1 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 cmc2007.ppt National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 應用數學系

2 Research collaborators Dr. I. L. Chen Dr. K. H. Chen Dr. I. L. Chen Dr. K. H. Chen Dr. S. Y. Leu Dr. W. M. Lee Dr. S. Y. Leu Dr. W. M. Lee Mr. Y. T. Lee Mr. Y. T. Lee Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao Mr. A. C. Wu Mr.P. Y. Chen Mr. A. C. Wu Mr.P. Y. Chen Mr. J. N. Ke Mr. H. Z. Liao Mr. J. N. Ke Mr. H. Z. Liao

3 URL: 海洋大學工學院河工所力學聲響振動實驗室 nullsystem2007.ppt` Elasticity & Crack Problem Laplace Equation Research topics of NTOU / MSV LAB on null-field BIEs ( ) Navier Equation Null-field BIEM Biharmonic Equation Previous research and project Current work (Plate with circulr holes ) BiHelmholtz Equation Helmholtz Equation (Potential flow) (Torsion) (Anti-plane shear) (Degenerate scale) (Inclusion) (Piezoleectricity) (Beam bending) Torsion bar (Inclusion) Imperfect interface Image method (Green function) Green function of half plane (Hole and inclusion) (Interior and exterior Acoustics) SH wave (exterior acoustics) (Inclusions) (Free vibration of plate) Indirect BIEM ASME JAM 2006 MRC,CMESEABE ASME JoM EABE CMAME 2007 SDEE JCA NUMPDE revision JSV SH wave Impinging canyons Degenerate kernel for ellipse ICOME 2006 Added mass 李應德 Water wave impinging circular cylinders Screw dislocation Green function for an annular plate SH wave Impinging hill Green function of`circular inclusion (special case:static) Effective conductivity CMC (Stokes flow) (Free vibration of plate) Direct BIEM (Flexural wave of plate)

4 Overview of numerical methods 4 PDE- variational IE DE Domain Boundary MFS,Trefftz method MLS, EFG 開刀 把脈把脈 針灸針灸 國科會專題報導 : 中醫式的工程分析法

5 Prof. C B Ling ( ) Fellow of Academia Sinica He devoted himself to solve BVPs with holes. PS: short visit (J T Chen) of Academia Sinica 2006 summer ` C B Ling (mathematician and expert in mechanics)

6 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Study of spurious solution Study of spurious solution SVD technique SVD technique Conclusions Conclusions

7 Motivation Motivation Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Meshless method BEM / BIEM (mesh required) Treatment of singularity and hypersingularity Boundary-layer effect Ill-posed model Convergence rate Mesh free for circular boundaries ?

8 Motivation and literature review Fictitious BEM BEM/BIEM Null-field approach Bump contour Limit process Singular and hypersingular Regular Improper integral CPV and HPV Ill-posed Fictitious boundary Collocation point

9 Present approach 1.No principal value 2. Well-posed 2. Well-posed 3. No boundary-layer effect 3. No boundary-layer effect 4. Exponetial convergence 4. Exponetial convergence 5. Meshless 5. Meshless Advantages of degenerate kernel Degenerate kernel Fundamental solution CPV and HPV No principal value

10 Engineering problem with arbitrary geometries Degenerate boundary Circular boundary Straight boundary Elliptic boundary (Fourier series) (Legendre polynomial) (Chebyshev polynomial) (Mathieu function)

11 Motivation and literature review Analytical methods for solving Laplace problems with circular holes Conformal mapping Bipolar coordinate Special solution Limited to doubly connected domain Lebedev, Skalskaya and Uyand, 1979, “Work problem in applied mathematics”, Dover Publications Chen and Weng, 2001, “Torsion of a circular compound bar with imperfect interface”, ASME Journal of Applied Mechanics Honein, Honein and Hermann, 1992, “On two circular inclusions in harmonic problem”, Quarterly of Applied Mathematics

