In this work, we introduce multipoint flux (MF) approximation method to the problem of conduction heat transfer in anisotropic media. In such media, the heat flux vector is no longer coincident with the temperature gradient vector. In this case, thermal conductivity is described as a second order tensor that usually requires, at least, six quantities to be fully defined in general three-dimensional problems. The two-point flux finite differences approximation may not handle such anisotropy and essentially more points need to be involved to describe the heat flux vector. In the framework of mixed finite element method (MFE), the MFMFE methods are locally conservative with continuous normal fluxes. We consider the lowest order Brezzi–Douglas–Marini (BDM) mixed finite element method with a special quadrature rule that allows for nodal velocity elimination resulting in a cell-centered system for the temperature. We show comparisons with some analytical solution of the problem of conduction heat transfer in anisotropic long strip. We also consider the problem of heat conduction in a bounded, rectangular domain with different anisotropy scenarios. It is noticed that the temperature field is significantly affected by such anisotropy scenarios. Also, the technique used in this work has shown that it is possible to use the finite difference settings to handle heat transfer in anisotropic media. In this case, heat flux vectors, for the case of rectangular mesh, generally require six points to be described.

References

References
1.
Wang
,
M.
,
Meng
,
F.
, and
Pan
,
N.
,
2007
, “
Transport Properties of Functionally Graded Materials
,”
J. Appl. Phys.
,
102
, p.
033514
.10.1063/1.2767629
2.
Wang
,
M.
,
Pan
,
N.
,
Wang
,
J.
, and
Chen
,
S.
,
2007
, “
Mesoscopic Simulations of Phase Distribution Effects on the Effective Thermal Conductivity of Microgranular Porous Media
,”
J. Colloids Interface Sci.
,
311
(
2
), pp.
562
570
.10.1016/j.jcis.2007.03.038
3.
Lu
,
Y.-F.
,
1992
, “
Transform of Dynamic Heat Equation in Anisotropic Media and Its Application in Laser-Induced Temperature Rise
,”
Appl. Phys. Lett.
,
61
, pp.
2482
2484
.10.1063/1.108159
4.
Ma
,
C.-C.
, and
Chang
,
S.-W.
,
2004
, “
Analytical Exact Solutions of Heat Conduction Problems for Anisotropic Multi-Layered Media
,”
Int. J. Heat Mass Transfer
,
47
, pp.
1643
1655
.10.1016/j.ijheatmasstransfer.2003.10.022
5.
Dong
,
C.
,
Sun
,
F.
, and
Meng
,
B.
,
2007
, “
A Method of Fundamental Solutions for Inverse Heat Conduction Problems in an Anisotropic Medium
,”
Eng. Anal. Boundary Elem.
,
31
, pp.
75
82
.10.1016/j.enganabound.2006.04.007
6.
Jin
,
B.
,
Zheng
,
Y.
, and
Marin
,
L.
,
2005
, “
The Method of Fundamental Solutions for Inverse Boundary Value Problems Associated With the Steady-State Heat Conduction in Anisotropic Media
,”
Int. J. Numer. Methods Eng.
,
65
, pp.
1865
1891
.10.1002/nme.1526
7.
Zhou
,
H.
,
Zhang
,
S.
, and
Yang
,
M.
,
2007
, “
The Effect of Heat-Transfer Passages on the Effective Thermal Conductivity of High Filler Loading Composite Materials
,”
Compos. Sci. Technol.
,
67
, pp.
1035
1040
.10.1016/j.compscitech.2006.06.004
8.
Bagchi
,
A.
, and
Nomur
,
S.
,
2006
, “
On the Effective Thermal Conductivity of Carbon Nanotube Reinforced Polymer Composites
,”
Compos. Sci. Technol.
,
66
, pp.
1703
1712
.10.1016/j.compscitech.2005.11.003
9.
Yu
,
B.
,
2008
, “
Analysis of Flow in Fractal Porous Media
,”
ASME Appl. Mech. Rev.
,
61
(
5
), p. 050801.10.1115/1.2955849
10.
Ma
,
Y.
,
Yu
,
B.
,
Zhang
,
D.
, and
Zou
,
M.
