[PDF] 1 How Multigrid works - Comsol


[PDF] 1 How Multigrid works - Comsol

Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Paris

Cristina Filip, Bertrand Garnier, Florin Danes

Prediction of the effective thermal conductivity of composites with

spherical particles of higher thermal conductivity than the one of

the polymer matrix

Abstract. The introduction of conductive filler

in polymer matrix is an effective way to increase

the thermal conductivity of the plastic materials as

needed by several industrial applications. The

design of composites requires to optimize heat

transfer while taking into account mechanical,

ecological concerns and this involves a lot of

experimental work. The prediction of the effective

conductivity of composites material is of great help

in order to reduce tests first to elaborate thermally

conductive polymer. Finite element modelling is

used in this study to compute the effective thermal

conductivity of spherical particles of metal

regularly arranged in the polymer matrix. A thermal

contact resistance is considered at the interface

between metal and polymer in order to take into

account the effects of asperities, defects.. on the

heat transfer.

Keywords FEMLAB – Composite- Thermal

conductivity-Contact Resistance –Filled Polymer

The effect of the introduction of conductive

particles on the effective conductivity of such

composites depends strongly on the filler amount:

with low or moderate amount (less than 30- 40 %

by volume, i.e. 60- 90 % by weight) the relative

effective conductivity E remains lower than 1.25

but by increasing the metallic filler amount, one

obtains a strongly increasing value of E which

looks like a well known phenomenon: “the


Furthermore as the filler amount is increased

(over 40% vol. filler amount), it becomes more and

more risky for the processing tools (especially the

mixing machine) and usually we obtain a drastic

drop of the impact strength of the injected parts.

Therefore, it is of great importance to control the

effect of different factors on the (relative)

conductivity E of composites especially for higher

amount, i.e. when the effect of filler amount is nonlinear


1 Challenge of thermally conductive


Plastics are used anytime when the conditions of

mechanical and small deformations does not require

the use of metals. However, when there are

significant heat sources the use of pure plastics is

not recommended because of their too low thermal

conductivity. This difficulty can be overcome by

incorporating within the polymer a high amount of

filler of higher thermal conductivity (e.g. metal).

The objective is to obtain a composite material with

a conductivity from 1.5 to 3 W/ (m⋅K), i.e. 7 to 15

times higher than the one of pure polymer.

C. Filip, B.Garnier, F Danes

Laboratoire de Thermocinétique de Nantes UMR CNRS6607

Tel: (33)2 40 68 31 42

Fax: (33)2 40 68 31 41

E-mail: bertrand.garnier@univ-nantes.fr, florin.danes@neuf.fr

2 Differential model of the effective thermal


The physical model chosen to realize the

previously cited objective is a structure composed

of spheres organized as a tetragonal network, the

conductivity of the sphere being 100 to 10000 times

more conductive (factor M) than the one of the

surrounding (i.e. the matrix). The distance between

consecutive spheres depends on the direction: it is

about 0 to 7.5% of the sphere diameter in the out of

plane direction (transversal one) -factor B- whereas

in the other two directions the spheres are tangent.

A third factor is the thermal contact resistance at

the interface between sphere and its surrounding

which relative value H (multiplied by the radius of

the spheres and divided by the matrix conductivity)

was varied from 0.001 to 1000. One was looking

for the dependence E(B,M,H).

Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Paris

The mathematical model describing steady state

heat transfer is the Laplace equation, the

temperature T being the unknown. For the chosen

configuration, the problem was solved by

considering an elementary cell as shown in fig. 1. -

a quadratic prism with Dirichlet conditions on the

two in- plane sides, of adiabatic conditions on the 4

lateral sides and of 3 rd type – the contact conditionat

the interface between the sphere centered in the

prism and its surroundings.


The problem ∇ T = 0 was solved in the prism of

dimension 2, 2 and 2+2B, with temperature T= 1 on

the side Z = 1+ B and T = −1 on the side Z = −1−

B, the sides X = 1, Y = −1, Y = 1 and Y = −1 being

adiabatic. The radius of the sphere is 1, the

conductivities are equal to M in the sphere and 1

outside. The double boundary condition on the

sphere is Gext = M⋅G int = H⋅(T int −T ext ), where G is

the temperature gradient in the normal direction,

and the indexes ext and int respectively the inside

and the outside of the sphere.

