An ACA-SBM for some 2D steady-state heat conduction problems. (English) Zbl 1403.80007

Summary: In this paper, an accelerated singular boundary method (SBM) incorporating adaptive cross approximation (ACA) is developed for the steady-state heat conduction problems. The SBM, a recently developed boundary collocation method, employs the fundamental solutions of the governing operators as the kernel functions, and desingularizes the source singularity with a concept of origin intensity factor. However, the SBM suffers fully-populated influence matrix which results in prohibitively expensive operation counts and memory requirements as the number of degrees of freedom increases. In this paper, the ACA is applied to accelerate the SBM meanwhile reducing the memory requirement. Furthermore, the ACA-SBM is robust to different fundamental solutions, which enables it to deal with different heat conduction problems. The effectiveness, feasibility and robustness of the proposed method are numerically tested on different heat conduction problems including isotropic homogeneous, anisotropic homogeneous and non-homogeneous media with quadratic material variation of thermal conductivity, highlighting the accuracy as well as the significant reduction in memory storage and analysis time in comparison with the traditional SBM.


80A20 Heat and mass transfer, heat flow (MSC2010)
80M15 Boundary element methods applied to problems in thermodynamics and heat transfer
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[1] Brebbia, C. A.; Telles, J. C.F.; Wrobel, L. C., Boundary element techniques: theory and applications in engineering, (1984), Springer-Verlag Berlin · Zbl 0556.73086
[2] Kupradze, V. D.; Aleksidze, M. A., The method of functional equations for the approximate solution of certain boundary value problems, USSR Comput Math Math Phys, 4, 4, 82-126, (1964) · Zbl 0154.17604
[3] Mathon, R.; Johnston, R., The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J Numer Anal, 14, 4, 638-650, (1977) · Zbl 0368.65058
[4] Chen, C. S., The method of fundamental solutions for non-linear thermal explosions, Commun Numer Methods Eng, 11, 8, 675-681, (1995) · Zbl 0839.65143
[5] Chen, C. S.; Karageorghis, A.; Smyrlis, Y. S., The method of fundamental solutions: a meshless method, (2008), Dynamic Publishers USA
[6] Chen, C. S.; Cho, H. A.; Golberg, M. A., Some comments on the ill-conditioning of the method of fundamental solutions, Eng Anal Bound Elem, 30, 5, 405-410, (2006) · Zbl 1187.65136
[7] Chen, W.; Tanaka, M., A meshless, integration-free, and boundary-only RBF technique, Comput Math Appl, 43, 3-5, 379-391, (2002) · Zbl 0999.65142
[8] Chen, J.; Chang, M.; Chen, K.; Chen, I., Boundary collocation method for acoustic eigen analysis of three-dimensional cavities using radial basis function, Comput Mech, 29, 4, 392-408, (2002) · Zbl 1146.76622
[9] Young, D. L.; Chen, K. H.; Lee, C. W., Novel meshless method for solving the potential problems with arbitrary domain, J Comput Phys, 209, 1, 290-321, (2005) · Zbl 1073.65139
[10] Šarler, B., Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions, Eng Anal Bound Elem, 33, 12, 1374-1382, (2009) · Zbl 1244.76084
[11] Chen, W., Singular boundary method: a novel, simple, meshfree, boundary collocation numerical method, Chin J Solid Mech, 30, 6, 592-599, (2009)
[12] Wei, X.; Chen, W.; Sun, L.; Chen, B., A simple accurate formula evaluating origin intensity factor in singular boundary method for two-dimensional potential problems with Dirichlet boundary, Eng Anal Bound Elem, 58, 151-165, (2015) · Zbl 1403.65263
[13] Gu, Y.; Chen, W.; He, X.-Q., Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media, Int J Heat Mass Transf, 55, 17-18, 4837-4848, (2012)
[14] Wei, X.; Chen, W.; Chen, B.; Sun, L., Singular boundary method for heat conduction problems with certain spatially varying conductivity, Comput Math Appl, 69, 3, 206-222, (2015) · Zbl 1364.65217
