1. October, 13-17, 2008 p.1 海大陳正宗終身特聘教授 A linkage of Trefftz method and method of fundamental solutions for annular Green’s functions using addition theorem Jeng-Tzong Chen Department of Harbor and River Engineering, National Taiwan Ocean University 08:00-08:20, Oct. 15, 2008 ICCES Special Symposium on Meshless & Other Novel Computational Methods in Suzhou, China

2. October 13-17, 2008 p.2 海大陳正宗終身特聘教授 Outline Introduction Problem statements Present method  MFS (image method)  Trefftz method Equivalence of Trefftz method and MFS Numerical examples Conclusions

3. October 13-17, 2008 p.3 海大陳正宗終身特聘教授 Trefftz method 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions is the j th T-complete function Interior problem: exterior problem:

4. October 13-17, 2008 p.4 海大陳正宗終身特聘教授 MFS 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Interior problem exterior problem

5. October 13-17, 2008 p.5 海大陳正宗終身特聘教授 Trefftz method and MFS MethodTrefftz methodMFS Definition Figure caption Base, (T-complete function), r=|x-s| G. E. Match B. C.Determine c j Determine w j D u(x) s D r is the number of complete functions is the number of source points in the MFS 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

6. October 13-17, 2008 p.6 海大陳正宗終身特聘教授 Optimal source location 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions MFS (special case) Image method Conventional MFS Alves CJS & Antunes PRS

7. October 13-17, 2008 p.7 海大陳正宗終身特聘教授 Problem statements a b Governing equation : BCs: 1.fixed-fixed boundary 2.fixed-free boundary 3.free-fixed boundary 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

8. October 13-17, 2008 p.8 海大陳正宗終身特聘教授 Present method- MFS (Image method) 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

9. October 13-17, 2008 p.9 海大陳正宗終身特聘教授 MFS-Image group 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

10. October 13-17, 2008 p.10 海大陳正宗終身特聘教授 Analytical derivation 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

11. October 13-17, 2008 p.11 海大陳正宗終身特聘教授 Numerical solution 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions a b

12. October 13-17, 2008 p.12 海大陳正宗終身特聘教授 Interpolation functions a b 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

13. October 13-17, 2008 p.13 海大陳正宗終身特聘教授 Trefftz Method PART 1 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

14. October 13-17, 2008 p.14 海大陳正宗終身特聘教授 Boundary value problem PART 2 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

15. October 13-17, 2008 p.15 海大陳正宗終身特聘教授 PART 1 + PART 2 : 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

16. October 13-17, 2008 p.16 海大陳正宗終身特聘教授 Equivalence of solutions derived by Trefftz method and MFS 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Equivalence Trefftz method Image method

17. October 13-17, 2008 p.17 海大陳正宗終身特聘教授 Equivalence of Trefftz method and MFS The same 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Trefftz method series expand Image method series expand

18. October 13-17, 2008 p.18 海大陳正宗終身特聘教授 Equivalence of solutions derived by Trefftz method and MFS 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Trefftz methodMFS Equivalence addition theorem

19. October 13-17, 2008 p.19 海大陳正宗終身特聘教授 Numerical examples-case 1 (a) Trefftz method (b) Image method Contour plot for the analytical solution (m=N). fixed-fixed boundary 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions m=20N=20

20. October 13-17, 2008 p.20 海大陳正宗終身特聘教授 Numerical examples-case 2 (a) Trefftz method (b) Image method Contour plot for the analytical solution (m=N). fixed-free boundary 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions m=20 N=20

21. October 13-17, 2008 p.21 海大陳正宗終身特聘教授 Numerical examples-case 3 (a) Trefftz method (b) Image method Contour plot for the analytical solution (m=N). free-fixed boundary 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions m=20 N=20

22. October 13-17, 2008 p.22 海大陳正宗終身特聘教授 Numerical and analytic ways to determine c(N) and d(N) Values of c(N) and d(N) for the fixed-fixed case. 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

23. October 13-17, 2008 p.23 海大陳正宗終身特聘教授 Numerical examples- convergence 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Pointwise convergence test for the potential by using various approaches.

24. October 13-17, 2008 p.24 海大陳正宗終身特聘教授 Numerical examples- convergence rate Image method Trefftz method Conventional MFS 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Best Worst

25. October 13-17, 2008 p.25 海大陳正宗終身特聘教授 Conclusions The analytical solutions derived by the Trefftz method and MFS were proved to be mathematically equivalent for the annular Green’s functions. We can find final two frozen image points (one at origin and one at infinity). Their singularity strength can be determined numerically and analytically in a consistent manner. Convergence rate of Image method(best), Trefftz method and MFS(worst) due to optimal source locations in the image method 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz and MFS 5.Numerical examples 6.Conclusions

26. October, 13-17, 2008 p.26 海大陳正宗終身特聘教授 Thanks for your kind attentions You can get more information from our website http://msvlab.hre.ntou.edu.tw/ The 32nd Conference on Theoretical and Applied Mechanics