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海洋大學力學聲響振動實驗室 1 Regularized meshless method for boundary value problems with multiply-connected domain Jeng-Hung Kao Advisor:

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1 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 1 Regularized meshless method for boundary value problems with multiply-connected domain Jeng-Hung Kao Advisor: Jeng-Tzong Chen, Kue-Hung Chen 6, 29, 2006 HRE2-307

2 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 2 Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-domain problems Application on multiply-connected-domain problems Conclusions Further research

3 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 3 Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-domain problems Application on multiply-connected-domain problems Conclusions Further research

4 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 4 Motivation and literature review Numerical Methods Mesh Methods Finite Difference Method Meshless Methods Finite Element Method Boundary Element Method (MFS)(RMM) Motivation

5 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 5 Meshless methods FEM BEM Chen et al. 2002 JSV Chen et al. 2002 JSV Continuous moving least square Continuous moving least square Continuous Kernel Boundary node method Boundary collocation method Belyschko et al. 1994 Monagh 1982 Liu et al. 1995 Monagh 1982 Liu et al. 1995 Mukherjee, Huang, Chen & Kang 2002 EABE, IJNME Mukherjee, Huang, Chen & Kang 2002 EABE, IJNME RMM MFS Kupradze 1964 CMMP Young and Chen 2005 JCP, JASA Young and Chen 2005 JCP, JASA Nonsingular kernel Nonsingular kernel Singular kernel Singular kernel Motivation and literature review literature review

6 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 6 Motivation and literature review literature review Helmholtz problem Laplace problem d=0 Chen,Tanaka BKM onsingular general solution 2002 JT Chen BEM imaginary-part 2002 Young and Chen RMM 2005 SW Kang NDIF imaginary-part 2002

7 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 7 Motivation and literature review Exact solution

8 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 8 Motivation and literature review d=0.1d=1.0 Convention MFS

9 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 9 Motivation and literature review RMM

10 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 10 Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-domain problems Application on multiply-connected-domain problems Conclusions Further research

11 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 11 Neumann problem Dirichlet problem Relation between MFS and RMM Interior problem Exterior problem Kernel functions Introduction of MFS

12 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 12 Relation between MFS and RMM Single-layer Potentials Double-layer Potentials Laplace problemHelmholtz problem Introduction of MFS

13 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 13 Relation between MFS and RMM d=0 Introduction of MFS Convention MFS RMM

14 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 14 Relation between MFS and RMM Introduction of RMM =0

15 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 15 Introduction of Method of Fundamental Solutions Introduction of RMM

16 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 16 Relation between MFS and RMM Source points Collocation points Kernel functions MFS RMM fictitious boundary Real boundary Single-layer potentials Double-layer potentials Double-layer potentials Compared RMM with MFS

17 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 17 Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-domain problems Application on multiply-connected-domain problems Conclusions Further research

18 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 18 RMM for solving multiply-connected-domain problems Source point Collocation point Laplace problem

19 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 19 RMM for solving multiply-connected-domain problems Source point Collocation point Laplace problem

20 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 20 RMM for solving multiply-connected-domain problems Source point Collocation point Laplace problem

21 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 21 RMM for solving multiply-connected-domain problems Construction of influence matrices

22 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 22 RMM for solving multiply-connected-domain problems Test cases Neumann problem

23 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 23 RMM for solving multiply-connected-domain problems Test cases

24 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 24 RMM for solving multiply-connected-domain problems Arbitrary-shape problem Test cases

25 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 25 RMM for solving multiply-connected-domain problems Test cases

26 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 26 RMM for solving multiply-connected-domain problems Source point Collocation point Helmholtz problem

27 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 27 RMM for solving multiply-connected-domain problems Source point Collocation point Helmholtz problem

28 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 28 RMM for solving multiply-connected-domain problems Source point Collocation point Helmholtz problem

29 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 29 RMM for solving multiply-connected-domain problems Source point Collocation point Helmholtz problem

30 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 30 RMM for solving multiply-connected-domain problems Construction of influence matrices

31 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 31 RMM for solving multiply-connected-domain problems Extracting out the eigenvalues Treatments of spurious eigenvalues SVD and SVD updating term SVD Singular values matrix

32 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 32 RMM for solving multiply-connected-domain problems 4.44 (T) 7.02 (T) 8.87 (T) 9.93 (T) 11.33 (T) Test case

33 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 33 Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-domain problems Application on multiply-connected-domain problems Conclusions Further research

34 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 34 Application on multiply-connected-domain problems 0 0 Antiplane piezoelectricity problem Antiplane shear problem

