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Semi-Analytical Approach for Surface Motion of Alluvial Valleys under SH-Waves

This report presents a semi-analytical method for solving surface motion in multiple alluvial valleys with incident SH-waves. The method involves adaptive observer systems, linear algebraic equations, and image techniques. Numerical examples validate the approach's effectiveness in solving scattering problems.

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Semi-Analytical Approach for Surface Motion of Alluvial Valleys under SH-Waves

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  1. A semi-analytical approach for solving surface motion of multiple alluvial valleys for incident plane SH-waves半解析法求解多山谷沉積物入射SH波之地表位移 Reporter: 陳柏源 Authors : 陳柏源、陳正宗博士 國立台灣海洋大學河海工程學系

  2. Outlines • Motivation • Present method • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Image technique for solving scattering problems of half-plane • Numerical examples • Conclusions

  3. Outlines • Motivation • Present method • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Image technique for solving scattering problems of half-plane • Numerical examples • Conclusions

  4. Conventional BEM Interior case Exterior case

  5. Motivation BEM / BIEM Improper integral Singularity & hypersingularity Regularity Fictitious BEM Bump contour Limit process Fictitious boundary Achenbach et al. (1988) Null-field approach Guiggiani (1995) Gray and Manne (1993) Collocation point CPV and HPV Ill-posed Waterman (1965)

  6. Present method Degenerate kernel Fundamental solution No principal value CPV and HPV • Advantages of degenerate kernel • No principal value • Well-posed • Exponential convergence • Free of boundary-layer effect

  7. Outlines • Motivation • Present method • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Image technique for solving scattering problems of half-plane • Numerical examples • Conclusions

  8. Separable form of fundamental solution (1D) Separable property continuous jump

  9. Degenerate (separate) form of fundamental solution (2-D)

  10. U(s,x) T(s,x)

  11. Jump behavior across the boundary

  12. Outlines • Motivation • Present method • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Image technique for solving scattering problems of half-plane • Numerical examples • Conclusions

  13. collocation point Adaptive observer system r2,f2 r0 ,f0 r1 ,f1 rk,fk

  14. Outlines • Motivation • Present method • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Image technique for solving scattering problems of half-plane • Numerical examples • Conclusions

  15. Linear algebraic equation Index of collocation circle Index of routing circle Column vector of Fourier coefficients (Nth routing circle)

  16. Explicit form of each submatrix and vector Truncated terms of Fourier series Number of collocation points Fourier coefficients

  17. Outlines • Motivation • Present method • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Image technique for solving scattering problems of half-plane • Numerical examples • Conclusions

  18. Image technique for solving half-plane problem SH-Wave Alluvial Free surface a h Matrix Matrix SH-Wave SH-Wave

  19. Inclusion SH-Wave Matrix Matrix SH-Wave Take free body SH-Wave Matrix SH-Wave

  20. Matrix field Inclusion field Two constrains Linear algebraic system of the inclusion problem

  21. Flowchart of present method Analytical Degenerate kernel Fourier series Numerical Collocation point and matching B.C. Adaptive observer system Stress field Vector decomposition Linear algebraic equation Potential of domain point Surface amplitude Fourier coefficients

  22. Outlines • Motivation • Present method • Expansions of fundamental solution and boundary density • Adaptive observer system • Linear algebraic equation • Image technique for solving scattering problems of half-plane • Numerical examples • Conclusions

  23. y Alluvial Matrix x a h SH-Wave A half-plane problem with a semi-circular alluvial valley subject to the SH-wave Governing equation Wave number Dimensionless frequency Velocity of shear wave

  24. Surface amplitudes of the alluvial valley problem Present method Manoogian’s results [60] vertical horizontal

  25. Limiting case of a canyon Present method Manoogian’s results [60] vertical horizontal

  26. Limiting case of a rigid alluvial valley Present method

  27. Soft-basin effect Present method 18 14 3

  28. No boundary-layer effect

  29. A half-plane problem with two alluvial valleys subject to the incident SH-wave Canyon Matrix SH-Wave 3a 房[93]將正弦和餘弦函數的正交特性使用錯誤，以至於推導出錯誤的聯立方程，求得錯誤的結果。 --亞太學報 曹2004

  30. Limiting case of two canyons Present method Tsaur et al.’s results [103] vertical horizontal

  31. Surface displacements of two alluvial valleys Present method

  32. A half-plane problem with a circular inclusion subject to the incident SH-wave y Inclusion x h Matrix a SH-Wave

  33. Surface displacements of a inclusion problem under the ground surface Present method Tsaur et al.’s results [102] When I solved this problem I could find no published results for comparison.  I also verified my results using the limiting cases.  I did not have the benefit of published results for comparing the intermediate cases.  I would note that due to precision limits in the Fortran compiler that I was using at the time. --Private communication Manoogian and Lee’s results [62]

  34. Limiting case of a cavity problem Present method Lee and Manoogian’s [53] for the cavity case. vertical horizontal

  35. A half-plane problem with two circular inclusions subject to the SH-wave y Matrix x Inclusion h a a D SH-Wave

  36. Limiting case of two-cavities problem Present method Jiang et al. result [95]

  37. Surface amplitudes of two-inclusions problem Present method

  38. Conclusions • A systematic way to solve the Helmholtz problems with circular boundaries was proposed successfully in this paper by using the null-field integral equation in conjunction with degenerate kernels and Fourier series. • The present method is more general for calculating the torsion and bending problems with arbitrary number of holes and various radii and positions than other approach. • Our approach can deal with the cavity problem as a limiting of inclusion problem with zero shear modulus.

  39. Some findings ?

  40. Thanks for your kind attentions. You can get more information from our website. http://msvlab.hre.ntou.edu.tw/

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