Simulation of two-dimensional steady-state heat conduction problems by a fast singular boundary method. (English) Zbl 1464.65238
Summary: The study presents a fast direct algorithm for solutions of systems arising from singular boundary method (SBM) in two-dimensional (2D) steady-state heat conduction problems. As a strong-form boundary discretization collocation technique, the SBM is mathematically simple, easy-to-implement, and free of mesh. Similar to boundary element method (BEM), the SBM generates a dense coefficient matrix, which requires \(O(N^3)\) operations and \(O(N^2)\) memory to solve the system with direct solvers. In this study, a fast direct solver based on recursive skeletonization factorization (RSF) is applied to solve the linear systems in the SBM. Due to the fact that the coefficient matrix is hierarchically block separable, we can construct a multilevel generalized LU decomposition that allows fast application of the inverse of the coefficient matrix. Combing the RSF and the SBM can dramatically reduce the operations and the memory to \(O(N)\). The accuracy and effectiveness of the RSF-SBM are tested through some 2D heat conduction problems including isotropic and anisotropic cases.
MSC:
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
Keywords:
singular boundary method; fast algorithms; heat conduction; interpolative decomposition; fast factorizationReferences:
| [1] | Chen, W., Singular boundary method: a novel, simple, meshfree, boundary collocation numerical method (in Chinese), Chin J Sol Mech, 30, 6, 592-599 (2009) |
| [2] | Gu, Y.; Hua, Q.; Zhang, C.; He, X., The generalized finite difference method for long-time transient heat conduction in 3D anisotropic composite materials, Appl Math Modell, 71, 316-330 (2019) · Zbl 1481.74032 |
| [3] | Li, J.; Chen, W.; Fu, Z., Numerical investigation on convergence rate of singular boundary method, Math Probl Eng, 2016, 2, 1-13 (2016) · Zbl 1400.65061 |
| [4] | Lin, J.; Chen, C. S.; Liu, C. S.; Lu, J., Fast simulation of multi-dimensional wave problems by the sparse scheme of the method of fundamental solutions, Comput Math Appl, 72, 3, 555-567 (2016) · Zbl 1359.65222 |
| [5] | Gu, Y.; Fan, C.-M.; Xu, R., Localized method of fundamental solutions for large-scale modelling of two-dimensional elasticity problems, Appl Math Lett, 93, 8-14 (2019) · Zbl 1458.74143 |
| [6] | Fu, Z.; Chen, W.; Chen, J. T.; Qu, W., Singular boundary method: three regularization approaches and exterior wave applications, Comput Modell Eng Sci, 99, 5, 417-443 (2014) · Zbl 1357.65279 |
| [7] | 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 Transfer, 9, 291-301 (2015) |
| [8] | Li, W.; Chen, W.; Fu, Z., Precorrected-FFT accelerated singular boundary method for large-scale three-dimensional potential problems, Commun Comput Phys, 22, 2, 460-472 (2017) · Zbl 1488.65691 |
| [9] | Wang, F.; Chen, W.; Hua, Q., A simple empirical formula of origin intensity factor in singular boundary method for two-dimensional Hausdorff derivative Laplace equations with Dirichlet boundary, Comput Math Appl, 76, 5, 1075-1084 (2018) · Zbl 1425.35205 |
| [10] | Li, J.; Chen, W., A modified singular boundary method for three-dimensional high frequency acoustic wave problems, Appl Math Modell, 54, 189-201 (2018) · Zbl 1480.74151 |
| [11] | Li, W.; Chen, W.; Pang, G., Singular boundary method for acoustic eigenanalysis, Comput Math Appl, 72, 3, 663-674 (2016) · Zbl 1359.76202 |
| [12] | Lin, J.; Chen, W.; Chen, C. S., Numerical treatment of acoustic problem with boundary singularities by the singular boundary method, J Sound Vib, 333, 14, 3177-3188 (2014) |
| [13] | Fu, Z.; Chen, W.; Gu, Y., Burton-Miller-type singular boundary method for acoustic radiation and scattering, J Sound Vib, 333, 16, 3776-3793 (2014) |
| [14] | Fu, Z.; Chen, W.; Wen, P.; Zhang, C., Singular boundary method for wave propagation analysis in periodic structures, J Sound Vib, 427, 5, 170-188 (2018) |
| [15] | Qu, W.; Chen, W., Solution of two-dimensional stokes flow problems using improved singular boundary method, Adv Appl Math Mech, 7, 1, 13-30 (2015) · Zbl 1488.76036 |
| [16] | Qu, W.; Chen, W.; Fu, Z.; Gu, Y., Fast multipole singular boundary method for stokes flow problems, Math Comput Simul, 146, 57-69 (2018) · Zbl 1482.76085 |
| [17] | Gu, Y.; Chen, W.; Gao, H.; Zhang, C., A meshless singular boundary method for three-dimensional elasticity problems, Int J Numer Methods Eng, 107, 2, 109-126 (2016) · Zbl 1352.74039 |
| [18] | Lin, J.; Zhang, C.; Sun, L.; Lu, J., Simulation of seismic wave scattering by embedded cavities in an elastic half-plane using the novel singular boundary method, Adv Appl Math Mech, 12, 2, 322-342 (2018) · Zbl 1488.65719 |
