Image method for the Green’s functions of annulus and half-plane Laplace problems Reporter: Shiang-Chih Shieh Authors: Shiang-Chih Shieh, Ying-Te Lee and Jeng-Tzong Chen Department of Harbor and River Engineering, National Taiwan Ocean University Oct.22, 2008

Outline Introduction Problem statements Analytical solution Method of Fundmental Solution (MFS) Trefftz method Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions

Trefftz method is the jth T-complete function Interior problem: Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Trefftz method is the jth T-complete function Interior problem: exterior problem:

Method of Fundamental Solution (MFS) Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Method of Fundamental Solution (MFS) exterior problem Interior problem

, (T-complete function) Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Trefftz method and MFS Method Trefftz method MFS Definition Figure caption Base , (T-complete function) , r=|x-s| G. E. Match B. C. Determine cj Determine wj s D u(x) r D u(x) is the number of complete functions is the number of source points in the MFS

Optimal source location Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Optimal source location Conventional MFS Alves CJS & Antunes PRS MFS (special case) Image method

Problem statements Case 1 Annular Governing equation : Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Problem statements Case 1 Annular a b Governing equation : t1=0 u2=0 Boundary condition : Fixed-Free boundary

Problem statements Case 2 half-plane problem Governing equation : Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Problem statements Case 2 half-plane problem Governing equation : u2=0 Dirichlet boundary condition : u1=0

Problem statements Case 3 eccentric problem Governing equation : Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Problem statements Case 3 eccentric problem Governing equation : u1=0 u2=0 a Dirichlet boundary condition : b e

Present method- MFS (Image method) Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Present method- MFS (Image method)

MFS-Image group

MFS-Image group

Analytical derivation Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Analytical derivation

Numerical solution t1=0 u2=0 a b Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Numerical solution t1=0 u2=0 a b

Numerical and analytic ways to determine c(N) and e(N) Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Numerical and analytic ways to determine c(N) and e(N) e(N)=-0.1 c(N)=-0.159

Trefftz Method PART 1 Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Trefftz Method PART 1

Boundary value problem Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Boundary value problem PART 2

PART 1 + PART 2 : Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions PART 1 + PART 2 :

Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Equivalence of solutions derived by using Trefftz method and MFS for annular problem MFS(Image method) The same Trefftz method

Trefftz method series expand Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Trefftz method series expand Without loss of generality

Image method series expansion Trefftz series expansion

Image method series expansion Trefftz series expansion

Image method series expansion Trefftz series expansion

Image method series expansion Trefftz series expansion

Equivalence of solutions derived by Trefftz method and MFS Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Equivalence of solutions derived by Trefftz method and MFS Trefftz method MFS Equivalence addition theorem

Semi-analytical solution-case 2 Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Semi-analytical solution-case 2 a b

MFS-Image group a Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions a

Successive images (20 points) Frozen image location Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions (0,-0.171) (frozen) (0,-5.828) (frozen) Successive images (20 points)

Analytical derivation of location for the two frozen points Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Analytical derivation of location for the two frozen points b a x y (0.171 & 5.828)

Series of images frozen The final two frozen images frozen Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions frozen a b The final two frozen images frozen

Rigid body term Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Rigid body term a b

Matching BCs to determine three coefficient

Numerical approach to determine c(N), d(N) and e(N) Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Numerical approach to determine c(N), d(N) and e(N)

Semi-analytical solution-case 3 Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Semi-analytical solution-case 3 y e x a b u2=0 u1=0

MFS-Image group Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions

Analytical derivation of location for the two frozen points Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Analytical derivation of location for the two frozen points e a b

Numerical approach to determine c(N), d(N) and e(N) Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Numerical approach to determine c(N), d(N) and e(N) d(N)=-0.1375 c(N)=-0.8624

Numerical examples-case 1 Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Numerical examples-case 1 Fixed-Free boundary for annular case m=20 (a) Trefftz method N=20 (b) Image method Contour plot for the analytical solution (m=N).

Numerical examples-case 2 Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Numerical examples-case 2 Dirichlet boundary for half-plane case Present method-image Null-field BIE approach (addition theorem and superposition technique) (M=50) 40+2 points

Numerical examples-case 3 Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Numerical examples-case 3 Dirichlet boundary for eccentric case image method analytical solution (bi-polar coordinate )

Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions 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. The image method can be seen as a special case for method of fundamental solution with optimal locations of sources.

Image method versus MFS Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Image method versus MFS large

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