×

zbMATH — the first resource for mathematics

Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method. (English) Zbl 0995.80010
Summary: The Green’s function for three-dimensional transient heat conduction (diffusion equation) for functionally graded materials (FGMS) is derived. The thermal conductivity and heat capacitance both vary exponentially in one coordinate. In the process of solving this diffusion problem numerically, a Laplace transform (LT) approach is used to eliminate the dependence on time. The fundamental solution in Laplace space is derived and the boundary integral equation formulation for the Laplace Transform boundary element method (LTBEM) is obtained. The numerical implementation is performed using a Galerkin approximation, and the time-dependence is restored by numerical inversion of the LT. Two numerical inversion techniques have been investigated: a Fourier series method and Stehfest’s algorithm, the latter being preferred. A number of test problems have been examined, and the results are in excellent agreement with available analytical solutions.

MSC:
80M25 Other numerical methods (thermodynamics) (MSC2010)
Software:
Algorithm 368
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chang, Y.; Kang, C.S.; Chen, D.J., The use of fundamental Green’s functions for the solution of problems of heat conduction in anisotropic media, Int J heat mass transf, 16, 1905-1918, (1973) · Zbl 0263.35041
[2] Shaw, R., An integral equation approach to diffusion, Int J heat mass transf, 17, 693-699, (1974)
[3] Curran, D.A.S.; Cross, M.; Lewis, B.A., A preliminary analysis of boundary element methods applied to parabolic partial differential equations, (), 179-190
[4] Wrobel, L.C.; Brebbia, C.A., The boundary element method for steady-state and transient heat conduction, (), 58-73
[5] Lesnic, D.; Elliott, L.; Ingham, D.B., Treatment of singularities in time-dependent problems using boundary element method, Engng anal boundary elem, 16, 65-70, (1995)
[6] Coda, H.B.; Venturini, W.S., A simple comparison between two 3D time domain elastodynamic boundary element formulations, Engng anal boundary elem, 17, 33-44, (1996)
[7] Coda, H.B.; Venturini, W.S., Further improvements on 3-D treatment BEM elastodynamic analysis, Engng anal boundary elem, 17, 231-243, (1996)
[8] Pasquetti, R.; Caruso, A., Boundary element approach for transient and non-linear thermal diffusion, Num heat transf, partb, 17, 83-99, (1990) · Zbl 0693.76091
[9] Pasquetti, R.; Caruso, A.; Wrobel, L.C., Transient problems using time-dependent fundamental solutions, (), 33-62
[10] Divo, E.; Kassab, A., A generalized BEM for steady and transient heat conduction in media with spatially varying thermal conductivity, (), 37-76 · Zbl 0945.65113
[11] Wrobel, L.C.; Brebbia, C.A., The dual reciprocity boundary element formulation for non-linear diffusion problems, Comput meth appl mech engng, 65, 147-164, (1987) · Zbl 0612.76094
[12] Nowak, A.J., The multiple reciprocity method of solving transient heat conduction problems, (), 81-95
[13] Rizzo, F.; Shippy, D.J.A., Method of solution for certain problems of transient heat conduction, Aiaa j, 8, 2004-2009, (1970) · Zbl 0237.65074
[14] Liggett, J.A.; Liu, P.L.F., Unsteady flow in confined aquifers: a comparison of boundary integral methods, Watr resour res, 15, 861-866, (1979)
[15] Moridis, G.J.; Reddell, D.L., The Laplace transform boundary element (LTBE) method for the solution of diffusion-type equations, (), 83-97
[16] Cheng, A.H.D.; Abousleiman, Y.; Badmus, T., A Laplace transform BEM for axysymmetric diffusion utilizing pre-tabulated Green’s function, Engng anal boundary elem, 9, 39-46, (1992)
[17] Zhu, S.P.; Satravaha, P.; Lu, X., Solving the linear diffusion equations with the dual reciprocity methods in Laplace space, Engin anal boundary elem, 13, 1-10, (1994)
[18] Zhu, S.P.; Satravaha, P., An efficient computational method for non-linear transient heat conduction problems, Appl math modeling, 20, 513-522, (1996) · Zbl 0856.73077
[19] Zhu, S.P., Time-dependent reaction diffusion problems and the LTDRM approach, (), 1-35 · Zbl 0942.65118
[20] Goldberg, M.A.; Chen, C.S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (), 103-176 · Zbl 0945.65130
[21] Cheng, A.H.-D., Darcy’s flow with variable permeability: a boundary integral solution, Watr resour res, 20, 980-984, (1984)
[22] Wu, T.W.; Lee, L., A direct boundary integral formulation for acoustic radiation in a subsonic uniform flow, J sound vibr, 175, 51-63, (1994) · Zbl 0945.76542
[23] Lacerda, L.A.; Wrobel, L.C.; Mansur, W.J., A boundary integral formulation for acoustic radiation in a subsonic uniform flow, J acoust soc am, 100, 98-107, (1996)
[24] Maillet, D.; Andre, S.; Batsale, J.C.; Degiovanni, A.; Moyne, C., Thermal quadrupoles solving the heat equation through integral transforms, (2000), Wiley New York · Zbl 0964.35004