| [19] | Qu, W.; Chen, W., Fast multipole singular boundary method for large-scale plane elasticity problems, Acta Mech Solida Sin, 28, 6, 629-638 (2015) |
| [20] | Sun, L.; Chen, W.; Cheng, A. H.D., Singular boundary method for 2D dynamic poroelastic problems, Wave Motion, 61, 40-62 (2016) · Zbl 1468.76053 |
| [21] | Zhang, A.; Gu, Y.; Hua, Q.; Chen, W.; Zhang, C., A regularized singular boundary method for inverse Cauchy problem in three-dimensional elastostatics, Adv Appl Math Mech, 10, 6, 1459-1477 (2018) · Zbl 1488.65590 |
| [22] | Wang, F.; Chen, W.; Zhang, C.; Lin, J., Analytical evaluation of the origin intensity factor of time-dependent diffusion fundamental solution for a matrix-free singular boundary method formulation, Appl Math Modell, 49, 647-662 (2019) · Zbl 1480.65243 |
| [23] | Wang, F.; Chen, W.; Tadeu, A.; Correia, C. G., Singular boundary method for transient convection-diffusion problems with time-dependent fundamental solution, Int J Heat Mass Transfer, 114, 114, 1126-1134 (2017) |
| [24] | Lin, J.; Reutskiy, S.; Lu, J., A novel meshless method for fully nonlinear advection-diffusion-reaction problems to model transfer in anisotropic media, Appl Math Comput, 339, 459-476 (2018) · Zbl 1429.65285 |
| [25] | Li, W.; Chen, W., Band gap calculations of photonic crystals by singular boundary method, J Comput Appl Math, 315, 1, 273-286 (2017) · Zbl 1421.65028 |
| [26] | Li, J.; Fu, Z.; Chen, W., Numerical investigation on the obliquely incident water wave passing through the submerged breakwater by singular boundary method, Comput Math Appl, 71, 1, 381-390 (2016) · Zbl 1443.65406 |
| [27] | Gu, Y.; Chen, W.; He, X., Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media, Int J Heat Mass Transfer, 55, 4837-4848 (2012) |
| [28] | Wei, X.; Chen, B.; Chen, S.; Yin, S., An ACA-SBM for some 2D steady-state heat conduction problems, Eng Anal Bound Elem, 71, 101-111 (2016) · Zbl 1403.80007 |
| [29] | 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 |
| [30] | Wei, X.; Sun, L. L.; Yin, S. H.; Chen, B., A boundary-only treatment by singular boundary method for two-dimensional inhomogeneous problems, Appl Math Modell, 62, 338-351 (2018) · Zbl 1460.65151 |
| [31] | Qu, W.; Chen, W.; Gu, Y., Fast multipole accelerated singular boundary method for the 3D Helmholtz equation in low frequency regime, Comput Math Appl, 70, 4, 679-690 (2015) · Zbl 1443.65411 |
| [32] | Qu, W.; Chen, W.; Zheng, C., Diagonal form fast multipole singular boundary method applied to the solution of high-frequency acoustic radiation and scattering, Int J Numer Methods Eng, 11, 803-815 (2017) · Zbl 07867125 |
| [33] | Li, W., A fast singular boundary method for 3D Helmholtz equation, Comput Math Appl, 77, 2, 525-535 (2018) · Zbl 1442.65415 |
| [34] | Hackbusch, W., A sparse matrix arithmetic based on H-matrices. Part I: introduction to H-matrices, Computing, 62, 2, 89-108 (1999) · Zbl 0927.65063 |
| [35] | Hackbusch, W.; Khoromskij, B. N., A sparse H-matrix arithmetic. Part II: application to multidimensional problems, Computing, 64, 1, 21-47 (2002) · Zbl 0962.65029 |
| [36] | Hackbusch, W.; Borm, S., Data-sparse approximation by adaptive H²-Matrices, Computing, 69, 1, 1-35 (2002) · Zbl 1012.65023 |
| [37] | Chandrasekaran, S.; Dewilde, P.; Gu, M.; Lyons, W.; Pals, T., A fast solver for HSS representations via sparse matrices, SIAM J Matrix Anal Appl, 29, 1, 67-81 (2006) · Zbl 1135.65317 |
| [38] | Chandrasekaran, S.; Gu, M.; Pals, T., A fast ULV decomposition solver for hierarchically semiseparable representations, SIAM J Matrix Anal Appl, 28, 3, 603-622 (2006) · Zbl 1120.65031 |
| [39] | Ho, K. L.; Greengard, L., A fast direct solver for structured linear systems by recursive skeletonization, SIAM J Sci Comput, 34, 5, A2507-A2A32 (2011) · Zbl 1259.65062 |
| [40] | Ho, K. L.; Li, L., Hierarchical interpolative factorization for elliptic operators: integral equations, Comm Pure Appl Math, 69, 7, 1-39 (2016) |
| [41] | 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 |
| [42] | Greengard, L.; Rokhlin, V., A fast algorithm for particle simulations, J Comput Phy, 73, 2, 325-348 (2015) · Zbl 0629.65005 |
| [43] | Cheng, H.; Gimbutas, Z.; Martinsson, P.; Rokhlin, V., On the compression of low rank matrices, SIAM J Sci Comput, 26, 4, 1389-1404 (2005) · Zbl 1083.65042 |
| [44] | Liberty, E.; Woolfe, F.; Martinsson, P. G.; Rokhlin, V.; Tygert, M., Randomized algorithms for the low-rank approximation of matrices, Proc Natl Acad Sci USA, 104, 51, 20167-20172 (2007) · Zbl 1215.65080 |
| [45] | 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 |
| [46] | Corona, E.; Martinsson, P. G.; Zorin, D., An \(O(N)\) direct solver for integral equations on the plane, Appl Comput Harmon Anal, 38, 2, 889-894 (2013) |
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