10 −4
10 −5
Amplitude [m/N]
10 −6
10 −7
10 −8
10 −9
10 −10
10 −11
0 100 200 300 400 500 600 700 800 900 1000
(b) Point 3 at location x = 140.10 −3 m, y = 150.10 −3 m.
Fig. 9.14. Comparison between ACTRAN (dashed line) and PUFEM (p=2 full, continuous line)
simulation for the OPTRION naked plate configuration with CFCF clamped boundary conditions
(only point 2 and 3).
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9.2 Correlation between PUFEM and ACTRAN software results
A good agreement is obtained between the PUFEM model and ACTRAN
simulation.
Patched plate PA1
For this PA1 configuration, the same mesh is used that previously for the
free-free conditions (96 elements in total).
Point 2
10 −3 Frequency [Hz]
ACTRAN model
PUFEM Mindlin p=2 full
10 −4
Amplitude [m/N]
10 −5
10 −6
10 −7
10 −8
0 100 200 300 400 500 600 700 800 900 1000
(a)Point2atlocationx =70.10 −3 m, y = 175.10 −3 m.
10 −3 Point 3
10 −4
PUFEM Mindlin p=2 full
ACTRAN model
10 −5
Amplitude [m/N]
10 −6
10 −7
10 −8
10 −9
10 −10
0 100 200 300 400 500 600 700 800 900 1000
Frequency [Hz]
(b) Point 3 at location x = 140.10 −3 m, y = 150.10 −3 m.
Fig. 9.15. Comparison between ACTRAN (dashed line) and PUFEM (p=2 full, continuous line)
simulation for the PA1 OPTRION plate configuration with CFCF clamped boundary conditions
(only point 2 and 3).
A good agreement is again obtained between the PUFEM model and
ACTRAN simulation; the damping effect is well apparent and well captured
by both numerical approaches.
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9.3 Correlation with experimental measurements
In this section, we present a correlation between numerical predictions obtained
with the PUFEM technique and the experimental measurements made
at OPTRION laboratory.
9.3.1 Free-Free setup (FFFF)
During the experimental measurement sessions, the free-free setup was used
in a frequency range from 50 to 2050 Hz. For all configuration, the quality
of the recorded signals was good in the whole range and the correlation is
therefore performed up to 2000 Hz.
Naked host plate
The correlation is fine up to approximately 1000 Hz. After that, there is
a decay in the predicted modal peeks and the experimental results; this
discrepancy is brutal and does not seem to increase regularly with frequency:
after 1000 Hz, the predicted modal peeks appear at lower frequencies than
those measured. Beside this phenomenon, the trend of the physical behaviour
is still perfectly predicted by the numerical model.
10 0 Point 1
10 −2
Experimental measurement [Hazard, 2006]
PUFEM Mindlin p=2 full
10 −4
Amplitude [m/N]
10 −6
10 −8
10 −10
10 −12
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(a)Point1atlocationx =35.10 −3 m, y =50.10 −3 m.
For point 3, the experimental curve is a noisy between 400 and 800 Hz.
This is due to the fact that this point should be on nodal lines for many modes
in this range of frequencies. The experiment still capture some movement in
the vicinity of the measurement point but not accurately.
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9.3 Correlation with experimental measurements
10 0 Point 2
Experimental measurement [Hazard, 2006]
PUFEM Mindlin p=2 full
10 −2
Amplitude [m/N]
10 −4
10 −6
10 −8
10 −10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(b) Point 2 at location x =70.10 −3 m, y = 175.10 −3 m.
10 −2 Point 3
10 −3
Experimental measurement [Hazard, 2006]
PUFEM Mindlin p=2full
10 −4
Amplitude [m/N]
10 −5
10 −6
10 −7
10 −8
10 −9
10 −10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(c)Point3atlocationx = 140.10 −3 m, y = 150.10 −3 m.
Fig. 9.16. Comparison between experimental measurement (Hazard, 2006) and numerical simulation
with PUFEM Mindlin elements (p=2 full), for the naked OPTRION configuration.
Globally, the agreement between PUFEM and experiments is very satisfactory
for the naked host plate configuration.
Patched plate PA1
Just like the naked plate, the correlation is fine up to approximately 1000 Hz.
After that, the same decay between the predicted modal peeks and the experimental
results appears. Beside this phenomenon, the trend of the physical
behaviour is perfectly predicted by the numerical model.
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10 −2 Point 1
10 −3
Amplitude [m/N]
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(a)Point1atlocationx =35.10 −3 m, y =50.10 −3 m.
10 −1 Point 2
10 −2
10 −3
Amplitude [m/N]
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(b)Point2atlocationx =70.10 −3 m, y = 175.10 −3 m.
For point 3, the experimental curve is again noisy between 400 and 800
Hz. This is due to the same reasons as for the naked plate alone.
Globally, the agreement between PUFEM and experiments is very satisfactory
for this PA1 plate configuration.
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9.3 Correlation with experimental measurements
10 −2 Point 3
10 −3
10 −4
Amplitude [m/N]
10 −5
10 −6
10 −7
10 −8
10 −9
10 −10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(c)Point3atlocationx = 140.10 −3 m, y = 150.10 −3 m.
Fig. 9.17. Comparison between experimental measurement (Hazard, 2006) and numerical simulation
with PUFEM Mindlin elements (p=2 full), for the PA1 OPTRION configuration.
Patched plate PA2
Again, the correlation is fine up to approximately 1000 Hz. After that, the
same decay between the predicted modal peeks and the experimental results
appears. Beside this phenomenon, the trend of the physical behaviour is
perfectly predicted by the numerical model. The damping is well represented
in the model and the amplitude of the peeks are similar to those measured.
10 0 Point 1
10 −1
10 −2
Amplitude [m/N]
10 −3
10 −4
10 −5
10 −6
10 −7
10 −8
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(a)Point1atlocationx =35.10 −3 m, y =50.10 −3 m.
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10 −1 Point 2
10 −2
10 −3
Amplitude [m/N]
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Fequency [Hz]
(b)Point2atlocationx =70.10 −3 m, y = 175.10 −3 m.
10 −2 Point 3
10 −3
10 −4
Amplitude [m/N]
10 −5
10 −6
10 −7
10 −8
10 −9
10 −10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(c) Point 3 at location x = 140.10 −3 m, y = 150.10 −3 m.
Fig. 9.18. Comparison between experimental measurement (Hazard, 2006) and numerical simulation
with PUFEM Mindlin elements (p=2 full), for the PA2 OPTRION configuration.
For point 3, the experimental curve is again noisy between 400 and 800
Hz. This is due to the same reasons as for the naked plate alone.
Globally, the agreement between PUFEM and experiments is very satisfactory
for this PA2 plate configuration.
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9.3 Correlation with experimental measurements
Patched plate PA3
Again, the correlation is fine up to approximately 1000 Hz. After that, the
same decay between the predicted modal peeks and the experimental results
appears. Beside this phenomenon, the trend of the physical behaviour is
perfectly predicted by the numerical model. This configuration exhibit the
highest damping of all tested specimens. The damping is well represented
in the model and the amplitude of the peeks are similar to those measured,
except for the frequencies above 1700 Hz.
10 0 Point 1
10 −1
10 −2
Amplitude [m/N]
10 −3
10 −4
10 −5
10 −6
10 −7
10 −8
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(a)Point1atlocationx =35.10 −3 m, y =50.10 −3 m.
10 −1 Point 2
10 −2
10 −3
Amplitude [m/N]
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(b) Point 2 at location x =70.10 −3 m, y = 175.10 −3 m.
In this configuration with lot of damping, the signal for point 3 is less
noisy than for the other tested specimens.
Globally, the agreement between PUFEM and experiments is again very
satisfactory for this PA3 plate configuration.
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10 −2 Point 3
10 −3
10 −4
Amplitude [m/N]
10 −5
10 −6
10 −7
10 −8
10 −9
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Frequency [Hz]
(c) Point 3 at location x = 140.10 −3 m, y = 150.10 −3 m.
Fig. 9.19. Comparison between experimental measurement (Hazard, 2006) and numerical simulation
with PUFEM Mindlin elements (p=2 full), for the PA2 OPTRION configuration.
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9.3 Correlation with experimental measurements
9.3.2 Clamped setup (CFCF)
In this clamped setup, the experimental results were noisy above approximately
1000 Hz. We choose therefore to limit the validation to the [0. Hz;
1000. Hz] frequency range, which still cover a larger spectrum than the published
validations in the literature.
Naked host plate
The presented results for the naked plate, with CFCF clamping boundary
conditions, are limited to point 2 and 3, following the availability of experimental
data.
Point 2
10 −3 Frequency [Hz]
PUFEM Mindlin p=2 full
Experimental measurement [Hazard, 2006]
10 −4
Amplitude [m/N]
10 −5
10 −6
10 −7
10 −8
0 100 200 300 400 500 600 700 800 900 1000
(a)Point2atlocationx =70.10 −3 m, y = 175.10 −3 m.
10 −3 Point 3
10 −4
PUFEM Mindlin p=2 full
Experimental measurement [Hazard, 2006]
10 −5
Amplitude [m/N]
10 −6
10 −7
10 −8
10 −9
10 −10
0 100 200 300 400 500 600 700 800 900 1000
Frequency [Hz]
(b) Point 3 at location x = 140.10 −3 m, y = 150.10 −3 m.
Fig. 9.20. Comparison between experimental measurement (Hazard, 2006) and numerical simulation
with PUFEM Mindlin elements (p=2 full), for the naked OPTRION configuration.
The correlation of the numerical results with experiments is excellent.
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Patched plate PA1
The correlation of numerical results with experimental data is again excellent
for this clamped PA1 configuration.
Point 3
10 −3 Frequency [Hz]
10 −4
10 −5
Amplitude [m/N]
10 −6
10 −7
10 −8
10 −9
10 −10
0 100 200 300 400 500 600 700 800 900 1000
Point 3 at location x = 140.10 −3 m, y = 150.10 −3 m.
Fig. 9.21. Comparison between experimental measurement (Hazard, 2006) and numerical simulation
with PUFEM Mindlin elements (p=2 full), for the PA1 OPTRION configuration.
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9.4 Summary
9.4 Summary
This chapter focused on the experimental validation of the numerical tools developed
in the previous chapters: PUFEM Mindlin plate element, viscoelastic
interface element and dynamical analysis tools.
We first presented the evaluated specimens and the experimental setups
(for free-free conditions and CFCF clamping). The methodology and equipment
has been extensively introduced. Special care was taken, especially for
the free-free conditions, in order to obtain clean data up to the highest possible
frequency (2050 Hz, for the FFFF condition, and 1000 Hz for the CFCF
condition). The evaluated frequency range is larger than the one already
avalaible in the publications for the validation of viscoelastic damping systems.
The validation was performed in two steps: we first confronted the
PUFEM simulation tool to a well known finite element software called AC-
TRAN, developed by FFT SA. Both numerical results are in good agreement,
for all specimen configurations and boundary conditions.
In a second step, we compare the PUFEM results to collected experimental
data. The agreement was again excellent.
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10
DAMPING SANDWICH DESIGN RULES
FROM PARAMETRIC STUDIES
In this section, parametric studies are performed using the two-dimensional
Q4-PUFEM element presented in chapter 3 (for 2D plane stress applications).
This element is well adapted to the modelling of sandwich beams.
The problem studied is the Oberst test, where a slender beam sample of the
sandwich material is clamped at one side and excited harmonically at the
other (see illustration 10.1). This test is widely used in industry to account
for the damping properties of sandwich assemblies and materials and is described
in details elsewhere (refer, for instance, to the American norm ASTM
E756-98 [AST98]).
