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Meshless local B-spline collocation method for heterogeneous heat conduction problems. (English) Zbl 07034716
Summary: Several numerical issues still pertain in the modeling of heterogeneous heat conduction, particularly from the viewpoints of material discontinuity and its handling and the presence of heat source. In this paper, a meshless local B-spline collocation method is presented for unsteady heat conduction problems of heterogeneous media. Unknown field variables are approximated by using B-spline basis functions within overlapped compact domains covering the geometry of materials. The present method is a truly meshless approach. The proposed approach is mainly coming with the following advantage that it is straightforward in dealing with discontinuity across the interface of heterogeneous materials. Treatment of discontinuity by using non-crossing interface compact domains allows material discontinuity to be handled geometrically without enforcing additional term/function at the interface. Several heat conduction problems in 2D and 3D heterogeneous media with arbitrary discontinuity shapes are considered. Attention is given for heterogeneous heat conduction problem accompanied by the presence of crack as well. The analysis is then completed by simulating effect of heat generation, in particular which produces high temperature rise inside a heterogeneous structure/component. Simulation results show that the proposed method is a simple and accurate numerical technique for solving unsteady heat conduction problems of heterogeneous media.

MSC:
80M22 Spectral, collocation and related (meshless) methods applied to problems in thermodynamics and heat transfer
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
80A19 Diffusive and convective heat and mass transfer, heat flow
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[1] Yang, P.; Qian, X., A B-spline-based approach to heterogeneous objects design and analysis, Comput-Aided Des, 39, 95-111, (2007)
[2] Zhou, H. M.; Liu, Z. G.; Lu, B. H., Heat conduction analysis of heterogeneous objects based on multi-color distance field, Mater Des, 31, 3331-3338, (2010)
[3] Gao, Y.; Cheng, H., Assembly of heterogeneous materials for biology and electronics: from bio-inspiration to bio-integration, J Electron Packag, 139, (2017)
[4] Batra, R. C.; Porfiri, M.; Spinello, D., Treatment of material discontinuity in two meshless local Petrov-Galerkin (MLPG) formulations of axisymmetric transient heat conduction, Int J Numer Methods Eng, 61, 2461-2479, (2004) · Zbl 1075.80001
[5] Ma, C. C.; Chang, S. W., Analytical exact solutions of heat conduction problems for anisotropic multi-layered media, Int J Heat Mass Transf, 47, 1643-1655, (2004) · Zbl 1057.80003
[6] Chao, C. K.; Chen, C. K.; Chen, F. M., Analytical exact solutions of heat conduction problems for a three-phase elliptical composite, Comput Model Eng Sci, 2, 3, 283-297, (2009) · Zbl 1231.80008
[7] Choi, D. K.; Nomura, S., Numerical Green’s function for steady-state heat conduction in heterogeneous media, Trans ASME, 120, 284-286, (1998)
[8] Shen, M. H.; Chen, F. M.; Hung, S. Y., Analytical solutions for anisotropic heat conduction problems in a trimaterial with heat sources, J Heat Transf, 132, (2010)
[9] Ozisik, M. N., Heat conduction, (1993), Wiley: Wiley New York, USA
[10] Cheng, A. H.D.; Cheng, D. T., Heritage and early story of the boundary element method, Eng Anal Boundary Elem, 29, 268-302, (2005) · Zbl 1182.65005
[11] Brebbia, C. A., The birth of the boundary element method from conception to application, Eng Anal Boundary Elem, 77, iii-iix, (2017) · Zbl 1403.65002
