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Hybrid Method Combines Transfinite Interpolation with Series Expansion to Simulate the Anti-Plane Response of a Surface Irregularity

Published online by Cambridge University Press:  05 June 2014

W.-S. Shyu  and
T.-J. Teng
W.-S. Shyu*
Affiliation:
Department of Civil Engineering, National Pingtung University of Science and Technology, Pingtung, Taiwan, 91201, R.O.C.
T.-J. Teng
Affiliation:
National Center for Research on Earthquake Engineering, Taipei, Taiwan, 10668, R.O.C.
*
*[email protected]
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Abstract

The responses to an incident plane SH wave on or near a surface irregularity which is embedded in an elastic half-plane are investigated. The surface irregularity represents a canyon, an alluvial valley or a hill. The wave function expansion method has been employed to solve surface irregularities, such as a semi-cylindrical canyon, a semi-cylindrical alluvial valley, or a semi-elliptical canyon and a semi-elliptical alluvial valley. These solutions to the scattering problem of SH wave can be used to test the accuracy of the other numerical methods. But solutions for surface irregularities with arbitrarily shapes cannot be found easily. A hybrid method combines the finite element method with series expansion is applied to solve scattering problems in this study. A subregion encloses the surface irregularity with a semi-circular auxiliary boundary can be meshed by the finite element method. By using the transfinite interpolation (TFI) produces excellent grid mesh on the subregion. The advantage of TFI is the flexibility to facilitate modeling of the subregion. On the other hand, the boundary data can be formulated by using a series representation with unknown coefficients. The Lamb's solution which satisfies the traction free condition and the radiation condition at infinity is implemented to be the basis function. The unknown coefficients can be obtained by satisfying the continuity conditions of the semi-circular auxiliary boundary between the subregion and the half-plane. The hybrid method that combines TFI with series expansion is successfully herein to solve the scattering problem by a surface irregularity. Numerical results in this study for special cases agree well with those in the published literatures. In this study, the steps and skills of hybrid method are described systematically and completely to solve the surface irregularity.

Keywords

SH wavescatteringsurface irregularityhybrid methodtransfinite interpolation
Type
Research Article
Information
Journal of Mechanics , Volume 30 , Issue 4 , August 2014 , pp. 349 - 360
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

