April, 8-12, 2009 p.1 海大陳正宗終身特聘教授 Bipolar coordinates, image method and method of fundamental solutions Jeng-Tzong Chen Department of Harbor and River Engineering, National Taiwan Ocean University 15:30-15:50, April 10, 2009 ICCES 09 in Phuket, Thailand

April, 8-12, 2009 p.2 海大陳正宗終身特聘教授 Prof. Wen Hwa Chen 60th birthday symposium My Ph.D. Committee member

April 8-12, 2009 p.3 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 (2-D and 3-D annular cases) Numerical examples Conclusions

April 8-12, 2009 p.4 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 exterior problem:

April 8-12, 2009 p.5 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

April 8-12, 2009 p.6 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

April 8-12, 2009 p.7 海大陳正宗終身特聘教授 b a New point of view of image location instead of Kelvin concept (Chen and Wu, IJMEST 2006)

April 8-12, 2009 p.8 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 Collocation point Source point True source point

April 8-12, 2009 p.9 海大陳正宗終身特聘教授 Equivalence of solutions derived by Trefftz method and image method (special 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 2-D 3-D True source

April 8-12, 2009 p.10 海大陳正宗終身特聘教授 Key point Addition theorem (Degenerate kernel) 2-D 3-D

April 8-12, 2009 p.11 海大陳正宗終身特聘教授 Annular cases (Green’s functions) 2-D EABE 2009 x y z 3-D a b a b u 1 =0 u 2 =0 True source

April 8-12, 2009 p.12 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 (u=0) m=20N=20 10 terms 100 terms Melnikov and Arman, 2001 Computer unfriendly

April 8-12, 2009 p.13 海大陳正宗終身特聘教授 Trefftz solutionImage solution Annular circle Annular sphere Analytical solutions Addition theorem Addition theorem

April 8-12, 2009 p.14 海大陳正宗終身特聘教授 Analytical and numerical approaches to determine the strength 2-D3-D

April 8-12, 2009 p.15 海大陳正宗終身特聘教授 Bipolar coordinates Eccentric annulusA half plane with a holeAn infinite plane with double holes focus

April 8-12, 2009 p.16 海大陳正宗終身特聘教授 Animation – eccentric case The final images terminate at the focus

April 8-12, 2009 p.17 海大陳正宗終身特聘教授 Animation - a half plane with a circular hole The final images terminate at the focus

April 8-12, 2009 p.18 海大陳正宗終身特聘教授 Animation- an infinite plane with double holes The final images terminate at the focus Multipole expansion and Multipoles

April 8-12, 2009 p.19 海大陳正宗終身特聘教授 Eccentric annulus Image method (50+2 points) u 1 =0 u 2 =0 a b

April 8-12, 2009 p.20 海大陳正宗終身特聘教授 A half plane with a circular hole Image method (40+2 points) u 1 =0 u 2 =0 a h

April 8-12, 2009 p.21 海大陳正宗終身特聘教授 An infinite plane with double holes a b h Image method ( point) t 1 =0 t 2 =0

April 8-12, 2009 p.22 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 (2D and 3D) after using addition theorem (degenerate kernel). We can find final two frozen image points which are focuses in the bipolar coordinates. The image idea provides the optimal location of MFS and only at most 4 by 4 matrix is required. 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz and MFS 5.Numerical examples 6.Conclusions

April, 8-12, 2009 p.23 海大陳正宗終身特聘教授 Thanks for your kind attentions You can get more information from our website

April 8-12, 2009 p.24 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

April 8-12, 2009 p.25 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

April 8-12, 2009 p.26 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

April 8-12, 2009 p.27 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

April 8-12, 2009 p.28 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

April 8-12, 2009 p.29 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

April 8-12, 2009 p.30 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

April 8-12, 2009 p.31 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

April 8-12, 2009 p.32 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

April 8-12, 2009 p.33 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

April 8-12, 2009 p.34 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

April 8-12, 2009 p.35 海大陳正宗終身特聘教授 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

April 8-12, 2009 p.36 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

April 8-12, 2009 p.37 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

April 8-12, 2009 p.38 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

April 8-12, 2009 p.39 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.