,
2003
, “
A Self-Similarity Model for Effective Thermal Conductivity of Porous Media
,”
J. Phys. D: Appl. Phys.
,
36
(
17
), pp.
2157
2164
.10.1088/0022-3727/36/17/321
11.
Sladek
,
J.
,
Sladek
,
V.
,
Hellmich
,
C.
, and
Eberhardsteiner
,
J.
,
2007
, “
Heat Conduction Analysis of 3-D Axisymmetric and Anisotropic FGM Bodies by Meshless Local Petrov-Galerkin Method,
Comput. Mech.
,
39
, pp.
323
333
.10.1007/s00466-006-0031-3
12.
Ang
,
W.-T.
, and
Clements
,
D. L.
,
2010
, “
Nonlinear Heat Equation for Nonhomogeneous Anistropic Materials: A Dual-Reciprocity Boundary Element Solution
,”
Numer. Methods Partial Differ. Equ.
,
26
, pp.
771
784
.10.1002/num.20452
13.
Shiah
,
Y.
, and
Shi
,
Y.
,
2006
, “
Heat Conduction Across Thermal Barrier Coatings of Anisotropic Substrates
,”
Int. Commun. Heat Mass Transfer
,
33
, pp.
827
835
.10.1016/j.icheatmasstransfer.2006.04.006
14.
Ishiguro
,
S.
, and
Tanaka
,
M.
,
2007
, “
Analysis of Two-Dimensional Nonlinear Transient Heat Conduction in Anisotropic Solids by Boundary Element Method Using Dual Reciprocity Method
,”
J. Environ. Eng.
,
2
, pp.
266
277
.10.1299/jee.2.266
15.
Choo
,
K.
,
2005
, “
Microscale Heat Conduction in Homogeneous Anisotropic Media: A Dual-Reciprocity Boundary Element Method and Polynomial Time Interpolation Approach
,”
Eng. Anal. Boundary Elem.
,
29
, pp.
1143
1152
.10.1016/j.enganabound.2005.07.002
16.
Shiah
,
Y.
, and
Lin
,
C.
,
2002
, “
Anisotropic Heat Conduction Involving Internal Arbitrary Volume Heat Generation Rate
,”
Int. Commun. Heat Mass Transfer
,
29
, pp.
1079
1088
.10.1016/S0735-1933(02)00436-0
17.
Wang
,
M.
,
Kang
,
Q.
, and
Pan
,
N.
,
2009
, “
Thermal Conductivity Enhancement of Carbon Fiber Composites
,”
Appl. Therm. Eng.
,
29
, pp.
418
421
.10.1016/j.applthermaleng.2008.03.004
18.
Padovan
,
J.
,
1974
, “
Semi-Analytical Finite Element Procedure for Conduction in Anisotropic Axisymmetric Solids
,”
Int. J. Numer. Methods Eng.
,
8
, pp.
295
310
.10.1002/nme.1620080209
19.
Krizek
,
M.
, and
Liu
,
L.
,
1998
, “
Finite Element Approximation of a Nonlinear Heat Conduction Problem in Anistropic Media
,”
Comput. Methods Appl. Mech. Eng.
,
157
, pp.
387
397
.10.1016/S0045-7825(97)00247-8
20.
Raviart
,
P. A.
, and
Thomas
,
J. M.
,
1975
, “
A Mixed Finite Element Method for 2nd Order Elliptic Problems
,”
Mathematical Aspects of Finite Element Methods, Proc. Conf. Consiglio Naz delle Ricerche CNR
,
Rome
.
21.
Brezzi
,
F.
,
Douglas
,
J.
, and
Marini
,
L. D.
,
1985
, “
Two Families of Mixed Finite Elements for Second Order Elliptic Problems
,”
Numer. Math.
,
47
, pp.
217
235
.10.1007/BF01389710
22.
Wheeler
,
M. F.
, and
Yotov
,
I.
,
2006
, “
A Multipoint Flux Mixed Finite Element Method
,”
SIAM J. Numer. Anal.
,
44
, pp.
2082
2106
.10.1137/050638473
23.
Zhang
,
X.
,
1990
, “
Steady-State Temperatures in an Anisotropic Strip
,”
ASME J. Heat Transfer
,
112
(1), pp.
16
20
.10.1115/1.2910340
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