Conformal mapping and bipolar coordinate for eccentric problems

Conformal mapping and bipolar coordinate for eccentric problems Reporter: Ming-Hong Tsai (蔡明宏) Advisor: Jeng-Tzong Chen and Chein-Shan Liu (陳正宗 特聘教授 與 劉進賢 教授) National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 中 會海洋大學分會學生論文競賽 2007. 03. 16 1

Outlines • Motivation • Conformal mapping & bipolar coordinate ◎ Conformal mapping technique using bilinear function ◎ Geometric characterization of bipolar coordinate ◎ Viewpoint of conformal mapping for the bipolar coordinate • An illustrative example ◎Geometry ◎ Analytic transformation solution of Dirichlet Laplace problems • Conclusions MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 2

Motivation Partial differential equation: Regular case MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 3

Motivation Literature review Conformal mapping [Eccentric domain] Carrier & Pearson Brown & Churchill Muskhelishvili Spiegel Farlow Chen & Weng Shen Bipolar coordinate [Curvilinear coordinate] MSVLAB Timoshenko Stephens & Casemore Lebedev Ling National Taiwan Ocean University Department of Harbor and River Engineering 4

Motivation Conformal mapping [Eccentric domain] Bipolar coordinate [Curvilinear coordinate] MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 5

Outlines • Motivation • Conformal mapping & bipolar coordinate ◎ Conformal mapping technique using bilinear function ◎ Geometric characterization of bipolar coordinate ◎ Viewpoint of conformal mapping for the bipolar coordinate • An illustrative example ◎Geometry ◎ Analytic transformation solution of Dirichlet Laplace problems • Conclusions MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 6

Conformal mapping technique using bilinear function z plane r 1 C 1 r 2 w plane ρ2 ρ1 C 2 MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 7

Geometric characterization of bipolar coordinate η 1 R 2 R 1 MSVLAB η 2 National Taiwan Ocean University Department of Harbor and River Engineering 8

Viewpoint of conformal mapping for the bipolar coordinate MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 9

Outlines • Motivation • Conformal mapping & bipolar coordinate ◎ Conformal mapping technique using bilinear function ◎ Geometric characterization of bipolar coordinate ◎ Viewpoint of conformal mapping for the bipolar coordinate • An illustrative example ◎Geometry ◎ Analytic transformation solution of Dirichlet Laplace problems • Conclusions MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 10

Geometry transformation MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 11

Analytic solution of Dirichlet Laplace problems MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 12

Table Irregular domain Regular domain MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 13

Outline • Motivation • Conformal mapping & bipolar coordinate ◎ Conformal mapping technique using bilinear function ◎ Geometric characterization of bipolar coordinate ◎ Viewpoint of conformal mapping for the bipolar coordinate • An illustrative example ◎Geometry ◎ Analytic transformation solution of Dirichlet Laplace problems • Conclusions MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 14

Conclusions Ø Various approaches including Carrier & Pearson, Muskhelishvili, Ling, Timoshenko and Goodier, and Lebedev et al. for solving Laplace problems were reviewed. Ø Based on the conformal mapping, all of them are unified together. Ø The relations among them are constructed through operations of translation, rotation, stretching, inversion and taking Log. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 15

The end Thanks for your kind attention. Your comments will be highly appreciated. Welcome to visit the web site of MSVLAB: http: //ind. ntou. edu. tw/~msvlab MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 16