ACME 2011 Proceedings of the 19 UK National Conference of the ...

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Pressure, P (mmHg) Flow Rate, Q (L/min) 140 120 100 80 60 30 20 10 0 -10 Normal Aneurysm Diameter of 5cm Aneurysm Diameter of 10cm Aneurysm Diameter of 15cm Ascending Aorta Aorta Arc B Thoracic Aorta A Thoracic Aorta B Abdominal Aorta A Abdominal Aorta D Ascending Aorta Aorta Arc B Thoracic Aorta A Thoracic Aorta B Abdominal Aorta A Abdominal Aorta D Figure 2b: Simulations of normal and various aneurysm sizes for one cardiac cycle (0.8s) where the medical terms can be shown in [3]. 4 CONCLUSIONS AND FUTURE WORK A 1D model of the human arterial system using the LCG finite element method is presented for normal ‘at rest’ state, for aneurysm sizes of 5cm, 10cm and 15cm. It should be noted that the aim of this paper is to identify arterial waveforms in the presence of a TAA. The example above showed that wave reflections occur when there is a sudden widening in an artery. Future work includes studying the influences of different aneurysm and stenosis shapes. Acknowledgement This work is partially funded by Levelhulme Trust grant F/00391/R. The first author acknowledges the financial support received from the College of Engineering. References [1] J.C. Lasheras. The biomechanics of arterial aneurysms. Annual Review of Fluid Mechanics, 39, 293- 319, 2007. [2] A. Swillens, L. Lanoye, J.D. Backer, N. Stergiopulos, P.R. Verdonck, F. Vermassen and P. Segers. Effect of an abdominal aortic aneurysm on wave reflection in the aorta. IEEE Transactions on Biomedical Engineering, 55(5), 1602-11, 2008. [3] K.S. Matthys, J. Alastruey, J. Peiro, A.W. Khir, P. Segers, P.R. Verdonck, K.H. Parker and S.J. Sherwin. Pulse wave propagation in a model human arterial network: assessment of 1-D numerical simulations against in vitro measurements. Journal of Biomechanics, 40(15), 3476-3486, 2007. [4] J.P. Mynard and P. Nithiarasu. A 1D arterial blood flow model incorporating ventricular pressure, aortic valve and regional coronary flow using the locally conservative Galerkin (LCG) method. Communications in Numerical Methods in Engineering, 24, 367-417, 2008. [5] C.G. Thomas and P. Nithiarasu. An element-wise, locally conservative Galerkin (LCG) method for solving diffusion and convection-diffusion problems. International Journal for Numerical Methods in Engineering, 73, 642-664, 2008. [6] C.G. Thomas, P. Nithiarasu and R. L. T. Bevan. The locally conservative Galerkin (LCG) method for solving the incompressible Navier-Stokes equations. International Journal for Numerical Methods in Engineering, 57, 1771-1792, 2008. [7] S.J. Sherwin, V. Franke and J. Peiro. One-dimensional modelling of a vascular network in spacenetwork variables. Journal of Engineering Mathematics, 47(3), 217-250, 2003. 32
Proceedings of the 19 th UK Conference of the Association for Computational Mechanics in Engineering 5 – 6 April 2011, Heriot-Watt University, Edinburgh RESPONSE OF A TUNNEL DEEPLY EMBEDDED IN A VISCOELASTIC MEDIUM *Thomas J. Birchall¹ and Ashraf S. Osman¹ 1 School of Engineering and Computing Sciences, Durham University, South Road, Durham, DH1 3LE *t.j.birchall@durham.ac.uk Key Words: viscoelasticity; creep; tunnels; energy-based solution. ABSTRACT This paper presents a three-dimensional energy-based solution for the time-dependent creep response of a deeply embedded and unsupported tunnel, of circular cross-section. Rock behaviour is described in the Laplace domain using Burger’s viscoelastic model. The differential equations governing the displacements of the tunnel-rock system and appropriate boundary conditions are obtained using the principle of minimum potential energy. The method produces results with accuracy comparable to that of a finite element analysis but requires much less computation effort. 1 INTRODUCTION When tunnelling in salt rock or potash in a mining environment, significant time-dependent tunnel closure can occur, as the deformation of these weak rocks is dominated by creep. Creep is also the prime mechanism causing ground squeezing in sheared or faulted rock masses containing mylonite or clay gouge. The efficient and accurate prediction of the time-dependent performance of tunnels located in such rock is a main concern in the design. However, the full three-dimensional (3D) interaction between a tunnel and the surrounding creeping rock is complex. Finite element (FE) analysis with advanced constitutive models can be used to accurately predict the time-dependent response (e.g. [1]), but such analyses are computationally expensive for routine practice. An alternative approach is to idealise the problem as a two-dimensional (2D) plane-strain problem using empirical relations based on field measurements or 3D FE analyses to predict the stresses and displacements in close proximity to the tunnel face (e.g. [2]). However, there are significant uncertainties in extrapolating these empirical relations to different design situations and rock behaviour. Here we present a 3D approximate solution for the time-dependent response of a semi-infinite and unsupported tunnel which is excavated quasi-instantaneously from a homogeneous, isotropic, viscoelastic, infinite rock body. The rock is assumed to behave volumetrically linear elastic and to exhibit exclusively deviatoric creep which is modelled by the classical Burger’s model. The initial in-situ stresses are taken to be isotropic. The potential energy of the tunnel-rock system is expressed in terms of an assumed displacement field. The principle of minimum potential energy is used to obtain the differential equations governing the ground deformation and appropriate boundary conditions. Burger’s model is the simplest linear viscoelastic model that can be used to trace primary and secondary creep deformation. According to Goodman [3] this model is preferable for many practical purposes. This model can be characterised by linear elastic volumetric behaviour and viscoelastic deviatoric behaviour. The relationship between the deviatoric stress s ij and deviatoric strain e ij in the Laplace domain is: 2 (η1s k1s) sˆ ij 2 êij (1) η1 2 G1 η1 G1 G 2 s 1 G 2 η 2 s η 2 33
Page 1: ACME 2011 Proceedings of the 19 th
Page 4 and 5: First published 2011 by Heriot-Watt
Page 7 and 8: CONTENTS Invited Lectures CHALLENGE
Page 9 and 10: A DAMAGE-MECHANICS-BASED RATE-DEPEN
Page 11: A PROJECTED QUASI-NEWTON METHOD FOR
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Page 18 and 19: (a) Finite element mesh (b) Gas mes
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Page 23 and 24: T E −1 ( ) ( ) T ∂s ∂T − du
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3 SIMULATING CONCRETE MESO-STRUCTUR
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Proceedings of the 19 th UK Confere