10.1 Sandwich Oberst beam
The sandwich beam is made of two steel plates separated by a dissipative
layer with a constant complex Young modulus E ∗ = 1794 × 10 3 (1 + η c )and
a loss factor η c =1.
We parameterise the section of the beam (see figure 10.2) as follows: the
thickness of each layer of steel equals 2e, while the thickness of the damping
polymer equals 2h. The study consists in varying the factor h , while keeping
e
the total thickness of the sandwich constant and equal to 1 mm. This factor
represents the ratio of polymer with respect to steel (in thickness).
Frequency response plots can be drawn to illustrate the influence of the
factor on the damping and on the amplitude displacement at the tip of the
sample beam (see figure 10.3). From such simple plots, it is apparent that
the increase of the thickness factor (i.e. increasing the polymer thickness
w.r.t. the steel thickness, for the same total thickness of 1 mm) improves the
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10 DAMPING SANDWICH DESIGN RULES FROM PARAMETRIC STUDIES
Fig. 10.1. Oberst test for a viscoelastic sandwich beam.
Fig. 10.2. Sandwich beam section used for Oberst test.
damping of the beam. However, only a parametric study can tell us if there
is an optimum value.
In figure 10.4, we show the evolution of the modal loss factor for each
modes and for different values of the thickness factor h . The modal loss factor
e
(or modal damping factor) is calculated using the expression 7.19 developed
in chapter 7.
This parametric study clearly shows that there is an optimal ratio that
leads to the most efficient use of the dissipative material, i.e. the largest
modal loss factor values are not obtained for the largest quantity of dissipative
material. For the problem considered, and for a constant total thickness
of 1 mm, the optimal factor is close to 3, which means that for a full sandwich
beam the thickness of the damping material should be three times the
thickness of one single steel layer. For larger thickness ratio, the modal loss
factor has a tendency to reach an asymptot or even to decrease.
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10.1 Sandwich Oberst beam
Fig. 10.3. Frequency response function of Oberst beam for two different ratios h e .
0.7
Modal loss factor w.r.t. thickness ratio (sandwich, total thickness = 1mm)
0.6
0.5
Global system loss factor
0.4
0.3
0.2
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
0.1
0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Factor h/e
Fig. 10.4. Modal loss factor for the first five modes of the full sandwich beam, as a function of
the thickness ratio h e . 163

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10 DAMPING SANDWICH DESIGN RULES FROM PARAMETRIC STUDIES
10.2 Partial sandwich Oberst beam
We now cover the case of partial damping treatments. Clearly, the damping
of such configurations will be dependent not only of the thickness of the
layers composing the patch, but also on the position of the patch itself on
the beam and on its dimensions.
We consider here only one configuration, with a central patch of half the
length of the beam and we would like to know if, like for the full sandwich
beam, there exists an optimal thickness ratio between the viscoelastic layer
and the constraining layer of the patch. The same material properties as for
the full sandwich beam are used in this parametric study.
The partial sandwich beam is represented in figure 10.5. A cut in the
central position of the beam shows a section that is illustrated in figure 10.6.
The mesh used for our numerical simulations is also plotted in figure 10.7.
Fig. 10.5. Partial sandwich beam for Oberst test.
Fig. 10.6. Sandwich beam section used for Oberst test (partial beam covering).
The parametric study was performed for three different total patch thicknesses
(i.e. thickness of the viscoelastic layer + thickness of constraining
layer). The thickness factor, in this case, is formed by the ratio of the polymer
thickness to the constraining plate thickness (again noted h , see figure
e
10.6).
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10.2 Partial sandwich Oberst beam
Fig. 10.7. Mesh of the partial sandwich beam used in Oberst test configuration.
For a total patch thickness of 4 mm (see figure 10.8) and considering only
the three first modes, there is clearly an optimum for the modal loss factor,
that is obtained for a much smaller value of the parameter h (around 0.5).
e
For lower values of the total patch thickness (see, respectively, figures 10.9
and 10.10 for total thicknesses of 2 mm and 1 mm), the optimum values of
decrease progressively.
h
e
0.09
0.08
Modal loss factor w.r.t. thickness ratio (base thickness=1 mm, patch thickness=4mm)
Mode 1
Mode 2
Mode 3
0.07
0.06
Modal loss factor
0.05
0.04
0.03
0.02
0.01
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Factor h/e (patch configuration)
Fig. 10.8. Modal loss factor for the first five modes of the partial sandwich beam, as a function
of the thickness ratio h . Total patch thickness = 4 mm.
e
It is therefore apparent that the optimum thickness ratio is tightly dependent
of the total patch thickness. Often, this total patch thickness will be
governed by constraints of manufacturing, aesthetics or space clearance constraints.
For given total patch thickness, it is then possible to perform such
simple parametric studies to find the optimal ratio of polymer to constraining
layer.
It is also worth remarking that the modal loss values obtained with this
partial sandwich treatment are a lot smaller than for the full sandwich configuration.
This is not only due to the smallest amount of material but depend
largely on the fact that, for the full sandwich beam, in Oberst configuration,
both the upper and lower plates are clamped in the tool: this generates
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10 DAMPING SANDWICH DESIGN RULES FROM PARAMETRIC STUDIES
0.05
0.045
Modal loss factor w.r.t. thickness ratio (base thickness=1 mm, patch thickness=2mm)
Mode 1
Mode 2
Mode 3
0.04
0.035
Modal loss factor
0.03
0.025
0.02
0.015
0.01
0.005
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Factor h/e (patch configuration)
Fig. 10.9. Modal loss factor for the first five modes of the partial sandwich beam, as a function
of the thickness ratio h . Total patch thickness = 2 mm.
e
0.06
Modal loss factor w.r.t. thickness ratio (base thickness=1 mm, patch thickness=1mm)
Mode 1
Mode 2
Mode 3
0.05
0.04
Modal loss factor
0.03
0.02
0.01
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Factor h/e (patch configuration)
Fig. 10.10. Modal loss factor for the first five modes of the partial sandwich beam, as a function
of the thickness ratio h . Total patch thickness = 1 mm.
e
higher shear in the viscoelastic core when sollicitated in vibration than for the
general case, where the patch constraining plate does not support boundary
conditions (floating constraining plate).
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10.3 Summary
10.3 Summary
In this chapter, we presented some parametric studies that can be performed
on simple 2D models, using the Q4-PUFEM element introduced in chapter
3. It was shown that, for a full sandwich configuration, these simple models
can effectively guide the engineer in his selection of the optimal thickness
ratio between the plates (host and constraining plates) and the viscoelastic
core.
In the case of partial covering, for a certain base plate thickness, it is also
possible to reach an optimal ratio between the thickness of the constraining
plate and the thickness of the core material. This optimal ratio is however
largely dependent on the allowed total thickness of the patch. We can also
remark that the study was done for one particular position and size of the
patch. The optimal thickness ratio will actually be dependent on these two
parameters.
This last remark lead us to the question of the optimisation of all the
parameters linked to the design of a single patch (or multiple patches) for a
structure: choice of materials (core, constraining layer), position, size, form
and thickness of each layers.
Material manufacturers often sell their damping material in the form of
tapes, such as the 3M ScotchDamp products or Avery-Dennison specialty
tapes. The choice of the viscoelastic material has already been covered in
chapter 2. For the constraining material, weight and stiffness considerations
often conduct these manufacturers to the choice of aluminium. Different types
of tapes exists, with different polymers (tuned for certain temperature-range
efficiency or certain environment characteristics such as humidity or chemical
aggressivity) and limited thickness ratio. These products should cover the
widest customer’s needs.
We therefore suggest the following global design strategy:
1. Select a tape product, from-the-shelf, based on simple design rules (focus
shear energy inside the core layer, 2D parametric studies illustrated in
this chapter) and other constraints (relations with manufacturer, costs,
availability, performance, manufacturability, etc.).
2. Perform optimisation study of position, size with the previous properties.
3. Once an optimal solution is found, eventually perform a sensibility study
for the thicknesses to see if the selected tape product is definitely well
adapted.
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Laurent Hazard 21/12/2006
11
OPTIMISATION
The word “optimisation” is defined, in common language, as the act of rendering
optimal, or the procedure used to make a design or a system as effective
or functional as possible. In our work, we are concerned with the optimal design
of structures on an acoustic level, with the help of viscoelastic sandwich
damping devices that we called damping patches (see chapters 1, 8 and 9).
This means that we want to bring to our product the best “acoustic quality”
possible, with limited expenses. Naturally, we should define accurately
what we call “acoustic quality”, since this terminology covers a broad range
of domains, often involving the physiological perception of sounds by humans
(see part on psychological acoustics in [Cro98]). Without entering into details,
it is clear that some sounds can be annoying to human ears (a flying
mosquito during a good night sleep, for instance), even at very low noise levels,
and that others are perceived as symbols of quality (car door slamming
noise can be modelled to reflect a good “old and heavy” door noise). In this
work, we only address the problem of making the structures less noisy, and
the acoustic quality is directly extrapolated from the radiated sound power
levels, without any filtering or weighting of the data.
11.1 Numerical optimisation in acoustics
In this chapter, we develop numerical optimisation strategies to solve such
industrial design optimisation applications as:
• “My actual product exhibits annoying noise levels when excited locally
around a certain frequency. I am allowed to spend a certain amount of
money to design a specific damping treatment to solve this issue. What
type of patch should I choose and where should I bond it on the product”
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11 OPTIMISATION
• “For a given (single) patch geometry (including thickness ratio) and a
given host structure geometry, where should I locate this patch in order
to reduce the radiated sound power the most, in a certain frequency range,
for a random excitation”
To answer these practical questions by numerical optimisation, one has
first to answer some crucial questions linked to modelling and computational
ressource issues.
We can roughly consider that there are three levels of acoustic optimisation.
The first approach avoids the use of dedicated acoustic models or tools,
and consists simply in making conclusions based on the vibrational behaviour
of structures. In simple words, we consider that if we lower the amplitudes
of vibration in a certain frequency range, the acoustic noise level will also
be decreased. This kind of approach is used when we address the problem of
defining optimal design rules for viscoelastic sandwich structures, with the
maximisation of the system loss factor as goal.
The second acoustic optimisation approach integrates the acoustic medium
in the model. This can be done with a fully coupled strategy, building models
that integrates the vibro-acoustic behaviour and solve the problem in one
step, for all solid and acoustic unknowns.
In this work, we developed an uncoupled strategy, adapted to our type of
problems and cheap to solve. We first solve the dynamic problem, including
the structure only, for the surface velocities and import these data into a
model of acoustic propagation. The validity of this strategy is limited to light
acoustic medium like air and heavy structures, for which there is a very small
influence of the fluid on the dynamic behaviour of the solid. These approaches
are more exact than the purely vibrational approach, in the sense that they
deliver directly acoustic indicators like sound pressures at some points above
the radiating product or total radiated sound power.
The third option for acoustic optimisation of products consists in using
the second approach (coupled or uncoupled vibro-acoustic calculation) and
post-processing the results using psychological and physiological rules to define
the acoustic quality. This refers to the domain of the psycho-acoustics
(part XII in [Cro98]) and will not be covered in this work.
11.1.1 Literature review
Numerical optimisation in the field of vibration and acoustics is not new and
we can find references to research publications dating back to the 70’s...
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11.1 Numerical optimisation in acoustics
To the author’s knowledge, the papers by Olhoff [Olh70][Olh74] concerning
the optimal design of vibrating plates, published in 1970 and 1974, were
the first one in the field of numerical optimisation of vibro-acoustics problems.
