2013/06/13 Page 1 複變數邊界積分方程推導 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering Date: June, 13, 2013 Student: Jia-Wei Lee ( 李家瑋 ) Advisor: Jeng-Tzong Chen ( 陳正宗 )

2013/06/13 Page 2 Outline 1. Revisit the boundary integral equation in R 2 2. Green’s third identity in C 3. Derivation of boundary integral equation in C 4. RVBIE vs. CVBIE

2013/06/13 Page 3 Outline 1. Revisit the boundary integral equation in R 2 2. Green’s third identity in C 3. Derivation of boundary integral equation in C 4. RVBIE vs. CVBIE

2013/06/13 Page 4 Green’s third identity Divergence theorem in R 2 F is a vector function Green’s third identity u and v are scalar functions

2013/06/13 Page 5 Boundary integral equation in R 2 2D Laplace problem Auxiliary system Unknown field Fundamental solution

2013/06/13 Page 6 BIE in real number Fundamental solution Singular boundary integral equation 2D Laplace equation

2013/06/13 Page 7 Outline 1. Revisit the boundary integral equation in R 2 2. Green’s third identity in C 3. Derivation of boundary integral equation in C 4. RVBIE vs. CVBIE

2013/06/13 Page 8 Complex analysis Cauchy-Riemann operator Complex number Complex function 2D Laplace operator Cauchy-Riemann operator (Conjugate form)

2013/06/13 Page 9 Holomorphic function & Harmonic function Harmonic function Holomorphic function (Analytic function) Cauchy-Riemann equation Exist 2D Laplace equation

2013/06/13 Page 10 Gauss theorem (Green theorem) in Green theorem in R 2

2013/06/13 Page 11 Gauss theorem (Green theorem) in

2013/06/13 Page 12 Gauss theorem (Green theorem) in Conjugate form

2013/06/13 Page 13 Green’s third identity in C

2013/06/13 Page 14 Outline 1. Revisit the boundary integral equation in R 2 2. Green’s third identity in C 3. Derivation of boundary integral equation in C 4. RVBIE vs. CVBIE

2013/06/13 Page 15 Boundary integral equation in C 2D Laplace problem Auxiliary system Two unknown fields Fundamental solution

2013/06/13 Page 16 BIE in complex number Singular boundary integral equation 2D Laplace equation

2013/06/13 Page 17 Outline 1. Revisit the boundary integral equation in R 2 2. Green’s third identity in C 3. Derivation of boundary integral equation in C 4. RVBIE vs. CVBIE

2013/06/13 Page 18 RVBIE vs. CVBIE RVBIE CVBIE

2013/06/13 Page 19 The end Thanks for your kind attentions Welcome to visit the web site of MSVLAB/NTOU

2013/06/13 Page 20 BIE in real number Fundamental solution Singular boundary integral equation 2D Helmholtz equation

2013/06/13 Page 21 Conventional CVBIE vs. Present CVBIE Conventional CVBIE Present CVBIE

2013/06/13 Page 22 Gauss theorem (Divergence theorem) in Green theorem in R F is a vector function

2013/06/13 Page 23 Q & A CVBIE 的優缺點。 有沒有實際的問題，一定要用新導到的 CVBIE 求解，假若使用傳統的 CVBIE 是無法求解的。 如何用 CVBIE 求解外域問題。 若 u 和 v 的邊界條件類型不一樣，還能保留 CVBIE 的優點嗎 ? 引入退化核，對於一些特別的幾何 外形，是否也可以得到半解析解。 可否將 CVBIE 推廣至 Helmholtz problem 。

2013/06/13 Page 24 Q & A( 克氏分析 ) Clifford BIE 的優缺點。 簡單的例題應用。 實際程式的運算。 廣義 stokes’ theorem 如何退回去一般在向 量微積分所學到的 Stokes’ theorem 。 可否將 Clifford BIE 推廣至 Helmholtz problem 。 若引入退化核，對於一些特別的幾何外 形，是否也可以得到半解析解。

2013/06/13 Page 25 Singular BIE from Borel-Pompeiu formula Borel-Pompeiu formula Singular BIE for

2013/06/13 Page 26 BIE in complex number Singular boundary integral equation 2D Laplace equation

2013/06/13 Page 27 BIE in Clifford number Singular boundary integral equation Fundamental solution

2013/06/13 Page 28 BIE in Clifford number Singular boundary integral equation Fundamental solution

2013/06/13 Page 29 and

2013/06/13 Page 30 What do I want to do ? Try to extend complex valued BIE to deal with 2D Helmholtz problem 2D Helmholtz equation Harmonic function OK Now Non harmonic function Analytical function ?? First

2013/06/13 Page 31 What do I want to do ? Try to extend complex valued BIE to Clifford BEM and deal with 2D Helmholtz problem by using the Clifford BEM Second