Null-field approach for multiple circular inclusion problems in anti-plane piezoelectricity

1. Null-field approach for multiple circular inclusion problems in anti-plane piezoelectricity Reporter: An-Chien Wu Advisor: Jeng-Tzong Chen Date: 2006/06/29 Place: HR2 307

2. Outline • Motivation and literature review • Unified formulation of null-field approach ◎Boundary integral equations and null-field integral equations ◎Adaptive observer system ◎Linear algebraic equation ◎Vector decomposition technique • Numerical examples • Conclusions • Further studies

3. Outline • Motivation and literature review • Unified formulation of null-field approach ◎Boundary integral equations and null-field integral equations ◎Adaptive observer system ◎Linear algebraic equation ◎Vector decomposition technique • Numerical examples • Conclusions • Further studies

4. Motivation Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Meshless method BEM / BIEM Treatment of singularity and hypersingularity Boundary-layer effect Convergence rate Ill-posed model

5. Motivation BEM / BIEM Improper integral Singularity & hypersingularity Regularity Fictitious BEM Bump contour Limit process Fictitious boundary Achenbach et al. (1988) Null-field approach Guiggiani (1995) Gray and Manne (1993) Collocation point CPV and HPV Ill-posed Waterman (1965)

6. Present approach Degenerate kernel Fundamental solution No principal value CPV and HPV • Advantages of degenerate kernel • No principal value • Well-posed • Exponential convergence • Free of boundary-layer effect

7. Engineering problem with holes, inclusions and cracks [Chebyshev polynomial] Degenerate boundary Straight boundary [Legendre polynomial] [Mathieu function] Elliptic hole Circular inclusion [Fourier series]

8. Literature review – analytical solutions for problems with circular boundaries Those analytical methods are only limited to doubly connected regions even to conformal connected regions.

9. Literature review - Fourier series approximation However, no one employed the null-field approach and degenerate kernel to fully capture the circular boundary.

10. Outline • Motivation and literature review • Unified formulation of null-field approach ◎Boundary integral equations and null-field integral equations ◎Adaptive observer system ◎Linear algebraic equation ◎Vector decomposition technique • Numerical examples • Conclusions • Further studies

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

12. Expansions of fundamental solution and boundary density Degenerate kernel – fundamental solution Fourier series expansion – boundary density

13. Convergence rate between present method and conventional BEM Conventional BEM Present method Degenerate kernel Two-point function Fundamental solution Constant, linear, quadratic elements Fourier series expansion Boundary density Convergence rate Exponential convergence Linear convergence

14. Degenerate (separate) form of fundamental solution (2-D)

15. Outline • Motivation and literature review • Unified formulation of null-field approach ◎Boundary integral equations and null-field integral equations ◎Adaptive observer system ◎Linear algebraic equation ◎Vector decomposition technique • Numerical examples • Conclusions • Further studies

16. collocation point Adaptive observer system r2,f2 r0 ,f0 r1 ,f1 rk,fk

17. Outline • Motivation and literature review • Unified formulation of null-field approach ◎Boundary integral equations and null-field integral equations ◎Adaptive observer system ◎Linear algebraic equation ◎Vector decomposition technique • Numerical examples • Conclusions • Further studies

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

19. Explicit form of each submatrix and vector Truncated terms of Fourier series Number of collocation points Fourier coefficients

20. Physical meaning of influence coefficients and mth collocation point on the jth circular boundary xm jth circular boundary fm cosnq, sinnq boundary distribution kth circular boundary

21. Outline • Motivation and literature review • Unified formulation of null-field approach ◎Boundary integral equations and null-field integral equations ◎Adaptive observer system ◎Linear algebraic equation ◎Vector decomposition technique • Numerical examples • Conclusions • Further studies

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

23. Numerical Flowchart of present method Degenerate kernel Fourier series Potential gradient Adaptive observer system Analytical Vector decomposition Continuity of displacement and equilibrium of traction Collocating point to construct compatible boundary data relationship Potential of domain point Linear algebraic system Fourier coefficients

