1. Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong Chen 以正規化無網格法求解含多孔洞拉普拉斯方程式 工學院 2005/04/01

2. 2 Outlines Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions

3. 3 Outlines Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions

4. 4 Motivation Numerical Methods Mesh Methods Finite Difference Method Meshless Methods Finite Element Method Boundary Element Method (MFS)(RMM)

5. 5 Outlines Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions

6. 6 Statement of problem Laplace equation with multiple holes : potential flow around cylinders electrostatic field of wires torsion bar with holes MZMZ

7. 7 Outlines Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions

8. 8 Method of fundamental solutions (MFS) Method of fundamental solutions (MFS) : Source point Collocation point — Physical boundary -- Off-set boundary d = off-set distance d Double-layer potential approach Single-layer Potential approach Dirichlet problem Neumann problem Dirichlet problem Neumann problem Distributed type

9. 9 The artificial boundary (off-set boundary) distance is debatable. The diagonal coefficients of influence matrices are singular when the source point coincides the collocation point.

10. 10 Outlines Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions

11. 11 Regularized meshless method (RMM) Source point Collocation point — Physical boundary Regularized meshless method (RMM) Double-layer potential approach Dirichlet problem Neumann problem where I = Inward normal vector O = Outward normal vector

12. 12 In a similar way,

13. 13

14. 14 Outlines Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation with multiple holes Numerical examples Conclusions

15. 15 Formulation with multiple holes Source point Collocation point — Physical boundary inner holes = m-1 outer hole = m th

16. 16 inner holes = m-1 outer hole = m th Source point Collocation point — Physical boundary P=1

17. 17 inner holes = m-1 outer hole = m th Source point Collocation point — Physical boundary

18. 18 inner holes = m-1 outer hole = m th Source point Collocation point — Physical boundary

19. 19 inner holes = m-1 outer hole = m th Source point Collocation point — Physical boundary P=m

20. 20 inner holes = m-1 outer hole = m th Source point Collocation point — Physical boundary P=m

21. 21 The linear algebraic systems s s x x

22. 22 Outlines Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions

23. 23 Numerical examples y x y x Case 1 Dirichlet B.C.Case 2 Mixed-type B.C.

24. 24 Contour of potential (case 1) Exact solutionRMM (360 points)BEM (360 elements)

25. 25 Contour of potential (case 2) Exact solutionRMM (400 points)BEM (800 elements)

26. 26 Error convergence (case 2)

27. 27 Outlines Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions

28. 28 Conclusions Only boundary nodes on the real boundary are required. Singularity of kernels is desingularized. The present results for multiply-hole cases were well compared with exact solutions and BEM.

29. 29 The end Thanks for your attention. Your comment is much appreciated.