Nation Taiwan Ocean University Department of Harbor and River June 18, 2015 p. 1 Null-field equation approach as a tool for computing Green ’ s function for Laplace operator with circular holes and/or inclusions Jeng-Tzong Chen Taiwan Ocean University 6G-4, 11:00-11:20, April 6, 2007 ICCM 2007, Hiroshima, Japan (ICCM2007.ppt)

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 2 Outlines  Motivation and literature review  Derivation of the Green ’ s function Null-field integral approach to construct the Green ’ s function Expansion of kernel function and boundary density Adaptive observer system Linear algebraic equation  Numerical examples Annular problem Eccentric ring A half -plane with an aperture (1) Dirichlet boundary (2) Robin boundary A half-plane problem with a circular hole and a half-circular inclusion  Conclusions

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 3 Outlines  Motivation and literature review  Derivation of the Green ’ s function Null-field integral approach to construct the Green ’ s function Expansion of kernel function and boundary density Adaptive observer system Linear algebraic equation  Numerical examples Annular problem Eccentric ring A half -plane with an aperture (1) Dirichlet boundary (2) Robin boundary A half-plane problem with a circular hole and a half-circular inclusion  Conclusions

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 4 Motivation  Green ’ s function has been studied and applied in many fields in recent years.  For complicated domain, series form Green ’ s function as well as closed-form is not easy to obtain.  To search a systematic method for Green ’ s function with several circular boundaries is not trivial.

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 5 Literature review  Analycal Greeen ’ s function Jaswon and Symm (1977) Melnikov (1977)  Numerical Greeen ’ s function Telles et al. (1995) Melnikov (2001)

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 6 Outlines  Motivation and literature review  Derivation of the Green ’ s function Null-field integral approach to construct the Green ’ s function Expansion of kernel function and boundary density Adaptive observer system Linear algebraic equation  Numerical examples Annular problem Eccentric ring A half -plane with an aperture (1) Dirichlet boundary (2) Robin boundary A half-plane problem with a circular hole and a half-circular inclusion  Conclusions

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 7 Null-field integral approach to construct the Green ’ s function Governing equation: Fundamental solution: Boundary condition: Governing equation: Original Problem Auxiliary system Green’s third identity BIE for Green’s function Subjected to given B. C.

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 8 Expansions of fundamental solution (2D)  Laplace problem-- Where O s x x

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 9 Boundary density discretization  Fourier series expansions - boundary density Fourier series Ex. constant element

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 10 Adaptive observer system Source point collocation point

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 11 Linear algebraic equation

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 12 Flowchart of present method Degenerate kernel Fourier series Null-field equation Algebraic system Fourier Coefficients Potential Analytical Numerical

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 13 Outlines  Motivation and literature review  Derivation of the Green ’ s function Null-field integral approach to construct the Green ’ s function Expansion of kernel function and boundary density Adaptive observer system Linear algebraic equation  Numerical examples Annular problem Eccentric ring A half -plane with an aperture (1) Dirichlet boundary (2)Robin boundary A half-plane problem with a circular hole and a half-circular inclusion  Conclusions

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 14 Annular problem subjected to a concentrated force a b 0u  Degenerate kernel Fourier series 0u 

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 15 Null-field integral equation

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 16 The explicit form for the unkown Fourier series can be obtained as

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 17 Analytic form for annular problem subjected to a concentrated force

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 18 Result of annular problem subjected to a concentrated force Green’s function for annular case M=50 (analytical solution) Green’s function for annular case M=50 (semi-analytical solution)

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 19 Outlines  Motivation and literature review  Derivation of the Green ’ s function Null-field integral approach to construct the Green ’ s function Expansion of kernel function and boundary density Adaptive observer system Linear algebraic equation  Numerical examples Annular problem Eccentric ring A half -plane with an aperture (1) Dirichlet boundary (2) Robin boundary A half-plane problem with a circular hole and a half-circular inclusion  Conclusions

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 20 Eccentric ring Present method Melnikov’s method Eccentric ring

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 21 Outlines  Motivation and literature review  Derivation of the Green ’ s function Null-field integral approach to construct the Green ’ s function Expansion of kernel function and boundary density Adaptive observer system Linear algebraic equation  Numerical examples Annular problem Eccentric ring A half -plane with an aperture (1) Dirichlet boundary (2) Robin boundary A half-plane problem with a circular hole and a half-circular inclusion  Conclusions

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 22 A Half plane with an aperture subjected to Dirichlet bound condition Present method Melnikov’s method Dirichlet bound condition

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 23 A Half plane with an aperture subjected to Robin bound condition Robin bound condition Melikov's approach Present method

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 24 Outlines  Motivation and literature review  Derivation of the Green ’ s function Null-field integral approach to construct the Green ’ s function Expansion of kernel function and boundary density Adaptive observer system Linear algebraic equation  Numerical examples Annular problem Eccentric ring A half -plane with an aperture (1) Dirichlet boundary (2) Robin boundary A half-plane problem with a circular hole and a half-circular inclusion  Conclusions

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 25 A half-plane problem with a circular hole and a half-circular inclusion Half -plane Inclusion Matrix A circular hole Source of Green’s function Material conductivity

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 26 A half plane →A full plane

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 27 Take free body Interface boundary

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 28 Linear algebraic equation where represent the material conductivity of matrix and inclusion, respectively. and

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 29 Result of a half-plane problem with a circular hole and a half-circular inclusion Contour plot by using the null-field integral equation approach Contour plot by using Melikov's approach (2006)

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 30 Outlines  Motivation and literature review  Derivation of the Green ’ s function Null-field integral approach to construct the Green ’ s function Expansion of kernel function and boundary density Adaptive observer system Linear algebraic equation  Numerical examples Annular problem Eccentric ring A half -plane with an aperture (1) Dirichlet boundary (2) Robin boundary A half-plane problem with a circular hole and a half-circular inclusion  Conclusions

Mechanics Sound Vibration Laboratory HRE, NTOU June 18, 2015 p. 31 Conclusions  A semi-analytical Green ’ s function was successfully derived using null-field integral equation.  Several cases, including the annular, eccentric, half-plane problems with circular cavity as well as inclusion were demonstrated to show the validity of our formulation.  A general-purpose program for constructing Green ’ s function of Laplace problems with circular boundaries was implemented.

Nation Taiwan Ocean University Department of Harbor and River June 18, 2015 p. 32 Thanks your kind attentions You can get more information on our website.