1. 1 Particular solutions for some engineering problems Chia-Cheng Tasi 蔡加正 Department of Information Technology Toko University, Chia-Yi County, Taiwan 2008 NTOU

2. 2 Motivation Method of Particular Solutions (MPS) Particular solutions of polyharmonic spline Numerical example I Particular solutions of Chebyshev polynomials Numerical example II Conclusions Overview

3. 3 Motivation BEM has evolved as a popular numerical technique for solving linear, constant coefficient partial differential equations. Other boundary type numerical methods: Treffz method, MFS… Advantage: Reduction of dimensionalities (3D->2D, 2D->1D) Disadvantage: domain integration for nonhomogeneous problem For inhomogeneous equations, the method of particular solution (MPS) is needed. In BEM, it is called the dual reciprocity boundary element method (DRBEM) (Partridge, et al., 1992).

4. 4 Motivation and Literature review

5. 5 Motivation RBF Golberg (1995) Chebyshev MFS MPS with Chebyshev Polynomials spectral convergence Golberg, M.A.; Muleshkov, A.S.; Chen, C.S.; Cheng, A.H.-D. (2003)

6. 6 Motivation

7. 7 We note that the polyharmonic and the poly-Helmholtz equations are encountered in certain engineering problems, such as high order plate theory, and systems involving the coupling of a set of second order elliptic equations, such as a multilayered aquifer system, or a multiple porosity system. These coupled systems can be reduced to a single partial differential equation by using the Hörmander operator decomposition technique. The resultant partial differential equations usually involve the polyharmonic or the products of Helmholtz operators. Hence My study is to fill an important gap in the application of boundary methods to these engineering problems.

8. 8 Method of particular solutions Method of fundamental solutions, Trefftz method, boundary element method, et al.

9. 9 Method of particular solutions

10. 10 Method of particular solutions (basis functions)

11. 11 Particular solutions for the engineering problems Method of particular solutions (Hörmander Operator Decomposition technique)

12. 12 Example

13. 13 Example

14. 14 Other examples Stokes flow Thermal Stokes flow

15. 15 Other examples Thick plate Solid deformation

16. 16 Remark Particular solutions for engineering problems Particular solutions for product operator H ö rmander operator decomposition technique

17. 17 Particular solutions for Partial fraction decomposition Particular solutions for product operator Method of particular solutions (Partial fraction decomposition)

18. 18 Partial fraction decomposition (Theorem)

19. 19 Partial fraction decomposition (Proof 1)

20. 20 Partial fraction decomposition (Proof 2)

21. 21 Example (1)

22. 22 Example (2)

23. 23 Remark Partial fraction decomposition

24. 24 Particular solutions of polyharmonic spline (APS)

25. 25 Particular solutions of polyharmonic spline (APS)

26. 26 Particular solutions of polyharmonic spline (Definition)

27. 27 Particular solutions of polyharmonic spline (Generating Theorem)

28. 28 Particular solutions of polyharmonic spline (Generating Theorem)

29. 29 Particular solutions of polyharmonic spline (Generating Theorem)

30. 30 Particular solutions of polyharmonic spline (2D Poly-Helmholtz Operator)

31. 31 Particular solutions of polyharmonic spline (2D Poly-Helmholtz Operator) Generating Theorem proof

32. 32 Particular solutions of polyharmonic spline (2D Poly-Helmholtz Operator)

33. 33 Particular solutions of polyharmonic spline (2D Poly-Helmholtz Operator)

34. 34 Particular solutions of polyharmonic spline (3D Poly-Helmholtz Operator)

35. 35 Particular solutions of polyharmonic spline (2D Poly-Helmholtz Operator) Generating Theorem proof

36. 36 Particular solutions of polyharmonic spline (Limit Behavior)

37. 37 Particular solutions of polyharmonic spline (Limit Behavior)

38. 38 Numerical example I

39. 39 Numerical example I

40. 40 Numerical example I (BC)

41. 41 Numerical example I (BC)

42. 42 Numerical example I (MFS)

43. 43 Numerical example I (results)

44. 44 Particular solutions of Chebyshev polynomials (why orthogonal polynomials) Fourier series: exponential convergence but Gibb’s phenomena Lagrange Polynomials: Runge phenomena Jacobi Polynomials (orthogonal polynomials): exponential convergence

45. 45 Particular solutions of Chebyshev polynomials (why Chebyshev)

46. 46 Chebyshev interpolation (1)

47. 47 Chebyshev interpolation (2)

48. 48 Particular solutions of Chebyshev polynomials

49. 49 Particular solutions of Chebyshev polynomials (poly-Helmholtz) Generating Theorem Golberg, M.A.; Muleshkov, A.S.; Chen, C.S.; Cheng, A.H.-D. (2003)

50. 50 Particular solutions of Chebyshev polynomials (polyharmonic)

51. 51 Particular solutions of Chebyshev polynomials (polyharmonic)

52. 52 Method of fundamental solutions Fig. 2: Geometry configuration of the MFS.

53. 53 Method of fundamental solutions (example)

54. 54 Example (2D modified Helmholtz) Numerical example II

55. 55 Example (2D Laplace) Numerical example II

56. 56 Example (3D modified Helmholtz) Numerical example II

57. 57 Example (3D Laplace) Numerical example II

58. 58 Example (2D polyharmonic) Numerical example II

59. 59 Example (2D product operator) Numerical example II

60. 60 1. MFS+APS => scattered data in right-hand sides 2. MFS+Chebyshev => spectral convergence 3. H ö rmander operator decomposition technique 4. Partial fraction decomposition 5. polyHelmholtz & Polyharmonic particular solutions 6. MFS for the product operator Conclusion

61. 61 Thank you