M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 1 國立台灣海洋大學河海工程學系 陳正宗 教授 河 海 工 程 概論結 構 工 程河 海 工 程 概論結 構 工 程 國立台灣海洋大學河海工程學系 Analysis of acoustic eigenfrequencies and eigenmodes by using the meshless method 指導教授 : 陳正宗 教授 學 生 : 張 銘 翰 15, June, 2001

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 2 Outlines I.Numerical Methods. II.The developments of meshless methods and radial basis functions. III.The approaching methods of the diagonal elements. IV.The technique for extracting true eigenvalues. V.The techniques for filtering out the spurious eigenvalues VI.Conclusions.

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 3 Numerical Methods Mesh Methods Finite Difference Method Meshless Method Finite Element Method Boundary Element Method Numerical methods

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 4 Meshless methods Variational methods (Galerkin methods) Variational methods (Galerkin methods) BIEM BEM Continuous moving least square Belyschko et al. (1994) Continuous moving least square Belyschko et al. (1994) Continuous Kernel Monagh (1982) Liu et al. (1995) Continuous Kernel Monagh (1982) Liu et al. (1995) Moving least square Lancaster & Salkauskas Belyschko et al. (1994) Nayroles et al. (1992) Moving least square Lancaster & Salkauskas Belyschko et al. (1994) Nayroles et al. (1992) Discrete Kernel Monaghan (1982) Liu et al. (1995) Discrete Kernel Monaghan (1982) Liu et al. (1995) Partitious of Unity Babuska and Melenk Duarte and Oden (1995) Partitious of Unity Babuska and Melenk Duarte and Oden (1995) Boundary node method Mukherjee, Huang, Chen & Kang Boundary node method Mukherjee, Huang, Chen & Kang Local BIE (unsymmetric) Atluri, Zhu, and Sladek Local BIE (unsymmetric) Atluri, Zhu, and Sladek Local Petrov-Galerkin approach (symmetric) Atluri, Zhu, and Liu Local Petrov-Galerkin approach (symmetric) Atluri, Zhu, and Liu Complete solution + Particular solution Complete solution + Particular solution Complete solution Complete solution Particular solution Particular solution Volume potential Volume potential RBF solution RBF solution Chen & Kang The developments of meshless methods

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 5 The developments of radial basis functions Radial basis function (RBFs) Mesh method Globally-supported RBFs DRBEM Nardini Brebbia DRBEM Nardini Brebbia Method of particular integral Ahmad & Banerjee Method of particular integral Ahmad & Banerjee Meshless method Globally-supported RBFs Compactly-supported RBFs Wu 、 Buhmann & Wendland Compactly-supported RBFs Wu 、 Buhmann & Wendland Volume potential Method of fundamental solution Method of fundamental solution (Potential theory) This thesis

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 6 Complex-valued BEM Half-effort computation Imaginary-part formulations Imaginary-part formulations Multiple Reciprocity Method (MRM) Multiple Reciprocity Method (MRM) Real-part BEM Differed by a complementary solution Half-effort computation Singular integrals Avoid singular integrals

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 7 Governing Eq.: Two-dimension One-dimension Three-dimension Imaginary – part formulation

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 8 Double-layer potential approach Single-layer Potential approach * NDIF method by Kang is the special case Dirichlet problem Neumann problem Dirichlet problem Neumann problem Distributed-type The distribute and concentrate-type Dirichlet problem Neumann problem Dirichlet problem Neumann problem Concentrated-type*

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 9 Influence matrices L ’ Hôpital ’ s rule Invariant method Indeterminate forms ( ) The derivation of indeterminate forms RBF for

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 10 Addition theorem of Bessel function The diagonal elements of U kernel

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 11 Addition theorem of Bessel function The diagonal elements of T kernel

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 12 Two-dimensional cavity Influence matrices U(s,x), L(s,x), T(s,x), M(s,x) + Indirect method RBF for Eigenvalues Eigenmodes

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 13 Single-layer potential approach Eigenfrequencies for Dirichlet B.C. Double-layer potential approach True eigenvalues Spurious eigenvalues

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 14 Eigenfrequencies for Neumann B.C. Single-layer potential approach Double-layer potential approach True eigenvalues Spurious eigenvalues

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 15 To extract the true eigenvalues Deficient constraints Spurious eigensolutions SVD updating terms To extract the true eigensolutions Additional constraints

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 16 Direct method for Dirichlet B. C. : SVD updating terms SVD updating terms Singular equation (UT method) Hypersingular equation (LM method)

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 17 SVD updating terms To extract the true eigenfrequencies

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 18 The degenerate kernels for interior and exterior problems: The degenerate kernels X Y Interior problem X Y Exterior problem Blue: field points Red: source points

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 19 The degenerate kernels

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 20 Indirect method for Dirichlet B. C. : SVD updating terms Singular-layer approach: Double-layer approach: To extract the true eigenvalues by the indirect method

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 21 Double-layer potential approach SVD updating-terms Examples for the SVD updating terms For Dirichlet B.C. True eigenvalues

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 22 Single-layer potential approach SVD updating-terms For Neumann B.C. Examples for the SVD updating terms True eigenvalues

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 23 Deficient constraints Spurious eigensolutions Mathematically explaining Fredholm alternative theorem Numerical technique SVD updating documents To filter out the spurious solutions