[15] Greengard, L., The rapid evaluation of potential fields in particle systems, (1988), the MIT Press Cambridge, Massachusetts · Zbl 1001.31500
[16] Liu, Y.; Nishimura, N.; Yao, Z., A fast multipole accelerated method of fundamental solutions for potential problems, Eng Anal Bound Elem, 29, 11, 1016-1024, (2005) · Zbl 1182.74256
[17] Nishimura, N.; Yoshida K-i; Kobayashi, S., A fast multipole boundary integral equation method for crack problems in 3D, Eng Anal Bound Elem, 23, 1, 97-105, (1999) · Zbl 0953.74074
[18] Gu, Y.; Gao, H.; Chen, W.; Liu, C.; Zhang, C.; He, X., Fast-multipole accelerated singular boundary method for large-scale three-dimensional potential problems, Int J Heat Mass Transf, 90, 291-301, (2015)
[19] Karageorghis, A.; Marin, L., Efficient MFS algorithms for problems in thermoelasticity, J Sci Comput, 1-26, (2012)
[20] Bebendorf, M.; Grzhibovskis, R.; Accelerating Galerkin, B. E.M., For linear elasticity using adaptive cross approximation, Math Methods Appl Sci, 29, 14, 1721-1747, (2006) · Zbl 1110.74054
[21] Phillips, J. R.; White, J. K., A precorrected-FFT method for electrostatic analysis of complicated 3-D structures, Comput-Aided Des Integr Circuits Syst IEEE Trans, 16, 10, 1059-1072, (1997)
[22] Xiao, J.; Ye, W.; Cai, Y.; Zhang, J.; Precorrected, F. F.T., Accelerated BEM for large-scale transient elastodynamic analysis using frequency-domain approach, Int J Numer Methods Eng, 90, 1, 116-134, (2012) · Zbl 1242.74186
[23] Sammis, I.; Strain, J., A geometric non uniform fast Fourier transform, J Comput Phys, 228, 18, 7086-7108, (2009) · Zbl 1175.65155
[24] Hackbusch, W.; Nowak, Z. P., On the fast matrix multiplication in the boundary element method by panel clustering, Numer Math, 54, 4, 463-491, (1989) · Zbl 0641.65038
[25] Hackbusch, W.; Khoromskij, B. N., A sparse H-matrix arithmetic part II: application to multi-dimensional problems, Computing, 64, 1, 21-47, (2000) · Zbl 0962.65029
[26] Hackbusch, W.; Khoromskij, B. N., A sparse H-matrix arithmetic: general complexity estimates, J Comput Appl Math, 125, 1-2, 479-501, (2000) · Zbl 0977.65036
[27] Hackbusch, W.; Khoromskij, B. N.; Sauter, S. A., On H^{2}-matrices, (2000), Springer Berlin · Zbl 0963.65043
[28] Goreinov, S. A.; Tyrtyshnikov, E. E.; Zamarashkin, N. L., A theory of pseudoskeleton approximations, Linear Algebra Appl, 261, 1-3, 1-21, (1997) · Zbl 0877.65021
[29] Bebendorf, M., Approximation of boundary element matrices, Numer Math, 86, 4, 565-589, (2000) · Zbl 0966.65094
[30] Fu, Z.-J.; Chen, W.; Gu, Y., Burton-Miller-type singular boundary method for acoustic radiation and scattering, J Sound Vib, 333, 16, 3776-3793, (2014)
[31] Gu, Y.; Chen, W.; Zhang, C.-Z., Singular boundary method for solving plane strain elastostatic problems, Int J Solids Struct, 48, 18, 2549-2556, (2011)
[32] Chen W, Gu Y. Recent advances on singular boundary method. Joint international workshop on Trefftz method VI and method of fundamental solution II, Taiwan; 2011.
[33] Wei, X.; Chen, W.; Chen, B., An ACA accelerated MFS for potential problems, Eng Anal Bound Elem, 41, 0, 90-97, (2014) · Zbl 1297.65198
[34] Woolfe, F.; Liberty, E.; Rokhlin, V.; Tygert, M., A fast randomized algorithm for the approximation of matrices, Appl Comput Harmon Anal, 25, 3, 335-366, (2008) · Zbl 1155.65035
[35] Bebendorf, M., Hierarchical matrices: a means to efficiently solve elliptic boundary value problems, (2008), Springer Verlag Berlin · Zbl 1151.65090
[36] Rjasanow S. Adaptive cross approximation of dense matrices. IABEM 2002, International Association for Boundary Element Methods; 2002.
[37] Rjasanow, S.; Weggler, L., ACA accelerated high order BEM for Maxwell problems, Comput Mech, 51, 4, 431-441, (2013) · Zbl 1312.78009
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