November, 28-29, 2008 p.1 A linkage of Trefftz method and method of fundamental solutions for annular Green’s functions using addition theorem Shiang-Chih Shieh Authors: Shiang-Chih Shieh, Ying-Te Lee, Shang-Ru Yu and Jeng-Tzong Chen Department of Harbor and River Engineering, National Taiwan Ocean University Nov.28, 2008 The 32nd Conference on Theoretical and Applied Mechanics

November, 28-29, 2008 p.2 Outline Introduction Problem statements Present method  MFS (image method)  Trefftz method Equivalence of Trefftz method and MFS Numerical examples Conclusions

November, 28-29, 2008 p.3 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 exterior problem:

November, 28-29, 2008 p.4 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

November, 28-29, 2008 p.5 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

November, 28-29, 2008 p.6 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 Not good Good

November, 28-29, 2008 p.7 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

November, 28-29, 2008 p.8 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 … …

November, 28-29, 2008 p.9 b a MFS-Image group 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

November, 28-29, 2008 p.10 MFS-Image group 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

November, 28-29, 2008 p.11 Analytical derivation 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

November, 28-29, 2008 p.12 Numerical solution 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions a b

November, 28-29, 2008 p.13 Interpolation functions a b 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

November, 28-29, 2008 p.14 Trefftz Method PART 1 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

November, 28-29, 2008 p.15 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

November, 28-29, 2008 p.16 PART 1 + PART 2 : 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

November, 28-29, 2008 p.17 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

November, 28-29, 2008 p.18 The same 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Equivalence of solutions derived by Trefftz method and MFS

November, 28-29, 2008 p.19 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

November, 28-29, 2008 p.20 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

November, 28-29, 2008 p.21 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

November, 28-29, 2008 p.22 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

November, 28-29, 2008 p.23 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

November, 28-29, 2008 p.24 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.

November, 28-29, 2008 p.25 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

November, 28-29, 2008 p.26 Optimal location of MFS Depends on loading Depends on geometry 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz and MFS 5.Numerical examples 6.Conclusions

November, 28-29, 2008 p.27 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

November, 28-29, 2008 p.28 Conclusions Optimal image group points depend on loading Frozen image point depends on geometry 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz and MFS 5.Numerical examples 6.Conclusions

November, 28-29, 2008 p.29 Thanks for your kind attentions You can get more information from our website The 32nd Conference on Theoretical and Applied Mechanics