Olhoff attempted to maximise the fundamental frequencies, in a given range,
of circular and rectangular simply supported plates, by acting on thickness.
Lyon et al. [LMP73] studied both theoretically and numerically the noise
radiated by helicopter rotor tips. The shape of the rotor blades was optimised
to minimise the radiated sound power, using a steepest-descent algorithm.
Early work also includes the articles by Lang and Dym [LD74] and Lalor
[Lal79]. The first authors used a pattern search algorithm to optimise transmission
loss properties of sandwich panels. The thicknesses and densities of
layers were used as design variables.
Recent vibro-acoustic optimisation research
The literature on the design of quiet structures is expanding rapidly.
Naghshineh et al. [NKB92] have developed expressions for the acoustic
power in function of surface velocities. Optimisation then leads to a set of
velocity profiles which radiates minimal sound power. Material tailoring variables
such as thickness or density are used. Lamancusa et al. [LH94] minimised
the sound radiation of a rectangular panel, with the thickness and
composite material distribution as design variables. Several other authors
have applied optimisation methods to reduce the noise radiation via material
tailoring, including Belegundu et al. [BSK94], Hambric [Ham95], Saint-Pierre
and Koopmann [SK95], or Constans et al. [CKB98], to name a few actors in
the field. All these researchers used different types of optimisation methods,
including various gradient-based techniques and heuristic algorithms such as
simulated annealing (SA) or genetic algorithms (GA).
Optimisation of damping treatments
A number of authors developed damping layers optimisation strategies. These
are however essentially focused on the optimisation of the structural vibration
problem and do not address the full vibro-acoustic problem. In the research
work by Lumsdaine et al.[LS95] [LS98], a sequential quadratic programming
(SQP) algorithm was used to monitor finite element parametric models to
optimise unconstrained damping layers on beams and plates. Lumsdaine did
not, however, consider partial treatments but instead used the thickness of
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11 OPTIMISATION
the damping layer as variable. The system loss factor of the structures was
chosen as objective function.
Marcelin et al. [MTS92] used optimisation techniques to determine the
location and length of partial coverage damping patches. The optimisation
technique was gradient-based (MMA, for method of moving asymptots),
while the beam model was based on the finite element method. The objective
function is build from the summation of modal loss factor in a range
of frequencies. Further research work by Marcelin and Trompette [MT94],
and [Mar03] addressed the optimal location of patches on plates by use of a
genetic algorithm.
Bandini et al. [BGP02a][BGP02b] considered the problem of optimisation
of unconstrained layer damping treatments on flat plates. A genetic algorithm
strategy was chosen which monitored a finite element code (Nastran). The
objective function definition was based on the integral of the surface velocities
in a range of frequencies. Partial damping treatments patterns were obtained
for different types of boundary conditions and loadings.
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11.2 An optimisation application
11.2 An optimisation application
The first objective in a numerical analysis is to characterise the problem so
that the appropriate modelling tools and algorithms are chosen. In the case of
optimisation, the same approach should be adopted and leads to ask typical
questions like [EBA + 06]:
• Are the design variables continuous, discrete or mixed
• Is the problem constrained or unconstrained
• How expensive are the objective functions to evaluate
• Will the objective functions behave smoothly as the design variables
change
• Are the objective functions likely to be multimodal
• Is gradient data available analytically Can it be evaluated numerically
with sufficient accuracy and at reasonable cost
Often, there is not sufficient information available at hand to answer these
questions and particular problem characterisation studies are necessary: prior
to tackle the optimisation problem, parameter space exploration should be
performed through parametric studies or design of experiments methods.
We propose to introduce the different features involved in vibro-acoustic
optimisations by studying a simple problem. A rectangular steel plate (2
mm thick) such as those considered in our experiments (see chapter 9), is
clamped at two opposite sides. The dimensions of the plate are 280 mm ×
200 mm(L × H). We consider the following problem: we are allowed to paste
a single patch on the naked plate. The patch itself is rectangular and has
limited dimensions of 20%L by 20%H.
Question :
What would be the most effective position of this patch in order
to minimise the acoustic sound power radiated, under an impact
loading
11.2.1 Preliminary undamped model study
Before facing the optimisation problem, interesting data can be gathered from
a simple modal analysis of the undamped (or unpatched) plate problem.
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11 OPTIMISATION
The first five modes are represented in figure 11.1. Looking at the mode
shapes, and recalling that the damping principle behind the patches is the
localisation of shear energy in the viscoelastic core layer, we can already conclude
that locations that do not deform during excitation at modal frequencies
are not optimal. This simple observation is correlated by the literature
on the optimal placement of damping patches. Ravish S. Mati and Sainsbury
[MS04][SM06] propose to distribute partial coating on cylindrical shells over
regions of high strain energy intensity. Strain energy intensity maps are obtained
for each modes within the range of interest and combined to obtain a
global broad range indicator.
Similarly, Henderson [Hen95] perform strain measurement on the surface
of beams to provide guidance in placement of damping patches. More recently,
Balmès et al. [BG04] proposed to use an evaluation of the surface
membrane strain energy to determine candidate positions for damping treatments
on an automotive floor panel.
The strain energy of each modes is plotted in figure 11.2. By combining
them (by simple summation, as proposed by [SM06]), we obtain the plot 11.3.
In this combined plot, high strain energy spots are located in the four corner
points of the plate. These spots can be explained by the proximity of the
clamping boundary. With a refinement at the clamped edges, it is possible to
see that these spots are limited to a narrow region close to the corners (much
smaller than the patch size). These locations are therefore less interesting for
placement of damping patches than could be thought initially. Secondly, it
should be noted that floating patches would not be able to exploit the benefit
of these positions because the constraining layer itself would not be clamped
(see also a similar remark in section 10.2).
With this remark in mind, we can determine four potentially interesting
zones for the placement of patches (close to points of coordinates (0.085; 0.),
(0.195; 0.), (0.085; 0.2) and (0.195; 0.2)). These four locations are explained
by the symmetry of the vibration problem. There is an additional potential
spot for the bonding of a patch at the center, which benefits from a high
combined strain energy value.
11.2.2 The parametric model of the patched structure
A parametric model of the patched plate problem should be built. A Matlab R○
routine is implemented that generates a new mesh for each position of the
patch. The patch is defined by the location of its center point (x c ,y c )and
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11.2 An optimisation application
x 10 −5
x 10 −5
x 10 −5
2
1.5
1.5
Mode shape − mode 1
1.5
1
0.5
Mode shape − mode 2
1
0.5
0
−0.5
−1
−1.5
Mode shape − mode 3
1
0.5
0
−0.5
−1
0
0.2
−2
0.2
−1.5
0.2
0.15
0.1
Y
0.05
0
0
0.1
X
0.2
0.3
0.4
0.15
0.1
Y
0.05
0
0
0.1
X
0.2
0.3
0.4
0.15
0.1
Y
0.05
0
0
0.1
X
0.2
0.3
0.4
f 1 = 140.79Hz
f 2 = 190.40Hz
f 3 = 388.44Hz
4
x 10 −6
4
x 10 −5
Mode shape − mode 4
2
0
−2
−4
Mode shape − mode 5
2
0
−2
−6
0.2
−4
0.2
0.15
0.1
Y
0.05
0
0
0.1
X
0.2
0.3
0.4
0.15
0.1
Y
0.05
0
0
0.1
X
0.2
0.3
0.4
f 4 = 411.97Hz
f 5 = 463.33Hz
Fig. 11.1. Mode shapes for undamped plate application.
4.4
4.8
0.2
0.18
0.16
3.8
3.6
3.4
0.2
0.18
0.16
4.2
4
0.2
0.18
0.16
4.6
4.4
0.14
3.2
0.14
3.8
0.14
4.2
0.12
3
0.12
3.6
0.12
4
0.1
0.08
0.06
2.8
2.6
2.4
0.1
0.08
0.06
3.4
3.2
3
0.1
0.08
0.06
3.8
3.6
0.04
0.02
0
0 0.05 0.1 0.15 0.2 0.25
2.2
2
1.8
0.04
0.02
0
0 0.05 0.1 0.15 0.2 0.25
2.8
2.6
2.4
0.04
0.02
0
0 0.05 0.1 0.15 0.2 0.25
3.4
3.2
3
0.2
4.8
0.2
5.2
5
0.18
0.16
4.6
0.18
0.16
4.8
0.14
4.4
0.14
4.6
0.12
0.1
4.2
0.12
0.1
4.4
4.2
0.08
0.06
4
0.08
0.06
4
0.04
3.8
0.04
3.8
0.02
0
0 0.05 0.1 0.15 0.2 0.25
3.6
0.02
0
0 0.05 0.1 0.15 0.2 0.25
3.6
3.4
Fig. 11.2. Elemental strain energy calculated for the first 5 modes of the untreated plate. The
strain energy scale is logarithmic.
its extension along the two host plate cardinal axes. In this study, there are
only two design variables for the optimisation problem which are x c and y c .
The size of the patch is fixed.
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11 OPTIMISATION
0.2
5.4
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0 0.05 0.1 0.15 0.2 0.25
5.2
5
4.8
4.6
4.4
Fig. 11.3. Elemental combined strain energy calculated for the untreated plate: Combined plot
of the first 5 modes (logarithmic scale).
11.2.3 The choice of the objective function
Objective function based on modal loss factor
All authors cited in our literature review 11.1.1 about the optimisation of
damping treatments choose the maximisation of the modal loss factor as
optimisation goal. The calculation of the quantity has already been covered
in chapter 7.
Inthecaseofbroad range frequency optimisation, the objective function
will be obtained by summation of the modal loss factors of each modes inside
the range.
Objective function based on radiated sound power
The sound power radiated from a structure depends on the excitation frequency.
In most cases, the excitation frequency varies over a certain range,
including some of the natural frequencies of the structure. Therefore, our
objective will be to minimise the total sound power radiated over the whole
range considered. Normally, the total power in the range should be obtained
by integrating the power versus frequency curve between the extremities of
the band. However, it is clear that the major contributions to the total radiated
power are mainly coming from the power at resonance frequencies.
Therefore, we can assume that the total power over the range of interest
can be approximated by the sum of the power at each resonance frequencies
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11.2 An optimisation application
within the range [BSK94]. Practically, it is not convenient to limit the optimisation
to a predefined, fixed, frequency range. In some cases, for particular
configurations of the variables, one mode might jump outside the predefined
range and suddenly will no more contribute to the total sound power. Therefore,
from an industrial and practical point of view, we find more adequate
to consider as an objective function, the sum of the contribution to the total
sound power of a fixed number of modes. The acoustic sound power radiated
over the plate can be obtained from the velocity amplitudes at its surface,
as developed in chapter 7. Hence, the first optimisation problem considered
is expressed as follows:
∑
Minimise L Combined
#modes
W =10log 10 (
vH n Rv n
),
¯W Ref
Subject to
∀x ∈ R n
x L ≤ x ≤ x U
(11.1)
using the notations of section 7.2.2.
The only constraints are defined through the definition of upper and lower
bounds on the design variables. These bounds avoid the patch to fall out of
the host plate surface.
The broad range acoustic sound power level, or combined sound power
level is defined by the sound power level of the summation of each modal
sound power (10log 10 ( W 1+W 2 +...
¯W ref
)), where ¯W ref is a reference sound power
(standardised at 10 −12 W).
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11 OPTIMISATION
11.2.4 Parametric studies
Since we have exposed the different levels of vibro-acoustic optimization in
section 11.1, we find interesting to oppose the different approaches on this
simple example.
