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a more accurate solution at a comparable computational cost. In addition the task of choosing suitable source points would be eliminated. APPENDIX.—REFERENCES 43. Almansi, E., "Un Teorema Sulle Deformazioni Eslastiche dei Solidi Isotropi," Atti della Reale Accademia Nazionale dei Lincei, Vol. 16, 1904, pp. 865-868. 44. Dasgupta, G., "Wellposedness of Substructure Deletion Formulation," Pro- ceedings of the 16th Midwestern Mechanics Conference, Kansas State Uni- versity, Manhattan, Kansas, Vol. 10, 1979. 45. Stroud, A. H., and Secrest, D., "Gaussian Quadrature Formulas," Prentice Hall, New York, N.Y., 1966. Closure by Marijan Dravinski 4 The writer wishes to thank Mr. Ralf Peek and Professor Eduardo Kau- sel for their interesting remarks on his paper. This reply may help to clarify some basic characteristics of the indirect boundary integral method (IBIEM) employed by the writer. First, the method is an approximate one, though the accuracy of the results is excellent for a wide class of problems. Second, the method has been chosen for its simplicity and numerical effectives in problems of interest in engineering and geo- physics. The procedure suggested by discussers is well documented in the literature on the boundary integral methods, e.g., Cruse and Rizzo (49), Brebbia (47), and it has been employed in a number of investiga- tions including one by the writer (50). A detailed account of the nu- merical aspects of the direct boundary integral equation methods (BIEM) is given by Cole, et al. (48). Although the BIEM's are numerically very efficient, still many problems of practical interest in geophysics and en- gineering preclude evaluation of corresponding solutions due to the lim- ited size of the memory core of digital computers. Therefore, there is a need to explore further a possibility of simplifying the existing BIEM approach. This led to an indirect BIEM employed in the paper by the writer (50). The basic idea of the method is to represent the scattered wave field due to interface between two different materials in terms of a finite number of line (point) sources located away from the interface. The advantage of this procedure is elimination of quadrature along the boundary (interface), the disadvantage being that the location of the sources remains to be determined. This basic idea was proposed by Ku- pradze (51). Oshaki (52) used an equivalent approach in elastostatics. Subsequent investigations by Sanchez-Sesma and Rosenblueth (54), San- chez-Sesma and Esquivel (53), Wong (55), and Apsel (46) employed the same idea successfully to problems ranging from antiplane strain model (53,54), plane strain model (55), to three dimensional multilayered vis- coelastic model (46). In all these investigations the method proved to be very accurate and numerically very efficient. Therefore, the need to in- vestigate the role of the number and location of the sources upon the accuracy of the method become apparent. This was the prime motivation 4 Asst. Prof, of Mech. Engrg., Univ. of Southern California, Los Angeles, Calif. 90007. 934 J. Eng. Mech. 1983.109:934-935. ed from ascelibrary.org by Duke University on 08/26/12. For personal use only. No other uses without permission. Copyright (c) 2012. American Society of Civil Engineers. All rights
Transcript of Closure to “Scattering of SH Waves by Subsurface Topography”
a more accurate solution at a comparable computational cost. In addition the task of choosing suitable source points would be eliminated.
APPENDIX.—REFERENCES
43. Almansi, E., "Un Teorema Sulle Deformazioni Eslastiche dei Solidi Isotropi," Atti della Reale Accademia Nazionale dei Lincei, Vol. 16, 1904, pp. 865-868.
44. Dasgupta, G., "Wellposedness of Substructure Deletion Formulation," Proceedings of the 16th Midwestern Mechanics Conference, Kansas State University, Manhattan, Kansas, Vol. 10, 1979.
