Null-field integral equations and engineering applications

1. Null-field integral equations and engineering applications I. L. Chen Ph.D. Department of Naval Architecture, National Kaohsiung Marine University Mar. 11, 2010

2. Research collaborators • Prof. J. T. Chen • Dr. K. H. Chen • Dr. S. Y. Leu Dr. W. M. Lee • Dr. 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 • Mr. Y. J. Lin • Mr. C. F. Wu Mr. J. W. Lee

3. Outline • Introduction of NTOU/MSV group

4. Previous research and project Current work (Interior and exterior Acoustics) SH wave (exterior acoustics) (Inclusions) Research topics of NTOU / MSV LAB on null-field BIE (2003-2010) 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:staic) Green function of half plane (Hole and inclusion) AOR 2009 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`

5. 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

6. 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 ?

7. 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

8. 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

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

10. 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

11. 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

12. 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.

13. 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

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

15. 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

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

17. How to separate the region

18. Expansions of fundamental solution and boundary density • Degenerate kernel - fundamental solution • Fourier series expansions - boundary density

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

20. Separable form of fundamental solution (2D)

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

22. 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

23. collocation point Adaptive observer system

24. 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

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

26. 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

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

28. 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

29. Comparisons of conventional BEM and present method

30. 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

31. Numerical examples • Laplace equation (EABE 2005, EABE 2007) (CMES 2005, ASME 2007, JoM2007) (MRC 2007, NUMPDE 2010) • Biharmonic equation (JAM, ASME 2006) • Plate eigenproblem (JSV ) • Membrane eigenproblem (JCA) • Exterior acoustics (CMAME, SDEE ) • Water wave (AOR 2009)

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

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

34. 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)

35. Torsional rigidity ?

36. Numerical examples • Biharmonic equation (exponential convergence)

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

38. 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)

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

40. 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)

41. An infinite plate with two inclusions

42. Distribution of dynamic moment concentration factors by using the present method and FEM( L/a = 2.1)

43. Membrane eigenproblem(Nonuniqueness problems) A confocal elliptical annulus

44. G. E.: B. Cs.: A confocal elliptical annulus

45. (42) True FEM mesh BEM mesh True and spurious eigenvalues Spurious UT equation Note: the data inside parentheses denote the spurious eigenvalue. Eigenvalues of an elliptical membrane (11)

46. (42) Mode shapes Even Odd Even Odd Even

47. Water wave(Nonuniqueness problems) Interaction of water waves with vertical cylinders

48. Trapped mode（nonuniqueness in physics ） M.S. Longuet-higgins JFM, 1967. A.E.H. Love 1966. Williams & Li OE, 2000.

49. Trapped and near-trapped modes Trapped mode Near-trapped mode

50. Numerical and physical resonance Physical resonance Fictitious frequency (BEM/BIEM) Present t(a,0)