12 Fourier series approximation Ling (1943) - torsion of a circular tube Ling (1943) - torsion of a circular tube Caulk et al. (1983) - steady heat conduction with circular holes Caulk et al. (1983) - steady heat conduction with circular holes Bird and Steele (1992) - harmonic and biharmonic problems with circular holes Bird and Steele (1992) - harmonic and biharmonic problems with circular holes Mogilevskaya et al. (2002) - elasticity problems with circular boundaries Mogilevskaya et al. (2002) - elasticity problems with circular boundaries

13 Contribution and goal However, they didn’t employ the null-field integral equation and degenerate kernels to fully capture the circular boundary, although they all employed Fourier series expansion. However, they didn’t employ the null-field integral equation and degenerate kernels to fully capture the circular boundary, although they all employed Fourier series expansion. To develop a systematic approach for solving Laplace problems with multiple holes is our goal. To develop a systematic approach for solving Laplace problems with multiple holes is our goal.

14 Outlines (Direct problem) Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

15 Boundary integral equation and null-field integral equation Interior case Exterior case Degenerate (separate) form

16 Outlines (Direct problem) Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Degenerate scale Degenerate scale Conclusions Conclusions

17 Gain of introducing the degenerate kernel Degenerate kernel Fundamental solution CPV and HPV No principal value? interior exterior

18 How to separate the region

19 Expansions of fundamental solution and boundary density Degenerate kernel - fundamental solution Degenerate kernel - fundamental solution Fourier series expansions - boundary density Fourier series expansions - boundary density

20 Separable form of fundamental solution (1D) Separable property continuous discontinuous

21 Separable form of fundamental solution (2D)

22 Boundary density discretization Fourier series Ex. constant element Present method Conventional BEM

23 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

24 Adaptive observer system collocation point

25 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

26 Vector decomposition technique for potential gradient Special case (concentric case) : Non-concentric case: True normal direction

27 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

28 Linear algebraic equation where Column vector of Fourier coefficients (Nth routing circle) Index of collocation circle Index of routing circle

29 Physical meaning of influence coefficient kth circular boundary xmxm mth collocation point on the jth circular boundary jth circular boundary Physical meaning of the influence coefficient cosnθ, sinnθ boundary distributions

30 Flowchart of present method Potential of domain point Analytical Numerical Adaptive observer system Degenerate kernel Fourier series Linear algebraic equation Collocation point and matching B.C. Fourier coefficients Vector decomposition Potential gradient

31 Comparisons of conventional BEM and present method BoundarydensitydiscretizationAuxiliarysystemFormulationObserversystemSingularityConvergenceBoundarylayereffect ConventionalBEMConstant,linear,quadratic…elementsFundamentalsolutionBoundaryintegralequationFixedobserversystem CPV, RPV and HPV LinearAppear PresentmethodFourierseriesexpansionDegeneratekernelNull-fieldintegralequationAdaptiveobserversystemDisappearExponentialEliminate

32 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

33 Numerical examples Laplace equation (EABE 2005, EABE 2007) Laplace equation (EABE 2005, EABE 2007) (CMES 2006, JAM-ASME 2007, JoM2007) (CMES 2006, JAM-ASME 2007, JoM2007) (CMA2007,MRC 2007, NUMPDE revision) (CMA2007,MRC 2007, NUMPDE revision) Membrane eigenproblem (JCA 2007) Membrane eigenproblem (JCA 2007) Exterior acoustics (CMAME 2007, SDEE 2007 ) Exterior acoustics (CMAME 2007, SDEE 2007 ) Biharmonic equation (JAM-ASME 2006 ) Biharmonic equation (JAM-ASME 2006 ) Plate eigenproblem (JSV 2007) Plate eigenproblem (JSV 2007)

34 Laplace equation A circular bar under torque A circular bar under torque (free of mesh generation) (free of mesh generation)

35 Torsion bar with circular holes removed The warping function Boundary condition where on Torque

36 Axial displacement with two circular holes Present method (M=10) Caulk’s data (1983) ASME Journal of Applied Mechanics Dashed line: exact solution Solid line: first-order solution