Parametric study based on modal loss factor
The purely vibrational approach is chosen: the patch (of dimensions 20%L by
20%H) is moved over the host plate and the modal loss factor is plotted for
each modes (figure 11.4). The objective function, in this case, is the modal
loss factor of the system and the goal would be to maximise its value for
a mode or the sum of its values for all modes considered in the frequency
range.
The result plot forms the response functions of the modal loss factor,
where for each evaluated patch location, we plot the corresponding loss factor
in ordinate against the position of the center of the patch in X− and
Y −abscisses. In our case, we face a broad range optimisation problem and
the response surface would be obtained by summation of the five modal response
surfaces, as illustrated in figure 11.5. We can see that the response
surface is non-convex and multimodal, while rather smooth. These aspects
will guide our choice of an optimisation strategy.
Another important conclusion that can already be drawn is that the optimal
locations for the patches appear to be the same as previously foreseen
from the undamped plate strain energy extraction (see section 11.2.1). This
confirms our feeling that the calculation of the strain energy distribution is
already a good indicator of the potential patch locations and can certainly
be used to reduce the design space for further optimisations. By doing so,
the multimodal aspect of the response surfaces would almost vanish.
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11.2 An optimisation application
x 10 −3
0.02
14
0.018
0.025
0.016
0.015
12
Modal loss factor − mode 1
0.02
0.015
0.01
0.005
0
−0.005
0.2
0.15
0.1
0.05
Position on Y−axis
0
0
0.05
0.2
0.15
0.1
Position on X−axis
0.25
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
Modal loss factor − mode 2
0.01
0.005
0
0.2
0.15
0.1
0.05
Position on Y−axis
0
0
0.05
0.2
0.15
0.1
Position on X−axis
0.25
10
8
6
4
2
0.02
0.02
0.025
0.018
0.025
0.018
Modal loss factor − mode 3
0.02
0.015
0.01
0.005
0.016
0.014
0.012
0.01
Modal loss factor − mode 4
0.02
0.015
0.01
0.005
0.016
0.014
0.012
0.01
0
0.2
0.008
0
0.2
0.008
0.15
0.1
0.05
Position on Y−axis
0
0
0.05
0.2
0.15
0.1
Position on X−axis
0.25
0.006
0.004
0.15
0.1
0.05
Position on Y−axis
0
0
0.05
0.2
0.15
0.1
Position on X−axis
0.25
0.006
0.004
x 10 −3
x 10 −3
14
16
Modal loss factor − mode 5
14
12
10
8
6
4
2
0.2
12
10
8
6
0.15
0.1
0.05
Position on Y−axis
0
0
0.05
0.2
0.15
0.1
Position on X−axis
0.25
4
Fig. 11.4. Response function of modal loss factors for the first five modes. The parameter space
was sampled with discrete positions (represented by red dots on the graphs) and a surface was
fit to match all the points.
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0.06
Combined modal loss factor − first five modes
0.07
0.06
0.05
0.04
0.03
0.02
0.2
0.15
0.1
Position on Y−axis
0.05
0
0
0.05
0.2
0.15
0.1
Position on X−axis
0.25
0.055
0.05
0.045
0.04
0.035
0.03
0.025
(a)
0.16
0.055
0.14
0.05
0.12
0.1
0.045
0.08
0.04
0.06
0.035
0.04
0.03
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
(b)
Fig. 11.5. (a) Response surface of combined modal loss factor (covering the first five modes).
The parameter space was sampled with discrete positions and a surface was fit to match all the
points. The sampling points are illustrated in figure 11.4. (b) Contour plot of the response surface
of combined modal loss factor.
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Parametric study based on acoustic sound power
We now examine the response surfaces obtained by choosing the radiated
sound power as objective function in our optimisation problem. This time,
the goal would be to minimise the acoustic sound power at one particular
modal frequency of excitation or the sum of the modal acoustic sound power
for the modal frequencies in the range considered. For each patch position,
the first five modes of the system are extracted and the surface velocities
are evaluated for excitations at these frequencies (point force acting on the
plate). The acoustic sound power is then calculated as exposed in 7.2.
The individual modal response function, obtained for an acoustic objective
function, are presented in figure 11.6. Clearly, these response functions
exhibit increased complexity than the ones obtained with the modal loss
factor objective function. This is an illustration of the radiation efficiencies
effect: some patch locations can add reasonable product damping while they
lead to high radiated sound power.
The same behaviour, but even sharper, is observed in the case of broad
range optimisation, while we consider the first five modes of vibration. The
combined acoustic response function, illustrated in figure 11.7 as to be compared
with its vibrational equivalent of figure 11.5, based only on modal
damping. The multimodal character of the response function, already present
in the damping response, is again increased by taking into account the
acoustic radiation. However, the optimal patch positions are the same 4+1
positions as already found previously.
From an acoustic design point of view, the figure 11.5 also underlines the
importance of optimal placement of the patch. The worst locations lead to
a product that radiates sound with a sound power level L W greater than
130dB, while the best locations give a design with values of L W smaller than
95dB. To get a more physical intuition of what this means, the reader can
refer to the table in appendix E. It can be seen that levels of 130dB are
obtained by a machine gun noise and are considered above the humain pain
limit. On the other hand, sound power levels of 95dB are found in loud speech
situations, or close to a car driven at high speed.
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90
130
Radiated Sound Power Levels [dB] − mode 1
95
90
85
80
75
0.2
0.15
0.1
0.05
Position on Y−axis
0
0.05
0.1
0.15
0.2
Position on X−axis
0.25
88
86
84
82
80
78
76
Radiated Sound Power Levels [dB] − mode 2
140
120
100
80
60
0.2
0.15
0.1
0.05
Position on Y−axis
0
0.05
0.1
0.15
0.2
Position on X−axis
0.25
120
110
100
90
80
70
90
110
Radiated Sound Power Levels [dB] − mode 3
91
90
89
88
87
86
85
0.2
0.15
0.1
0.05
Position on Y−axis
0
0.05
0.1
0.15
0.2
Position on X−axis
0.25
89.5
89
88.5
88
87.5
87
86.5
86
Radiated Sound Power Levels [dB] − mode 4
120
110
100
90
80
70
60
0.2
0.15
0.1
0.05
Position on Y−axis
0
0.05
0.1
0.15
0.2
Position on X−axis
0.25
105
100
95
90
85
80
75
70
115
Radiated Sound Power Levels [dB] − mode 5
120
110
100
90
80
0.2
0.15
0.1
0.05
Position on Y−axis
0
0.05
0.1
0.15
0.2
Position on X−axis
0.25
110
105
100
95
90
85
Fig. 11.6. Response function of radiated sound power level L W , for each of the first five modes.
The parameter space was sampled with discrete positions (represented by red dots on the graphs)
and a surface was fit to match all the points.
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130
Objective function (Equally weighted Radiated sound power level [dB])
140
135
130
125
120
115
110
105
100
95
90
0.16
0.14
0.12
0.1
Position on Y−axis
0.08
0.06
0.04
0.04
0.06
0.08
0.1
0.16
0.14
0.12
Position on X−axis
0.18
0.2
0.22
0.24
125
120
115
110
105
100
95
(a)
0.16
0.14
130
125
0.12
120
115
0.1
110
0.08
105
0.06
100
95
0.04
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
(b)
Fig. 11.7. (a) Response surface of combined radiated sound power level L W (covering the first
five modes). The parameter space was sampled with discrete positions (represented by red dots
on the graphs) and a surface was fit to match all the points. (b) Contour plot of the response
surface of combined radiated sound power level L W .
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11 OPTIMISATION
11.2.5 Partial conclusions
At the beginning of this section, we asked one simple question:
What would be the most effective position of this patch in order
to minimise the acoustic sound power radiated, under an impact
loading
We have seen that both parametric studies and the preliminary strain
energy distribution simulation oriented the answer towards the same conclusions.
Due to the problem symmetry, there are in fact four optimal positions
of the patch, and an additional fifth position in the center of the plate.
This means that, if only one patch is to be placed, one may choose its
location from one of the four equivalent positions, denoted by the number 1
in figure 11.8. But the best would be to position four patches at these four
positions.
If no cost or manufacturing constraints are taken into account, the best
would be to use an additional patch located in position number 2 of the same
figure.
Fig. 11.8. The optimal patch location are surimposed to the strain energy plot. Locations with
a number 1 are the most optimal and are all equivalent. The additional patch location (number
2) is the second coming optimisation choice.
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11.2.6 Optimisation strategy
We can now build an optimisation strategy that would be adapted to the
problem properties:
• the response surfaces are multimodal and smooth;
• the optimisation problem is bound-constrained, with only upper and lower
bounds on the design variables;
• the design variables are continuous;
• the function evaluation are relatively expensive.
The multimodal character suggests a global optimisation strategy.
We have shown that a preliminary study of the distribution of strain energy
could point out the potentially interesting area of the parameter space
and therefore partially avoid this difficulty. Nevertheless, the optimisation
sequence of choice should be able to handle such behaviour.
Additional useful information that will guide the choice is the lack of analytic
gradient data. Gradient-based optimisation methods are well-known,
highly efficient techniques (see [HG92], [EBA + 06]), with among the best convergence
rates of all the optimisation algorithms. If analytic gradient and
Hessian information can be obtained from the application code, a full Newton
method will give quadratic convergence rates close to the solution. When
only gradient information is available and that the Hessian is approximated
from accumulation of gradient data, superlinear convergence rates can be
obtained. These gradient-based techniques are however not well suited when
the problem exhibits non-smooth, discontinuous or multimodal behaviour.
Gradient-data can be analytically calculated for vibration and acoustic
models build from application of the finite element method [HG92]. The
reader can, for instance, refer to the section on analytical sensivities of sound
power levels in [BSK94] and its references. We did not develop such features
in our work and choosed to use numerical gradients, based on finite difference
approaches instead. The implementation of analytic sensitivities could
reduce the optimisation time by eliminating a number of function evaluations
devoted to the numerical gradient calculation.
Nongradient-based methods exhibit slower rates of convergence and therefore
tend require more function evaluations. The computational cost of each
evaluation must be relatively low in order to obtain an optimal solution in
a reasonable amount of time. Such methods can be applied when gradient
calculations are too expensive or unreliable.
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Optimisation strategies
The surrogate-based optimisation (SBO) strategy is an approach that combines
the efficiency of gradient-based optimisation methods and the application
to non-smooth or noisy response. This technique smooth noisy or discontinuous
response results through the use of a data fitted surrogate model (e.g.
a quadratic polynomial) and optimise the smooth surrogate using gradientbased
techniques. The multivel hybrid optimisation (MHO) [EBA + 06] strategy
is an attempt to bring the efficiency of gradient-based methods to global
optimisation problems. The name points out the fact that different optimisation
algorithms are used at different stages of the optimisation run. Typically,
a global non-gradient based is used for a few cycles to locate promising regions
and then local gradient-based optimisation is used to converge on one
optima.
We propose to tackle our optimisation problem with a particular version of
the MHO strategy that we propose to call Multi-Fidelity Hybrid Optimisation
strategy (MFHO). The MFHO strategy sequence is illustrated in figures 11.9
and 11.10.
The Multi-Fidelity character of the strategy comes from the use of two
physical models of different fidelity. We exploit a major advantage of our
modelling technique: different model accuracies can be obtained, based on the
same coarse support mesh, by adapting the enrichment of the approximation
fields in the PUFEM Mindlin elements. The Low Fidelity (LF) model is based
on a 5 × 5 mesh of the host plate and a p = 2 reduced enrichment set. The
High Fidelity (HF) is constructed with the same support mesh and a p =4
reduced enrichment set.
