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Particular solutions for some engineering problems

2008 NTOU. Particular solutions for some engineering problems. Chia-Cheng Tasi 蔡加正 Department of Information Technology Toko University, Chia-Yi County, Taiwan. Overview. Motivation Method of Particular Solutions (MPS) Particular solutions of polyharmonic spline Numerical example I

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Particular solutions for some engineering problems

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

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

  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. Motivation and Literature review

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

  6. Motivation

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

  9. Method of particular solutions

  10. Method of particular solutions (basis functions)

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

  12. Example

  13. Example

  14. Other examples Stokes flow Thermal Stokes flow

  15. Other examples Thick plate Solid deformation

  16. Remark Particular solutions for product operator Particular solutions for engineering problems Hörmander operator decomposition technique

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

  18. Partial fraction decomposition (Theorem)

  19. Partial fraction decomposition (Proof 1)

  20. Partial fraction decomposition (Proof 2)

  21. Example (1)

  22. Example (2)

  23. Remark Partial fraction decomposition

  24. Particular solutions of polyharmonic spline (APS)

  25. Particular solutions of polyharmonic spline (APS)

  26. Particular solutions of polyharmonic spline (Definition)

  27. Particular solutions of polyharmonic spline (Generating Theorem)

  28. Particular solutions of polyharmonic spline (Generating Theorem)

  29. Particular solutions of polyharmonic spline (Generating Theorem)

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

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

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

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

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

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

  36. Particular solutions of polyharmonic spline (Limit Behavior)

  37. Particular solutions of polyharmonic spline (Limit Behavior)

  38. Numerical example I

  39. Numerical example I

  40. Numerical example I (BC)

  41. Numerical example I (BC)

  42. Numerical example I (MFS)

  43. Numerical example I (results)

  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. Particular solutions of Chebyshev polynomials (why Chebyshev)

  46. Chebyshev interpolation (1)

  47. Chebyshev interpolation (2)

  48. Particular solutions of Chebyshev polynomials

  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. Particular solutions of Chebyshev polynomials (polyharmonic)

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