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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Presentation transcript:

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: Jeng-Tzong Chen, Kue-Hung Chen 6, 29, 2006 HRE2-307

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 4 Motivation and literature review Numerical Methods Mesh Methods Finite Difference Method Meshless Methods Finite Element Method Boundary Element Method (MFS)(RMM) Motivation

海洋大學力學聲響振動實驗室 5 Meshless methods FEM BEM Chen et al JSV Chen et al JSV Continuous moving least square Continuous moving least square Continuous Kernel Boundary node method Boundary collocation method Belyschko et al Monagh 1982 Liu et al Monagh 1982 Liu et al 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

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 7 Motivation and literature review Exact solution

海洋大學力學聲響振動實驗室 8 Motivation and literature review d=0.1d=1.0 Convention MFS

海洋大學力學聲響振動實驗室 9 Motivation and literature review RMM

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 11 Neumann problem Dirichlet problem Relation between MFS and RMM Interior problem Exterior problem Kernel functions Introduction of MFS

海洋大學力學聲響振動實驗室 12 Relation between MFS and RMM Single-layer Potentials Double-layer Potentials Laplace problemHelmholtz problem Introduction of MFS

海洋大學力學聲響振動實驗室 13 Relation between MFS and RMM d=0 Introduction of MFS Convention MFS RMM

海洋大學力學聲響振動實驗室 14 Relation between MFS and RMM Introduction of RMM =0

海洋大學力學聲響振動實驗室 15 Introduction of Method of Fundamental Solutions Introduction of RMM

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 18 RMM for solving multiply-connected-domain problems Source point Collocation point Laplace problem

海洋大學力學聲響振動實驗室 19 RMM for solving multiply-connected-domain problems Source point Collocation point Laplace problem

海洋大學力學聲響振動實驗室 20 RMM for solving multiply-connected-domain problems Source point Collocation point Laplace problem

海洋大學力學聲響振動實驗室 21 RMM for solving multiply-connected-domain problems Construction of influence matrices

海洋大學力學聲響振動實驗室 22 RMM for solving multiply-connected-domain problems Test cases Neumann problem

海洋大學力學聲響振動實驗室 23 RMM for solving multiply-connected-domain problems Test cases

海洋大學力學聲響振動實驗室 24 RMM for solving multiply-connected-domain problems Arbitrary-shape problem Test cases

海洋大學力學聲響振動實驗室 25 RMM for solving multiply-connected-domain problems Test cases

海洋大學力學聲響振動實驗室 26 RMM for solving multiply-connected-domain problems Source point Collocation point Helmholtz problem

海洋大學力學聲響振動實驗室 27 RMM for solving multiply-connected-domain problems Source point Collocation point Helmholtz problem

海洋大學力學聲響振動實驗室 28 RMM for solving multiply-connected-domain problems Source point Collocation point Helmholtz problem

海洋大學力學聲響振動實驗室 29 RMM for solving multiply-connected-domain problems Source point Collocation point Helmholtz problem

海洋大學力學聲響振動實驗室 30 RMM for solving multiply-connected-domain problems Construction of influence matrices

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 32 RMM for solving multiply-connected-domain problems 4.44 (T) 7.02 (T) 8.87 (T) 9.93 (T) (T) Test case

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 34 Application on multiply-connected-domain problems 0 0 Antiplane piezoelectricity problem Antiplane shear problem

海洋大學力學聲響振動實驗室 35 Application on multiply-connected-domain problems Decomposition of the problem

海洋大學力學聲響振動實驗室 36 Application on multiply-connected-domain problems Inclusion Matrix Antiplane piezoelectricity problems Antiplane shear problems

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 38 Application on multiply-connected-domain problems Influence matrices Piezoelectricity problem Antiplane shear problem 0 Compared antiplane piezoelectric with antiplane shear problems

海洋大學力學聲響振動實驗室 39 Application on multiply-connected-domain problems Nm -2 Cm -2 CV -1 m -1 Nm -2 Antiplane piezoelectric problems with multiple inclusions

海洋大學力學聲響振動實驗室 40 Application on multiply-connected-domain problems Case 1: Single inclusion

海洋大學力學聲響振動實驗室 41 Application on multiply-connected-domain problems Case 1: Single inclusion

海洋大學力學聲響振動實驗室 42 Application on multiply-connected-domain problems Nm -2 Cm -2 CV -1 m -1 Nm -2 Antiplane piezoelectric problems with multiple inclusions

海洋大學力學聲響振動實驗室 43 d Application on multiply-connected-domain problems Case 2: Two inclusions

海洋大學力學聲響振動實驗室 44 Application on multiply-connected-domain problems d=10 d=1d=0.1 Case 2: Two inclusions

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

海洋大學力學聲響振動實驗室 46 Application on multiply-connected-domain problems Nm -2 Antiplane shear problems with multiple inclusions

海洋大學力學聲響振動實驗室 47 Application on multiply-connected-domain problems Case 1: Two inclusions

海洋大學力學聲響振動實驗室 48 Application on multiply-connected-domain problems Case 1: Two inclusions

海洋大學力學聲響振動實驗室 49 Application on multiply-connected-domain problems Antiplane shear problems with multiple inclusions

海洋大學力學聲響振動實驗室 50 Application on multiply-connected-domain problems Case 2: Three inclusions

海洋大學力學聲響振動實驗室 51 Application on multiply-connected-domain problems True eigenequations Dirichlet type Neumann type Spurious eigenequation Acoustic problems

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 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

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

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 56 Application on multiply-connected-domain problems Case 2: A circular domain with two equal holes

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 58 Application on multiply-connected-domain problems Case 3: A circular domain with four equal holes

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 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.

海洋大學力學聲響振動實驗室 62 Conclusions The RMM successfully are applied on three engineering problems. (antiplane, piezoelectricity, acoustics)

海洋大學力學聲響振動實驗室 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

海洋大學力學聲響振動實驗室 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.

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

海洋大學力學聲響振動實驗室 66 Regularized meshless method for Helmholtz problems with multiply-connected domain where Source point Collocation point

海洋大學力學聲響振動實驗室 67 Formulation