1. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU1 A study on half-plane Laplace problems with a circular hole 日期： 2004/08/22-24 地點：桃園大溪 報告者：沈文成

2. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU2 Outlines Motivations Boundary integral equations Degenerate kernels Numerical examples Conclusions

3. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU3 Motivations Heat conduction Pipe design Tunnels

4. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU4 Drawbacks of available methods Methodsdrawback FEMmesh BEM (truncated boundary) mesh BEM (image method) mesh MFS (Single-layer)node MFS (Double-layer)node

5. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU5 Boundary integral equations null-field integral equation x ： source point s ： field point

6. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU6 Degenerate kernels x ： source point ； s ： field point s x O R s r x

7. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU7 Image concept interior integral equation null-field integral equation

8. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU8 Degenerate kernels

9. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU9 Modified degenerate kernels homogeneous Neumann boundary condition homogeneous Dirichlet boundary condition

10. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU10 Present method Degenerate kernelFourier series Null-field equation Algebraic equation Fourier Coefficients Potential Analytical Numerical

11. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU11 Collocation points By choosing M terms of Fourier series, we have 2M+1 collocation points on the circle. 2M+1 terms

12. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU12 Problem statements Dirichlet Mixed-type

13. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU13 Boundary flux on the circle

14. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU14 Illustrations (Dirichlet) MFS (Double-layer potential method) BEM with truncated boundary Present method MFS (Single-layer potential method) BEM using image method Analytical solution (Lebedev)

15. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU15 Illustrations (Mixed-type) MFS (Double-layer potential method) BEM with truncated boundary Present method MFS (Single-layer potential method) BEM using image method Analytical solution (Lebedev)

16. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU16 Conclusions A novel method using degenerate kernels, Fourier series expansion and null-field equation has been developed. Boundary flux were obtained to converge to exact solution for only ten terms of Fourier series. The results of present method agree with analytical solution of Lebedev.

17. The 7th National Conference on Structure EngineeringMSVLAB, HRE, NTOU17 The End Thanks for your attentions. You can get more information on our website. http://msvlab.hre.ntou.edu.tw