Derivation of the Greens function for Laplace and

Derivation of the Green’s function for Laplace and Helmholtz problems with circular boundaries by using the null-field integral equation approach Reporter : Ke J. N. Advisor : Chen J. T. Committee members : Chen I. L. , Lee W. M. , Leu S. Y. & Chen K, H. 95學年度第 2學期碩士論文口試 1

Outlines Motivation and literature review Derivation of the Green’s function n n n n Numerical examples n n Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Take free body Image technique for solving half-plane problems Green’s function for Laplace problems Green’s function for Helmholtz problems Conclusions National Taiwan Ocean University Department of Harbor and River Engineering 2

Outlines Motivation and literature review Derivation of the Green’s function n n n n Numerical examples n n Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Take free body Image technique for solving half-plane problems Green’s function for Laplace problems Green’s function for Helmholtz problems Conclusions National Taiwan Ocean University Department of Harbor and River Engineering 3

Motivation Numerical methods for engineering problems FDM / FEM / BIEM / Meshless method BEM / BIEM Treatment of singularity and hypersingularity Boundary-layer effect Convergence rate National Taiwan Ocean University Department of Harbor and River Engineering Ill-posed model 4

Engineering problem with arbitrary geometries Straight boundary (Legendre polynomial) (Fourier series) Degenerate boundary (Chebyshev polynomial) Circular boundary Elliptic boundary National Taiwan Ocean University Department of Harbor and River Engineering (Mathieu function) 5

Literature review Derivation of the Green’s function Successive iteration method Modified potential method Boley, 1956, “A method for the construction of Green’s functions, ”, Quarterly of Applied Mathematics Melnikov, 2001, “Modified potential as a tool foor computing Green’s functions in continuum mechanics”, Computer Modeling in Engineering Science Trefftz bases Wang and Sudak, 2007, “Antiplane timeharmonic Green’s functions for a circular inhomogeneity with an imperfect interface”, Mechanics Research Communications National Taiwan Ocean University Department of Harbor and River Engineering 6

Outlines Motivation and literature review Derivation of the Green’s function n n n n Numerical examples n n Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Take free body Image technique for solving half-plane problems Green’s function for Laplace problems Green’s function for Helmholtz problems Conclusions National Taiwan Ocean University Department of Harbor and River Engineering 7

Null-field integral approach to construct the Green’s function Original Problem Governing equation: Boundary condition: Subjected to given B. C. Green’s third identity BIE for Green’s function Auxiliary system Governing equation: Fundamental solution National Taiwan Ocean University Department of Harbor and River Engineering 8

Boundary integral equation and null-field integral equation Exterior case Interior case Degenerate (separate) form National Taiwan Ocean University Department of Harbor and River Engineering 9

Expansions of fundamental solution (2 D) Laplace problem-- x x s O Helmholtz problem-Neumann factor National Taiwan Ocean University Department of Harbor and River Engineering 10

U(s, x) T(s, x) L(s, x) M(s, x) x x s Laplace problem-O Helmholtz problem-- National Taiwan Ocean University Department of Harbor and River Engineering 11

Boundary density discretization Fourier series expansions - boundary density Fourier series Ex. constant element National Taiwan Ocean University Department of Harbor and River Engineering 12

Adaptive observer system Source point Collocation point National Taiwan Ocean University Department of Harbor and River Engineering 13

Vector decomposition technique for potential gradient True normal direction Non-concentric case: Source point Collocation point Concentric case (special case) : National Taiwan Ocean University Department of Harbor and River Engineering 14

Linear algebraic equation Collocation point National Taiwan Ocean University Department of Harbor and River Engineering 15

Take free body National Taiwan Ocean University Department of Harbor and River Engineering 16

Image technique for solving half-plane problems Neumannboundarycondition Dirichlet National Taiwan Ocean University Department of Harbor and River Engineering Mirror 17

Flowchart of present method Degenerate kernel Fourier series Null-field equation Potential Analytical Numerical Algebraic system National Taiwan Ocean University Department of Harbor and River Engineering Fourier Coefficients 18

Outlines Motivation and literature review Derivation of the Green’s function n n n n Numerical examples n n Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Take free body Image technique for solving half-plane problems Green’s function for Laplace problems Green’s function for Helmholtz problems Conclusions National Taiwan Ocean University Department of Harbor and River Engineering 19

Numerical examples n n Laplace problems n Eccentric ring n A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition n A half-plane problem with a circular hole and a halfcircular inclusion Helmholtz problems n An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion n Special cases and parameter study n An infinite matrix containing two circular inclusions with a concentrated force in the matrix National Taiwan Ocean University Department of Harbor and River Engineering 20

Present study for Laplace equation u=0 a b u=0 Annular Green’s case (Chenfunction P. Y. ) Analytical Semi-Analytical Green’s function Eccentric ring (Ke J. N. ) National Taiwan Ocean University Department of Harbor and River Engineering 21