The Hybrid character of the strategy is due to the use of various optimisation
algorithms at the differents levels of exploration of the design space
(see 11.9). We introduce, somewhat arbitrarily, three levels of exploration.
The first level corresponds to the use of a GLOBAL optimisation algorithm
and is called the GLOBAL level. As shown in the previous sections, this level
exhibit a highly multimodal response; the choice of a gradient-free algorithm
is hence recommended. We opt for the DIRECT algorithm, whose principle
and properties will be detailed later.
The second level of exploration is called SEMI-LOCAL and corresponds
to an intermediate level between global and local. At this point of the optimisation
sequence, the response can still show a multimodal response. A this
level, the PATTERN SEARCH gradient-free algorithm is applied, with the
best solution of the upper level as starting point.
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The third level of exploration is called LOCAL and matches the use of a
local optimisation algorithm. A gradient-based method (CONMIN library)
is chosen to converge to the final optimum.
Fig. 11.9. The different levels of observation of the design space in the MFHO strategy. The
GLOBAL level extends to all the parameter space and corresponds potentially to a highly multimodal
response. The SEMI-LOCAL level is restricted to a subset of the parameter space and
generally exhibits a less multimodal character. The LOCAL level is the lowest level of optimisation
of the sequence.
Choice of algorithms
The DIRECT optimisation algorithm was introduced by Perttunen et al. in
[PJS93]. It is implemented into the COLINY package of the DAKOTA framework
[EBA + 06]. It was created to tackle difficult global optimisation problems
with bound constraints and real-valued objective functions. DIRECT
is a gradient-free algorithm and requires no information about the gradient
of the objective function. It is based on a particular sampling strategy that
can be resumed by the phrase DIviding RECTangles, which gives his name
to the algorithm and describes the way it operates towards the optimum. An
advantage of this algorithm is that it balances local searches in promising
regions with global searches in yet unexplored regions of the design space.
Within our multifidelity hybrid strategy, we use DIRECT to quickly identify
potential regions of interest. The best point found by DIRECT after a
limited number of iteration (or after the minimum acceptable box size is
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11 OPTIMISATION
Fig. 11.10. Organigram of the MFHO strategy. The sequence starts by using a gradient-free
algorithm (DIRECT) to explore the GLOBAL level. At this level, we operate through the Low
Fidelity model. After a fixed number of iterations (or minimal box size is reached), we swith to
the second algorithm (PATTERN SEARCH) to explore the SEMI-LOCAL level. This algorithm
still monitors the Low Fidelity model. Again, after a fixed number of iteration, we switch to
the last optimisation algorithm. We choose a gradient-based (conjugate gradient method from
the CONMIN library) algorithm that is now operating through the High Fidelity model, at the
LOCAL level.
reached) is then passed to another gradient-free algorithms, in order to explore
the design landscape at the SEMI-LOCAL level.
The PATTERN SEARCH algorithm, used at the SEMI-LOCAL level, is
an nongradient-based method which uses a set of offsets from the current
iterate to locate promising points in the design space. We use a traditional
pattern search method, whith a fixed pattern of search directions to try to
identify improvements to the current point. The pattern can contract and expand
during design space exploration. An example of pattern search method
is the simplex algorithm. Details concerning the options and DAKOTA implementation
can be found in [EBA + 06].
Finally, a gradient-based algorithm from the CONMIN library is applied.
It is an implementation of the Fletcher-Reeves conjugate gradient (CG)
method for unconstrained optimisation. For a general description of the CG
method and convergence properties, the reader is invited to refer to Ciar-
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11.2 An optimisation application
let [Cia90]. The DAKOTA implementation of the Fletcher-Reeves version of
the CG method is described in [EBA + 06], while specific details about the
public-domain CONMIN library can be found in [Van73].
This method requires the calculation of gradient data (but not Hessians).
This data can be calculated numerically by application of a finite differencing
scheme. Forward and central differencing schemes can be chosen. However,
central differencing produces, in general, more reliable gradients than forward
differencing, at twice the expense. Since the quality of the gradient
information is of great importance for the convergence of the CG algorithm,
we prefer to use the central differencing technique.
A DAKOTA input file corresponding to the described MFHO strategy is
presented in figure 11.11. Methods are applied sequentially in the strategy
procedure, each new method starting with the best found design point as
starting point.
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-
# DAKOTA INPUT FILE: dakota_multilevel_LHA.in
# STRATEGY --------------------------------------------------------------
strategy, \
multi_level uncoupled \
method_list = ’COL_DIRECT’ ’PS’ ’CONMIN’
# METHODS --------------------------------------------------------------
method, \
id_method = ’COL_DIRECT’ \
model_pointer = ’MODEL_LOWFI’ \
coliny_direct \
global_balance_parameter = 0.0 \
local_balance_parameter = 1.e-8 \
max_boxsize_limit = 0.0 \
min_boxsize_limit = 0.01 \
max_function_evaluations = 20
method, \
id_method = ’PS’ \
model_pointer = ’MODEL_LOWFI’ \
coliny_pattern search stochastic \
seed = 1234 \
initial_delta = 0.1 \
threshold_delta = 1.e-4 \
solution_accuracy = 1.e-5 \
exploratory_moves basic_pattern \
max_function_evaluations = 20
method, \
id_method = ’CONMIN’ \
model_pointer = ’MODEL_HIFI’ \
conmin_frcg, \
gradient_tolerance = 1.e-4 \
convergence_tolerance = 1.e-5
# MODELS --------------------------------------------------------------
model, \
id_model = ’MODEL_LOWFI’ \
single \
variables_pointer = ’VARIABLES’ \
interface_pointer = ’INT_LOWFI’ \
responses_pointer = ’RESPONSE_1’ \
model, \
id_model = ’MODEL_HIFI’ \
single \
variables_pointer = ’VARIABLES’ \
interface_pointer = ’INT_HIFI’ \
responses_pointer = ’RESPONSE_2’
Fig. 11.11. (a) First part of the DAKOTA input file; it defines the three methods and the two
models properties. Each method block contains the choice of optimisation algorithm, options and
a pointer to the model of choice (LF or HF).
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-
# VARIABLES --------------------------------------------------------------
variables, \
id_variables = ’VARIABLES’ \
continuous_design = 2 \
cdv_initial_point 52.E-3 40.E-3 \
cdv_lower_bounds 52.E-3 40.E-3 \
cdv_upper_bounds 228.E-3 160.E-3 \
cdv_descriptor ’XC’ ’YC’
# INTERFACES --------------------------------------------------------------
interface, \
id_interface = ’INT_LOWFI’ \
system \
analysis_driver = ’./simulator_script_central_acou_5modes_LF_ASYM’ \
parameters_file = ’params.in’ \
results_file = ’results.out’ \
file_tag
interface, \
id_interface = ’INT_HIFI’ \
system \
analysis_driver = ’./simulator_script_central_acou_5modes_HF_ASYM’ \
parameters_file = ’params.in’ \
results_file = ’results.out’ \
file_tag
# RESPONSES --------------------------------------------------------------
responses, \
id_responses = ’RESPONSE_1’ \
num_objective_functions = 1 \
no_gradients \
no_hessians
responses, \
id_responses = ’RESPONSE_2’ \
num_objective_functions = 1 \
numerical_gradients \
method_source dakota \
interval_type central \
fd_gradient_step_size = 1.e-4 \
no_hessians
Fig. 11.11. (b) Second part of the DAKOTA input file; it defines the variables of the models
(bounds and starting point), the two model interfaces (LF and HF) and the two response handling
techniques.
11.2.7 Application of MFHO strategy to the first optimisation
problem
We apply here the MFHO strategy to our first optimisation problem. Since
the DIRECT algorithm initially samples the whole design space, the starting
point is of no importance. Figure 11.12 illustrates graphically the progressive
exploration of the design space by each algorithms. The contour was
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11 OPTIMISATION
obtained from the parametric study performed on this application, with the
combined radiated sound power level as objective function (for the first five
modes of the structure). The DIRECT algorithms operates by sampling the
global design space, with a limited number of function evaluations (determined
by the minimal box size in in this case, max. 13 iterations). The best
captured evaluation point (iteration 11) is handled to the next algorithm.
ThePSmethodisthenappliedforafixednumberofiterations.Thebest
point found at this level of the exploration is used as starting point for the
local exploration by the CONMIN CG algorithm.
The evolution of function evaluations in the optimisation strategy are
plotted in figure 11.13.
Response function of combined sound power level
0.16
130
0.14
0.12
125
120
Position in Y
0.1
115
110
0.08
0.06
105
100
0.04
95
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
Position in X
Fig. 11.12. Progressive parameter space exploration with the MFHO strategy. Each point represents
one function evaluation. Different colors are used for each single algorithm in the strategy.
The contour of combined radiated sound power level (in dB, based on the first five modes) is
plotted in the background. The cyan dots in the plot corresponds to the first sequence in the
strategy, with application of the DIRECT method to the GLOBAL level. The green dots are the
function evaluations monitored by the PS method, at the SEMI-LOCAL level. Finally, at the
LOCAL level, the red dots are the successive evaluations in the application of the CG method.
The yellow dot represents the final optimum found.
As summarised in table 11.1, the optimisation run leads to a grand total of
58 point explorations, including 53 new evaluations and 5 duplicates. Among
these new evaluations, 24 lead to a call to the low fidelity model (during the
GLOBAL and SEMI-LOCAL level exploration) and 29 lead to a call to the
high fidelity model. The total run consumed 1722s of CPU time. Ignoring the
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115
110
Response
105
100
95
90
0 5 10 15 20 25 30 35 40 45
Number of function evaluation
Fig. 11.13. Plot of the evolution of the response function (combined sound power level, in dB)
for the sequence of function evaluations, during application of the MFHO strategy to the first
optimisation problem. The same color codes are used as in the previous figure (11.12).
DAKOTA overhead, this time is split as follow: 30% is spend in LF model
evaluations (DIRECT+PS) and 70% in HF model evaluations (CONMIN-
CG). On an heavily charged machine, and with low priorities, the whole
optimisation ran for 176 min 10 s (wall clock or elapsed time).
The optimum found corresponds to a combined radiated sound power
level of 91.04 dB, and is obtained for a patch centred at point (1.9218 ×
10 −1 , 1.6×10 −1 ). As foreseen, this location is one of the four potential optimal
location described earlier (see 11.2.5).
Table 11.1. Details of function evaluations for MFHO strategy on first optimisation problem.
Note that the estimates best values for the first two methods correspond to the low fidelity model
and should therefore not be compared with the final value (obtained with the high fidelity model).
Model Function evaluations Best found
Method type Total New Duplicates value
DIRECT LF 13 13 0 93.10 dB
PS LF 12 11 1 91.19 dB
CG HF 33 29 4 91.04 dB
Grand total 58 53 5 91.04 dB
In order to get an idea of the gain of such an optimisation, we present
in table 11.2, the results obtained with three configurations: naked plate,
plate with a single central patch and optimised version. The optimised patch
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11 OPTIMISATION
configuration is obviously less noisy than the two others. Looking at the naked
plate results, modes 3, 2 and 1 are the most noisy modes, in degressive order.
The central patch configuration successfully damps out the first mode but
fail at correcting the second and third modes. The optimised patch location
damps the three first modes and is globally more efficient than the central
patch location.
Table 11.2. Comparison of modal sound power levels (L W ) and combined sound power level
(first five modes) for three configurations: naked plate, plate with a central patch and optimised
configuration.
