A study on the method of fundamental solutions using the image concept

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A study on the method of fundamental solutions using the image concept The 33rd Conference of Theoretical and Applied Mechanics A study on the method of fundamental solutions using the image concept Jhen-Jyun Tsai Authors: Jhen-Jyun Tsai, Hung-Chih Shieh, Jia-Wei Lee and Jeng-Tzong Chen Department of Harbor and River Engineering National Taiwan Ocean University pm 3:20~3:50, Nov. 13, 2009 行政大樓4樓第二會議室 2009/11/13

Outline Introduction Present method Numerical examples Conclusions 2009/11/13

Outline Introduction Present method Numerical examples Conclusions 2009/11/13

Method of Fundamental Solutions (MFS) This method was proposed by Kupradze in 1964. where Interior problem Exterior problem 2009/11/13

Two critical issues of MFS 1. Optimal source location Conventional MFS Good Not Good Alves & Antunes ? Image idea 2009/11/13

Two critical issues of MFS 2. Is a free constant necessary? Conventional MFS Saavedra and Power References: I. Saavedra and H. Power, “Multipole fast algorithm for the least-squares approach of the method of fundamental solutions for three-dimensional harmonic problems,” Numer Methods Partial Differ Equ, vol. 19, pp. 825-845, 2003. 2009/11/13

Outline Introduction Present method Numerical examples Conclusions 2009/11/13

2-D Degenerate kernel (addition theorem ) 2009/11/13

Image concept Mirror Dirichlet boundary condition Neumann boundary condition 2009/11/13

Image location Rigid body term u=0 u=0 a a (image) (True) (image) 2009/11/13

Outline Introduction Present method Numerical examples Conclusions 2009/11/13

Illustrative examples Analytical solution Semi-analytical solutions Numerical solution Image method Image method + MFS MFS Green’s function BVP 2009/11/13

Annular (analytical solution) Problem 1 Annular (analytical solution) B2 Governing equation : B1 a Fixed-free B.C. b 2009/11/13

Present method - MFS (Image method) Outside Inside Frozen points 2009/11/13

Analytical derivation Frozen points Analytical expression 2009/11/13

Numerical approach to determine c(N) and e(N) b 2009/11/13

The coefficients of c(N) and e(N) versus N Analytical expression c(N) 2009/11/13

Problem 2-A (annular → eccentric) Governing equation : b a Fixed-fixed B. C. B2 2009/11/13

Problem 2-A (annular → eccentric) Rc Rd Governing equation : B1 Fixed-fixed B.C. Frozen points B2 2009/11/13

The coefficients of c(N), d(N) and e(N) versus N 2009/11/13

(b) bipolar coordinates Contour (a) image method (b) bipolar coordinates (J.F. Heyda, J. Franklin Inst., 1959 ) 2009/11/13

Analytical derivation of location for the two frozen points b a 2009/11/13

Problem 2-B (eccentric → half plane) y Governing equation : B2 a x Fixed-fixed B. C. B1 2009/11/13

Problem 2-B (eccentric → half plane) y Governing equation : B2 a x Rc Rd=2b-Rc b Frozen points Fixed-fixed B. C. B1 2009/11/13

The coefficients of c(N), d(N) and e(N) versus N 2009/11/13

Null-field BIE approach A half-plane problem with an aperture subjected to the Dirichlet boundary condition Null-field BIE approach Ke-Hsun Chou Mech. Res. Comm. (2009) Melnikov’s method Melnikov and Melnikov CMES (2001) Image method 2009/11/13

Problem 3 (Green’s function → BVP) Governing equation : B2 fixed-fixed B. C. b=1 a=0.4 Two focuses B1 2009/11/13 27

Analytical solution a b d Reference: x (c, 0) a (-c, 0) b d Reference: N.N. Lebedev, I.P. Skalskaya and Y.S. Uflyand, Worked Problems in Applied Mathematics, Dover Publications, New York; 1979. 2009/11/13

Four cases Source Focus (b) Including two focuses (a) Including no focuses Source Focus (c) Including the left (outer) focus (d) Including the right (inner) focus 2009/11/13

Error distribution (without a free constant) (a) Including no focuses (b) Including two focuses (-0.5,0) (-2,0) (c) Including the left (outer) focus (-2,0) (d) Including the right (inner) focus (-0.5,0) 2009/11/13

Error distribution (add a free constant ) The best (a) Including no focuses (b) Including two focuses (-0.5,0) (-2,0) (d) Including the right (inner) focus (-0.5,0) (c) Including the left (outer) focus (-2,0) 2009/11/13

Compare the unknown coefficients for two approach Rigid body term c0=2.000 (B=2.000) d16=1.441452056277 (1.442695068254) c16=1.442695389383 (1.442695068254) 0.19236047619 (－) d1 d16 c1 c16 d1=－1.250342510472 (A－1.442695068254) c1= －1.442702295857 (A=－1.442695068254) －1.250342510472－ 0.19236047619 = －1.442702986 Analytical solution: ( ) 2009/11/13

Constant field u(x)=2 Uniform (0.19236047619) Constant field? 2009/11/13

Outline Introduction Present method Numerical examples Conclusions 2009/11/13

Conclusions An image method is an optimal MFS Image method x large N y a Image method large N (Optimal) MFS 2009/11/13

Conclusions Image locations terminate at the two focuses of the bipolar coordinates s s focus Eccentric annulus A half plane with a hole An infinite plane with double holes 2009/11/13

Conclusions A free constant is not so necessary since a uniform field can be superimposed by using external source distributions Including two focuses (-0.5,0) (-2,0) Including two focuses (-0.5,0) (-2,0) 2009/11/13

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/ 2009/11/13

Complex Variable Real Variable Present method Reference: Carrier & Pearson Muskhelishvili Chih-Bing Ling Timoshenko Present method r1 r2 ln r1-ln r2 = C Reference: J. T. Chen, M. H. Tsai and C. S. Liu, Comput Appl Eng Educ, 2009 2009/11/13

Bipolar coordinates x 2009/11/13

Frozen points of the image method and focuses in the bipolar coordinates 2009/11/13

Optimal location of MFS Depends on loading (image location) Depends on geometry (frozen image point) 2009/11/13

Final images to bipolar (bispherical) focus 2-D 3-D Bipolar coordinates Bispherical coordinates 2009/11/13

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) 2009/11/13

Alves & Antunes 2009/11/13

Closed-form Green’s function (Interior problem) Image point a 2009/11/13

2009/11/13

The simplest MFS 1-D Rod where U(x,s) is the fundamental solution. l l where U(x,s) is the fundamental solution. 2009/11/13

Optimal locations of MFS Frozen points (depends on geometry) Depends on the loading location 2009/11/13

Contour plot for the analytical solution (m=N). Numerical examples Image method Contour plot for the analytical solution (m=N). 2009/11/13

Boundary Element Method Numerical methods Numerical methods Finite Element Method Boundary Element Method Meshless Method 2009/11/13

Interpolation functions b 2009/11/13