Numerical examples n n Laplace problems n Eccentric ring n A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition n A half-plane problem with a circular hole and a halfcircular inclusion Helmholtz problems n An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion n Special cases and parameter study n An infinite matrix containing two circular inclusions with a concentrated force in the matrix National Taiwan Ocean University Department of Harbor and River Engineering 22

Eccentric ring National Taiwan Ocean University Department of Harbor and River Engineering 23

Eccentric ring Potential contour using the Melnikov’s method Potential contour using the present method (M=50) National Taiwan Ocean University Department of Harbor and River Engineering 24

Numerical examples n n Laplace problems n Eccentric ring n A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition n A half-plane problem with a circular hole and a halfcircular inclusion Helmholtz problems n An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion n Special cases and parameter study n An infinite matrix containing two circular inclusions with a concentrated force in the matrix National Taiwan Ocean University Department of Harbor and River Engineering 25

A half plane with an aperture subjected to Dirichlet boundary condition National Taiwan Ocean University Department of Harbor and River Engineering 26

Result of a half-plane problem with an aperture subjected to Dirichlet boundary condition Potential contour using the Melnikov’s method Potential contour using the present method (M=50) National Taiwan Ocean University Department of Harbor and River Engineering 27

A half plane with an aperture subjected to Robin boundary condition National Taiwan Ocean University Department of Harbor and River Engineering 28

Result of a half-plane problem with an aperture subjected to Robin boundary condition Potential contour using the Melnikov’s method Potential contour using the present method (M=50) National Taiwan Ocean University Department of Harbor and River Engineering 29

Numerical examples n n Laplace problems n Eccentric ring n A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition n A half-plane problem with a circular hole and a halfcircular inclusion Helmholtz problems n An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion n Special cases and parameter study n An infinite matrix containing two circular inclusions with a concentrated force in the matrix National Taiwan Ocean University Department of Harbor and River Engineering 30

A half-plane problem with a circular hole and a half-circular inclusion Material conductivity A circular hole Material conductivity Matrix Source of Green’s function Inclusion Half -plane National Taiwan Ocean University Department of Harbor and River Engineering 31

Result of a half-plane problem with a circular hole and a half-circular inclusion Contour plot by using the Melikov's approach (2006) Contour plot by using the null-field integral equation approach National Taiwan Ocean University Department of Harbor and River Engineering 32

Numerical examples n n Laplace problems n Eccentric ring n A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition n A half-plane problem with a circular hole and a halfircular inclusion Helmholtz problems n An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion n Special cases and parameter study n An infinite matrix containing two circular inclusions with a concentrated force in the matrix National Taiwan Ocean University Department of Harbor and River Engineering 33

Present study for Helmholtz equation Perfect interface boundary Imperfect interface boundary SH-Wave Inclusion Matrix SH-Wave SH-wave problem (Chen P. Y. ) Green’s function problem (Ke J. N. ) National Taiwan Ocean University Department of Harbor and River Engineering 34

Numerical examples n n Laplace problems n Eccentric ring n A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition n A half-plane problem with a circular hole and a halfcircular inclusion Helmholtz problems n An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion n Special cases and parameter study n An infinite matrix containing two circular inclusions with a concentrated force in the matrix National Taiwan Ocean University Department of Harbor and River Engineering 35

An infinite matrix containing a circular inclusion with a concentrated force at in the matrix Interface condition is the shear modulus is the wave speed National Taiwan Ocean University Department of Harbor and River Engineering is the imperfect interface parameter 36

Take free body National Taiwan Ocean University Department of Harbor and River Engineering 37

Distribution of for the quasi-static solution along the circular boundary 2. 5 (quasi-static) 2 (static) 1. 5 1 0. 5 0 0 50 100 150 200 250 300 350 (Deg) Wang and Sudak’s solution The present solution National Taiwan Ocean University Department of Harbor and River Engineering 38

Parameter study of stress response for the Bonding behavior Ideal bonding Wang and Sudak’s solution The present solution National Taiwan Ocean University Department of Harbor and River Engineering 39

The distribution of displacement the circular boundary for the case along Dynamic effect Wang and Sudak’s solution The present solution National Taiwan Ocean University Department of Harbor and River Engineering 40

Test of convergence for the Fourier series Terms of Fourier series (M) real part Parseval’s sum of imaginary part for Parseval’s sum of real solution for with a concentrated force in the inclusion Parseval’s sum Terms of Fourier series (M) National Taiwan Ocean University Department of Harbor and River Engineering imaginary part 41

An infinite matrix containing a circular inclusion with a concentrated force at in the inclusion is the shear modulus is the wave speed is the imperfect interface parameter National Taiwan Ocean University Department of Harbor and River Engineering 42

Distribution of for the quasi-static solution along the circular boundary (quasi-static) (Deg) National Taiwan Ocean University Department of Harbor and River Engineering 43