Naked plate Central patch Optimised patch
Modes L W [dB] Modes L W [dB] Modes L W [dB]
140.79 Hz 217.55 137.59 Hz 76.66 138.06 Hz 78.05
190.40 Hz 223.08 189.33 Hz 93.81 186.52 Hz 76.33
388.44 Hz 227.11 384.70 Hz 90.86 383.68 Hz 88.45
411.97 Hz 196.60 400.53 Hz 80.83 407.93 Hz 82.93
463.33 Hz 209.85 462.95 Hz 92.56 454.07 Hz 84.27
Combined L W : 228.95 Combined L W : 97.48 Combined L W : 91.04
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11.2 An optimisation application
11.2.8 Application of MFHO strategy to a second optimisation
problem
We consider the same plate problem than previously, but this time with
asymmetric boundary conditions. The plate is clamped at the left edge and
is fixed at the lower right plate corner. The corresponding modal patterns
are illustrated in figure 11.14.
2
400
60
0
200
40
−2
0
20
−4
−6
−200
0
−8
−400
−20
−10
0.2
−600
0.2
−40
0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
f 1 =42.67 Hz
f 2 = 117.36 Hz
f 3 = 171.88 Hz
2000
600
1500
1000
500
0
−500
400
200
0
−200
−1000
0.2
−400
0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
f 4 = 269.01 Hz
f 5 = 351.52 Hz
Fig. 11.14. Modes shapes for asymmetric plate problem.
We introduced before the utility of a preliminar estimation of the strain
energy distribution in the naked host plate. The results of this study are
presented in figures 11.15 and 11.16. As foreseen, the combined strain energy
contour is no more symmetric. This time, there is apparently one area
where the strain energy is concentrated. This area is, potentially, the most
interesting spot to position a single patch.
A parametric study was performed, with the combined radiated sound
power level (first five modes) as objective function. Clearly, this study is not
necessary for the optimisation but it is useful to illustrate the progress of the
MFHO strategy. The response plot of combined sound power level is given
in figure 11.17.
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3
4
0.2
0.18
0.16
0.14
2.8
2.6
2.4
0.2
0.18
0.16
0.14
3.5
3
0.2
0.18
0.16
0.14
4
3.5
0.12
2.2
0.12
2.5
0.12
0.1
2
0.1
0.1
3
0.08
1.8
0.08
2
0.08
0.06
0.04
1.6
0.06
0.04
1.5
0.06
0.04
2.5
0.02
0
1.4
1.2
0.02
0
1
0.02
0
2
0 0.05 0.1 0.15 0.2 0.25
1
0 0.05 0.1 0.15 0.2 0.25
0.5
0 0.05 0.1 0.15 0.2 0.25
4.5
0.2
0.2
4.5
0.18
0.16
4
0.18
0.16
0.14
0.14
4
0.12
3.5
0.12
0.1
0.08
0.1
0.08
3.5
0.06
3
0.06
0.04
0.04
3
0.02
0
2.5
0.02
0
0 0.05 0.1 0.15 0.2 0.25
0 0.05 0.1 0.15 0.2 0.25
2.5
Fig. 11.15. Elemental strain energy calculated for the first 5 modes of the untreated plate with
asymmetric boundary conditions. The strain energy scale is logarithmic.
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0 0.05 0.1 0.15 0.2 0.25
4.8
4.6
4.4
4.2
4
3.8
3.6
3.4
Fig. 11.16. Elemental combined strain energy calculated for the untreated plate: Combined plot
of the first 5 modes (logarithmic scale). The area which concentrates the most strain energy is
located close to the lower edge, to the right.
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11.2 An optimisation application
Objective function− Combined radiated sound power level [dB]− 5 modes
180
160
140
120
100
80
0.2
0.15
0.1
Position on Y−axis
0.05
0
0.05
0.1
0.15
0.2
Position on X−axis
0.25
170
160
150
140
130
120
110
100
90
(a)
0.16
170
0.14
160
150
0.12
140
0.1
130
120
0.08
110
0.06
100
0.04
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
90
(b)
Fig. 11.17. (a) Response surface of combined radiated sound power level [dB] (covering the first
five modes). The parameter space was sampled with discrete positions (represented by red dots
on the graphs) and a surface was fit to match all the points. (b) Contour plot of the response
surface of combined radiated sound power level [dB].
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11 OPTIMISATION
The response function ranges from approximately 85 dB to up to 170 dB
(on a combined sound power level scale). The multimodal character of the
response is again apparent.
We now apply the MFHO strategy to the asymmetric plate optimisation
problem. Figure 11.18 shows the exploration of the design space by each
algorithms. The evolution of function evaluations are plotted in figure 11.19.
Response function of combined sound power level
0.16
0.14
160
150
Position in Y
0.12
0.1
0.08
140
130
120
0.06
0.04
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22
Position in X
110
100
90
Fig. 11.18. Progressive parameter space exploration with the MFHO strategy for the asymmetric
problem. Each point represents one function evaluation. Different colors are used for each single
algorithm in the strategy. The contour of combined radiated sound power level (first five modes)
is plotted in the background. The cyan dots in the plot corresponds to the first sequence in the
strategy, with application of the DIRECT method to the GLOBAL level. The green dots are the
function evaluations monitored by the PS method, at the SEMI-LOCAL level. Finally, at the
LOCAL level, the red dots are the successive evaluations in the application of the CG method
(lower right corner). The yellow dot represents the final optimum found.
The optimisation run leads to a grand total of 75 point explorations,
including 57 new evaluations and 18 duplicates (see table 11.3). Among
these new evaluations, 29 lead to a call to the low fidelity model (during
the GLOBAL and SEMI-LOCAL level exploration) and 28 lead to a call to
the high fidelity model.
The total run consumed 1523s of CPU time. Ignoring the DAKOTA overhead,
this time is split as follow: 38% is spend in LF model evaluations
(DIRECT+PS) and 62% in HF model evaluations (CONMIN-CG). On an
heavily charged machine, and with low priorities, the whole optimisation ran
for 122 min 9 s (wall clock or elapsed time).
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11.2 An optimisation application
130
125
120
115
Response
110
105
100
95
90
85
80
0 10 20 30 40 50 60
Number of function evaluation
Fig. 11.19. Plot of the successive function evaluations during application of the MFHO strategy
to the asymmetric optimisation problem. The same color codes are used as in the previous figure
(11.18).
The optimum found corresponds to a combined radiated sound power
level of 89.12 dB, and is obtained for a patch centred at point (2.2507 ×
10 −1 , 4.1807 × 10 −2 ). This location is compatible with our preliminary estimation
based on the strain energy distribution is the naked plate.
Table 11.3. Details of function evaluations for MFHO strategy on first optimisation problem.
Note that the estimates best values for the first two methods correspond to the low fidelity model
and should therefore not be compared with the final value (obtained with the high fidelity model).
Model Function evaluations Best found
Method type Total New Duplicates value
DIRECT LF 17 17 0 87.52 dB
PS LF 15 12 3 84.30 dB
CG HF 43 28 15 89.12 dB
Grand total 75 57 18 89.12 dB
The results obtained with three configurations (naked plate, plate with a
single central patch and optimised version) are presented in table 11.4.
The optimised patch configuration is again (fortunately !) less noisy than
the two others. Looking at the naked plate results, modes number 4 and 5
are the most noisy modes, in degressive order. The optimised patch location
successfully outperforms the central patch configuration for all modes.
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11 OPTIMISATION
Table 11.4. Comparison of modal sound power levels (L W ) and combined sound power level
(first five modes) for three configurations of the asymmetric plate problem: naked plate, plate
with a central patch and optimised configuration.
Naked plate Central patch Optimised patch
Modes L W [dB] Modes L W [dB] Modes L W [dB]
42.67 Hz 156.74 41.71 Hz 90.44 41.85 Hz 71.65
117.36 Hz 193.88 115.11 Hz 84.01 114.20 Hz 78.51
171.88 Hz 181.02 171.96 Hz 84.25 171.27 Hz 81.99
269.01 Hz 262.28 272.57 Hz 109.54 269.72 Hz 80.01
351.52 Hz 207.06 349.52 Hz 89.67 348.00 Hz 86.91
Combined L W : 262.28 Combined L W : 109.66 Combined L W : 89.12
11.3 Summary
The use of optimisation tools can offer an interesting alternative to the design
of silent products by application of experience-based rule-of-thumb or to the
trial-and-error prototyping. Coupled to our efficient modelling methodology
for the simulation of passive viscoelastic damping in structures, numerical
optimisation tools bring a new dimension to the design procedure.
We started this chapter with a review of the literature on the structuralacoustics
optimisation. Particular trends concerning the optimisation of passive
damping devices were covered and we showed that the maximisation of
damping was always chosen as objective function in the optimisation problem
formulation. Recalling some conclusions drawn in chapter 7, we prefer
to develop objective functions directly on acoustic quantities. The radiated
acoustic sound power level is chosen for single frequency studies. We introduce
the combined radiated sound power level for broad range studies.
Parametric studies are performed with both modal damping and acoustic
sound power levels as objective functions. A list of characteristics for the
structural-acoustics optimisation problem are deduced from these studies.
The strain energy distribution in the naked structure was also presented and
we showed that it was a useful preliminary analysis, giving potentially interesting
positions for the damping patches (at least, to maximise damping).
After a short review of the modern numerical optimisation tools, we propose
a particular strategy, inspired by both surrogate-based (SBO) and multilevel
techniques and able to cope with the problem specificities. The proposed
strategy, called Multifidelity Hybrid Optimisation (MFHO), is also specifically
designed to benefit from our main PUFEM-based modelling approach:
the low fidelity model and the high fidelity model share the same support
mesh, but different polynomial enrichment basis.
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11.3 Summary
The applicability and performance of the MFHO strategy is demonstrated
on two academic examples. The approach enables product improvements
with a limited amount of computational resources. The multifidelity concept
clearly outperformed some attempts previously made with standard SBO
strategies, because the low fidelity model naturally exhibits most of the trends
of the high fidelity model, at a much lower cost than the cost needed to build
surrogate functions.
The results of the optimisation were very promising, and clearly demonstrated
the added value brought by optimisation.
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Laurent Hazard 21/12/2006
12
CONCLUSIONS
Passive damping by viscoelastic sandwich or patches is a well-known efficient
mean for the noise and vibration control of structures. The PUFEM
technique, developed in chapters 3 to 5 of this dissertation, provides an interesting
alternative to the classical finite element method for the simulation
of structures involving passive damping by viscoelastic material (VEM) layers.
The approach, coupled with an interface element concept (chapter 6) for
the modelling of the VEM, was applied to both modal and direct frequency
analyses (chapter 7). Numerous validation tests were conducted (chapter 8),
confronting the methodology to real experimental measurements or to numerical
approaches (FEM, Assumed modes method). The previously published
experimental data were however limited to relatively low frequencies (or first
few modes). We developed our own experimental set-up and pursued the
validation of our technique up to higher frequencies (chapter 9). Finally, we
found interesting for our industrial partner to illustrate how our development
could answer specific design needs. First, we showed in chapter 10 that simple
design rules could already be defined using bidimensional models (based
on our Q4-PUM element). Chapter 11 covered the problem of optimal location
of patches on structures. In the literature, this problem is solved with a
damping objective, namely: “How to make better use of a limited amount of
VEM surface to maximise the damping brought to the structure” Inspired
by the work on active noise control, we proposed to directly address the
acoustic optimisation problem and to answer the question: “Where to place
the patch(es) in order to minimise the radiated sound power level”
In the following, we discuss the work that was accomplished in the three
different directions of this thesis (modelling, validation and optimisation),
and give our recommendations for further use of these developments. We end
up with the perspectives for future research and developments.
