National Taiwan Ocean University MSVLAB Department of Harbor
National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering True and spurious eigensolutions of elliptical membranes by using null-field boundary integral equations Jai-Wei Lee, Jeng-Tzong Chen and Shyue-Yuh Leu Date: Dec. , 10, 2009 Time: 10: 30~10: 50 am Place: Lunghwa University of Science and Technology 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics Page 1
Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 2
Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 3
2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics Keelung 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 4
NTOU/MSV Group members (2009) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics Page 5
陳清祥 程宏達 陳俊賢 (C. S. Chen, USA) (J S Chen, UCLA) (Alex H. -D. Cheng, USA) (黃晉, China) 姚振漢 (Yao Z H, China) 美國 中國 NTOU/MSV visitors 吳漢津 杜慶華 (Q. H. Du, China) (H C Wu, Iowa, USA) 日 本 南 韓 吳鼎文 (T. W. Wu, USA) 2009/12/10 陳 鞏(USA, Texas A M) Jeong-Guon Ih (KAIST, Korea) (M. Tanaka, Japan) 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 余德浩 中國科學院 7 祝家麟 (J. L. Zhu, China)
Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 8
Free vibration of a membrane G. E. : Helmholtz equation displacement wave number domain Eigenproblems of Laplace operator 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 9
Advantages and disadvantages of FEM, BEM and BIEM Eigenproblems FEM BEM/BIEM Advantages No spurious eigenvalues Mesh reduction Disadvantages Mesh generation Spurious eigenvalues 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 10
Extension to the eigenproblems with elliptical boundaries OK Extension Key point Degenerate kernel (Polar coordinates) 2009/12/10 Degenerate kernel (Elliptic coordinates) 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 11
Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 12
Boundary integral equation and null-field boundary integral equation Interior case Exterior case Degenerate (separable) form 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 13
Degenerate (separable) form of fundamental solution (2 D) Circle 2009/12/10 Extension Ellipse 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 14
Contour plots of the closed-form fundamental solution and the degenerate kernel Abs Re Im Closed-form fundamental solution Degenerate kernel 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 15
Relationship of kernel functions U(s, x) T(s, x) (Dual system) L(s, x) 2009/12/10 M(s, x) 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 16
Four degenerate kernels 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 17
Boundary densities Expand boundary densities by using the eigenfunction expansion is a constants along the elliptical boundary 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 18
Keypoint for solving the problem with elliptical boundaries The orthogonal relations are reserved 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 19
Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 20
Illustrative examples • Case 1: An elliptical membrane • Case 2: A confocal elliptical annulus 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 21
Case 1: An elliptical membrane G. E. : B. Cs. : 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 22
True and spurious eigenvalues Complex-valued kernel Dirichlet BC (11) Real-part kernel True Spurious (11) Note: the data inside parentheses denote the spurious eigenvalue. 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 23
Mode shapes Even Odd (11) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 24 Even
A confocal elliptical annulus Simply-connected Multiply-connected G. E. : B. Cs. : 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 25
True and spurious eigenvalues UT equation Spurious (42) Note: the data inside parentheses denote the spurious eigenvalue. True Eigenvalues of an elliptical membrane (case 1) (11) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 26
Mode shapes Even Odd (42) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 27 Even
Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 28
Conclusions (Simply-connected domain) 1. Elliptic coordinates 2. Mathieu function 3. Modified Mathieu function 1. Polar coordinates 2. Fourier series 3. Bessel function Kernel Real-part Imaginary-part Kuo et al. Int. J. Numer. Meth. Engng. 2000 Present, 2009 Spurious eigenequations depend on 1. The real-part kernel used 2. The imaginary-part used 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 29
Conclusions (Multiply-connected domain) 1. Polar coordinates 2. Fourier series 3. Bessel function 1. Elliptic coordinates 2. Mathieu function 3. Modified Mathieu function Complex-valued kernel UT or LM Inner boundary Chen et al. Proc. R. Soc. Lond. , Ser. A, 2002 & 2003 Present, 2009 Spurious eigenequations depend on 1. The geometry of inner boundary 2. The approach used (Singular or Hypersingular) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 30
The end Thanks for your kind attentions Welcome to visit the web site of MSVLAB/NTOU http: //msvlab. hre. ntou. edu. tw/ 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 31
Successful experiences in 2 -D eigenproblems with circular boundaries UT equation (Singular) LM equation (Hypersingular Spurious eigenvalues ) Spurious eigenvalues Complex-valued kernel UT or LM Inner boundary Kernel Real-part Imaginary-part Kuo et al. Int. J. Numer. Meth. Engng. 2000 Key point 2009/12/10 Chen et al. Proc. R. Soc. Lond. , Ser. A, 2002 & 2003 (Found and treated) Degenerate kernel (Polar coordinates) 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 32
Elliptic coordinates and Mathieu function Modified Mathieu function angular coordinate radial coordinate 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 33
Degenerate kernels Addition theorem (Morse and Feshbach’s book) Methods of Theoretical Physics, 1953, p. 1421 Modified Mathieu functions of the third kind Orthogonal relations Analytical study 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 34 (norm)
True and spurious eigenequations True UT Even (Singular) Complex-valued Odd Spurious Even UT (Singular) Odd Real-part 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 35
