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Reporter: H.C. Shieh Adviser: Dr. J.T. Chen Department of Harbor and River Engineering,

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## Reporter: H.C. Shieh Adviser: Dr. J.T. Chen Department of Harbor and River Engineering,

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**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**Frame**Motivation and literature review Two-dimensional Green’s function MFS (Image method) Green’s Trefftz method function BVP without sources Conclusions**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**Conventional MFS**Alves & Antunes Optimal source location Not Good Good ?**The simplest image method**Mirror Neumann boundary condition Dirichlet boundary condition**P**P a a O R’ R O Conventional method to determine the image location P r’ a r R’ A B O Lord Kelvin(1824~1907) (1949,相似三角形) R Greenberg (1971,取巧法)**a**a Image location (Chen and Wu, 2006) u=0 Rigid body term 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**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 Trefftz method and MFS • Boundary value problem without source • Conclusions**Eccentric annulus**Case 1 Governing equation: B2 B1 a Dirichletboundary condition: b d**Eccentric problem**Case 2 Governing equation: B2 B1 a Dirichletboundary condition: b d**Half plane with circular hole problem**Case 3 Governing equation: u2=0 Dirichletboundary condition: B1 u1=0 B2**x**Bipolar coordinates**Bipolar coordinates**focus Eccentric annulus A half plane with a hole An infinite plane with double holes**+**Image point - s2 s6 s4 s1 s5 s3 Annular (EABE, 2009) to eccentric case …. Source point …. s**The final images**sc1 sc2 Series of images**sc1**sc2 Numerical approach to determine c1(N), c2(N) and e(N) Coefficients**Contour plot of eccentric annulus problem**Dirichlet boundary for the eccentric case 1 Image method Analytical solution (bipolar coordinates )**Eccentric case**True source Image sources**Contour plot of eccentric annulus**Image method Null-field BIE approach (addition theorem and superposition technique)**Contour plot of half plane problem**Image method Null-field BIE approach (addition theorem and superposition technique)**Linking of MFS and image method**Conventional MFS MFS (special case) s**Image method versus MFS**Image method 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 Trefftz method and MFS • 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**r**s u(x) u(x) D D Trefftz method and MFS 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**z z a 1 y y 1 a x x**The image group**Obtain image weighted Obtain image location**a**b Interpolation functions**a**b Numerical solution**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 2 PART 1 PART 1**Boundary value problem**Interior: Exterior: 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 Trefftz method and MFS • Boundary value problem without sources • Conclusions**Trefftz solution**Without loss of generality**s10**s10 s4 s4 s2 s2 s3 s3 s8 s8 s6 s6 s1 s1 s s s4 s4 s2 s2 s1 s1 s5 s5 s7 s7 s s s3 s3 s9 s9 Mathematical equivalencethe Trefftz method and MFS Trefftz method series expand Image method series expand**Equivalence of solutions derived by Trefftz method and image**method (special MFS) 3-D 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 Trefftz method and MFS • Boundary value problem without sources • Conclusions**An infinite plane with two circular holes (anti-symmetric**BC) a=1.0 d=10 y u=V=V1=-1 u=V=V2=1 2c a a x B1 d B2**s4**s2 s1 s3 sc1 sc2 Animation - An infinite plane with two circular holes u=1 u=-1**Numerical approach to determine q(N), c1(N), c2(N) and e(N)**q(N)=e(N)=0 Coefficients