24. Comparisons of conventional BEM and present method

25. Outline • Motivation and literature review • Unified formulation of null-field approach ◎Boundary integral equations and null-field integral equations ◎Adaptive observer system ◎Linear algebraic equation ◎Vector decomposition technique • Numerical examples • Conclusions • Further studies

26. Numerical examples • Anti-plane piezoelectricity problems (EABE, 2006, accepted) • In-plane electrostatics problems (??) • Anti-plane elasticity problems (ASME-JAM, 2006, accepted)

27. Numerical examples • Anti-plane piezoelectricity problems • In-plane electrostatics problems • Anti-plane elasticity problems

28. Problem statement = + +

29. Analogy between anti-plane deformation and in-plane electrostatics for anti-plane piezoelectricity Coupling effect szi= c44gzi – e15 Ei Di = e15gzi + e11 Ei Shear modulus c44 Piezoelectric constant e15 Dielectric constant e11

30. For the exterior problem of matrix For the interior problem of each inclusion The continuity of displacement The equilibrium of traction Linear algebraic system

31. Two circular inclusions embedded in a piezoelectric matrix under such loadings

32. Tangential stress distribution for different ratios d/r1 with r2=2r1, e15M/e15I=3.0 and b=90° Present method (L=20) Chao & Chang’s data (1999)

33. Tangential electric field distribution for different ratios d/r1 with r2=2r1, e15M/e15I=3.0 and b=90° Present method (L=20) Chao & Chang’s data (1999)

34. Tangential stress distribution for different ratios d/r1 with r2=2r1, e15M/e15I=-5.0 and b=90° Present method (L=20) Chao & Chang’s data (1999)

35. Tangential electric field distribution for different ratios d/r1 with r2=2r1, e15M/e15I=-5.0 and b=90° Present method (L=20) Chao & Chang’s data (1999)

36. Parseval’s sum for r2=2r1, d/r1=0.01, b=90° and e15M/e15I=5.0 Parseval’s sum

37. Parseval’s sum for r2=2r1, d/r1=0.01, b=90° and e15M/e15I=5.0 Parseval’s sum

38. Tangential stress distribution for different ratios d/r1 with r2=2r1, e15M/e15I=-5.0 and b=0° Present method (L=20) Chao & Chang’s data (1999)

39. Tangential electric field distribution for different ratios d/r1 with r2=2r1, e15M/e15I=-5.0 and b=0° Present method (L=20) Chao & Chang’s data (1999)

40. Stress concentrations as a function of the ratio of piezoelectric constants with b=0° Present method (L=20) Chao & Chang’s data (1999)

41. Electric field concentrations as a function of the ratio of piezoelectric constants with b=0° Present method (L=20) Chao & Chang’s data (1999)

42. Stress concentrations as a function of the ratio of piezoelectric constants with b=0° Present method (L=20) Chao & Chang’s data (1999)

43. Electric field concentrations as a function of the ratio of piezoelectric constants with b=0° Present method (L=20) Chao & Chang’s data (1999)

44. Contour of shear stress szx when d/r1=0.01 Present method (L=20) Wang & Shen’s data (2001)

45. Contour of shear stress szy when d/r1=0.01 Present method (L=20) Wang & Shen’s data (2001)

46. Contour of electric potential F when d/r1=0.01 Present method (L=20)

47. Stress distributionwith r2=2r1 and d/r1=0.01 in two-directions loadings Present method (L=20) Wang & Shen’s data (2001)

48. Electric displacement distributionwith r2=2r1 and d/r1=0.01 in two-directions loadings Present method (L=20) Wang & Shen’s data (2001)

49. Stress distributionwith r2=2r1 and d/r1=0.01 in two-directions loadings Present method (L=20) Wang & Shen’s data (2001)

50. Electric displacement distributionwith r2=2r1 and d/r1=0.01 in two-directions loadings Present method (L=20) Wang & Shen’s data (2001)