Recent Results of the Null-Field Integral Equation Approach for Engineering Problems with Circular Boundaries

1. National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering Some recent results of the null-field integral equation approach for engineering problems with circular J. T. Chen Ph.D. Taiwan Ocean University Keelung, Taiwan November, 14-18, 2006 icome2006heife.ppt

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

3. Previous research and project Future work 陳佳聰05’ (Interior and exterior Acoustics) 陳柏源05’ SH wave (exterior acoustics) (Inclusions) Research topics of NTOU / MSV LAB on null-field BIE (2003-2006) Null-field BIEM Navier Equation Laplace Equation Helmholtz Equation Biharmonic Equation BiHelmholtz Equation EABE MRC,CMES ASME Elasticity & Crack Problem 李應德 沈文成05’ (Potential flow) (Torsion) (Anti-plane shear) (Degenerate scale) 蕭嘉俊05’ (Plate with circulr holes) (Stokes’ flow) 李為民、李應德 (Plate vibration) JoM ASME 吳安傑06’ (Inclusion) (Piezoleectricity) 陳柏源06’ (Beam bending) EABE 李應德 Torsion bar (Inclusion) 廖奐禎、李應德 Image method (Green function) 柯佳男 Green function of half plane (Hole and inclusion) 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 ``

4. 哲人日已遠 典型在宿昔(1909-1993) 林致平院士(數學力學家) 省立中興大學第一任校長 林致平校長(民國五十年~民國五十二年) 林致平所長(中研院數學所) 林致平院士(中研院) PS: short visit (J T Chen) of Academia Sinica 2006 summer `

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

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. NTUCE Ó Definitions of R.P.V., C.P.V. and H.P.V.using bump approach • R.P.V. (Riemann principal value) • C.P.V.(Cauchy principal value) • H.P.V.(Hadamard principal value)

16. Principal value in who’s sense • Riemann sense (Common sense) • Lebesgue sense • Cauchy sense • Hadamard sense (elasticity) • Mangler sense (aerodynamics) • Liggett and Liu’s sense • The singularity that occur when the base point and field point coincide are not integrable. (1983)

17. Two approaches to understand HPV Differential first and then trace operator (Limit and integral operator can not be commuted) Trace first and then differential operator (Leibnitz rule should be considered)

18. Bump contribution (2-D) U T s s 0 x x 0 L s M s x x

19. Bump contribution (3-D) 0` s s x x 0 s s x x

20. 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

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

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

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

24. Separable form of fundamental solution (2D)

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

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. collocation point Adaptive observer system

28. 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

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

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

32. 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

33. 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

34. Comparisons of conventional BEM and present method

35. 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

36. Numerical examples • Laplace equation (EABE 2005, EABE 2007) (CMES 2005, ASME 2007) (JoM 2007,MRC, 2007) • Eigen problem • Exterior acoustics • Biharmonic equation (JAM, ASME 2006) • Plate vibration (JSV revision, 2006)

37. Laplace equation • Steady state heat conduction problems • Electrostatic potential of wires • Flow of an ideal fluid pass cylinders • A circular bar under torque • An infinite medium under antiplane shear • Half-plane problems

38. Steady state heat conduction problems Case 1 Case 2

39. Case 1: Isothermal line Exact solution (Carrier and Pearson) FEM-ABAQUS (1854 elements) Present method (M=10) BEM-BEPO2D (N=21)

40. Relative error of flux on the small circle

41. Convergence test - Parseval’s sum for Fourier coefficients Parseval’s sum

42. Laplace equation • Steady state heat conduction problems • Electrostatic potential of wires • Flow of an ideal fluid pass cylinders • A circular bar under torque • An infinite medium under antiplane shear • Half-plane problems

43. Electrostatic potential of wires Two parallel cylinders held positive and negative potentials Hexagonal electrostatic potential

44. Contour plot of potential Exact solution (Lebedev et al.) Present method (M=10)

45. Contour plot of potential Onishi’s data (1991) Present method (M=10)

46. Laplace equation • Steady state heat conduction problems • Electrostatic potential of wires • Flow of an ideal fluid pass cylinders • A circular bar under torque • An infinite medium under antiplane shear • Half-plane problems

47. Flow of an ideal fluid pass two parallel cylinders is the velocity of flow far from the cylinders is the incident angle

48. Velocity field in different incident angle Present method (M=10) Present method (M=10)

49. Laplace equation • Steady state heat conduction problems • Electrostatic potential of wires • Flow of an ideal fluid pass cylinders • A circular bar under torque • An infinite medium under antiplane shear • Half-plane problems

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