[25] Gray, L.J.; Kaplan, T.; Richardson, D.; Paulino, G.H., Green’s functions and boundary integral analysis for exponentially graded materials: heat conduction, ASME J appl mech, (2002), (in press) · Zbl 1110.74461
[26] Tanigawa, Y., Some basic thermoelastic problems for non-homogeneous structural materials, Appl mech rev, 48, 6, 287-300, (1995)
[27] Noda, N., Thermal stresses in functionally graded material, J thermal stresses, 22, 477-512, (1999)
[28] Suresh, S.; Mortensen, A., Fundamentals of functionally graded materials, (1998), Institute of Materials, IOM Communications Ltd London
[29] Miyamoto, Y.; Kaysser, W.A.; Rabin, B.H.; Kawasaki, A.; Ford, R.G., Functionally graded materials: design, processing and applications, (1999), Kluwer Academic Publishers Dordrecht
[30] Bonnet, M., Boundary integral equation methods for solids and fluids, (1995), Wiley New York
[31] Carslaw, H.S.; Jaeger, J.C., Conduction of heat in solids, (1959), Clarendon Press Oxford · Zbl 0972.80500
[32] Li, B.Q.; Evans, J.W., Boundary element solution of heat convection – diffusion problems, J comput phys, 93, 255-272, (1991) · Zbl 0724.65106
[33] Ikeuchi, M.; Onishi, K., Boundary elements in transient convective diffusive problems, (), 275-282
[34] Ramanchandran, P.A., Boundary element methods in transport phenomena, (1994), Elsevier London
[35] Singh, K.M.; Tanaka, M., On exponential variable transformation based boundary element formulation fro advection – diffusion problems, Engng anal boundary elem, 24, 225-235, (2000) · Zbl 0942.80003
[36] Partridge, P.W.; Brebbia, C.A.; Wrobel, L.C., The dual reciprocity boundary element method, (1992), Computational Mechanics Publications Southampton · Zbl 0758.65071
[37] Brebbia, C.A.; Telles, J.C.F.; Wrobel, L.C., Boundary element techniques: theory and applications in engineering, (1984), Springer Berlin · Zbl 0556.73086
[38] Gray, L.J., Evaluation of singular and hypersingular Galerkin integrals: direct and symbolic computation, ()
[39] Sudicky, E.A., The Laplace transform Galerkin technique: a time-continuous finite element theory and application to mass transport in groundwater, Water resource res, 25, 1833-1846, (1989)
[40] Modiris, G.L.; Reddell, D.L., The Laplace transform finite difference (LTFD) numerical method for the simulation of solute transport in groundwater, 1990 AGU fall meet, San Francisco, EOS trans am geophys union, 71, 43, (1990)
[41] Chen, H.T.; Chen, C.K., Hybrid Laplace transform/finite difference method for transient heat conduction problems, Int J numer meth engng, 26, 1433-1447, (1988) · Zbl 0635.73120
[42] Chen, H.T.; Lin, J-Y., Application of the Laplace transform to non-linear transient problems, Appl math model, 15, 144-151, (1991) · Zbl 0731.65106
[43] Talbot, A., The accurate numerical inversion of Laplace transforms, J inst math appl, 23, 97-120, (1979) · Zbl 0406.65054
[44] Dubner, H.; Abate, J., Numerical inversions of Laplace transforms by relating them to the finite Fourier cosine transform, J assoc comput Mach, 15, 115-223, (1968) · Zbl 0165.51403
[45] Durbin, F., Numerical inversion of Laplace transforms: efficient improvement to dubner and Abate’s method, Computer J, 17, 371-376, (1974) · Zbl 0288.65072
[46] Crump, K.S., Numerical inversion of Laplace transforms using a Fourier series approximation, J assoc comput Mach, 23, 89-96, (1976) · Zbl 0315.65074
[47] Stehfest, H., Algorithm 368: numerical inversion of Laplace transform, Commun assoc comput Mach, 13, 47-49, (1970)
[48] Stehfest, H., Remarks on algorithm 368. numerical inversion of Laplace transform, Commun assoc comput Mach, 13, 624, (1970)
[49] Davies, B.; Martin, B., Numerical inversion of the Laplace transform: a survey and comparison of methods, J comput phys, 33, 1-32, (1979) · Zbl 0416.65077
[50] Duffy, D.G., On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications, ACM trans math software, 19, 3, 333-359, (1993) · Zbl 0892.65079
[51] Weeks, W.T., Numerical inversion of Laplace transforms using Laguerre functions, J assoc comput Mach, 13, 3, 419-429, (1966) · Zbl 0141.33401
[52] Gaver, D.P., Observing stochastic processes, and approximate transform inversion, Oper res, 14, 3, 444-459, (1966)
[53] D’Amore, L.; Laccetti, G.; Murli, A., An implementation of a Fourier series method for the numerical inversion of the Laplace transform, ACM trans math software, 25, 3, 279-305, (1999) · Zbl 0962.65109
[54] Bruch, J.C.; Zyvoloski, G., Transient two dimensional heat conduction problems solved by the finite element method, Int J numer meth engng, 8, 481-494, (1974) · Zbl 0281.65060
[55] Paulino, G.H.; Shi, F.; Mukherjee, S.; Ramesh, P., Nodal sensitivities as error estimates in computational mechanics, Acta mech, 121, 191-213, (1997) · Zbl 0878.73078
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.