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Null field integral equation approach for engineering problems with circular boundaries

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

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Null field integral equation approach for engineering problems with circular boundaries

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

  2. Research collaborators • 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

  3. Previous research and project Current work (Interior and exterior Acoustics) SH wave (exterior acoustics) (Inclusions) Research topics of NTOU / MSV LAB on null-field BIEs (2003-2007) Null-field BIEM NUMPDE revision Navier Equation Laplace Equation Helmholtz Equation Biharmonic Equation BiHelmholtz Equation ASME JAM 2006 JSV EABE MRC,CMES Elasticity & Crack Problem (Plate with circulr holes) (Potential flow) (Torsion) (Anti-plane shear) (Degenerate scale) (Free vibration of plate) Indirect BIEM Screw dislocation (Stokes flow) JCA CMAME 2007 JoM ASME (Free vibration of plate) Direct BIEM EABE (Inclusion) (Piezoleectricity) (Beam bending) Green function for an annular plate SDEE ICOME 2006 SH wave Impinging canyons (Flexural wave of plate) Degenerate kernel for ellipse Torsion bar(Inclusion) Imperfect interface CMC Image method (Green function) Added mass SH wave Impinging hill Green function of`circular inclusion (special case:static) Green function of half plane (Hole and inclusion) Effective conductivity 李應德 Water wave impinging circular cylinders URL: http://ind.ntou.edu.tw/~msvlab E-mail: jtchen@mail.ntou.edu.tw海洋大學工學院河工所力學聲響振動實驗室 nullsystem2007.ppt`

  4. Overview of numerical methods 國科會專題報導:中醫式的工程分析法 Domain Boundary MFS,Trefftz method MLS, EFG IE DE PDE- variational 針 灸 把 脈 開刀 4

  5. 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 `

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

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

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

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

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

  11. Motivation and literature review Analytical methods for solving Laplace problems with circular holes Special solution Bipolar coordinate Conformal mapping 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 Lebedev, Skalskaya and Uyand, 1979, “Work problem in applied mathematics”, Dover Publications Limited to doubly connected domain

  12. Fourier series approximation • 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

  13. Contribution and goal • However, they didn’t employ the null-field integral equationand 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.

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

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

  18. How to separate the region

  19. Expansions of fundamental solution and boundary density • Degenerate kernel - fundamental solution • 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 • Mathematical formulation • Expansions of fundamental solution and boundary density • Adaptive observer system • Vector decomposition technique • Linear algebraic equation • Numerical examples • Conclusions

  24. collocation point Adaptive observer system

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

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

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

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

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

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

  31. Comparisons of conventional BEM and present method

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

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

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

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

  36. 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)

  37. Torsional rigidity ?

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

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

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

  41. Numerical examples • Biharmonic equation (exponential convergence)

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

  43. 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)

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

  45. BIE (Kelmanson) Present method Analytical solution Comparison for (160) (28) Algebraic convergence u1 (320) (640) (36) Exponential convergence (∞) (44) DOF of BIE (Kelmanson) DOF of present method

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

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

  48. Disclaimer (commercial code) • 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. inherent weakness ? misinterpretation ? User 當自強

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

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

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