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 24 Fredholm ’ s alternative theorem: For solving an algebraic system: Fredholm alternative theorem Adjoint homogenous sol. Alternative theorem : the transpose conjugate matrix of

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 25 For double-layer potential approach: SVD updating documents A b x

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 26 SVD updating documents Dirichlet problem: Neumann problem: For double-layer potential approach: SVD updating documents Spurious modes

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 27 Double-layer potential approach SVD updating documents For Dirichlet B.C. Examples for the SVD updating documents Spurious eigenvalues

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 28 Double-layer potential approach SVD updating documents For Dirichlet B.C. Examples for the SVD updating document Spurious eigenvalues

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 29 The interior modes of a circular cavity Single-layer protential approach Double-layer protential approach Analytical solution

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 30 Single-layer protential approach Double-layer protential approach SVD techniques for a square cavity SVD updating terms True eigenvalues SVD updating documents Spurious eigenvalues For Dirichlet B.C.

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 31 Single-layer protential approach Double-layer protential approach Analytical solution The interior modes of square cavity

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 32 Three-dimensional spherical cavity Indirect method RBF for Continuous system Degenerate kernel X Z Y X Z Y Blue ： source points Red ： field points Eigenvalues Eigenmodes Eigensolutions

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 33 Eigenfrequencies for Dirichlet B.C. Single-layer potential approach Double-layer potential approach True eigenvalues Supurious eigenvalues

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 34 Single-layer potential approach Double-layer potential approach True eigenvalues Spurious eigenvalues Eigenfrequencies for Neumann B.C.

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 35 Single-layer protential approach Double-layer protential approach SVD techniques for a spherical cavity Neumann B.C. SVD updating terms True eigenvalues SVD updating documents Spurious eigenvalues

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 36 n =0, k=0.0 Analytical modes Single-layer approach Double-layer approach m =0 Contours for the boundary modes

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 37 i th minimum eigenvalue Analytical eigenvalue Numerical eigenvalue Test 1 st ˇ 2 nd ˇ 3 rd ˇ 4 th  Multiplicity of eigenvalues 4 th : Multiplicity=3

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 38 m =0m =-1m =1 n =0 n =1 Spherical harmonics P. A. NELSON Y. KAHANA

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 39 Single-layer approach n =1, k=2.082 Analytical modes Double-layer approach m =-1m =0m =-1 Contours of the boundary modes

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 40 I.Only the boundary nodes are required such that the influence matrices can be determinated by the two-point function. II.The diagonal elements can be derived by invariant method.invariant method. III.Based on the imaginary-part formulations, the deficient constraints cause the spurious eigensolutions.spurious eigensolutions IV.The SVD techniques can solve the true or spurious eigensolutions well.SVD techniques Conclusions

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 41 To overcome the ill-posed problem, the generalized singular value decomposition (GSVD) could be applied.(GSVD) The extension to several problems, degenerate boundary, crack and solid corner may be considered. The meshless method based on the imaginary-part formulation can be extented to structure vibration problems. Further research

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 42 The End Thank you for your kind attention 感謝委員們辛勤指導

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 43 Boundary Value Problem Eigen- solution Density functionTrue and spurious eigenvalues Single-layer potential Double layer potential Direct methodIndirect method Dirichlet Problem True SVD updating term SVD updating term Spurious SVD updating document SVD updating document Neumann Problem True SVD Updating term SVD updating term Spurious SVD updating document SVD updating document SVD techniques for eigensolutions

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 44 To overcome the ill-posed problems 克 服 病 態 問 題 分 析 步 驟 三 維 球 形 聲場二 維 圓 形 聲 場 病 態 指 標 - 條 件 數 病 態 問 題 SVD 補 充 行 SVD 補 充 列 Tikhonov 正規化法 GSVD 克服病態 Preconditioner

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 45 Distributed & concentrated type Distributed-typeConcentrated-type* Single-layer potential approach Double-layer potential approach Dirichlet problem Neumann problem Dirichlet problrm Neumann problrm * NDIF method by Kang is the special case

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 46 One-dimensional duct --- Dirichlet problem Single-layer potential approach Double-layer potential approach

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 47 One-dimensional duct --- Dirichlet problem Single-layer potential approach Double-layer potential approach Analytical mode

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 48 One-dimensional duct --- Single-layer potential approach Double-layer potential approach Neumann problem

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 49 One-dimensional duct --- Neumann problem Single-layer potential approach Double-layer potential approach Analytical mode

M M S S V V 海洋大學力學聲響振動實驗室 MSV LAB HRE NTOU 50 張銘翰 口試問題記錄 Analysis of acoustic eigenfrequencies and eigenmodes by using the meshless method 1. 無網格可否做大空間問題 ? 2. 此種方法有何限制 ? 病態問題是否加重 ? 3.RBF 取法是否唯一 ?RBF 隱含於 kernel? 4. 與李慶鋒有何不同 ? 5. 為何取虛部 ? 若 kernel 為非奇異時，則 free term 能產生嗎 ? 跳 還是 u? 6.Eq. (2-37) & Eq. (2-38) 為何為零 ? 7. 將 mode 相差調正 ? 8. 模態可否疊加實驗之 data 結果 ? 9.GSRBF & CSRBF 何者適合大空間問題 ? 何者適合病態問題 ? 10.RBF 的定義問題 ?