The exact solution for this problem does not exist

for this configuration. The only one being

comparable is a simple cubic array - Cheng &

Torquato 1997 [2] and furthermore the solution is

only partially analytic. Indeed, because of the

complexity of computation, the authors have

provided results in terms of a linear and infinite

system where one can compute coefficients

function of B, H and M. Only the numerical

solution of this system after some truncations can

provide the conductivity E.

3 Finite elements modeling

We have chosen to vary the factors M, H and B

with 3, 7 and 13 levels between the indicated limits

previously chosen. A second set of runs was

performed with the factors M, B and BH (instead of

H) with 3, 12 and 5 levels. Then we have obtained

432 values for E (as 21 combinations were

identical) in the analyzed sets.

The prism volume including the sphere was divided

in 50000 to 100000 tri-linear (tetrahedral) elements

of different sizes by the FEMLAB automatic


The computation time using a PC computer

(Pentium IV, 3 GHz, 512 Mo) was between 2 to 30

minutes, according to the combination of levels of

the 3 factors and for an iteration numbers of 50.

Numerical results are presented in Table 1.

-A correct convergence of results for B larger than

0.001. For B=0, we did not succeed to compute E

except for the case M=1000 (with 7 levels of H)

-An accuracy (quadratic average of random errors)

of 0.2 to 0.6 % in the E value. This is a rough

estimation based on the residuals analysis of the

successive differences in the increasing order.

-The exactness is not always satisfactory: a great

bias seems to appear for small B values. As we did

not found any satisfying analytical solution, this

effect was highlighted only by comparing the finite

element results with a simplified physical model

« ADIABWALL »(double infinity of in- plane

adiabatic walls. This involves a transversal heat

flux, the associated computations being quite


One can show that the effective conductivity of the

ADIABWALL model should remain inferior to the

exact physical one; this is not always the case for

our finite element computations.

The (negative) bias is at least about 5 to 10 % for B

=0 and probably overcome 2% for B=0.001. For the

B level (B = 0.025), the bias is within the limits of

the inaccuracy.

5 Conclusions

The FEMLAB package provides satisfying

results except for the case when the layer between

particles is very thin (lower than 1/40 of the size of

the sphere). Unfortunately this case is the most

important one in the study of the thermal

conductivity in a quasi- percolative state.

We expect improvement in this situation in

particular by changing the mesh: element of more

elaborated shape or of smaller size or of lower size

especially for large temperature gradients and

strong variations of the gradient – the matrix layer

closed to the poles of the sphere.

The improvement of the computation (keeping

constant the number of elements) could probably

become totally satisfactory if one reduce the

elementary size to 1/16 of its volume by taking into

account the different symmetries of this


3 Characteristics of the results provided by

the finite element solver

In our computation, we have noticed:


Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Paris

1. Danès F., Garnier B., Dupuis T.

Predicting, measuring, and tailoring the

transverse thermal conductivity of

composites from polymer matrix and metal

filler. Int J Thermophysics 24:771-784


2. Cheng H., Torquato S; The effective

conductivity of periodic arrays of spheres

with interfacial resistance Proc. R. Soc.

Lond. A, 453 145-161 (1997)

T 1




T 2

a) b)

Figure 1: Elementary cell a) 3D view b) 2D cross section and boundary conditions

Table 1: Relative effective conductivity E = E (B, B⋅H) for M = 1000 computed by the finite element package


B⋅H 500 50 5 0.5 0.05 0.005 0.0005


0.0750 3.87994 3.85072 3.58842 2.31555 0.82392 0.41354 0.36275

0.0625 4.10723 4.07807 3.81590 2.51308 0.89635 0.42175 0.36132

0.0500 4.40357 4.37443 4.11075 2.77016 0.99947 0.43523 0.36054

0.0375 4.78359 4.75480 4.49335 3.12177 1.15753 0.45916 0.36118

0.0250 5.31885 5.29041 5.03198 3.64035 1.43096 0.50770 0.36452

0.0200 5.60987 5.58240 5.33231 3.94797 1.60956 0.54290 0.36701

0.0150 6.07852 5.95431 5.70582 4.32980 1.86430 0.60183 0.37371

0.0100 6.49260 6.46526 6.22217 4.86959 2.27477 0.71173 0.38633

0.0075 6.94224 6.91598 6.67733 5.31387 2.60275 0.81425 0.40004

0.0050 7.41056 7.37959 7.11647 5.79521 3.08416 0.99508 0.42455

0.0025 8.09982 8.07846 7.88668 6.76932 4.04853 1.44586 0.50226

0.0010 8.98579 8.96909 8.82488 7.92141 5.37781 2.31606 0.70927

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