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On the spurious eigensolutions for the real-part boundary element method

National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering. On the spurious eigensolutions for the real-part boundary element method. 學生 : 李家瑋 海洋大學河海工程系. 指導教授 : 陳正宗終身特聘教授. Dec, 24, 2007. Outlines. Introduction Problem statement, formulation and seven indicators

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On the spurious eigensolutions for the real-part boundary element method

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  1. National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering On the spurious eigensolutions for the real-part boundary element method 學生:李家瑋 海洋大學河海工程系 指導教授:陳正宗終身特聘教授 Dec, 24, 2007

  2. Outlines • Introduction • Problem statement, formulation and seven indicators • Filtering out spurious eigenvalues in the real-part BEM by using the CHEEF concept • Updating terms and updating documents to extract out true and spurious eigenvalues • Extensions to 2-D and 3-D eigenproblems • Conclusions 2007.12.24

  3. Outlines • Introduction • Problem statement, formulation and seven indicators • Filtering out spurious eigenvalues in the real-part BEM by using the CHEEF concept • Updating terms and updating documents to extract out true and spurious eigenvalues • Extensions to 2-D and 3-D eigenproblems • Conclusions 2007.12.24

  4. Why spurious solution occurs • FDM for ODE • Real-part BEM & MRM (Simply-connected problem) • Complex-valued BEM (Multiply-connected problem) 2007.12.24

  5. Literature review 2007.12.24

  6. Outlines • Introduction • Problem statement, formulation and seven indicators • Filtering out spurious eigenvalues in the real-part BEM by using the CHEEF concept • Updating terms and updating documents to extract out true and spurious eigenvalues • Extensions to 2-D and 3-D eigenproblems • Conclusions 2007.12.24

  7. Governing equation︰ Fundamental solution︰ The BIE ︰ t(L)=0 u(0)=0 Problem statement and formulation 2007.12.24

  8. Seven indicators Indicators • 1. Complex determinant using the complex-valued kernel BEM ( det[C]) • 2. Absolute value of determinant using the complex-valued kernel BEM (Abs det[C]) • 3. Imaginary-part of determinant using the complex-valued kernel BEM (Im det[C]) • 4. Real-part of determinant using the complex-valued kernel BEM (Re det[C]) • 5. Determinant using the imaginary-part kernel BEM (det[I]) • 6. Determinant using the real-part kernel BEM (det[R]) • 7. Determinant using the MRM (MRM) 2007.12.24

  9. Eigensolutions using seven indicators 2007.12.24

  10. Outlines • Introduction • Problem statement, formulation and seven indicators • Filtering out spurious eigenvalues in the real-part BEM by using the CHEEF concept • Updating terms and updating documents to extract out true and spurious eigenvalues • Extensions to 2-D and 3-D eigenproblems • Conclusions 2007.12.24

  11. t(L)=0 0 L c -∞ ∞ u(0)=0 t(L)=0 u(0)=0 0 c -∞ ∞ L Filtering out spurious eigenvalues in the real-part BEM by using the CHEEF concept . null-field BIE 2007.12.24

  12. true mode null field null field true node spurious mode nontrivial field nontrivial field Failure CHEEF point Filtering out spurious eigenvalues in the real-part BEM by using the CHEEF concept . 2007.12.24

  13. Outlines • Introduction • Problem statement, formulation and seven indicators • Filtering out spurious eigenvalues in the real-part BEM by using the CHEEF concept • Updating terms and updating documents to extract out true and spurious eigenvalues • Extensions to 2-D and 3-D eigenproblems • Conclusions 2007.12.24

  14. Updating terms and updating documents to extract out true and spurious eigenvalues The hypersingular integral equation (LM equation) ︰ UT equation LM equation 2007.12.24

  15. Updating terms and updating documents to extract out true and spurious eigenvalues true eigenvalues Extraction of true eigenvalues using SVD updating terms Spurious eigenvalues Extraction of spurious eigenvalues using SVD updating terms 2007.12.24

  16. Outlines • Introduction • Problem statement, formulation and seven indicators • Filtering out spurious eigenvalues in the real-part BEM by using the CHEEF concept • Updating terms and updating documents to extract out true and spurious eigenvalues • Extensions to 2-D and 3-D eigenproblems • Conclusions 2007.12.24

  17. a a a Extensions to 2-D and 3-D eigenproblems One-dimensional case (rod). Two-dimensional case (circle membrane). Three-dimensional case (spherical cavity). 2007.12.24

  18. Extensions to 2-D and 3-D eigenproblems Occurring mechanism of 1-D true and spurious eigenequations by using the real-part and imaginary-part BEM. Occurring mechanism of 2-D true and spurious eigenequations by using the real-part and imaginary-part BEM. Occurring mechanism of 3-D true and spurious eigenequations by using the real-part and imaginary-part BEM. 2007.12.24

  19. Outlines • Introduction • Problem statements, formulation and seven indicators • Filtering out spurious eigenvalues in the real-part BEM by using the CHEEF concept • Updating terms and updating documents to extract out true and spurious eigenvalues • Extensions to 2-D and 3-D eigenproblems • Conclusions 2007.12.24

  20. Conclusions • A simple example of one dimensional eigenproblem was demonstrated to show that seven indicators of determinant by using the direct searching scheme can obtain the possible solution. • Spurious eigensolutions in the real-part BEM as well as the MRM have been studied analytically. • The CHEEF concept in conjunction with the SVD updating technique was applied to filter out spurious eigenvalues. Possible failure points were also examined. • Extensions to two and three dimensional cases were also made. • The sequence of one-dimensional examples gives a real good insight in the topic of spurious eigensolutions. 2007.12.24

  21. The end Thanks for your kind attentions • You can get more information from our website: • http://msvlab.hre.ntou.edu.tw/ 2007.12.24

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