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¯K i is an approximation of the co
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h = 5.36 matrix 4.3 z P E f νf 0.9
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From the total hoop strainε θ , t
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3 EXTERIOR POINT ESHELBY BASED CRAC
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is applied uniformly on the edge to
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(a) (b) (c) (d) Figure 1: Influence
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On each subdomain, we then have to
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( i�1) ( i�1) �Un�1 � ε
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� ~ � 1 ψ ~ � � ~ (2) �
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For the case of uni-axial tension,
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3 Numerical results Preliminary num
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The coefficients An cannot be deriv
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The spatial semi-discretisation is
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3 CRITICAL TIME STEP INCREASED WITH
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v/vo 2E-05 1.5E-05 1E-05 5E-06 0 -3
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3 RESULTS AND DISCUSSION Fig. 1a sh
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Model updating from eigenvalue and
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1. If R = ω 2 n, there is no eigen
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F = ´ Ω STv CSv dΩ A = ´ Γ U
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Fig. 3 Wall normal pressure in the
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contact contact (ii). f ≈ f ( d,
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In the analysis of the incident P-w
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elow a certain threshold. In both t
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K/d 2 Effective Permeability 0.0032
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METHOD 2 (Mellor and Herring Type)
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(a) Free Surface Profile from Janos
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2 IMMERSED FLUID SOLVER Consider th
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References [1] R. Mittal and G. Iac
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a second impact on the forward fuse
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Acknowledgement L. Papagiannis is f
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acceptable probability throughout t
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algorithm was then responsible for
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strategy of saving the computationa
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(a) (b) Figure 5. Comparison of (a)
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Thermal load step i i i i Update ,
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Displacement (mm) Conclusions 45 40
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load carrying of the structure to b
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compressive stress field between co
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Unbonded flexible pipes and risers
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Figure 2: (a) Riser configuration (
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Furthermore the total strain varies
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Displacement 14 12 10 8 6 4 2 0 Mem
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As this project was in the conceptu
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kg and additional mass to represent
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1 Step make box 2 Step make cylinde
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element method. The solid model is
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just new element classes) without t
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Fig.4. Frame model and the displace
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capacities to meet with such demand
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esponses. Severe cases are to be co
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2 Numerical results In this section
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the frequency intially increases un
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and F (R) = ⎡ ⎢ ⎣ ... UαT (R
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Merit function g 10 1 10 0 10 -1 10
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2 STRUCTURAL CHARACTERISTICS OF PTM
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Fig. 5. Simplified FE model of a PT
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general the computational results o
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References [1] Ainsworth M, Oden JT
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Error Cause Interpolation Error Ele
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(a) Before Improvement (b) After Im
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een applied to FE fluid flow proble
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Acknowledgements Figure 2: Flowchar
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1 1 ′′ 1 ′′′ 3 ′′ 2
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3 COMBINING SHAPE FUNCTIONS The exp
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electron current). For the case of
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electron concentration (1/cm 3 ) x
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2 CONSERVATION LAW FOR A MOVING CV
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5 NUMERICAL RESULTS A 1-D cellular
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FEM can be found in [3]. In this pa
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h Y 1 0 −1 −2 −3 −4 −5
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applications. The partition of unit
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1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 2
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2 NURBS BASIS FUNCTIONS NURBS are a
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4 RESULTS To illustrate the isoBEM
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290
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1 1 2 { σ ( ξ, s) } = [ D] { ε (
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4.4 Example 4 - A single edge crack
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