[12] LeVeque, R. J., Finite volume methods for hyperbolic problems, (2004), Cambridge University Press: Cambridge University Press UK
[13] Peng, H. F.; Bai, Y. G.; Yang, K.; Gao, X. W., Three-step multi-domain BEM for solving transient multi-media heat conduction problems, Eng Anal Boundary Elem, 37, 1545-1555, (2013) · Zbl 1287.80009
[14] Bazyar, M. H.; Talebi, A., Scaled boundary finite-element method for solving non-homogeneous anisotropic heat conduction problems, Appl Math Model, 39, 7583-7599, (2015)
[15] Mierzwiczak, M.; Chen, W.; Fu, Z. J., The singular boundary method for steady-state nonlinear heat conduction problem with temperature-dependent thermal conductivity, Int J Heat Mass Transf, 91, 205-217, (2015)
[16] Feng, W. Z.; Gao, X. W., An interface integral equation method for solving transient heat conduction in multi-medium materials with variable thermal properties, Int J Heat Mass Transf, 98, 227-239, (2016)
[17] Lu, S.; Liu, J.; Lin, G.; Zhang, P., Modified scaled boundary finite element analysis of 3D steady-state heat conduction in anisotropic layered media, Int J Heat Mass Transf, 108, 2462-2471, (2017)
[18] Mohamad, A. A.; Tan, Q. W.; He, Y. L.; Bawazeer, S., Treatment of transport at the interface between multilayers via the lattice Boltzmann method, Numer Heat Transf: Part B, 67, 124-134, (2015)
[19] Rihab, H.; Moudhaffar, N.; Sassi, B. N.; Patrick, P., Enthalpic lattice Boltzmann formulation for unsteady heat conduction in heterogeneous media, Int J Heat Mass Transf, 100, 728-736, (2016)
[20] Gao, D.; Chen, Z.; Chen, L.; Zhang, D., A modified lattice Boltzmann model for conjugate heat transfer in porous media, Int J Heat Mass Transf, 105, 673-683, (2017)
[21] Gong, J.; Xuan, L.; Ming, P.; Zhang, W., An unstructured finite-volume method for transient heat conduction analysis of multilayer functionally graded materials with mixed grids, Numer Heat Transf: Part B, 63, 222-247, (2013)
[22] Han, S., Finite volume solution of 2-D hyperbolic conduction in a heterogeneous medium, Numer Heat Transf: Part A, 70, 7, 723-737, (2017)
[23] Gong, J.; Xuan, L.; Ming, P.; Zhang, W., Application of the staggered cell-vertex finite volume method to thermoelastic analysis in heterogeneous materials, J Thermal Stress, 37, 506-531, (2014)
[24] Shiah, Y. C.; Chaing, Y. C.; Matsumoto, T., Analytical transformation of volume integral for the time-stepping BEM analysis of 2D transient heat conduction in an isotropic media, Eng Anal Boundary Elem, 64, 101-110, (2016) · Zbl 1403.80027
[25] Lucy, L. B., A numerical approach to the testing of the fission hypothesis, Astron J, 82, 1013-1024, (1977)
[26] Gingold, R. A.; Monaghan, J. J., Smoothed particle hydrodynamics: theory and application to non-spherical stars, Mon Not R Astron Soc, 181, 375-389, (1977) · Zbl 0421.76032
[27] Atluri, S. N.; Shen, S. P., The meshless local Petrov-Galerkin (MLPG) method, (2002), Tech Science Press: Tech Science Press USA · Zbl 1012.65116
[28] Liu, G. R., Meshfree methods: moving beyond the finite element method, (2009), CRC Press: CRC Press USA
[29] Gerace, S.; Erhart, K.; Matsumoto, T., A model-integrated localized collocation meshless method for large scale three-dimensional heat transfer problems, Eng Anal Boundary Elem, 64, 101-110, (2016)
[30] Sadat, H.; Dubus, N.; Gbahoué, L.; Sophy, T., On the solution of heterogeneous heat conduction problems by a diffuse approximation meshless method, Numer Heat Transf: Part B, 50, 491-498, (2006)
[31] Fang, J.; Zhao, G. F.; Zhao, J.; Parriaux, A., On the truly meshless solution of heat conduction problems in heterogeneous media, Numer Heat Transf: Part B, 55, 1-13, (2009)
[32] Singh, I. V.; Tanaka, M., Heat transfer analysis of composite slabs using meshless element free Galerkin method, Comput Mech, 38, 521-532, (2006) · Zbl 1168.80309
[33] Ahmadi, I.; Sheikhy, N.; Aghdam, M. M.; Nourazar, S. S., A new local meshless method for steady-state heat conduction in heterogeneous materials, Eng Anal Boundary Elem, 34, 1105-1112, (2010) · Zbl 1244.80018
[34] Zhang, X.; Zhang, P., Heterogeneous heat conduction problems by an improved element-free Galerkin method, Numer Heat Transf: Part B, 65, 359-375, (2014)
[35] Biswas, A.; Shapiro, V.; Tsukanov, I., Heterogeneous material modeling with distance fields, Comput Aided Geom Des, 21, 215-242, (2004) · Zbl 1069.65508
[36] Tsukanov, I.; Shapiro, V., Meshfree modeling and analysis of physical fields in heterogeneous media, Adv Comput Math, 23, 95-124, (2005) · Zbl 1137.76454
[37] Freytag, M.; Shapiro, V.; Tsukanov, I., Field modeling with sampled distances, Comput-Aided Des, 38, 87-100, (2006)
[38] Hidayat, M. I.P.; Wahjoedi, B. A.; Parman, S.; Megat Yusoff, P. S.M., Meshless local B-spline-FD method and its application for 2D heat conduction problems with spatially varying thermal conductivity, Appl Math Comput, 242, 236-254, (2014) · Zbl 1334.80012
[39] Hidayat, M. I.P.; Wahjoedi, B. A.; Parman, S., A new meshless local B-spline basis functions-FD method for two-dimensional heat conduction problems, Int J Numer Methods Heat Fluid Flow, 25, 2, 225-251, (2015) · Zbl 1356.80064
[40] Hidayat, M. I.P.; Wahjoedi, B. A.; Parman, S.; Rao, T. V.V. L.N, Meshless local B-spline collocation method for two-dimensional heat conduction problems with nonhomogenous and time-dependent heat sources, ASME J Heat Transf, 139, 7, (2017)
[41] Cox, M., The numerical evaluation of B-spline, J Inst Math Appl, 10, 134-149, (1972) · Zbl 0252.65007
[42] de Boor, C., On calculating with B-splines, J Approx Theory, 6, l, 50-62, (1972) · Zbl 0239.41006
[43] de Boor, C., A practical guide to splines, (2001), Springer: Springer New York · Zbl 0987.65015
[44] Farin, G., Curves and surfaces for computer aided geometric design, (2002), Academic Press: Academic Press San Diego, CA
[45] Piegl, L.; Tiller, W., The NURBS book, (1995), Springer: Springer New York · Zbl 0828.68118
[46] Liew, K. M.; Huang, Y. Q.; Reddy, J. N., Moving least squares dierential quadrature method and its application to the analysis of shear deformable plates, Int J Numer Methods Eng, 56, 2331-2351, (2003) · Zbl 1062.74658
[47] Shu, C.; Ding, H.; Yeo, K. S., Local radial basis function-based differential quadrature method and its application to solve two dimensional incompressible Navier-Stokes equations, Comput Methods Appl Mech Eng, 192, 941-954, (2003) · Zbl 1025.76036
[48] Oterkus, S.; Madenci, E.; Agwai, A., Peridynamic thermal diffusion, J Comput Phys, 265, 71-96, (2014) · Zbl 1349.80020
[49] Khosravifard, A.; Hematiyan, M. R.; Marin, L., Nonlinear transient heat conduction analysis of functionally graded materials in the presence of heat sources using an improved meshless radial point interpolation method, Appl Math Model, 35, 4157-4174, (2011) · Zbl 1225.74033
[50] Touloukian, Y. S., Thermophysical properties of high temperature solid materials, (1976), McMillan: McMillan New York
[51] Lei, M.; Li, M.; Wen, P. H.; Bailey, C. G., Moving boundary analysis in heat conduction with multilayer composites by finite block method, Eng Anal Boundary Elem, 89, 36-44, (2018) · Zbl 1403.80020
[52] Yang, D. S.; Ling, J.; Wang, H. Y.; Chen, T. Y.; Du, Y. D.; Chen, S. G.; Zhang, K. Q., Calculating the multi-domain transient heat conduction with heat source problem by virtual boundary meshfree Galerkin method, Numer Heat Transf: Part B: Fundam, 74, 1, 465-479, (2018)
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