National Taiwan Ocean University MSVLAB Department of Harbor
National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 應用數學系 Null field integral equation approach for engineering problems with circular boundaries J. T. Chen Ph. D. 陳正宗 終身特聘教授 Taiwan Ocean University Keelung, Taiwan June 22 -23, 2007 中山大學 高雄 cmc 2007. ppt 1
Research collaborators n n n Dr. I. L. Chen Dr. K. H. Chen Dr. S. Y. Leu Dr. W. M. Lee Mr. Y. T. Lee Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao Mr. A. C. Wu Mr. P. Y. Chen Mr. J. N. Ke Mr. H. Z. Liao 2
Research topics of NTOU / MSV LAB on null-field BIEs (2003 -2007) Null-field BIEM NUMPDE revision Navier Equation Laplace Equation Elasticity & Crack Problem EABE (Inclusion) (Piezoleectricity) (Interior and exterior Acoustics) Screw dislocation Bi. Helmholtz Equation JCA CMAME 2007 (Plate with circulr holes) JSV (Free vibration of plate) Indirect BIEM (Stokes flow) Jo. M SH wave (exterior acoustics) (Inclusions) (Beam bending) SDEE ICOME 2006 Torsion bar (Inclusion) Imperfect interface Biharmonic Equation ASME JAM 2006 MRC, CMES (Potential flow) (Torsion) (Anti-plane shear) (Degenerate scale) ASME Helmholtz Equation Degenerate kernel for ellipse (Free vibration of plate) Direct BIEM Green function for an annular plate SH wave Impinging canyons (Flexural wave of plate) CMC Image method (Green function) Added mass Green function of half plane (Hole and inclusion) Effective conductivity SH wave Impinging hill 李應德 Water wave impinging circular cylinders Green function of`circular inclusion (special case: static) Previous research and project Current work URL: http: //ind. ntou. edu. tw/~msvlab E-mail: jtchen@mail. ntou. edu. tw 海洋大學 學院河 所力學聲響振動實驗室 nullsystem 2007. ppt` 3
國科會專題報導: 中醫式的 程分析法 Overview of numerical methods Domain Boundary DE 開刀 4 IE PDE- variational 把 脈 MFS, Trefftz method MLS, EFG 針 灸 4
Prof. C B Ling (1909 -1993) Fellow of Academia Sinica C B Ling (mathematician and expert in mechanics) He devoted himself to solve BVPs with holes. PS: short visit (J T Chen) of Academia Sinica 2006 summer ` 5
Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n Numerical examples Study of spurious solution SVD technique Conclusions 6
Motivation Numerical methods for engineering problems FDM / FEM / BIEM / Meshless method BEM / BIEM (mesh required) Treatment of singularity and hypersingularity Boundary-layer effect Convergence rate Mesh free for circular boundaries ? Ill-posed model 7
Motivation and literature review BEM/BIEM Singular and hypersingular Bump contour Improper integral Limit process Regular Fictitious BEM Null-field approach CPV and HPV Fictitiou s boundar y Collocation point Ill-posed 8
Present approach Degenerate kernel Fundamental solution No principal CPV and HPV value Advantages of degenerate 1. No principal kernel value 2. Well-posed 3. No boundary-layer effect 4. Exponetial convergence 9
Engineering problem with arbitrary geometries Straight boundary (Legendre polynomial) (Fourier series) Degenerate boundary (Chebyshev polynomial) Circular boundary Elliptic boundary (Mathieu function) 10
Motivation and literature review Analytical methods for solving Laplace problems with circular holes Conformal mapping Chen and Weng, 2001, “Torsion of a circular compound bar with imperfect interface”, ASME Journal of Applied Mechanics Bipolar coordinate Special solution Lebedev, Skalskaya and Uyand, 1979, “Work problem in applied mathematics”, Dover Publications Honein, Honein and Hermann, 1992, “On two circular inclusions in harmonic problem”, Quarterly of Applied Mathematics Limited to doubly connected domain 11
Fourier series approximation n n Ling (1943) - torsion of a circular tube Caulk et al. (1983) - steady heat conduction with circular holes Bird and Steele (1992) - harmonic and biharmonic problems with circular holes Mogilevskaya et al. (2002) - elasticity problems with circular boundaries 12
Contribution and goal However, they didn’t employ the nullfield integral equation and degenerate kernels to fully capture the circular boundary, although they all employed Fourier series expansion. n To develop a systematic approach for solving Laplace problems with multiple holes is our goal. n 13
Outlines (Direct problem) n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 14
Boundary integral equation and null-field integral equation Interior case Exterior case Degenerate (separate) form 15
Outlines (Direct problem) n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n Numerical examples Degenerate scale Conclusions 16
Gain of introducing the degenerate kernel Degenerate kernel Fundamental solution interior CPV and HPV exterior No principal value? 17
How to separate the region 18
Expansions of fundamental solution and boundary density n Degenerate kernel - fundamental solution n Fourier series expansions - boundary density 19
Separable form of fundamental solution (1 D) Separable property continuous discontinuo us 20
Separable form of fundamental solution (2 D) 21
Boundary density discretization Fourier series Ex. constant element Present method Conventional BEM 22
Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 23
Adaptive observer system collocation point 24
Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 25
Vector decomposition technique for potential gradient True normal direction Non-concentric case: Special case (concentric case) : 26
Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 27
Linear algebraic equation wher e Index of collocation circle Index of routing circle Column vector of Fourier coefficients (Nth routing circle) 28
