1. 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
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2. Outline Introduction Present method Numerical examples Conclusions
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3. Outline Introduction Present method Numerical examples Conclusions
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4. Method of Fundamental Solutions (MFS)
This method was proposed by Kupradze in 1964.
where
Interior problem
Exterior problem
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5. Two critical issues of MFS
1. Optimal source location
Conventional MFS
Good
Not Good
Alves & Antunes ?
Image idea
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6. 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 , 2003.
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7. Outline Introduction Present method Numerical examples Conclusions
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8. 2-D Degenerate kernel (addition theorem )
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9. Image concept Mirror Dirichlet boundary condition
Neumann boundary condition
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10. Image location Rigid body term u=0 u=0 a a (image) (True) (image)
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11. Outline Introduction Present method Numerical examples Conclusions
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12. Illustrative examples
Analytical solution
Semi-analytical solutions
Numerical solution
Image method
Image method + MFS
MFS
Green’s function
BVP
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13. Annular (analytical solution)
Problem 1
Annular (analytical solution) B2
Governing equation : B1 a
Fixed-free B.C. b
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14. Present method - MFS (Image method)
Outside
Inside
Frozen points
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15. Analytical derivation
Frozen points
Analytical expression
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16. Numerical approach to determine c(N) and e(N)b
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17. The coefficients of c(N) and e(N) versus N
Analytical expression
c(N)
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18. Problem 2-A (annular → eccentric)
Governing equation : b a
Fixed-fixed B. C. B2
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19. Problem 2-A (annular → eccentric)Rc Rd
Governing equation : B1
Fixed-fixed B.C.
Frozen points B2
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20. The coefficients of c(N), d(N) and e(N) versus N
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21. (b) bipolar coordinates
Contour
(a) image method
(b) bipolar coordinates
(J.F. Heyda, J. Franklin Inst., )
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22. Analytical derivation of location for the two frozen pointsb a
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23. Problem 2-B (eccentric → half plane)y
Governing equation : B2 a x
Fixed-fixed B. C. B1
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24. Problem 2-B (eccentric → half plane)y
Governing equation : B2 a x Rc
Rd=2b-Rc b
Frozen points
Fixed-fixed B. C. B1
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25. The coefficients of c(N), d(N) and e(N) versus N
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26. 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
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27. Problem 3 (Green’s function → BVP)
Governing equation : B2
fixed-fixed B. C.
b=1
a=0.4
Two focuses B1
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28. 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.
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29. 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
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30. 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)
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31. 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)
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32. Compare the unknown coefficients for two approach
Rigid body term
c0=2.000
(B=2.000)
d16=
( )
c16=
( )
(－) d1
d16 c1
c16
d1=－
(A－ )
c1= －
(A=－ )
－ －
= －
Analytical solution: ( )
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33. Constant field u(x)=2 Uniform (0.19236047619) Constant field?
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34. Outline Introduction Present method Numerical examples Conclusions
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35. Conclusions An image method is an optimal MFS Image method x large Ny a
Image method
large N
(Optimal) MFS
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36. 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
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37. 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)
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38. Thanks for your kind attentions
The end
Thanks for your kind attentions
You can get more information from our website
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39. 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
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40. Bipolar coordinates x
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41. Frozen points of the image method and focuses in the bipolar coordinates
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42. Optimal location of MFS
Depends on loading (image location)
Depends on geometry (frozen image point)
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43. Final images to bipolar (bispherical) focus
2-D
3-D
Bipolar coordinates
Bispherical coordinates
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44. Conventional method to determine the image locationP 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)
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45. Alves & Antunes
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46. Closed-form Green’s function (Interior problem)
Image point a
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47. 2009/11/13

48. The simplest MFS 1-D Rod where U(x,s) is the fundamental solution. ll
where U(x,s) is the fundamental solution.
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49. Optimal locations of MFS
Frozen points
(depends on geometry)
Depends on
the loading location
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50. Contour plot for the analytical solution (m=N).
Numerical examples
Image method
Contour plot for the analytical solution (m=N).
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51. Boundary Element Method
Numerical methods
Numerical methods
Finite Element Method
Boundary Element Method
Meshless Method
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52. Interpolation functionsb
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