Parameter study of stress response for the Bonding behavior National Taiwan Ocean University Department of Harbor and River Engineering 44

The distribution of displacement along the circular boundary for the case of 6 Dynamic effect 5 4 k. M a = 1 k. M a = 2 3 k. M a = 4 k. M a = 5 2 1 0 0 50 100 150 200 250 300 National Taiwan Ocean University Department of Harbor and River Engineering 350 45

Numerical examples n n Laplace problems n Eccentric ring n A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition n A half-plane problem with a circular hole and a halfircular inclusion Helmholtz problems n An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion n Special cases and parameter study n An infinite matrix containing two circular inclusions with a concentrated force in the matrix National Taiwan Ocean University Department of Harbor and River Engineering 46

Special case of an ideally bonded case Imperfect bonding Ideal bonding is the shear modulus is the wave speed is the imperfect interface parameter National Taiwan Ocean University Department of Harbor and River Engineering 47

The absolute amplitude of displacement by the present method National Taiwan Ocean University Department of Harbor and River Engineering 48

Special case of cavity Imperfect bonding Cavity is shear modulus is wave speed is the imperfect interface parameter National Taiwan Ocean University Department of Harbor and River Engineering 49

The absolute amplitude of displacement by the present method Cavity National Taiwan Ocean University Department of Harbor and River Engineering Cavity 50

Parameter study bonding for ideal Fundamental solution is the shear modulus National Taiwan Ocean University Department of Harbor and River Engineering is the imperfect interface parameter 51

Stress contours of and for the static solutions (a concentrated force in the matrix) National Taiwan Ocean University Department of Harbor and River Engineering 52

Stress contours of and for the dynamic solutions (a concentrated force in the matrix) National Taiwan Ocean University Department of Harbor and River Engineering 53

Stress contours of and for the static solutions (a concentrated force in the inclusion) National Taiwan Ocean University Department of Harbor and River Engineering 54

Stress contours of and for the dynamic solutions (a concentrated force in the inclusion) National Taiwan Ocean University Department of Harbor and River Engineering 55

Series-form & closed-form solutions for the static case (ideally bonded interface) Concentrated force in the matrix Concentrated force in the inclusion 2. 5 2 Stress distribution along the interface (static) 1. 5 1 0. 5 0 0 50 100 150 200 250 300 350 (Deg) Closed-form solution Seires-form solution (Wang andseries Sudak, 2007) Fourier Degenerate (Poisson integral kernel formula) (easy ) (not easy) National Taiwan Ocean University Department of Harbor and River Engineering 56

Numerical examples n n Laplace problems n Eccentric ring n A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition n A half-plane problem with a circular hole and a halfircular inclusion Helmholtz problems n An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion n Special cases and parameter study n An infinite matrix containing two circular inclusions with a concentrated force in the matrix National Taiwan Ocean University Department of Harbor and River Engineering 57

An infinite matrix containing two circular inclusions with a concentrated force at in the matrix is the shear modulus is the wave speed National Taiwan Ocean University Department of Harbor and River Engineering is the imperfect interface parameter 58

Distribution of of various of the matrix at the position National Taiwan Ocean University Department of Harbor and River Engineering 59

The contour of the displacement for an infinite matrix containing two inclusions with a concentrated force at in the matrix for ideal bonding Potential contour using the present method (M=30) National Taiwan Ocean University Department of Harbor and River Engineering 60

Outlines n n Motivation and literature review Derivation of the Green’s function n n n Numerical examples n n n Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Take free body Image technique for solving half-plane problems Green’s function for Laplace problems Green’s function for Helmholtz problems Conclusions National Taiwan Ocean University Department of Harbor and River Engineering 61

Conclusions n n After introducing the degenerate kernel, the BIE is nothing more than the linear algebra. We derived the analytic Green’s function for one inclusion problem by using the null-field integral equation. Also, the present approach can be utilized to construct semi-analytic Green’s functions for several circular inclusions. National Taiwan Ocean University Department of Harbor and River Engineering 62

Conclusions n n Several examples, Laplace and Helmholtz problems were demonstrated to check the validity of the present formulation and the results match well with available solutions in the literature. A general-purpose program for deriving the Green’s function of Laplace or Helmholtz problems with arbitrary number of circular apertures and/or inclusions of arbitrary radii and various positions involving Dirichlet or Neumann or mixed boundary condition was developed. National Taiwan Ocean University Department of Harbor and River Engineering 63

Further studies n n The imperfect circular interface is homogeneous nonhomogeneous. According to our successful experiences for half-plane problems, it is straightforward to quarter-plane problems. National Taiwan Ocean University Department of Harbor and River Engineering 64

The end Thanks for your attentions. You can get more information on our website. http: //msvlab. hre. ntou. edu. tw National Taiwan Ocean University Department of Harbor and River Engineering 65