Regularized meshless method for solving Laplace equation with multiple holes

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

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

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

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

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

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

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

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

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. Outlines • Motivation • Statement of problem • Method of fundamental solutions • Regularized meshless method • Formulationfor multiple holes • Numerical examples • Conclusions

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

12. In a similar way,

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

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

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

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

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

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

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

21. The linear algebraic systems s x s x

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

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

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

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

26. Error convergence (case 2)

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

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

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