A new local meshless method for steady-state heat conduction in heterogeneous materials. (English) Zbl 1244.80018

Summary: In this paper a truly meshless method based on the integral form of energy equation is presented to study the steady-state heat conduction in the anisotropic and heterogeneous materials. The presented meshless method is based on the satisfaction of the integral form of energy balance equation for each sub-particle (sub-domain) inside the material. Moving least square (MLS) approximation is used for approximation of the field variable over the randomly located nodes inside the domain. In the absence of heat generation, the domain integration is eliminated from the formulation of presented method and the computational efforts are reduced substantially with respect to the conventional MLPG method. A direct method is presented for treatment of material discontinuity at the heterogeneous material in the presented meshless method. As a practical problem the heat conduction in fibrous composite material is studied and the steady-state heat conduction in unidirectional fiber-matrix composites is investigated. The solution domain includes a small area of the composite system called representative volume element (RVE). Comparison of numerical results shows that the presented meshless method is simple, effective, accurate and less costly method for micromechanical analysis of heat conduction in heterogeneous materials.


80M25 Other numerical methods (thermodynamics) (MSC2010)
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI


[1] Nayroles, B.; Touzot, B.; Villon, P., Generalizing the finite element method: diffuse approximation and diffuse elements, Comput mech, 10, 307-318, (1992) · Zbl 0764.65068
[2] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int J numer methods eng, 37, 229-256, (1994) · Zbl 0796.73077
[3] Liu, W.K.; Chen, Y.; Chang, C.T.; Belytschko, T., Advances in multiple scale kernel particle methods, Comput mech, 18, 73-111, (1996) · Zbl 0868.73091
[4] Zhu, T.; Zhang, J.D.; Atluri, S.N., A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach, Comput mech, 21, 223-235, (1998) · Zbl 0920.76054
[5] Atluri, S.N.; Zhu, T., A new meshless local petrov – galerkin (MLPG) approach in computational mechanics, Comput mech, 22, 117-127, (1998) · Zbl 0932.76067
[6] Singh, A.; Singh, I.V.; Prakash, R., Numerical solution of temperature dependent thermal conductivity problems using a meshless method, Numer heat transfer, part A, 50, 125-145, (2006)
[7] Singh, I.V.; Sandeep, K.; Prakash, R., The element free Galerkin method in three-dimensional steady state heat conduction, Int J comput eng sci, 3, 291-303, (2002)
[8] Singh, I.; Prakash, R., The numerical solution of three-dimensional transient heat conduction problems using element free Galerkin method, Int J heat technol, 21, 73-80, (2003)
[9] Singh, I.V., A numerical solution of composite heat transfer problems using meshless method, Int J heat mass transfer, 47, 2123-2138, (2004) · Zbl 1050.80006
[10] Liu, Y.; Zhang, X.; Liu, M.W., A meshless method based on least squares approach for steady-and unsteady state heat conduction problems, Numer heat transfer, 47, 257-275, (2005)
[11] Tan, J.Y.; Lin, L.H.; Li, B.X., Least-squares collocation meshless approach for coupled radiative and conductive heat transfer, Numer heat transfer, part B, 49, 179-195, (2006)
[12] Sadat, H.; Dubus, N.; Gbahoue, L.; Sophy, T., On the solution of heterogeneous heat conduction problems by a diffuse approximation meshless method, Numer heat transfer, part B, 50, 491-498, (2006)
[13] Qian, L.F.; Batra, R.C., Three dimensional transient heat conduction in a functionally graded thick plate with a high order plate theory and a meshless local Petrov Galerkin method, Comput mech, 35, 214-226, (2005) · Zbl 1143.74321
[14] Sladek, J.; Sladek, V.; Atluri, S.N., Meshless local petrov – galerkin method for heat conduction problem in an anisotropic medium, CMES: comput model eng sci, 6, 309-318, (2004) · Zbl 1084.80002
[15] Wu, X.H.; Tao, W.Q., Meshless method based on the local weak-forms for steady-state heat conduction problems, Int J heat mass transfer, 51, 3103-3112, (2008) · Zbl 1144.80356
[16] Han, L.S.; Cosner, A.A., Effective thermal conductivities of fibrous composites, J heat transfer, 103, 387-392, (1981)
[17] Gordon, F.H.; Turner, S.P.; Taylor, R.; Clyne, T.W., The effect of the interface on the thermal conductivity of titanium-based composites, Composites, 25, 583-592, (1994)
[18] Ning, Q.G.; Chou, T.W., Closed-form solutions of the in-plane effective thermal conductivities of woven-fabric composites, Comput sci technol, 55, 41-48, (1995)
[19] Hatta, H.; Taya, M.; Kulacki, F.A.; Harder, J.F., Thermal diffusivities of composites with various types of filler, J compos mat, 26, 612-625, (1992)
[20] Hamada, Y.; Otsu, W.; Fukai, J.; Morozumi, Y.; Miyatake, O., Anisotropic heat transfer in composites based on high-thermal conductive carbon fibers, Energy, 30, 221-233, (2005)
[21] Sevostianov, I.; Kachanov, M., Connection between elastic moduli and thermal conductivities of anisotropic short fiber reinforced thermoplastics: theory and experimental verification, Mater sci eng A, 360, 339-344, (2003)
[22] Milton, G.W., Mechanics of composites, (2000), Cambridge University Press Cambridge
[23] Progelhof, R.C.; Throne, J.L.; Ruetsch, R.R., Methods for prediction the thermal conductivity of composite systems: a review, Polym eng sci, 9, 615-625, (1976)
[24] Zhou, H.; Zhang, S.; Yang, M., The effect of heat-transfer passages on the effective thermal conductivity of high filler loading composite materials, Comput sci technol, 67, 1035-1040, (2007)
[25] Chen, C.H.; Wang, Y.C., Effective thermal conductivity of misoriented short-fiber reinforced thermoplastics, Mech mater, 23, 217-228, (1996)
[26] Hatta, H.; Taya, M., Equivalent inclusion method for steady state heat conduction in composites, Int J eng sci, 24, 1159-1172, (1986) · Zbl 0587.73179
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.