True and spurious eigenequations True Even UT (Singular) LM Odd (Hypersingular) Spurious Even UT Odd Even Odd (Singular) LM (Hypersingular) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 36
Confocal elliptical annulus 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 37
True and spurious eigenequations 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 38
True and spurious eigenequations Dirichlet BC 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics Neumann BC 39
True and spurious eigenequations True Even Spurious Even Odd B. C. fixed-fixed UT LM 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 40
Successful experiences in 2 -D problems with circular boundaries using the present approach Degenerate kernel Fundamental solution (Laplace) (Helmholtz) Advantages of present approach: 1. No principal value 2. Well-posed model 3. Exponential convergence 4. Free of mesh generation The proposed approach will be extended to deal with 2 -D problem with elliptic boundaries 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 41
Why spurious solution occurs • FDM for ODE • Real-part BEM & MRM (Simply-connected problem) • Complex-valued BEM (Multiply-connected problem) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 42
Separation of variables in the elliptic coordinates Cartesian coordinates Elliptic coordinates separation of variables 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 43
Addition theorem Q r a b P = + Subtraction theorem O 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 44
A circular membrane with an elliptical hole Note: the data inside parentheses denote the spurious eigenvalue. 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 45
An elliptical membrane with a circular hole Note: the data inside parentheses denote the spurious eigenvalue. 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 46
Note: the data inside parentheses denote the spurious eigenvalue. 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 47
Nonunique solution Non-unique solution: Near-trapped mode (physical) Fictitious frequency (Numerical) (1) CHIEF method (Schenck, JASA , 1968) t(a, 0) Additional constraint (CHIEF point) (2) Burton and Miller method (Burton and Miller, PRS , 1971) (3) SVD updating term technique (Chen et al. , JSV, 2002) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 48
SVD updating technique (去蕪[ ]存菁( )術) The same U T L M The same true mode, rigid body mode (physics) spurious mode, fictitious mode (mathematics) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 49
Degenerate cases in mathematics and mechanics 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 50
Jump behavior across the boundary 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 51
Other applications Water wave 1. Harbor resonance Acoustics 1. Hermetic compressor 2. Small automotive muffler Electromagnetics 1. Waveguides 2009/12/10 2. TM mode (Dirichlet BC) 3. TE mode (Neumann BC) 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 52
Literature review 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 53
Literature review 1. Tai and Shaw 1974 (complex-valued BEM) 2. De Mey 1976, Hutchinson and Wong 1979 (real-part kernel) 3. Wong and Hutchinson (real-part direct BEM program) 4. Shaw 1979, Hutchinson 1988, Niwa et al. 1982 (realpart kernel) 5. Tai and Shaw 1974, Chen et al. Proc. Roy. Soc. Lon. Ser. A, 2001, 2003 (multiply-connected problem) 6. Chen et al. (dual formulation, domain partition, SVD updating technique, CHEEF method) Mathematical analysis. Workshop and numerical study. Exchanges for free of vibration of 2009 International on Students’ 2009/12/10 Nano and plate Computational Mechanics using BEM- 54
The orthogonality of vector and function vectors functions and are orthogonal Mathieu function Orthogonal relations 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics (norm) 55
Jacobian Polar coordinates Elliptic coordinates arc length (Area) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 56
Adaptive observer system r 20 k 1 , f 20 k 1 collocation point 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 57
Linear algebraic equation Index of collocation circle Index of routing circle 2009/12/10 Column vector of Fourier coefficients (Nth routing circle) 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 58
Literature review (Degenerate kernel ) Author Degenerate kernel approximation Applications Sloan et al. (1975) Prove that it is equivalent to iterated Petrov. Galerkin approximation Kress (1989) Prove that the integral equation combined with degenerate kernel has convergence of exponential order Chen et al. (2005) Applied it to solve engineering problems with circular boundaries Chen et al. (Schaback) (2007) Link Trefftz method and method of fundamental solutions However, its applications in practical problems seem to have taken a back seat to other methods. ~ M. A. Golberg 1979 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 59
Hypersingular integral H. P. V. (Hadamard principal value) Principle value version Series summability version 1 D 2 D NTOU/MSV 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics D. H. Yu 60
Other degenerate kernels-1 (2 -D circular Laplace problem) (2 -D circular Helmholtz problem) (2 -D circular biharmonic problem) (2 -D circular bi. Helmholtz problem) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 61
Other degenerate kernels-2 (3 -D spherical Lalace problem) (3 -D spherical Helmholtz problem) (2 -D elliptical Laplace problem) (2 -D elliptical Helmholtz problem) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 62
Other degenerate kernels-3 (Two circles(bipolar) Laplace problem) (Two spheres(bispherical) Laplace problem) (Circular Navier problem) 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 63
Orthogonal coordinate systems 2 D 3 D Cartesian Polar Sphere Elliptic Bipolar Oblate spheroidal Prolate spheroidal Bispherical Toroidal Parabolic 2009/12/10 2009 International Workshop on Students’ Exchanges of Nano and Computational Mechanics 64
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