National Taiwan Ocean University MSVLAB Department of Harbor
National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering Null-field boundary integral equation approach for hydrodynamic scattering by multiple circular and elliptical cylinders Jai-Wei Lee and Jeng-Tzong Chen Date: Jan. 13, 2010 Time: 11: 50~12: 10 Place: Lectrue Theater F 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 1
Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 2
Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 3
The 8 th ACFD Conference in HK, 2010. 1. 10~14 Keelung HKUST NTOU 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 4
NTOU/MSV Group members (2010) 1962 1983 1955 1978 2010/01/13 1959 1985 1962 1971 1985 1987 1972 1986 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 1975 1976 1987 1988 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) 2010/01/13 陳 鞏(USA, Texas A M) Jeong-Guon Ih (KAIST, Korea) (M. Tanaka, Japan) The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 余德浩 中國科學院 祝家麟 (J. L. Zhu, China) Page 6
Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 7
Introduction of water wave problem (single cylinder) Analytical solution circular Mac. Camy and Fuchs (1954) 2010/01/13 elliptical Goda and Yoshimura (1972) The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 8
Introduction of water wave problem (multiple cylinders) Analytical solutions are not available Semi-analytical methods Spring and Monkmeyer (1974) Multipole expansion Linton and Evans (1990) Chatjigeorgio and Mavrakos (2009) y 1 2 b 37 2 4 x AOR (2009) Present method (Null-field BIEM) 2010/01/13 Meshless method Boundary type The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 9
Introduction of water wave problem (multiple cylinders) OK Multipole expansion (Null-field BIEM) Multipole expansion OK ? To the authors’ best knowledge 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 10
Problem statement (3 D) Governing equation Linearized wave theory and method of separation variables water depth 2010/01/13 constant The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 11
Reduction to 2 D Problem Governing equation Boundary condition Incident wave field Radiation field 2010/01/13 Governing equation Boundary condition The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 12
Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 13
Boundary integral equation and null-field boundary integral equation Interior case Exterior case Degenerate (separable) form 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 14
Degenerate (separable) form of fundamental solution (2 D) Circle 2010/01/13 Extension Ellipse The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 15
Degenerate kernels Addition theorem (Morse and Feshbach’s book) Methods of Theoretical Physics, 1953, p. 1421 Modified Mathieu functions of the third kind Normalized constants (norm) Analytical study 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 16
Contour plots of the closed-form fundamental solution and the degenerate kernel Abs Re Im Closed-form fundamental solution Degenerate kernel 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 17
Degenerate kernels (polar coordinates) (elliptic coordinates) 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 18
Expansions of boundary densities and incident plane wave for circular boundaries Boundary densities Fourier series Incident plane wave Polar coordinates 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 19
Expansions of boundary densities and incident plane wave for elliptical boundaries Boundary densities Eigenfunction expansion (Mathieu functions) Incident plane wave Elliptic coordinates 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 20
Keypoint for solving the problem with elliptical boundaries Orthogonal relations are reserved 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 21
Adaptive observer systems and linear algebraic equations Collocation point Boundary contour integration 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 22
Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 23
Illustrative examples • Case 1: A single elliptical cylinder • Case 2: Two parallel identical elliptical cylinders • Case 3: One circular and one elliptical cylinders 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 24
Case 1: A single elliptical cylinder 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 25
Resultant forces of an elliptical cylinder Number of degree of freedom [3] Au M. C. and Brebbia C. A. , “Diffraction of water waves for vertical cylinders using boundary elements”, Applied Mathematical Modelling, Vol. 7, (1983), pp 106 -114. 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 26
Resultant forces of an elliptical cylinder Number of degree of freedom [3] Au M. C. and Brebbia C. A. , “Diffraction of water waves for vertical cylinders using boundary elements”, Applied Mathematical Modelling, Vol. 7, (1983), pp 106 -114. 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 27
Case 2: Two parallel identical elliptical cylinders 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 28
Resultant forces of two parallel identical elliptical cylinders 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 29
Case 3: One circular and one elliptical cylinders 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 30
Resultant forces of two cylinders containing one circular and one elliptical cylinder 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 31
Outline • Introduction of NTOU/MSV group • Motivation and problem statement • Method of solution • Illustrative examples • Conclusions 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 32
Conclusions 1. The higher accurate and faster convergence rate of the present method over the EBM is observed 2. Null-field BIEM in conjunction with adaptive observer system and the degenerate kernel can solve water wave problems containing circular and elliptical cylinders in a semi-analytical way. 3. This method also belongs to a meshless method since collocation points on the boundaries are only required. 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 33
