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Equivalence of Trefftz Method and MFS for Laplace Equation

Explore the connection and numerical examples of Trefftz method and Method of Fundamental Solutions for Laplace equation, with further research.

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Equivalence of Trefftz Method and MFS for Laplace Equation

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  1. Meshless Method J. T. Chen Department of Harbor and River Engineering National Taiwan Ocean University May 6, 2003 1

  2. Topics • Part 1: • Equivalence of method of fundamental • solutions and Trefftz method • Part 2: • Membrane eigenproblem • Part 3: • Plate eigenproblem 2

  3. Part 1 • Description of the Laplace problem • Trefftz method • Method of fundamental solutions (MFS) • Connection between the Trefftz method and the MFS for Laplace equation • Numerical examples • Concluding remarks • Further research 3

  4. Description of the Laplace problem • Trefftz method • Method of fundamental solutions (MFS) • Connection between the Trefftz method • and the MFS for Laplace equation • Numerical examples • Concluding remarks • Further research 4

  5. Description of the Laplace problem Engineering applications: • Seepage problem • Heat conduction • Electrostatics • Torsion bar 5

  6. D a B denotes the Laplacian operator is the potential function is the radius of the field point is the angle along the field point Two-dimensional Laplace problem with a circular domain G.E. : B.C. : where 6

  7. Analytical solution where are the Fourier coefficients Field Solution: where Boundary Condition: Dirichlet type 7

  8. Description of the Laplace problem • Trefftz method • Method of fundamental solutions (MFS) • Connection between the Trefftz method • and the MFS for Laplace equation • Numerical Examples • Concluding remarks • Further research 8

  9. Trefftz method where is the number of complete functions is the unknown coefficient is the T-complete function which satisfies the Laplace equation Representation of the field solution : 9

  10. T-complete set T-complete set functions : 10

  11. By matching the boundary condition at 11

  12. Description of the Laplace problem • Trefftz method • Method of fundamental solutions (MFS) • Connection between the Trefftz method • and the MFS for Laplace equation • Numerical Examples • Concluding remarks • Further research 12

  13. Method of Fundamental Solutions Field solution : where R is the fundamental solution is the number of source points in the MFS is the complementary domain is the unknown coefficient B B’ is the source point is the collocation point 13

  14. Green’s function Wmeans Wronskin determinant 14

  15. Degenerate kernel : Symmetry property for kernel : 15

  16. Derivation of degenerate kernel Use the Complex Variablemethod to derive the degenerate kernel: Motivation: 16

  17. Derivation of degenerate kernel x Not important s Due to: 17

  18. By matching the boundary condition 18

  19. Description of the Laplace problem • Trefftz method • Method of fundamental solutions (MFS) • Connection between the Trefftz method • and the MFS for Laplace equation • Numerical Examples • Concluding remarks • Further research 19

  20. On the equivalence of Trefftz method and MFS for Laplace equation We can find that the T-complete functions of Trefftz method are imbedded in the degenerate kernels of MFS : MFS: Trefftz: 20

  21. By setting Trefftz MFS 21

  22. in which where 22

  23. 23

  24. Matrix 24

  25. Matrix ill-posed problem Degenerate scale problem 25

  26. Papers of degenerate scale (Taiwan) • J. T. Chen, S. R. Kuo and J. H. Lin, 2002, Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two-dimensional elasticity,Int. J. Numer. Meth. Engng., Vol.54, No.12, pp.1669-1681. (SCI and EI) • J. T. Chen, C. F. Lee, I. L. Chen and J. H. Lin, 2002 An alternative method for degenerate scale problem in boundary element methods for the two-dimensional Laplace equation,Engineering Analysis with Boundary Elements, Vol.26, No.7, pp.559-569. (SCI and EI) • J. T. Chen, J. H. Lin, S. R. Kuo and Y. P. Chiu, 2001, Analytical study and numerical experiments for degenerate scale problems in boundary element method using degenerate kernels and circulants, Engineering Analysis with Boundary Elements, Vol.25, No.9,pp.819-828. (SCI and EI) • J. T. Chen, S. R. Lin and K. H. Chen, 2003, Degenerate scale for Laplace equation using the dual BEM, Int. J. Numer. Meth. Engng, Revised. 26

  27. Papers of degenerate scale (China) • 胡海昌, 平面調和函數的充要的邊界積分方程,1992,中國科學學報, Vol.4, pp.398-404 • 胡海昌, 調和函數邊界積分方程的充要條件, 1989, 固體力學學報, Vol.2, No.2, pp.99-104 • W. J. He, H. J. Ding and H. C. Hu, Nonuniqueness of the conventional boundary integral formulation and its elimination for two-dimensional mixed potential problems, Computers and Structures, Vol.60, No.6, pp.1029-1035, 1996. 27

  28. The efficiency between the Trefftz method and the MFS We propose an example for exact solution: MFS : NM<50 Trefftz method : NT=50 N < 101 terms N=101 terms 28

  29. Numerical Examples B y R=1 R=1 x D D B Exact solution: 1. Trefftz method for simply-connected problem 2. MFS for simply-connected problem 29

  30. Numerical Examples Y R=2.5 R=1 X 小圓半徑為1;大圓半徑為2.5 Exact solution: 1. Trefftz method for multiply-connected problem 2. MFS for multiply-connected problem 30

  31. Numerical Example 1 31

  32. Numerical Example 2 32

  33. Numerical Example 3 33

  34. Numerical Example 4 34

  35. Description of the Laplace problem • Trefftz method • Method of fundamental solutions (MFS) • Connection between the Trefftz method • and the MFS for Laplace equation • Numerical Examples • Concluding remarks • Further research 35

  36. Concluding Remarks • The proof of the mathematical equivalence between the Trefftz method and MFS for Laplace equation was derived successfully. • The T-complete set functions in the Trefftz method for interior and exterior problems are imbedded in the degenerate kernels of the fundamental solutions as shown in Table 1 for 1-D, 2-D and 3-D Laplace problems. • The sources of degenerate scale and ill-posed behavior in the MFS are easily found in the present formulation. • It is found that MFS can approach the exact solution more efficiently than the Trefftz method under the same number of degrees of freedom. 36

  37. Comparison between the Trefftz method and MFS 37

  38. Description of the Laplace problem • Trefftz method • Method of fundamental solutions (MFS) • Connection between the Trefftz method • and the MFS for Laplace equation • Numerical Examples • Concluding remarks • Further research 38

  39. Further research Presented Laplace problem (exterior) Laplace problem (interior) Helmholtz problem (exterior) Helmholtz problem (interior) Simply-connected Multiply-connected? Numerical examples ? 39

  40. The End Thanks for your kind attention 40

  41. Basis of the Laplace equation for Trefftz method Laplace Equation where 41

  42. Fundamental solution where 42

  43. Degenerate kernel (step1) x Step 1 x: variable s: fixed r S 43

  44. Degenerate kernel (Step 2, Step 3) Step 2 x Step 3 s s RA RB A A B B x 44

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