博碩士論文 105221601 詳細資訊




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姓名 盧斯非(Ivan Luthfi Ihwani)  查詢紙本館藏   畢業系所 數學系
論文名稱
(A Multiscale Finite Element Method with Adaptive Bubble Function Enrichment for the Helmholtz Equation)
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摘要(中) 亥姆霍茲方程(Helmholtz equation)是描述許多物理現象(如散射和波傳播)的數學 模型之一,使用數值方法去求解亥姆霍茲方程存在著一些困難。首先,缺乏魯棒 性被稱為污染效應。其次,當在無邊界域的外域中定義問題時,很難找到一個有 效的迭代求解器,隨著波數的增加時,這些迭代求解器可以收斂並且得到少量的 迭代次數。本文提出了一個多尺度有限元方法(MsFEM)的新框架作為這類問題 的迭代求解器。該方法通過所謂的自適應氣泡函數來改進,使得該方法被稱為具 有自適應氣泡函數富集的多尺度有限元方法(MsFEM bub)。各種波數的數值實 驗中表現出該方法的穩健性和效率。
摘要(英) The Helmholtz equation is one of mathematical model to describe many physical phenomena, such as scattering and wave propagation. There are difficulties of solving Helmholtz equation numerically. First, the lack of robustness which is called pollution effect. Second, when the problem defined in an unbounded domain exterior domain, it is hard to find an efficient iterative solver that converge with a few number of iterations as the wavenumber increasing. This thesis presents a new framework of multiscale finite element method (MsFEM) as an iterative solver for such problems. The method is improved by so-called adaptive bubble function such that the method is called multiscale finite element method with adaptive bubble function enrichment (MsFEM bub). Numerical experiments for various wavenumbers indicate the robustness and the efficiency of the method.
關鍵字(中) ★ Helmholtz
★ pollution effect
★ MsFEM
★ bubble function
★ MsFEM bub
關鍵字(英) ★ Helmholtz
★ pollution effect
★ MsFEM
★ bubble function
★ MsFEM bub
論文目次 Contents
Tables......................................... vi Figures......................................... vii Nomenclature..................................... x
1 Introduction ................................... 1
2 Modelofthe2DHelmholtzproblem ...................... 4
2.1 Mathematicalmodel ............................ 4 2.2 Galerkinfiniteelementmethod....................... 4 2.3 Pollutioneffect ............................... 7 2.4 Stabilizedfiniteelementmethod ...................... 7
3 Reviewofsomeiterativesolvers......................... 15 3.1 Krylovsubspacemethods.......................... 15 3.2 Algebraicmultigrid(AMG) ........................ 22
4 The multiscale finite element method with adaptive bubble function enrich- ment........................................ 25
4.1 Motivation.................................. 25 4.2 MsFEMbubandsomenotations...................... 25 4.3 Coarsegridproblem ............................ 27 4.4 Thelocalproblem.............................. 28 4.5 Fundamentalidea.............................. 30 4.6 Smoothingstep ............................... 31
4.7 MsFEMbubalgorithm ........................... 31 5 Numericalexperimentanddiscussion ..................... 33 5.1 Testcases .................................. 33 5.2 Convergencestudies ............................ 37 5.3 Efficiencystudies .............................. 50 5.4 Comparisonwithothermethods ...................... 61 6 Conclusion .................................... 63 Bibliography ..................................... 64 AppendixA:ThederivationofHelmholtzequation . . . . . . . . . . . . . . . . . 66
Appendix B: Review of the iteratively adaptive multiscale finite element method (i-ApMsFEM)................................... 67
AppendixC:Additionalcomparisonwithothermethods . . . . . . . . . . . . . . 73
參考文獻 Bibliography
[1] W. Arnoldi. The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math., 9:17–29, 1951.
[2] I. Babuˇska, F. Ihlenburg, E. T. Paik, and S. A. Sauter. A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollu- tion. Comput. Methods Appl. Mech. Engrg., 128:325–359, 1995.
[3] R. Barrett, M. Berry, and T. F. Chan. Templates for The Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia, 1994.
[4] A. Bayliss, C.I. Goldstein, and E. Turkel. The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics. Comp. and Maths. with Appls., 11:655–665, 1985.
