×

zbMATH — the first resource for mathematics

An improved implementation of triple reciprocity boundary element method for three-dimensional steady state heat conduction problems. (English) Zbl 07110370
Summary: The domain integrals caused by heat generation appearing in the boundary integral equation (BIE) of steady state heat conduction problems can be converted into boundary integrals by triple reciprocity method (TRM). However, the current triple reciprocity approximation (TRA) of domain functions is time-consuming and its accuracy is not stable in different geometric models, due to the need to solve a combination equation, whose degree of freedom is twice the number of boundary nodes plus the number of domain interpolation points. Therefore, a new formula of TRA is proposed in the current study to save computing time and storage space and improve accuracy. In the new proposed TRA formula, the combination equation is transferred into three equations that are solved in sequence. Four numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method. Results show that although the new TRA formulation is equivalent to the original one, the approximation results are quite different, especially for non-thin structures. Particularly, the improved formula of TRA can save computing time and storage space, and get better accuracy. The improved triple reciprocity BEM is a useful approach to analyze the three-dimensional steady heat conduction problems more efficiently and accurately.

MSC:
80M15 Boundary element methods applied to problems in thermodynamics and heat transfer
65N38 Boundary element methods for boundary value problems involving PDEs
80A19 Diffusive and convective heat and mass transfer, heat flow
PDF BibTeX Cite
Full Text: DOI
References:
[1] Zienkiewicz, O. C.; Taylor, R. L.; Nithiarasu, P., The finite element method[M], (1977), McGraw-hill: McGraw-hill London
[2] Gu, Y.; Hua, Q.; Zhang, C., The generalized finite difference method for long-time transient heat conduction in 3D anisotropic composite materials[J], Appl Math Model, 71, 316-330, (2019)
[3] Li, Zeng-Yao, Coupled MLPG-FVM simulation of steady state heat conduction in irregular geometry, Eng Anal Bound Elem, 100, 265-275, (2019) · Zbl 07014430
[4] Singh, Rituraj; Singh, Krishna Mohan, ”Interpolating meshless local Petrov-Galerkin method for steady state heat conduction problem, Eng Anal Bound Elem, 101, 56-66, (2019) · Zbl 07034714
[5] Dong, C. Y., Shape optimizations of inhomogeneities of two dimensional (2D) and three dimensional (3D) steady state heat conduction problems by the boundary element method, Eng Anal Bound Elem, 60, 67-80, (2015) · Zbl 1403.80016
[6] Wang, Fajie, A BEM formulation in conjunction with parametric equation approach for three-dimensional Cauchy problems of steady heat conduction, Eng Anal Bound Elem, 63, 1-14, (2016) · Zbl 1403.80028
[7] Wei, Xing, An ACA-SBM for some 2D steady-state heat conduction problems, Eng Anal Bound Elem, 71, 101-111, (2016) · Zbl 1403.80007
[8] Cui, Miao, A new radial integration polygonal boundary element method for solving heat conduction problems, Int J Heat Mass Transf, 123, 251-260, (2018)
[9] Gu, Y.; Fan, C. M.; Xu, R. P., Localized method of fundamental solutions for large-scale modeling of two-dimensional elasticity problems[J], Appl Math Lett, 93, 8-14, (2019) · Zbl 1458.74143
[10] Zhang, A.; Gu, Y.; Hua, Q., A regularized singular boundary method for inverse Cauchy problem in three-dimensional elastostatics[J], Adv Appl Math Mech, 10, 6, 1459-1477, (2018)
[11] Partridge, P. W.; Brebbia, C. A., Dual reciprocity boundary element method, (2012), Springer Science & Business Media · Zbl 0712.73094
[12] Nowak, A. J.; Neves, A. C., The multiple reciprocity boundary element method, (1994), Springer · Zbl 0868.73006
[13] Guo, S. P.; Zhang, J. M.; Li, G. Y.; Zhou, F. L., Three-dimensional transient heat conduction analysis by Laplace transformation and multiple reciprocity boundary face method, Eng. Anal. Boundary Elem., 37, 15-22, (2013) · Zbl 1351.80016
[14] Gao, X. W., The radial integration method for evaluation of domain integrals with boundary-only discretization, Eng Anal Bound Elem, 26, 905-916, (2002) · Zbl 1130.74461
[15] Peng, H. F.; Yang, K.; Cui, M.; Gao, X. W., Radial integration boundary element method for solving two-dimensional unsteady convection – diffusion problem, Eng Anal Bound Elem, 102, 39-50, (2019) · Zbl 07063020
[16] Najarzadeh, L.; Movahedian, B.; Azhari, M., Numerical solution of scalar wave equation by the modified radial integration boundary element method, Eng Anal Bound Elem, 105, 267-278, (2019) · Zbl 07063076
[17] Ochiai, Yoshihiro; Sekiya, Tsuyoshi, ”Steady heat conduction analysis by improved multiple-reciprocity boundary element method, Eng Anal Bound Elem, 18, 2, 111-117, (1996)
[18] Ochiai, Y.; Sladek, V.; Sladek, J., ”Transient heat conduction analysis by triple-reciprocity boundary element method, Eng Anal Bound Elem, 30, 3, 194-204, (2006) · Zbl 1195.80030
[19] Ochiai, Y., Three-dimensional thermo-elastoplastic analysis by triple-reciprocity boundary element method, Eng Anal Bound Elem, 35, 3, 478-488, (2011) · Zbl 1259.74058
[20] Ochiai, Y., Three‐dimensional steady thermal stress analysis by triple‐reciprocity boundary element method, Int J Numer Methods Eng, 63, 12, 1741-1756, (2005) · Zbl 1131.74343
[21] Ochiai, Y., Meshless large plastic deformation analysis considering with a friction coefficient by triple-reciprocity boundary element method, Boundary Elem Other Mesh Reduct Methods, 27, 989-999, (2018) · Zbl 1416.74100
[22] Ochiai, Y., Numerical integration to obtain moment of inertia of nonhomogeneous material, Eng Anal Bound Elem, 101, 149-155, (2019) · Zbl 07034722
[23] Shuaiping, G.; Xuejun, L.; Jianming, Z., A triple reciprocity method in laplace transform boundary element method for three-dimensional transient heat conduction problems[J], Int J Heat Mass Transf, 114, 258-267, (2017)
[24] Zhang, J. M.; Qin, X. Y.; Han, X.; Li, G. Y., A boundary face method for potential problems in three dimensions, Int. J. Numer. Meth. Eng., 80, 320-337, (2009) · Zbl 1176.74212
[25] Qin, X.; Zhang, J.; Liu, L., Steady-state heat conduction analysis of solids with small open-ended tubular holes by BFM[J], Int J Heat Mass Transf, 55, 23-24, 6846-6853, (2012)
[26] Brebbia, C. A.; Telles, J. C.P.; Wrobel, L. C., Boundary element techniques: theory and applications in engineering, (1984), Springer-Verlag: Springer-Verlag Berlin and New York · Zbl 0556.73086
[27] Divo, E.; Kassab, A. J., Boundary element method for heat conduction: with applications in non-homogenous media, (2003), WIT Press: WIT Press Southampton · Zbl 1012.80013
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.