The end Thanks for your kind attentions Welcome to visit the web site of MSVLAB/NTOU http: //msvlab. hre. ntou. edu. tw/ 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 34
Extension (circle to ellipse) Expand fundamental solution by using the degenerate kernel 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 35
Degenerate kernels (polar coordinates) 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 36
Four degenerate kernels (elliptic coordinates) 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 37
Adaptive observer systems and linear algebraic equations Collocation point Boundary contour integration 2010/01/13 Boundary contour integration The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 38
Elliptic coordinates and Mathieu function Modified Mathieu function angular coordinate radial coordinate 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 40
Resultant forces of a circular cylinder Number of degree of freedom 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 41
Degenerate (separable) form of fundamental solution (1 D) continuous 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 jump Page 42
Elliptic coordinates and Mathieu function Modified Mathieu function angular coordinate radial coordinate 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 43
Difference between the 33 rd CTAM and present work Eigenproblems Interior problem Water wave problems Interior problem 33 rd CTAM Present work 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 44
Resultant forces of an elliptical cylinder [3] Au M. C. and Brebbia C. A. , “Diffraction of water waves for vertical cylinders using boundary elements”, Applied Mathematical Modelling, Vol. 7, (1983), pp 106 -114. 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 45
Boundary densities Expand boundary densities by using the Fourier series and eigenfunction expansion Circular boundaries Elliptical boundaries 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 46
Expansions of incident plane wave using the polar and the elliptic coordinates Circular boundaries Elliptical boundaries 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 47
Boundary densities Expand boundary densities by using the eigenfunction expansion is a constants along the elliptical boundary 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 48
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 2010/01/13 Chen et al. Proc. R. Soc. Lond. , Ser. A, 2002 & 2003 (Found and treated) Degenerate kernel (Polar coordinates) The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 49
Elliptic coordinates and Mathieu function Modified Mathieu function angular coordinate radial coordinate 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 50
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 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 51 (norm)
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 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 52
Why spurious solution occurs • FDM for ODE • Real-part BEM & MRM (Simply-connected problem) • Complex-valued BEM (Multiply-connected problem) 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 53
Separation of variables in the elliptic coordinates Cartesian coordinates Elliptic coordinates separation of variables 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 54
Addition theorem Q r a b P = + Subtraction theorem O 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 55
Degenerate (separable) form of fundamental solution (2 D) Circle 2010/01/13 Extension Ellipse The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 56
Degenerate cases in mathematics and mechanics 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 57
Jump behavior across the boundary 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 58
Other applications Water wave 1. Harbor resonance Acoustics 1. Hermetic compressor 2. Small automotive muffler Electromagnetics 1. Waveguides TM mode (Dirichlet BC) TE mode (Neumann BC) 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 59
Literature review 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 60
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 and numerical study for free vibration of The 8 th Asian Computational Fluid Dynamics Conference 2010/01/13 Hong Kong, 10 -14 January, plate using BEM-2010 Page 61
The orthogonality of vector and function vectors functions and are orthogonal Mathieu function Orthogonal relations 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 (norm) Page 62
Jacobian Polar coordinates Elliptic coordinates arc length (Area) 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 63
Adaptive observer system r 20 k 1 , f 20 k 1 collocation point 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 64
Linear algebraic equation Index of collocation circle Index of routing circle 2010/01/13 Column vector of Fourier coefficients (Nth routing circle) The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 65
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 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 66
Hypersingular integral H. P. V. (Hadamard principal value) Principle value version Series summability version 1 D NTOU/MSV 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 2 D D. H. Yu Page 67
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) 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 68
Other degenerate kernels-2 (3 -D spherical Lalace problem) (3 -D spherical Helmholtz problem) (2 -D elliptical Laplace problem) (2 -D elliptical Helmholtz problem) 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 69
Other degenerate kernels-3 (Two circles(bipolar) Laplace problem) (Two spheres(bispherical) Laplace problem) (Circular Navier problem) 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 70
Orthogonal coordinate systems 2 D 3 D Cartesian Polar Sphere Elliptic Bipolar Oblate spheroidal Prolate spheroidal Bispherical Toroidal Parabolic 2010/01/13 The 8 th Asian Computational Fluid Dynamics Conference Hong Kong, 10 -14 January, 2010 Page 71
- Slides: 71