碩士學位論文口試報告 Study on the Green’s functions for Laplace problems with circular and spherical boundaries by using the image method Reporter: H.C. Shieh Adviser: Dr. J.T. Chen Department of Harbor and River Engineering, National Taiwan Ocean University July 25, 2009

Motivation and literature review Two-dimensional Green’s function Frame Motivation and literature review Two-dimensional Green’s function MFS (Image method) Green’s function Trefftz method BVP without sources Conclusions

Boundary Element Method Numerical methods Numerical methods Finite Element Method Boundary Element Method Meshless Method

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

Optimal source location Not Good Good Conventional MFS Alves & Antunes ?

The simplest image method Mirror Neumann boundary condition Dirichlet boundary condition

Conventional method to determine the image location P r’ a r R’ A O B R Lord Kelvin(1824~1907) (1949, 相似三角形) a O R’ R P Greenberg (1971, 取巧法)

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

2-D Degenerate kernal References: W. C. Chen, A study of free terms and rigid body modes in the dual BEM, NTOU Master Thesis, 2001. C. S. Wu,Degenerate scale analysis for membrane and plate problems using the meshless method and boundary element method, NTOU Master Thesis, 2004

Addition theorem & degenerate kernel Subtraction theorem Degenerate kernel for Laplace problem 1-D 2-D x s

3-D degenerate kernel s x interior x exterior

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 Boundary value problem without source Conclusions

Eccentric annulus Case 1 Governing equation: B2 B1 a Dirichlet boundary condition: b d

Eccentric problem Case 2 Governing equation: Dirichlet boundary condition: b d

Half plane with circular hole problem Case 3 Governing equation: u2=0 Dirichlet boundary condition: B1 u1=0 B2

Bipolar coordinates x

Bipolar coordinates focus Eccentric annulus A half plane with a hole An infinite plane with double holes

Annular (EABE, 2009) to eccentric case Image point + - Source point s5 s6 …. s4 s2 …. s3 s1 s

Series of images The final images sc1 sc2

Numerical approach to determine c1(N), c2(N) and e(N) Coefficients sc1 sc2

Contour plot of eccentric annulus problem Dirichlet boundary for the eccentric case 1 Image method Analytical solution (bipolar coordinates )

Analytical derivation of location for the two frozen points

Eccentric case True source Image sources

Contour plot of eccentric annulus Image method Null-field BIE approach (addition theorem and superposition technique)

A half plane with a circular hole

Contour plot of half plane problem Image method Null-field BIE approach (addition theorem and superposition technique)

Linking of MFS and image method MFS (special case) Conventional MFS s

Image method versus MFS Conventional MFS All the strength need to be determined. Only three coefficients are required to be determined. large

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 Boundary value problem without sources Conclusions

Trefftz method The method was proposed by Trefftz in 1926. is the jth T-complete function Interior case Exterior case

, (T-complete function) Trefftz method and MFS Method Trefftz method MFS Definition Figure sketch 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

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

The weighting of the image source in the 3-D problem Interior problem Exterior problem z z a 1 y y 1 a x x

The image group Obtain image weighted Obtain image location

Interpolation functions b

Analytical derivation

Numerical solution a b

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

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

Boundary value problem Interior: Exterior: PART 2

PART 1 + PART 2 :

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

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 Boundary value problem without sources Conclusions

Trefftz solution Without loss of generality

Mathematical equivalence the Trefftz method and MFS Trefftz method series expand Image method series expand s s1 s3 s2 s4 s6 s8 s10 s s1 s3 s2 s4 s6 s8 s10 s s1 s2 s4 s3 s5 s9 s7 s s1 s2 s4 s3 s5 s9 s7

Equivalence of solutions derived by Trefftz method and image method (special MFS) True source Trefftz method MFS (image method) Equivalence addition theorem linkage

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 Boundary value problem without sources Conclusions

An infinite plane with two circular holes (anti-symmetric BC) d=10 y u=V=V1=-1 u=V=V2=1 2c a a x B1 d B2

Animation - An infinite plane with two circular holes sc1 sc2 s4 s2

Numerical approach to determine q(N), c1(N), c2(N) and e(N) Coefficients q(N)=e(N)=0

Contour plot of an infinite plane with two circular holes (antisymmetric case) Image solution bipolar coordinates null-field BIEM

An infinite space with two cavities (anti-symmetric BC) z u=V1=-1 u=V2=1 a=1.0 d=5.0 y B1 B2 x

Numerical approach to determine q(N), c1(N) and c2(N) Coefficients

Contour plot of an infinite space with two spherical cavities y z x x-y plane Bispherical coordinates Image method Null-field BIE

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 Boundary value problem without sources Conclusions

Optimal location of MFS Depends on loading (image location) Depends on geometry (frozen image point)

Final images to bipolar (bispherical) focus 2-D 3-D Bipolar coordinates Bispherical coordinates

Equivalence of Trefftz method and MFS MFS (image method)

Image solution for BVP without sources y x y x

Thanks for your kind attentions The end Thanks for your kind attentions You can get more information from our website http://msvlab.hre.ntou.edu.tw/

A half plane with a circular hole 2007, Ke J. N. 2009, Image method a b

An infinite plane with two circular holes subject to Neumann boundary

Extra terms of complementary solutions Source point Frozen point Two complementary solutions

The method provide of JW. Lee Frozen point d1 d2 complementary solutions

The total potential Where the N=M

Null-field BIE approach (addition theorem and superposition technique) Results Image method) Null-field BIE approach (addition theorem and superposition technique)

Conclusions The analytical solutions derived by 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

Numerical examples 1: Eccentric annulus b Image method (50+2 points)

Numerical examples 3: An infinite plane with double holes Image method (20+4+10 point)

Animation- an infinite plane with double holes Multipole expansion t2=0 t1=0 and Multipoles The final images terminate at the focus

Equivalence of solutions derived by Trefftz method and MFS Trefftz solution The same Image solution

The simplest MFS 1-D Rod l where U(x,s) is the fundamental solution.

Present method- MFS (Image method) … …

An infinite space with two cavities (symmetric BC) Obtain image location z u=V1=1 u=V2=1 a a q q y d Obtain image weighting B1 B2 x

The strength of two frozen points and q(N) The strength of c1(N), c2(N) and q(N)

Contour plot of an infinite space with two spherical cavities (symmetric case) z x x-y plane 3-D Bipolar coordinates Bispherical coordinates Image method Null-field BIE

Illustrative examples – An eccentric annulus u=V1=0 u=V2=1 B1 B2 c r1 r2

Numerical approach to determine q(N), c1(N), c2(N) and e(N) e(N)=2 (exact) c1(N)=1.44 (exact) q(N)=0 (exact) c2(N)=-1.44 (exact)

Contour plot of eccentric annulus Image solution bipolar coordinates null-field BIEM

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 Boundary value problem without source Conclusions

Optimal source location Not good Good Conventional MFS Alves CJS & Antunes PRS

MFS-Image location and weighting-interior (Chen and Wu, 2006) z u=0 y 1 b The weighting of the image point x

MFS-Image location and weighting-exterior (Chen and Wu, 2006) 1 The weighting of the image point

Chen and Wu-image method (2006)

Analytical derivation of location for the two frozen points b a x y a=1, b=3 (0.171 & 5.828)

Numerical approach to determine c1(N), c2(N) and e(N)

Analytical derivation of location for the two frozen points b 2c