Degenerate kernel for ellipse Reporter HuanZhen Liao M

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Degenerate kernel for ellipse Reporter: Huan-Zhen Liao (M 9452056) Advisor: Jeng-Tzong Chen Date: 2007/01/04

Degenerate kernel for ellipse Reporter: Huan-Zhen Liao (M 9452056) Advisor: Jeng-Tzong Chen Date: 2007/01/04 Place: HR 2 307 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1

Outline • Motivation • Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some

Outline • Motivation • Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some properties of ellipse ◎ Derivation of degenerate kernel ◎ Convergence of degenerate kernel • A numerical example • Conclusions • Further studies MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 2

Outline • Motivation • Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some

Outline • Motivation • Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some properties of ellipse ◎ Derivation of degenerate kernel ◎ Convergence of degenerate kernel • A numerical example • Conclusions • Further studies MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 3

Motivation [Chebyshev polynomial] Degenerate boundary Straight boundary [Legendre polynomial] [Mathieu function] Elliptic hole Circular

Motivation [Chebyshev polynomial] Degenerate boundary Straight boundary [Legendre polynomial] [Mathieu function] Elliptic hole Circular inclusion [Fourier series] MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 4

Present approach Degenerate kernel Fundamental solution No principal value CPV and HPV Advantages of

Present approach Degenerate kernel Fundamental solution No principal value CPV and HPV Advantages of degenerate kernel 1. No principal value 2. Well-posed 3. Exponential convergence 4. Free of boundary-layer effect MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 5

Outline • Motivation • Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some

Outline • Motivation • Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some properties of ellipse ◎ Derivation of degenerate kernel ◎ Convergence of degenerate kernel • A numerical example • Conclusions • Further studies MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 6

Conformal mapping technique z plane MSVLAB w plane National Taiwan Ocean University Department of

Conformal mapping technique z plane MSVLAB w plane National Taiwan Ocean University Department of Harbor and River Engineering 7

Some properties of ellipse z plane MSVLAB When R>1 w plane National Taiwan Ocean

Some properties of ellipse z plane MSVLAB When R>1 w plane National Taiwan Ocean University Department of Harbor and River Engineering 8

Some properties of ellipse z plane MSVLAB When R<1 w plane National Taiwan Ocean

Some properties of ellipse z plane MSVLAB When R<1 w plane National Taiwan Ocean University Department of Harbor and River Engineering 9

Some properties of ellipse MSVLAB National Taiwan Ocean University Department of Harbor and River

Some properties of ellipse MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 10

Derivation of degenerate kernel MSVLAB National Taiwan Ocean University Department of Harbor and River

Derivation of degenerate kernel MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 11

Convergence of degenerate kernel Interior MSVLAB Exterior National Taiwan Ocean University Department of Harbor

Convergence of degenerate kernel Interior MSVLAB Exterior National Taiwan Ocean University Department of Harbor and River Engineering 12

Outline • Motivation • Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some

Outline • Motivation • Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some properties of ellipse ◎ Derivation of degenerate kernel ◎ Convergence of degenerate kernel • • • A numerical example Conclusions Further studies MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 13

Numerical examples w plane MSVLAB z plane National Taiwan Ocean University Department of Harbor

Numerical examples w plane MSVLAB z plane National Taiwan Ocean University Department of Harbor and River Engineering 14

Numerical examples MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 15

Numerical examples MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 15

Numerical examples w plane MSVLAB z plane National Taiwan Ocean University Department of Harbor

Numerical examples w plane MSVLAB z plane National Taiwan Ocean University Department of Harbor and River Engineering 16

Numerical examples MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 17

Numerical examples MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 17

Outline • Motivation • Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some

Outline • Motivation • Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some properties of ellipse ◎ Derivation of degenerate kernel ◎ Convergence of degenerate kernel • A numerical example • Conclusions • Further studies MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 18

Conclusions • In the future, we will choose the correct mapping points to calculate

Conclusions • In the future, we will choose the correct mapping points to calculate degenerate kernels for ellipse. • According to numerical results, only few terms of Fourier series can achieve accurate solutions. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 19

Outline • Motivation • Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some

Outline • Motivation • Degenerate kernel for ellipse ◎ Conformal mapping technique ◎ Some properties of ellipse ◎ Derivation of degenerate kernel ◎ Convergence of degenerate kernel • A numerical example • Conclusions • Further studies MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 20

Further studies • Application on many engineering problems using degenerate kernel for ellipse. •

Further studies • Application on many engineering problems using degenerate kernel for ellipse. • Extension to general boundaries. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 21

The end Thanks for your kind attention. Your comments will be highly appreciated. Welcome

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 22