National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Conformal mapping and bipolar coordinate for eccentric problems Ming-Hong Tsai ( 蔡明宏 ) Reporter: Ming-Hong Tsai ( 蔡明宏 ) Jeng-Tzong Chen Advisor: Jeng-Tzong Chen ( 陳正宗 特聘教授 ) ( 陳正宗 特聘教授 ) 中國工程師學會 學生論文競賽

MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 2 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 transformation ◎ Analytic solution of Dirichlet Laplace problems Conclusions

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

MSVLAB 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 Motivation Conformal mapping [Eccentric domain] Bipolar coordinate [Curvilinear coordinate] Carrier & Pearson Muskhelishvili Brown & Churchill Spiegel Shen Farlow Chen & Weng Ling Timoshenko Stephens & Casemore Lebedev Literature review

MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 6 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 transformation ◎ Analytic solution of Dirichlet Laplace problems Conclusions

MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 7 Conformal mapping technique using bilinear function z planew plane r1r1 r2r2 ρ1ρ1 ρ2ρ2 C2C2 C1C1

MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 8 Geometric characterization of bipolar coordinate R1R1 R2R2 η1η1 η2η2 x y

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

MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 10 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 transformation ◎ Analytic solution of Dirichlet Laplace problems Conclusions

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

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

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

MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 14 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 transformation ◎ Analytic solution of Dirichlet Laplace problems Conclusions

MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 15 Conclusions VVarious approaches including Carrier & Pearson, Muskhelishvili, Ling, Timoshenko and Goodier, and Lebedev et al. for solving Laplace problems were reviewed. BBased on the conformal mapping, all of them are unified together. T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 16 The end Thanks for your kind attention. Your comments will be highly appreciated. Welcome to visit the web site of MSVLAB: