zbMATH — the first resource for mathematics

Truly meshless localized type techniques for the steady-state heat conduction problems for isotropic and functionally graded materials. (English) Zbl 1259.80029
Summary: A numerical solution of steady-state heat conduction problems is obtained using the strong form meshless point collocation (MPC) method. The approximation of the field variables is performed using the moving least squares (MLS) and the local form of the multiquadrics radial basis functions (LRBF). The accuracy and the efficiency of the MPC schemes (with MLS and LRBF approximations) are investigated through variation (i) of the nodal distribution type used, i.e. regular or irregular, ensuring the so-called positivity conditions, (ii) of the number of nodes in the total spatial domain (TD), and (iii) of the number of nodes in the support domain (SD). Numerical experiments are performed on representative case studies of increasing complexity, such as, (a) a regular geometry with a constant conductivity and uniformly distributed heat source, (b) a regular geometry with a spatially varying conductivity and non-uniformly distributed heat source, and (c) an irregular geometry in case of insulation of vapor transport tubes, as well. Steady-state boundary conditions of the Dirichlet-, Neumann-, or Robin-type are assumed. The results are compared with those calculated by the Finite Element Method with an in-house code, as well as with analytical solutions and other literature data. Thus, the accuracy and the efficiency of the method are demonstrated in all cases studied.

80M25 Other numerical methods (thermodynamics) (MSC2010)
80A20 Heat and mass transfer, heat flow (MSC2010)
PDF BibTeX Cite
Full Text: DOI
[1] Bejan, A., Heat transfer, (1993), John Wiley & Sons New York · Zbl 0989.80012
[2] Holman, J.P., Heat transfer, (1989), McGraw-Hill
[3] Incropera, F.P.; Dewitt, D.P., Fundamentals of heat and mass transfer, (1990), John Wiley & Sons New York
[4] Sukhatme, S.P.A., Text book on heat transfer, (1992), Orient Longman Publishers
[5] Mahjoob, S.; Vafai, k., Analytical characterization of heat transport through biological media incorporating hyperthermia, Int J heat mass, 52, 1608-1618, (2009) · Zbl 1157.80353
[6] Forsythe, G.; Wasow, W., Finite-difference methods for partial differential equations, (1960), John Wiley New York · Zbl 0099.11103
[7] Collatz, L., The numerical treatment of differential equations, (1966), Springer Berlin · Zbl 0221.65088
[8] Versteeg, HK; Malalasekera, W., An introduction to computational fluid dynamics. the finite volume method, (1995), Longman Scientific & Technical
[9] Zienkiewicz, O.C.; Taylor, R.L., The finite element method, (1989), McGraw-Hill New York
[10] Brebbia, C.A.; Telles, J.C.F.; Wrobel, L.C., Boundary element techniques, (1984), Springer-Verlag · Zbl 0556.73086
[11] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput methods appl mech eng, 139, 3-47, (1996) · Zbl 0891.73075
[12] Bourantas, G.C.; Skouras, E.D.; Loukopoulos, V.C.; Nikiforidis, G.C., An accurate, stable and efficient domain-type meshless method for the solution of MHD flow problems, J comput phys, 228, 8135-8160, (2009) · Zbl 1391.76510
[13] Buhmann, M.D., Radial basis function: theory and implementations, (2003), Cambridge University Press Cambridge · Zbl 1038.41001
[14] Kansa, E.J., Multiquadrics: a scattered data approximation scheme with applications to computational fluid dynamics I. solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput math appl, 19, 147-161, (1990) · Zbl 0850.76048
[15] Fasshauer GE. Solving partial differential equations by collocation with radial basis functions. In: Mehaute AL, Rabut C, Schumaker LL, editors. Surface fitting and multiresolution methods, 1997. · Zbl 0938.65140
[16] Power, H.; Barraco, W.A., Comparison analysis between unsymmetric and symmetric RBFCMs for the numerical solution of pdes, Comput math appl, 43, 551-583, (2002) · Zbl 0999.65135
[17] Chen, W., New RBF collocation schemes and kernel RBFs with applications, Lect notes comput sci eng, 26, 75-86, (2002) · Zbl 1016.65094
[18] Mai-Duy, N.; Tran-Cong, T., Indirect RBPN method with thin plate splines for numerical solution of differential equations, Comput model eng sci, 4, 85-102, (2003) · Zbl 1148.76351
[19] Sarler, B., A radial basis function collocation approach in computational fluid dynamics, Comput model eng sci, 7, 185-194, (2005) · Zbl 1189.76380
[20] Mai-Duy, N.; Tran-Cong, T., Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks, Neural networks, 14, 185-199, (2001) · Zbl 1047.76101
[21] Sarler, B.; Perko, J.; Chen, C.S., Radial basis function collocation method solution of natural convection in porous media, Int J numer methods heat fluid flow, 14, 187-212, (2004) · Zbl 1103.76361