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References

1.Trifunac, M. D., “Scattering of Plane SH Waves by a Semi-Cylindrical Canyon,” Earthquake Engineering & Structural Dynamics, 1, pp. 267–281 (1973).Google Scholar
2.Wong, H. L. and Trifunac, M. D., “Scattering of Plane SH Waves by a Semi-Elliptical Canyon,” Earthquake Engineering & Structural Dynamics, 3, pp. 157–169 (1974).Google Scholar
3.Trifunac, M. D., “Surface Motion of a Semi-Cylindrical Alluvial Valley for Incident Plane SH Waves,” Bulletin of the Seismological Society of America, 61, pp. 1755–1770 (1971).CrossRefGoogle Scholar
4.Wong, H. L. and Trifunac, M. D., “Surface Motion of a Semi-Elliptical Alluvial Valley for Incident Plane SH Waves,” Bulletin of the Seismological Society of America, 64, pp. 1389–1408 (1974).CrossRefGoogle Scholar
5.Todorovska, M. I. and Lee, V. W., “Surface Motion of Shallow Cylindrical Alluvial Valleys for Incident Plane SH Waves-Analytical Solution,” Soil Dynamics and Earthquake Engineering, 10, pp. 192–200 (1991).Google Scholar
6.Yuan, X. M. and Men, F. L., “Scattering of Plane SH Waves by a Semi-Cylindrical Hill,” Earthquake Engineering & Structural Dynamics, 21, pp. 1091–1098 (1992).Google Scholar
7.Yuan, X. M. and Liao, Z. P., “Surface Motion of a Cylindrical Hill of Circular-Arc Cross-Section for Incident SH Waves,” Soil Dynamics and Earthquake Engineering, 15, pp. 189–199 (1996).Google Scholar
8.Zhou, H. and Chen, X. F., “A New Approach to Simulate Scattering of SH Waves by an Irregular Topography,” Geophysical Journal of International, 164, pp. 449–459 (2006).Google Scholar
9.Chen, J. T., Chen, P. Y. and Chen, C. T., “Surface Motion of Multiple Alluvial Valleys for Incident Plane SH-Waves by Using a Semi-Analytical Approach,” Soil Dynamics and Earthquake Engineering, 28, pp. 58–72 (2008).CrossRefGoogle Scholar
10.Tsaur, D. H. and Chang, K. H., “An Analytical Approach for the Scattering of SH Waves by a Symmetrical V-Shaped Canyon: Shallow Case,” Geophysical Journal of International, 174, pp. 255–264 (2008).Google Scholar
11.Tsaur, D. H., Chang, K. H. and Hsu, M. S., “An Analytical Approach for the Scattering of SH Waves by a Symmetrical V-Shaped Canyon: Deep Case,” Geophysical Journal of International, 183, pp. 1501–1511 (2010).CrossRefGoogle Scholar
12.Tsaur, D. H. and Chang, K. H., “Scattering of SH Waves by Truncated Semicircular Canyon,” Journal of Engineering & Mechanics, ASCE, 135, pp. 862–870 (2009).Google Scholar
13.Tsaur, D. H. and Chang, K. H., “Scattering and Focusing of SH Waves by a Convex Cylindrical-Arc Topography,” Geophysical Journal of International, 177, pp. 222–234 (2009).Google Scholar
14.Chen, J. T., Lee, J. W., Wu, C. F. and Chen, I. L., “SH-Wave Diffraction by a Semi-Circular Hill Revisited: A Null-Field Boundary Integral Equation Method Using Degenerate Kernels,” Soil Dynamics and Earthquake Engineering, 31, pp. 729–736 (2011).Google Scholar
15.Chen, J. T., Lee, J. W. and Shyu, W. S., “SH-Wave Scattering by a Semi-Elliptical Hill Using a Null-Field Boundary Integral Equation Method and a Hybrid Method,” Geophysical Journal of International, 188, pp. 177–194 (2012).Google Scholar
16.Gao, Y. C., Zhang, N. Y., Li, D., Liu, H., Cai, Y. and Wu, Y., “Effects of Topographic Amplification Induced by a U-Shaped Canyon on Seismic Waves,” Bulletin of the Seismological Society of America, 102, pp. 1748–1763 (2012).Google Scholar
17.Mei, C. C., Boundary Layer and Finite Element Techniques Applied to Wave Problem, Acoustic, Electromagnetic and Elastic Wave Scattering – Focus on the T-Matrix Approach, Varadan, V. V. and Vardan, V. K. Eds., Pergamon, New York, pp. 507–525 (1980).Google Scholar
18.Yeh, C. S., Teng, T. J., Shyu, W. S. and Liao, W. I., “A Hybrid Method to Solve the Half-Plane Radiation Problem-Numerical Verification,” 2000 ASME Pressure Vessels and Piping Conference, Seattle, 2, pp. 145–149 (2000).Google Scholar
19.Yeh, C. S., Shyu, W. S., Teng, T. J. and Liao, W. I., “SH Wave Scattering at a Semi-Elliptical Canyon by Hybrid Method,” The 20th KKCNN Symposium on Civil Engineering, Jeju, Korea, pp. 25–30 (2007).Google Scholar
20.Knupp, P. and Steinberg, S., Fundamentals of Grid Generation, CRC Press Inc., U.S.A.(1994).Google Scholar
21.Shyu, W. S., Teng, T. J., Yeh, C. S. and Liao, W. I., “SH Wave Scattering at an Irregular Canyon by Hybrid Method,” The 32th National Conference on Theoretical and Applied Mechanics, National Chung Cheng University, Chiayi, Taiwan, G013 (2008). (in Chinese)Google Scholar
22.Shyu, W. S., “SH Wave Scattering by an Irregular Canyon Using a New Hybrid Method,” Journal of Science and Innovation, 2, pp. 207–214 (2012).Google Scholar
23.Pao, Y. H., “Betti'S Identity and Transition Matrix for Elastic Waves,” Journal of the Acoustical Society of America, 64, pp. 302–310 (1978).Google Scholar
24.Yeh, C. S., Teng, T. J. and Liao, W. I., “On Evaluation of Lamb's Integrals for Waves in a Two-Dimensional Elastic Half-Space,” The Chinese Journal of Mechanics, 16, pp. 109–124 (2000).Google Scholar