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12 CONCLUSIONS
12.1 Conclusions & discussions
Structural-acoustic toolbox
Our objective in this part was the modelling of both the dynamic and acoustic
behaviour of thin planar structures including layers of viscoelastic materials,
in the form of sandwich or in the form of damping patches (chapter 2). In
the literature, numerical models of such assemblies are almost always based
on the finite element method. The major drawback of the FEM approach is
the pollution error, which limits its applicability to relatively low frequencies.
The search for alternative techniques is an important subject of research
but did not, until now, give birth to a viable replacement tool. Our effort,
in this dissertation, was concentrated on application of polynomial PUFEM
to the development of Mindlin plates. The technique, similar in result to
the p-FEM method, is however simple to implement and very flexible in its
use. The performance of high order polynomial shape functions for acoustic
or vibrational problems was already studied by Ainsworth [Ain03] and is
not original. Nevertheless, the application of the Mindlin PUFEM elements
to practical vibration problems and comparison to commercial FEM implementations
confirmed the interest of the approach: the computational effort
required by the solution step, to reach a certain error level is significantly
reduced. The related drawback, common to all high order techniques, lies
in the preparation of the problem matrices: the numerical integration of the
high order polynomials is expensive. However, in a direct frequency study,
the large number of single system resolutions quickly balances the longer
time spent in matrix pre-processing. The proposed method is very interesting
since the choice of polynomial enrichment order can be adapted to the
wave complexity of the problem and it can be applied to any geometrical
complexity, just like FEM. The computer implementation is easy because
the general isoparametric framework is kept. As such, the technique is more
suitable to medium frequency simulations than FEM, while building on top
of the capital advantages of FEM.
The interface element formulation is introduced for the representation
of the viscoelastic material layer in sandwich or patch configurations. This
formulation simplifies dramatically the modelling efforts linked to the tridimensional
brick element technique (see [BG04]) and is also compatible with
the Mindlin PUFEM elements. The assumptions behind the interface element
concept are restricted to a thin viscoelastic layer; this is not an important
limitation since most actual products (sandwich or damping tapes) involve
very thin VEM layers. This fact is not only based on damping considerations,
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12.1 Conclusions & discussions
but also on other design constraints such as cost, space clearance or stiffness
(for sandwich structures). We also demonstrated the validity of the approach
in chapter 8.
The acoustic part of the model is based on the Rayleigh integral approach
and is therefore limited to the radiation of baffled planar structures. In
this uncoupled implementation, the velocities at the radiating surface of the
product are obtained from a dynamic analysis and imported in the acoustic
module for the calculation of acoustic pressures or sound power levels. The
concepts of radiation efficiency and weak radiators [CC94], introduced in
chapter 7, motivated the addition of this acoustic propagation module in the
toolbox. Most literature on the passive damping subject is focused on the
maximisation of damping and authors often consider that if a patch configuration
optimises the VEM use for damping, then the configuration is also
optimal at the acoustic level. This is not rigorously true because damping
can affect the shape of structural modes and therefore their radiation efficiency.
When acoustic quality is the design objective, it is recommended to
perform a full structural-acoustic simulation and to estimate acoustic quality
indicators such as the sound power level.
Validations and experiments
We performed a large number of validations, addressing differents aspects
of our modelling approach. We showed the convergence and performance
aspects of the Mindlin PUFEM elements (chapter 5). We checked the interface
element formulation for structural bonding (chapter 8, section 8.1), full
sandwich configurations (section 8.2) and patch configurations (8.3). These
applications covered both modal analysis and direct frequency analysis approaches.
The calculation of modal loss factors, as developed in chapter 7, was
validated in section 8.4. These test were limited to low frequencies. We found
useful to perform our own experiments in a broader range of frequencies.
An experimental test bench was set up in the OPTRION SA laboratory.
This installation was designed for the measurement of frequency response
functions of plate samples equipped with different configurations of damping
patches. Two types of boundary conditions were considered: free and partially
clamped. In these measurements, we used both a laser vibrometer and
the holographic interferometry (HI) camera device developed by OPTRION
SA. To answer the demand of our industrial partner, we suggested a full
experimental strategy for the optimal positionning of damping patches. This
procedureisbasedontheuseoftheHIcameratotrackthemodeshapesof
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12 CONCLUSIONS
modes situated in critical frequency ranges. Patches should then be designed
to damp specifically these modes; the validity of the patched configuration
can then be verified by new FRF measurements and estimation of the modal
damping factor by the half-power bandwidth method, for instance.
We presented, in chapter 2, different models to take into account the
frequency- and temperature-dependence of viscoelastic materials. We introduced
the superposition principle and the method of reduced variables to
cope with the temperature dependence. We have shown that both tabulated
experimental data and parametric models could be derived for the description
of complex modulus data. For the applications involving the 3M ISD112
material covered in chapter 8, we used the parametric model presented in section
2.3.3 because we did not have at the time full experimental data from
our own tests. All our experimental measurements at OPTRION SA were
realised with a recent 3M Scotchdamp damping tape (details can be found
in chapter 9, section 9.1.1). We performed experimental characterisation of
the VEM in the Scotchdamp tape by performing DMTA measurements at
Solvay Central Laboratory (Neder-over-Hembeek). Results from these tests
are given in chapter 2. This recent experimental ISD112 data was used in all
simulation runs, in the form of tabulated complex moduli. The advantages of
this representation are that we are sure that the material data corresponds
to the 3M tape used for the plate samples and that no modelling error is
introduced by an approximate parameter determination. To conclude this
topic, the quality of the experimental correlation confirmed the validity of
the whole approach.
Design optimisation
In the design of damping patches for a specific application, there are many
questions to answer: which VEM What thickness for the VEM and for
the constraining layer Where to place the patches and what is their ideal
geometry
Generally, the damping patches will be made of commercially available
damping tapes and will be cut to size and bonded at the optimal position.
In this case, the basic design rules given in chapter 10 can help choosing
the damping tape in the manufacturer catalogues. The specific choice of the
VEM is simple if we only consider the damping objective: the material should
have the highest and widest possible loss modulus peak in the standard domain
of temperature and frequency covered by the final application. However,
this choice quickly becomes tricky when other constraints are added to the
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12.1 Conclusions & discussions
equation: costs, manufacturing constraints (like temperature or mechanical
sollicitations), aging (due to humidity, temperature, chemical agents, etc.).
The problem of optimal positionning of damping patches is studied in
chapter 11. The objective was to consider the different options for the determination
of the ideal position of patches. In the literature on passive damping
in structural-acoustic optimisation problems, most authors consider the
maximisation of damping as sufficient to guarantee an acoustically optimal
product. When this approach is followed, we showed first that the calculation
of strain energy distribution in the host product could already guide the
placement of damping patches, by pointing out the areas where the shear generated
in the viscoelastic layer should be the greatest. Secondly, the modal
loss factors, within a considered frequency range or for a given number of
modes, can be used to build a damping objective function that can be maximised
in a certain optimisation strategy.
We already mentionned earlier that, due to potential modification of the
radiation efficiency, the optimal damping objective was not ideal for the
structural-acoustic problem. We therefore proposed a new acoustic objective
function based on the combined sound power level, that is calculated using
the acoustic module of our toolbox.
Parametric studies of both types of objective functions (based on modal
loss factors and based on combined sound power level) were performed, with
two aims. First, we pointed out the difference in response surfaces linked
to the choice of objective functions, and secondly these studies guided the
choice of the optimisation strategy by giving the main characteristics of the
response surfaces. The differences in response surfaces is another illustration
of the importance to address the acoustic optimisation problem when acoustic
quality is the target of the design. The optimisation strategy should be able
to cope with multimodal and smooth response functions. Another aspect
which guided our choice was our decision not to develop analytical gradient
expressions in our toolbox.
We proposed a Multi-Fidelity Hybrid optimisation strategy implemented
in the DAKOTA optimisation framework [EBA + 06] and exploiting the flexibility
of the PUFEM for the development of solutions of different accuracies.
A low fidelity model (LF), built on a given mesh support and based on a low
order polynomial enrichment, is used to explore the design parameter space
at the global level with two different gradient-free algorithms (DIRECT and
PATTERN SEARCH). These two methods can cope with the multimodal
character of the response and demonstrated on two academic examples their
ability to find areas of high interest. The best design point found in the LF
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12 CONCLUSIONS
sequence is used as starting point at the local level. A gradient-based optimisation
algorithm (conjugate gradient from the CONMIN library) is applied
to converge towards the optimal point, and interact with the high fidelity
model (HF). This HF model is based on the same support mesh, but with
a higher order enrichment polynomial. The gradient data are generated by
application of a numerical central differencing technique. This strategy was
demonstrated to succeed in finding optimal patch positions for both types
of response functions, within a reasonable running time. The use of the LF
model for the global exploration is very effective and low cost. The HF model
allows the determination of the optimum with increased accuracy. Based on
the two academic examples treated, the total CPU time is attributed for
one-third to the global exploration by the LF model and for two-thirds to
the final optimisation with the HF model.
12.2 Perspectives
Structural-acoustic toolbox
The PUFEM Mindlin element formulation that we developed is currently
limited to the simulation of plates of any geometries. Shell structures could
be also modelled with this element by using the well-known faceted shell
approach. Three modifications should be done to the plate elements in order
to be able to model both flat planar structures and shell structures:
• Transformation of coordinate system. Currently, the plate element is defined
in the local coordinate system, corresponding to the modelled plate
structure. A simple transformation matrix should be introduced to switch
between global and local element coordinate systems.
• Our plate elements have five degrees of freedom (three translations and
two rotations around x- and y-axis). In a faceted shell model, it is necessary
to introduce an additional degree of freedom related to the rotation
around the z-axis. At the element level, this rotational degree of freedom
has no influence on the displacements, but it is necessary for the global
matrix assembly.
• The enrichment polynomial set, defined at each node, can no more be
expressed in the plate coordinate system. We propose to define a pseudonormal
direction at each node, based on the orientation of each neighbour
elements. We would then define the enrichment polynomial at each node
in a unique coordinate system attached to this pseudo-normal direction.
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12.2 Perspectives
The main drawbacks of the PUFEM technique, as opposed to FEM, are
the expensive numerical integration procedure and the linear dependencies of
the matrices. In this thesis, a Gauss-Legendre integration rule was adopted
and the number of integration points was determined from the order of
the generalised shape functions. Improvements in the numerical integration
methodology would clearly benefit to the PUFEM competitivity. The linear
dependencies lead to nearly-singular matrices and we adopted a dedicated
direct solver to accomodate this property. Modern iterative solvers could
eventually be tested to see if they can lead to performance improvements.
For complex geometries, p-adaptivity could bring a performance advantage
to the PUFEM, by allowing low enrichment order to be kept at nodes
with low wave complexity, while higher enrichment order would be developed
progressively in area with more modal details.
The acoustic module of the toolbox could also be developed to allow
coupled structural-acoustic simulations. For instance, polynomial PUFEM
acoustic elements can be implemented.
Validations and experiments
Clearly, the validations available in the literature were limited to academic
tests and very low frequencies. In our experimental measurements, we perfomed
correlations up to higher frequencies (or a larger number of modes),
on similar academic plate samples and for different types of boundary conditions.
The validation should be continued on more real-life applications such
as non-rectangular plates and curved structures.
Design optimisation
The proposed optimisation strategy should be validated more extensively
on practical applications. The development of analytical gradient within the
toolbox should help to reduce the time spent within the last optimisation
sequence, because the convergence rate of the conjugate gradient method
depends on the quality of the gradient information. Our choice of the central
differencing technique was motivated by this aspect, but corresponds to twice
the expense (in function evaluations) of a forward differencing method.