[22] Kovacevid, I.; Poredo, A.; Sarler, B., Solving the Stefan problem by the RBFCM, Numer heat transfer, part B: fundamentals, 44, 1-24, (2003)
[23] Chen, C.S.; Ganesh, M.; Golberg, M.A.; Cheng, AH-D, Multilevel compact radial basis functions based computational scheme for some elliptic problems, Comput math appl, 43, 359-378, (2002) · Zbl 0999.65143
[24] Mai-Duy, N.; Tran-Cong, T., Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson’s equations, Eng anal bound elem, 26, 133-156, (2002) · Zbl 0996.65131
[25] Lee, C.K.; Liu, X.; Fan, S.C., Local multiquadric approximation for solving boundary value problems, Comput mech, 30, 396-409, (2003) · Zbl 1035.65136
[26] Onate, E.; Idelsohn, S.; Zienkiewicz, O.; Taylor, R.L., A finite point method in computational mechanics application to convective transport and fluid flow, Int J numer methods eng, 39, 3839-3866, (1995) · Zbl 0884.76068
[27] Wu, X.H.; Tao, W.Q., Meshless method based on the local weak-forms for steady-state heat conduction problems, Int J heat mass trans, 51, 3103-3112, (2008) · Zbl 1144.80356
[28] Sladek, V.; Sladek, J.; Tanaka, M.; Zhang, C., Local integral equation method for potential problems in functionally graded materials, Eng anal bound elem, 29, 829-843, (2005) · Zbl 1182.74237
[29] Sladek, V.; Sladek, J.; Tanaka, M.; Zhang, C., Transient heat conduction in anisotropic and functionally graded media by local integral equations, Eng anal bound elem, 29, 1047-1065, (2005) · Zbl 1182.80016
[30] Sladek, J.; Sladek, V.; Tan, C.L.; Atluri, S.N., Analysis of transient heat conduction in 3D anisotropic functionally graded solids, by the MLPG method, CMES: comput modelling eng sci, 32, 161-174, (2008) · Zbl 1232.80006
[31] Chorin, A.J., Numerical study of slightly viscous flow, J fluid mech, 57, 785-796, (1973)
[32] Bernard, P.S., A deterministic vortex sheet method for boundary layer flow, J comput phys, 117, 132-145, (1995) · Zbl 0818.76063
[33] Girault, V., Theory of a GDM on irregular networks, SIAM J num anal, 11, 260-282, (1974) · Zbl 0296.65049
[34] Pavlin, V.; Perrone, N., Finite difference energy techniques for arbitrary meshes, Comp struct, 5, 45-58, (1975)
[35] Liszka T, Orkisz J. Finite difference methods of arbitrary irregular meshes in non-linear problems of applied mechanics. In: Proceedings of the fourth international conference on structural mechanism. In Reactor Tech, San Francisco. USA; 1977. · Zbl 0453.73086
[36] Liszka, T.; Orkisz, J., The finite difference methods at arbitrary irregular grids and its applications in applied mechanics, Comp struct, 11, 83-95, (1980) · Zbl 0427.73077
[37] Kansa, E.J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics, Comput math appl, 19, 127-145, (1990) · Zbl 0692.76003
[38] Wu, Z., Hermite – birkhoff interpolation of scattered data by radial basis functions, Approx theory appl, 8, 1-10, (1992) · Zbl 0757.41009
[39] Kim, D.W.; Liu, W.K., Maximum principle and convergence analysis for the meshfree point collocation method, SIAM J numer anal, 44, 515-539, (2006) · Zbl 1155.65090
[40] Liu, G.R.; Gu, Y.T., An introduction to meshfree methods and their programming, (2005), Springer
[41] Bourantas, G.C.; Skouras, E.D.; Nikiforidis, G.C., Adaptive support domain implementation on the moving least squares approximation for mfree methods applied on elliptic and parabolic PDE problems using strong-form description, CMES—comp model eng, 43, 1-25, (2009) · Zbl 1232.65153
[42] Arnold, D.N.; Liu, X., Local error estimates for finite element discretizations of the Stokes equations, Math model numer anal, 29, 367-389, (1995) · Zbl 0832.65117
[43] Taylor, C.; Hood, P., A numerical solution of the navier – stokes equations using the finite element technique, Comput fluid, 1, 73-100, (1973) · Zbl 0328.76020
[44] Goldberg, M.A.; Chen, C.S.; Karur, S.R., Improved multiquadrics approximation for partial differential equations, Eng anal bound elem, 18, 9-17, (1996)
[45] Goldberg, M.A.; Chen, C.S., A bibliography on radial basis functions approximations, Bound elem commun, 7, 155-163, (1996)
[46] Jin, X.; Li, G.; Aluru, N.R., Positivity conditions in meshless collocation methods, Comput methods appl mech eng, 193, 1171-1202, (2004) · Zbl 1060.74667
[47] Gosz, J.; Liu, W.K., Admissible approximations for essential boundary conditions in the reproducing kernel particle method, Comput mech, 19, 120-135, (1996) · Zbl 0889.73078
[48] Sellountos, E.J.; Polyzos, D.A., MLPG (LBIE) method for solving frequency domain elastic problems, CMES: comput modelling eng sci, 4, 619-636, (2003) · Zbl 1064.74170
[49] Xiao-Wei, Gao, A meshless BEM for isotropic heat conduction problems with heat generation and spatially varying conductivity, Int J numer meth eng, 66, 1411-1431, (2006) · Zbl 1116.80021
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.