45. Stroud, A. H., and Secrest, D., "Gaussian Quadrature Formulas," Prentice Hall, New York, N.Y., 1966.
Closure by Marijan Dravinski4
The writer wishes to thank Mr. Ralf Peek and Professor Eduardo Kau-sel for their interesting remarks on his paper. This reply may help to clarify some basic characteristics of the indirect boundary integral method (IBIEM) employed by the writer. First, the method is an approximate one, though the accuracy of the results is excellent for a wide class of problems. Second, the method has been chosen for its simplicity and numerical effectives in problems of interest in engineering and geophysics. The procedure suggested by discussers is well documented in the literature on the boundary integral methods, e.g., Cruse and Rizzo (49), Brebbia (47), and it has been employed in a number of investigations including one by the writer (50). A detailed account of the numerical aspects of the direct boundary integral equation methods (BIEM) is given by Cole, et al. (48). Although the BIEM's are numerically very efficient, still many problems of practical interest in geophysics and engineering preclude evaluation of corresponding solutions due to the limited size of the memory core of digital computers. Therefore, there is a need to explore further a possibility of simplifying the existing BIEM approach. This led to an indirect BIEM employed in the paper by the writer (50). The basic idea of the method is to represent the scattered wave field due to interface between two different materials in terms of a finite number of line (point) sources located away from the interface. The advantage of this procedure is elimination of quadrature along the boundary (interface), the disadvantage being that the location of the sources remains to be determined. This basic idea was proposed by Ku-pradze (51). Oshaki (52) used an equivalent approach in elastostatics. Subsequent investigations by Sanchez-Sesma and Rosenblueth (54), San-chez-Sesma and Esquivel (53), Wong (55), and Apsel (46) employed the same idea successfully to problems ranging from antiplane strain model (53,54), plane strain model (55), to three dimensional multilayered vis-coelastic model (46). In all these investigations the method proved to be very accurate and numerically very efficient. Therefore, the need to investigate the role of the number and location of the sources upon the accuracy of the method become apparent. This was the prime motivation
4Asst. Prof, of Mech. Engrg., Univ. of Southern California, Los Angeles, Calif. 90007.
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of the writer's work. Investigation provided some insight in the mechanism of the method in different problem models and overall influence of the source surfaces upon the accuracy of the results. This should help in application of the method in a more general class of problems for which no solution seems available at the present time.
APPENDIX.—-REFERENCES
46. Apsel, R., "Dynamic Green's Functions for Layered Media and Applications to Boundary Value Problems," thesis presented to the University of California, at San Diego, Calif., in 1979, in partial fulfillment for the requirements of the degree of Doctor of Philosophy.
47. Brebbia, C. A., The Boundary Element Method for Engineers, Halsted Press, New York, N.Y., 1978.
48. Cole, D. M., Kosloff, D. D., and Minster, J. B., "A Numerical Boundary Integral Equation Method for Elastodynamics. I," Bulletin of Seismological Society of America, Vol. 68, No. 5, Oct., 1978, pp. 1331-1357.
49. Cruse, T. A., and Rizzo, F. J., "A Direct Formulation and Numerical Solution of the General Transient Elastodynamic Problem. I," Journal of Mathematical Analysis and Applications, Vol. 22, 1968, pp. 244-259.
50. Dravinski, M. and Udwadia, F. E., "Scattering of Horizontally Polarized Shear Waves by a Periodic Boundary," Proceedings, 15th International Congress of Theoretical and Applied Mechanics, Toronto, Canada, 1980.
51. Kupradze, V. P., Dynamical Problems in Elasticity, North Holland, Amsterdam, The Netherlands, 1963.
52. Oshaki, Y., "On Movements of a Rigid Body in Semi-Infinite Elastic Medium," Proceedings, Japanese Earthquake Engineering Symposium, 1973, Tokyo, Japan.
53. Sanchez-Sesma, F. J., and Esquivel, J. A., "Ground Motion on Alluvial Valleys Under the Incident Plane SH Waves," Bulletin of Seismological Society of America, Vol. 69, No. 4, 1979, pp. 1107-1120.
54. Sanchez-Sesma, F. J. and Rosenblueth, E., "Ground Motions at Cantions of Arbitrary Shapes Under Incident SH Waves," Earthquake Engineering and Structural Dynamics, Vol. 7, No. 5, 1979, pp. 441-450.
55. Wong, H. L., "Effect of Surface Topography on the Diffraction of P, SV, and Rayleigh Waves," Bulletin of Seismological Society of America, Vol. 72, No. 4, 1982, pp. 1167-1183.
STATIONARY AND TRANSIENT RESPONSE STATISTICS3
Errata
The following corrections should be made to the original paper:
Page 629, Eq. 46: Should read Re [H,(o»)Hy(a>)] = |H,(a>)|2 CO? iiUij + (CO?
"August 1982, by Peter H. Madsen and Steen Krenk (Paper 17257).
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