As stated in the conclusions of chapter 10, the design optimisation of
damping patches is an iterative process, because the ideal patch thickness
ratio (between VEM and constraining layer) and their ideal position on the
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12 CONCLUSIONS
structure are linked. Therefore, the optimisation of the position should first
be performed with a pre-determined patch (based on simple design rules) and
should be followed by a sensibility analysis on the relative thickness ratio.
This was not done in this dissertation but would be interesting to test in
practice.
We considered only rectangular patches of fixed geometries. Another optimisation
approach would be to consider a shape optimisation problem. We
could for instance imagine to start with the full sandwich configuration and
suppress progressively the VEM material that is not used optimally. With
this kind of procedure, we would generate patterns that would be similar in
form to the contour of strain energy distributions, for instance. The problem
is then to filter the optimal patterns or to alter them, in order to be
compatible with industrial requirements and manufacturing constraints. For
instance, even if the shape optimisation algorithm generates holes in the
patch in order to minimise its surface, industrially speaking it would not be
interesting because the material within the holes should be cut out and would
not be reusable. This simple remark motivates the implementation of special
industrial constraints directly in the shape optimisation procedure. On the
other hand, our own approach directly defines these constraints inside the
parametric model and ensure that such constraints are naturally prescribed.
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A
ISD112 data
Table A.1. ISD112 material parameters from [dS03].
Complex modulus (see eq. 2.11)
B 1 [MPa] B 2 [MPa] B 3 [MPa] B 4 B 5 B 6
0.4307 1200 0.1543 × 10 7 0.6847 3.241 0.18
Shift factor (see eq. 2.12)
a =(D BC C − C BD C)/D E
b =(C AD C − D AC C)/D E
C A =(1/T L − 1/T 0) 2 C B =(1/T L − 1/T 0) C C = S AL − S AZ
D A =(1/T H − 1/T 0) 2 D B =1/T H − 1/T 0 D C = S AH − S AZ D E = D BC A − C BD A
T 0 [K] T L [K] T H [K] S AZ [K − 1] S AL [K − 1] S AH[K − 1]
290 210 360 0.5956 × 10 −1 0.1474 0.9725 × 10 −2
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B
Bending relationships for simply supported
rectangular plates
The Lévy solutions can be developed for rectangular plates with two opposite
edges simply supported and the other edges having any combination of free,
simply supported or clamped boundary conditions [WRL00][Red99]. For an
isotropic plate under Kirchhoff assumptions, the transverse displacement is
expressed as the solution of a fourth order differential equation
D∇ 4 w 0 = q
(B.1)
where ∇ 4 is a differential operator and D is the flexural rigidity
∇ 4 = ∂4
∂x 4 +
∂4
∂x 2 ∂y + ∂4
,D =
2 ∂y 4
Eh 3
12 (1 − ν 2 ) . (B.2)
Fig. B.1. A rectangular plate with simply supported edges at y =0,b.Source[Red99].
The Lévy solution can be represented in terms of single Fourier series as
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Laurent Hazard 21/12/2006
B Bending relationships for simply supported rectangular plates
w K 0
∞ = ∑ ( nπy
)
W n (x)sin , (B.3)
b
n=1
which satisfies the following simply supported boundary conditions on edges
y =0andy = b:
w 0 (x, 0) = w (x, b) =0 M yy (x, 0) = M yy (x, b) =0. (B.4)
Similarly, the load q is represented as
where q n (x) is defined by
q (x, y) =
q n (x) = 2 b
∞∑ ( nπy
)
q n (x)sin , (B.5)
b
n=1
∫ b
0
q (x, y)sin( nπy
b )dy.
(B.6)
The complete solution is expressed as the sum of the homogeneous solution
and a particular solution, as given by
w K 0
(x, y) = ∞
∑
n=1
(
W
K,h
n (x)+Wn K,p (x) ) sin (β n y). (B.7)
For isotropic plates, the homogeneous solution is given by
W K,h
n =(A n + B n x)cosh(β n x)+(C n + D n x)sinh(β n x) (B.8)
where
β n = nπ b .
(B.9)
The four constants A n , B n , C n and D n present in the expression for Wn
K,h
are determined by using the four boundary conditions associated with the
boundary points x =0,a (two for each point). For a uniformly distributed
load, the particular solution Wn
K,p of the problem’s fourth order differential
equation is
Wn K,p (x) =ˆq n = 4q 0b 4
n 5 π 5 D .
Therefore, the complete Kirchhoff solution is
w K 0
∞ (x, y) = ∑[ (An + B n x)cosh(β n x)+(C n + D n x)sinh(β n x)
n=1
214
+ˆq n
]
sin (βn y)
(B.10)
(B.11)

Laurent Hazard 21/12/2006
B Bending relationships for simply supported rectangular plates
where
A n = −ˆq n , B n = 1 ( )
1 − cosh (βn a)
β nˆq n ,
2 sinh (β n a)
C n = 1 ( )( )
1 − cosh (βn a) βn a − 2sinh(β n a)
ˆq n ,
2 sinh (β n a) sinh (β n a)
D n = 1 2 β nˆq n .
(B.12)
The development of a Lévy solution for Mindlin rectangular plates is explained
in [WRL00]. For simply supported plates, the Mindlin deflection can
be obtained from the Kirchhoff solution by the simple relation
w0 M = w0 K + MK
(B.13)
K s Gh
where w0
K is the Kirchhoff solution and the second term involves M K ,the
moment sum for Kirchhoff plates, G the plate shear modulus and K s the
shear correction factor.
The moment sum for Kirchhoff plates is
∞∑
[ ∂W
M K K
= −D
n
− (β
∂y 2 n ) 2 Wn
K
n=1
]
sin (β n y) .
(B.14)
Finally,we obtain the complete transverse deflection of a Mindlin rectangular
plate with simply supported edges and a uniformly distributed load by
replacing the expression B.11 into B.13. With a few terms reorganization, we
obtain
∞∑
w0 M (x, y) = Wn M (x)sin(β n y). (B.15)
with
W M n
n=1
( ) 4 {
(x) =q n 1
D β 1+ĥ2
n
[
1 ( ) [cosh ]
−
cosh (â)+1
1+ĥ2 (ˆx)+cosh(â − ˆx)
+ 1 ] [ˆx ]}
sinh (â − ˆx)+(â − ˆx)sinh(â − ˆx)
2
(B.16)
and
q n = 4q 0
nπ , h E
s = K s h, G =
2(1+ν) ,
(B.17)
√
D
ĥ = β n , ˆx = β n x, â = β n a. (B.18)
Gh s
215

Laurent Hazard 21/12/2006

Laurent Hazard 21/12/2006
C
Bending relationships for circular plates
These analytical expression are used for the convergence study in chapter 5.
They were presented by Wang et al. in [WRL00].
Fig. C.1. Axisymmetric circular plate.
The expression C.1 was established for Kirchhoff circular plates supporting
an axisymmetric partial uniform load over some inner portion. The case
of uniformly distributed load on the whole surface is obtained immediately
by replacing α = 1 in the expression:
w K 0 = q 0R 4
64D
[ ( r
) 4 ( + α
2
4 − 3α 2 +4α 2 log α )
R
−2α 2 ( r
R) 2 (
α 2 − 4logα )] . (C.1)
The expression of the deflection w0
K (under Kirchhoff assumptions) is only
valid for a clamped plate. The corresponding Mindlin plate deflection is obtained
the following relation :
217

Laurent Hazard 21/12/2006
C Bending relationships for circular plates
w0 M = w0 K + q [
0R 2
( ] r 2
α 2 (1 − 2logα) − . (C.2)
4K s Gh
R)
The Mindlin deflection w0
M is the obtained by adding a term due to the transverse
shear deformation to w0 K . This term applies to both simply supported
and clamped plates; the deflection component caused by shear is the same
regardless of the edge conditions.
218

Laurent Hazard 21/12/2006
D
Solver choice and computational complexity
A crucial step in many numerical simulation schemes consists in the resolution
of the system of equations obtained after mathematically putting down
the original physical problem. A key factor here is the choice of the solver.
The application of the PUFEM leads to moderatly large, sparse, semi-positive
definite linear systems of equations with real (or, sometimes, complex) variables.
The semi-positive character comes from the polynomial enrichment
procedure which often bring linear dependencies between equations. These
properties of the system matrix pushes us to choose a robust direct solver
adapted to these type of matrix. After a review of some existing codes (Matlab
solver, MA47 from HSL library, etc) and some tests, we choose the last
version of the UMFPACK software designed by Tim Davis.
We are interested in the estimation of the number of computational operations
needed for the factorization and solution steps of a square matrix
problem of the type Ax = b [Axe96]. The term “Flops”(floating point operations)
is the number of elementary computational operations of the type:
one multiplication followed by an addition (including the fetching of necessary
data in memory).
For a sparse symmetric matrix A [N × N], where the entries are organised
to form a bandmatrix, we can define the semibandwidth β (a positive integer)
by the following property: a ij =0if|i − j| >β.
LU factorization
The asymptotic operation count is ≈ 1 2 N (β +1)2 for the factorisation of the
symmetric matrix A.
219

Laurent Hazard 21/12/2006
D Solver choice and computational complexity
Forward and back substitution
After the LU factorization of A, wesolve
Ly = b (forward substitution step) (D.1)
and
Ux = y (back substitution step). (D.2)
Each of these steps requires ≈ N (β + 1) flops for a bandmatrix.
The total computational complexity for a single linear symmetric system
is ≈ 1N (β 2 +1)2 +2Nβ,forN →∞. To finish, we must note that the
resolution of indefinite system requires some more steps, linked to the fact
that some black-magic pivoting algorithms are needed. These steps are however
marginal in the operations count and we choose to ignore them in the
complexity total.
220

Laurent Hazard 21/12/2006
E
Sound power levels L W
Table E.1: Sound power and sound power levels L W
of some typical sound sources. Adapted from [Cro98],
[BV92] and [Eng06].
Situation and
Sound power Sound power
sound source [Watts] level L w
[dB re 10 −12 W ]
Rocket engine 10 6 W 180 dB
Turbojet engine 10 5 W 160 dB
Short period
of exposure can
cause hearing loss
Siren, Jet plane take-off 10 3 W 150 dB
Heavy truck engine
or loudspeaker rock concert 10 2 W 140 dB
Machine gun, large pipe organ 10 1 W 130 dB
Deafening,
humain pain limit
Jackhammer, Siren
Small aircraft engine 10 0 W 120 dB
Threshold of disconfort
Chain saw, accelerating motorcycle,
band music 10 −1 W 110 dB
Very loud
Chain saw, high pressure gaz leak,
car at highway speed 10 −2 W 100 dB
Loud speech, vivid children,
heavy city traffic, 10 −3 W 90 dB
221

Laurent Hazard 21/12/2006
E Sound power levels L W
Situation and
Sound power Sound power
sound source [Watts] level L w
[dB re 10 −12 W ]
Loud- intorable for phone use,
Alarm clock,
Dishwasher 10 −4 W 80 dB
Loud voice conversation,
Noisy office, car interior noise 10 −5 W 70 dB
Loud- unusual background,
Large department store, restaurant 10 −6 W 60 dB
Moderate,
Quiet office, average home 10 −7 W 50 dB
Low voice,
small electric clock, quiet home 10 −8 W 40 dB
Very quiet,
Soft whispers, home at night 10 −9 W 30 dB
Quietest audible sound,
for persons under normal conditions,
Human breath 10 −11 W 10 dB
Threshold of hearing,
Quietest audible sound for persons with
excellent hearing under laboratory conditions 10 −12 W 0dB
222

Laurent Hazard 21/12/2006
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