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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
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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
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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:
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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
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, (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
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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
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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
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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
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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
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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)
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MFS-Image group
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MFS-Image group
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Analytical derivation
Introduction Problem statements Analytical solution Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions Analytical derivation
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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
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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
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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
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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
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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 :
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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
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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
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Image method series expansion
Trefftz series expansion
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Image method series expansion
Trefftz series expansion
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Image method series expansion
Trefftz series expansion
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Image method series expansion
Trefftz series expansion
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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
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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
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MFS-Image group a Introduction Problem statements Analytical solution
Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions a
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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)
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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)
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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
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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
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Matching BCs to determine three coefficient
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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)
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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
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MFS-Image group Introduction Problem statements Analytical solution
Equivalence of Trefftz method and MFS Semi-analytical solution Numerical examples Conclusions
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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
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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)= c(N)=
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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).
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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
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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 )
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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.
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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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Thanks for your kind attentions
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