35 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 35 Application on multiply-connected-domain problems Decomposition of the problem

36 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 36 Application on multiply-connected-domain problems Inclusion Matrix Antiplane piezoelectricity problems Antiplane shear problems

37 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 37 Application on multiply-connected-domain problems G.E. Continuous conditions Shear stress Electric displacements Piezoelectricity problem Antiplane shear problem absent Compared antiplane piezoelectric with antiplane shear problems

38 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 38 Application on multiply-connected-domain problems Influence matrices Piezoelectricity problem Antiplane shear problem 0 Compared antiplane piezoelectric with antiplane shear problems

39 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 39 Application on multiply-connected-domain problems Nm -2 Cm -2 CV -1 m -1 Nm -2 Antiplane piezoelectric problems with multiple inclusions

40 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 40 Application on multiply-connected-domain problems Case 1: Single inclusion

41 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 41 Application on multiply-connected-domain problems Case 1: Single inclusion

42 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 42 Application on multiply-connected-domain problems Nm -2 Cm -2 CV -1 m -1 Nm -2 Antiplane piezoelectric problems with multiple inclusions

43 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 43 d Application on multiply-connected-domain problems Case 2: Two inclusions

44 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 44 Application on multiply-connected-domain problems d=10 d=1d=0.1 Case 2: Two inclusions

45 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 45 Application on multiply-connected-domain problems d=1 d=10 d=0.1 d=0.01 d=0.02 Case 2: Two inclusions

46 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 46 Application on multiply-connected-domain problems Nm -2 Antiplane shear problems with multiple inclusions

47 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 47 Application on multiply-connected-domain problems Case 1: Two inclusions

48 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 48 Application on multiply-connected-domain problems Case 1: Two inclusions

49 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 49 Application on multiply-connected-domain problems Antiplane shear problems with multiple inclusions

50 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 50 Application on multiply-connected-domain problems Case 2: Three inclusions

51 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 51 Application on multiply-connected-domain problems True eigenequations Dirichlet type Neumann type Spurious eigenequation Acoustic problems

52 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 52 Application on multiply-connected-domain problems 2.05 (T) 2.22 (T) 2.66 (T) 3.21 (T) 3.80 (T) 4.27 (T) 4.39 (T) 4.57 (T) 4.97 (T) 3.68 (S) 4.16 (T) (T): True eigenvalue (S): Spurious eigenvalue :Analytical soluition Case 1: Dirichlet BC

53 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 53 Application on multiply-connected-domain problems 0.83(T) 1.52(T) 2.12(T) 2.25(T) 2.52(T) 2.68(T) 3.20(T) 3.24(T) 3.76(S) 3.95(T) (T): True eigenvalue (S): Spurious eigenvalue :Analytical soluition Case 1: Neumann BC

54 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 54 Application on multiply-connected-domain problems 3.68 (S) Case 1: RMM+SVD updating term approached

55 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 55 Application on multiply-connected-domain problems Case 2: A circular domain with two equal holes Case 3: A circular domain with four equal holes Acoustic problems

56 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 56 Application on multiply-connected-domain problems Case 2: A circular domain with two equal holes

57 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 57 Application on multiply-connected-domain problems RMM BEM Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Case 2: A circular domain with two equal holes

58 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 58 Application on multiply-connected-domain problems Case 3: A circular domain with four equal holes

59 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 59 Application on multiply-connected-domain problems RMM BEM Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Case 3: A circular domain with four equal holes

60 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 60 Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-domain problems Application on multiply-connected-domain problems Conclusions Further research

61 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 61 Conclusions Only the boundary nodes on the physical boundary are required by using proposed method. The proposed method can regularize singularity by using subtracting and adding-back technique. A systematic approach to solve the Laplace and eigenproblems with multiply-connected domain was proposed successfully by using the regularized meshless method.

62 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 62 Conclusions The RMM successfully are applied on three engineering problems. (antiplane, piezoelectricity, acoustics)

63 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 63 Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-domain problems Application on multiply-connected-domain problems Conclusions Further research

64 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 64 Further research Three-dimensional problems with inclusions. Plane problems with multiple inclusions in an anisotropic medium. Piezoelectric inclusions subject to an incident wave and a harmonic inplane electric field. Multiple scattering problems.

65 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 65 The end Thanks for your attentions. Your comment is much appreciated. You can get more information on our website. http://msvlab.hre.ntou.edu.tw

66 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 66 Regularized meshless method for Helmholtz problems with multiply-connected domain where Source point Collocation point

67 海洋大學力學聲響振動實驗室 http://msvlab.hre.ntou.edu.tw 67 Formulation

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