37 Torsional rigidity ?

38 Extension to inclusion Extension to inclusion Anti-plane elasticity problems Anti-plane elasticity problems (free of boundary layer effect) (free of boundary layer effect)

39 Two circular inclusions with centers on the y axis Honein et al.’sdata (1992) Present method (L=20) Equilibrium of traction

40 Convergence test and boundary-layer effect analysis boundary-layer effect

41 Numerical examples Biharmonic equation Biharmonic equation (exponential convergence) (exponential convergence)

42 Plate problems Geometric data: and on Essential boundary conditions: (Bird & Steele, 1991)

43 Contour plot of displacement Present method (N=101)Bird and Steele (1991) FEM (ABAQUS) FEM mesh (No. of nodes=3,462, No. of elements=6,606)

44 Stokes flow problem Governing equation: Boundary conditions: and on and on Eccentricity: Angular velocity: (Stationary)

45 Comparison for DOF of BIE (Kelmanson) DOF of present method BIE (Kelmanson) Present method Analytical solution (160) (320) (640) u1u1 (28) (36) (44) (∞) Algebraic convergence Exponential convergence

46 Contour plot of Streamline for Present method (N=81) Kelmanson (Q=0.0740, n=160) Kamal (Q=0.0738) e Q/2 Q Q/5 Q/20 -Q/90 -Q/30 0 Q/2 Q Q/5 Q/20 -Q/90 -Q/30 0

47 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Discussions of spurious eigenvalues Discussions of spurious eigenvalues SVD SVD Conclusions Conclusions

48 Disclaimer (commercial code) Disclaimer (commercial code) The concepts, methods, and examples using our software are for illustrative and educational purposes only. The concepts, methods, and examples using our software are 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. 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 ? inherent weakness ? misinterpretation ? User 當自強 misinterpretation ? User 當自強

49 Eccentric membrane (true and spurious eignevalues) U T formulation Singular integral equations L M formulation Hypersingular formulation spurious Chen et al., 2001, Proc. Royal Soc. London Ser. A

50 SVD Technique (Google searching) [C] SVD decomposition [U] and [V} left and right unitary vectors

51 Physical meaning of SVD 變形前 變形後 假根真根 Chen et al., 2002, Int. J. Comp. Numer. Anal. Appl. 先拉再轉 先轉再拉

52 (SVD updating terms) Find true eigenvalue SVD updating terms SVD updating document 52 (SVD updating document) Find spurious eigenvalue Chen et al., 2003, Proc. Royal Soc. London Ser. A

53 Eccentric membrane (SVD updating for true eigenvalues) Dirichelet case Neumann case T M ULUL

54 Eccentric membrane (SVD updating for spurious eigenvalues) U T L M

55 Eccentric plate R1R1 R2R2 e o1o1 o2o2 Case 1: Geometric data: R 1 =1m R 2 =0.4m e=0.0 ～ 0.5m thickness=0.002m 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

56 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)

57 True boundary eigenmode Figure 5. Real and imagine part of Fourier coefficients for first true boundary mode ( =6.1716, e = 0.2, R 2 = 0.4m) R1R1 R2R2 e o1o1 o2o2 R1R1 R2R2 e o1o1 o2o2

58 Spurious boundary eigenmode Figure 6. Real and imagine part of Fourier coefficients for first spurious boundary mode ( =7.9906, e = 0.2, R2 = 0.4m) R1R1 R2R2 e o1o1 o2o2 R1R1 R2R2 e o1o1 o2o2

59 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

60 Conclusions 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. 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. Numerical results agree well with available exact solutions and FEM (ABAQUS) for only few terms of Fourier series. Spurious eigenvalues are examined. Spurious eigenvalues are examined.

61 Conclusions Free of boundary-layer effect Free of boundary-layer effect Free of singular integrals Free of singular integrals Well posed Well posed Exponetial convergence Exponetial convergence Mesh-free approach Mesh-free approach

62 The End Thanks for your kind attentions. Your comments will be highly appreciated. URL:

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