Physical meaning of influence coefficient kth circular boundary mth collocation point xm on the jth circular boundary cosnθ, sinnθ boundary distributions Physical meaning of the influence coefficient 29
Flowchart of present method Potential gradient Degenerate kernel Analytical Fourier series Adaptive observer system Vector decompositio n Potential of domain point Collocation point and matching B. C. Linear algebraic Fourier coefficients equation Numerical 30
Comparisons of conventional BEM and present method Boundary density discretization Auxiliary system Formulation Observer system Singularity Convergence Boundary layer effect Conventional BEM Constant, linear, quadratic… elements Fundamental solution Boundary integral equation Fixed observer system CPV, RPV and HPV Linear Appear Present method Fourier series expansion Degenerate kernel Null-field integral equation Adaptive observer system Disappear Exponential Eliminate 31
Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 32
Numerical examples n n n Laplace equation (EABE 2005, EABE 2007) (CMES 2006, JAM-ASME 2007, Jo. M 2007) (CMA 2007, MRC 2007, NUMPDE revision) Membrane eigenproblem (JCA 2007) Exterior acoustics (CMAME 2007, SDEE 2007) Biharmonic equation (JAM-ASME 2006) Plate eigenproblem (JSV 2007) 33
Laplace equation n A circular bar under torque (free of mesh generation) 34
Torsion bar with circular holes removed Torque The warping function Boundary condition on where 35
Axial displacement with two circular holes Dashed line: exact solution Solid line: first-order solution Caulk’s data (1983) ASME Journal of Applied Mechanics Present method (M=10) 36
Torsional rigidity ? 37
Extension to inclusion n Anti-plane elasticity problems (free of boundary layer effect) 38
Two circular inclusions with centers on the y axis Honein et al. ’sdata (1992) Equilibrium of traction Present method (L=20) 39
Convergence test and boundary-layer effect analysis boundary-layer effect 40
Numerical examples n Biharmonic equation (exponential convergence) 41
Plate problems Geometric data: Essential boundary conditions: on and and on on (Bird & Steele, 1991) 42
Contour plot of displacement Present method (N=101) Bird and Steele (1991) (No. of nodes=3, 462, No. of elements=6, 606) FEM mesh FEM (ABAQUS) 43
Stokes flow problem Governing equation: Angular velocity: Boundary conditions: and on (Stationary) Eccentricity : 44
Comparison for (160) (28) BIE (Kelmanson) Present method Analytical solution Algebraic convergence u 1 (320) (640) (36) Exponential convergence (∞) (44) DOF of BIE (Kelmanson) DOF of present method 45
Contour plot of Streamline for 0 Q/20 Q/5 Q/2 -Q/90 -Q/30 Q Present method (N=81) 0 Q/20 Q/5 Q/2 -Q/90 -Q/30 Q Kelmanson (Q=0. 0740, n=160) e Kamal (Q=0. 0738) 46
Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n Numerical examples Discussions of spurious eigenvalues SVD Conclusions 47
Disclaimer (commercial code) n n The concepts, methods, and examples using our software for illustrative and educational purposes only. Our cooperation assumes no liability or responsibility to any person or company for direct or indirect damages resulting from the use of any information contained here. inherent weakness ? misinterpretation ? User 當自強 48
Eccentric membrane (true and spurious eignevalues) Chen et al. , 2001, Proc. Royal Soc. London Ser. A U T formulation Singular integral equations spurious L M formulation Hypersingular formulation 49
SVD Technique (Google searching) [C] SVD decomposition [U] and [V} left and right unitary vectors 50
Physical meaning of SVD Chen et al. , 2002, Int. J. Comp. Numer. Anal. Appl. 假根 變形後 先拉再轉 真根 變形前 先轉再拉 51
SVD updating terms SVD updating document (SVD updating terms) Find true eigenvalue (SVD updating document) Find spurious eigenvalue Chen et al. , 2003, Proc. Royal Soc. London Ser. A 52 52
Eccentric membrane (SVD updating for true eigenvalues) Dirichelet case U L Neumann case T M 53
Eccentric membrane (SVD updating for spurious eigenvalues) U T L M 54
Eccentric plate R 1 e o 1 o 2 R 2 Case 1: Geometric data: R 1=1 m R 2=0. 4 m e=0. 0~ 0. 5 m thickness=0. 002 m Boundary condition: Inner circle : clamped Outer circle: clamped Figure 1. A clamped-clamped annular-like plate with one circular hole of radius 0. 4 m 55
Eigenvalue versus eccentricity Figure 2. Effect of the eccentricity e on the natural frequency parameter for the clamped- clamped annular-like plate (R 1=1. 0, R 2 = 0. 4) 56
True boundary eigenmode R 1 o 1 o 2 R 2 e e Figure 5. Real and imagine part of Fourier coefficients for first true boundary mode ( =6. 1716, e = 0. 2, R 2 = 0. 4 m) 57
Spurious boundary eigenmode R 1 o 1 o 2 R 2 e e Figure 6. Real and imagine part of Fourier coefficients for first spurious boundary mode ( =7. 9906, e = 0. 2, R 2 = 0. 4 m) 58
Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 59
Conclusions n n n A systematic approach using degenerate kernels, Fourier series and null-field integral equation has been successfully proposed to solve boundary value problems with circular boundaries. Numerical results agree well with available exact solutions and FEM (ABAQUS) for only few terms of Fourier series. Spurious eigenvalues are examined. 60
Conclusions n n n Free of boundary-layer effect Free of singular integrals Well posed Exponetial convergence Mesh-free approach 61
The End Thanks for your kind attentions. Your comments will be highly appreciated. URL: http: //msvlab. hre. ntou. edu. tw/ 62
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