1 Equivalence between the Trefftz method and the method of fundamental solutions for the Green’s function of concentric spheres using the addition theorem and image concept J.T. Chen Life-time Distinguished Professor Department of Harbor and River Engineering, National Taiwan Ocean University Sep. 2-4, 2009 New Forest, UK BEM/MRM 31

2 Outline Numerical methods Trefftz method and MFS Image method (special MFS) Trefftz method Equivalence of solutions derived by Trefftz method and MFS Conclusions

3 Numerical methods Boundary Element MethodFinite Element MethodMeshless Method

4 Method of fundamental solutions is the fundamental solution Interior caseExterior case This method was proposed by Kupradze in 1964.

5 Optimal source location Conventional MFSAlves CJS & Antunes PRS Not goodGood

6 Optimal source location Conventional MFSAlves & Antunes Good Not Good ?

7 The simplest image method Neumann boundary condition Dirichlet boundary condition Mirror

8 Conventional method to determine the image location R R’ O ar r’ P A B a a O R’ R O P P Lord Kelvin(1824~1907) Greenberg (1971 )

9 Image location using degenerate kernel (Chen and Wu, 2006) a a Rigid body term u=0

10 Degenerate kernel-2D (addition theorem)

11 Addition theorem & degenerate kernel Addition theorem Subtraction theorem Degenerate kernel for Laplace problem 1-D 2-D sx 3-D next page

12 3-D degenerate kernel s x exterior x interior

13 Trefftz method and MFS MethodTrefftz methodMFS Definition Figure sketch Base, (T-complete function), r=|x-s| G. E. Match B. C.Determine c j Determine w j D u(x)u(x) s D u(x)u(x) r is the number of complete functions is the number of source points in the MFS

14 Derivation of 3-D Green’s function by using the image method Interior problem Exterior problem

15 Weightings of the image source in the 3-D problem y z 1 a x y 1 a x z Interior problemExterior problem True source Image source

16 Weighting and locations of succesive images Weighting of successive images Location of successive images

17 Derivation of analytical solution using interpolation functions a b

18 Derivation of analytical solution using complementary solutions

19 Numerical approach by collocation BCs a b

20 Numerical and analytic ways to determine c(N) and d(N) Coefficients c(N) d(N)

21 Derivation of 3-D Green’s function by using the Trefftz Method PART 1 PART 2 PART 1

22 Boundary value problem Interior: Exterior: PART 2

23 PART 1 + PART 2 :

24 Results Trefftz method (x-y plane) Image method (x-y plane)

25 Outline Motivation and literature review Derivation of 2-D Green’s function by using the image method Trefftz method and MFS Image method (special MFS) Trefftz method Equivalence of solutions derived by Trefftz method and MFS Boundary value problem without sources Conclusions

26 Trefftz solution Without loss of generality

27 Mathematical equivalence the Trefftz method and MFS Trefftz method series expansion Image method series expansion s s1s1 s2s2 s4s4 s3s3 s5s5 s9s9 s7s7 s s1s1 s3s3 s2s2 s4s4 s6s6 s8s8 s 10 s s1s1 s2s2 s4s4 s3s3 s5s5 s9s9 s7s7 s s1s1 s3s3 s2s2 s4s4 s6s6 s8s8

28 Equivalence of solutions derived by Trefftz method and image method (special MFS) Trefftz methodMFS (image method) Equivalence Addition theorem Linkage 3-D True source

29 Equivalence of Trefftz method and MFS 3-D Trefftz method MFS (image method)

30 Conclusions The analytical solutions derived by using the Trefftz method and MFS were proved to be mathematically equivalent for Green’s functions of the concentric sphere. In the concentric sphere case, 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. It is found that final image points terminate at the two focuses of the bipolar (bispherical) coordinates for all the cases.

31 References J. T. Chen, Y. T. Lee, S. R. Yu and S. C. Shieh, 2009, Equivalence between Trefftz method and method of fundamental solution for the annular Green’s function using the addition theorem and image concept, Engineering Analysis with Boundary Elements, Vol.33, pp J. T. Chen and C. S. Wu, 2006, Alternative derivations for the Poisson integral formula, Int. J. Math. Edu. Sci. Tech, Vol.37, No.2, pp

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