[5] A. Brandt, S. F. McCormick, and J. W. Ruge. Algebraic Multigrid (AMG) for Sparse Matrix Equations, chapter Sparsity and Its Application, pages 257–284. Cambridge University Press, Cambridge, 1984.
[6] S. C. Eisenstat, H. C. Elman, and M. H. Schultz. Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 20:345–357, 1983.
[7] H. C. Elman, O. G. Ernst, and D. P. O’leary. A multigrid method enhanced by krylov subspace iteration for discrete Helmholtz equations. SIAM J. Sci. Comput., 23:1291–1315, 2001.
[8] B. Engquist and L. Ying. Sweeping preconditioner for the Helmholtz equation: hierarchical matrix representation. Commun. Pure Appl. Math., 64:6, 2011.
[9] Y. A. Erlangga, C. Vuik, and C. W. Oosterlee. On a class of preconditioners for solving the Helmholtz equation. Appl. Numer. Math., 50:409–425, 2004.
[10] O. G. Ernst and M. J. Gander. Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods, pages 325–363. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012.
[11] R. D. Falgout. An introduction to algebraic multigrid. J. Comp. in Sci. and Eng., 8:24–33, November 2006.
[12] L. P. Franca, C. Farhat, A. P. Macedo, and M. Lesoinne. Residual-free bubbles for the Helmholtz equation. Int. J. Numer. Meth. Eng., 40:4003–4009, 1997.
[13] L. P. Franca and A. P. Macedo. A two-level finite element method and its application to the Helmholtz equation. Int. J. Numer. Meth. Eng., 43:23–32, 1998.
[14] C.I. Goldstein. A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains. Math. Comput., 39:303–324, 1982.
[15] I. Harari and T. J. R. Hughes. Finite element methods for the Helmholtz equation in an exterior domain: model problems. Comp. Methods Appl. Mech. Eng., 87:59–96, 1991.
[16] T. Y. Hou, F.-N. Hwang, P. Liu, and C.-C. Yao. An iteratively adaptive multi-scale finite element method for elliptic PDEs with rough coefficients. J. Comput. Phys., 336:375–400, 2017.
[17] F.-N. Hwang, Y.-Z. Su, and C.-C. Yao. An iteratively adaptive multiscale finite element method for elliptic interface problems. App. Num. Maths., 127:211–225, 2018.
[18] F. Ihlenburg. Finite Element Analysis of Acoustic Scattering. Springer, New York, 1998.
[19] R.C. Maccamy and S.P. Marin. A finite element method for exterior interface prob- lems. Int. J. Math. and Math. Sci., 3:311–350, 1980.
[20] Y. Notay. An aggregation-based algebraic multigrid method. Electronic Transac- tions on Numerical Analysis, 37:123–146, 2010.
[21] Y.Notay.Aggregation-basedalgebraicmultigridforconvection-diffusionequations. SIAM J. Sci. Comput., 34:A2288–A2316, 2012.
[22] J. W. Ruge and K. Stu ?ben. Algebraic multigrid (AMG), volume 3, chapter Multigrid Methods, pages 73–130. SIAM, Philadelphia, 1987.
[23] Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, 2 edi- tion, 2003.
[24] Y. Saad and M. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 7:856–869, 1986.
[25] K. Stu ?ben. Algebraic multigrid (amg): experiences and comparisons. Appl. Math. Comput., 13:419–45, 1983.
[26] K. Stu ?ben. A review of algebraic multigrid. J. Comput. and App. Math., 128:281– 309, March 2001.
[27] L. L. Thompson and P. M. Pinsky. A Galerkin least squares finite element method for the two-dimensional Helmholtz equation. Int. J. Numer. Meth. Eng., 38:371–397, 1995.
[28] U. Trottentberg, C. W. Oosterlee, and A. Schu ?ller. Multigrid, chapter An Introduc- tion to Algebraic Multigrid, pages 413–532. Academic Press, San Diego, 2001. Appendix A.
[29] H. A. Van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Numer. Linear Algebra Appl., 1:369–386, 1994.
指導教授 黃楓南(Feng-Nan Hwang) 審核日期 2018-7-25
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