Applications of generalized stress in elastodynamics

Meade, Douglas B.

doi:10.1090/qam/1096236

Author: Douglas B. Meade
Journal: Quart. Appl. Math. 49 (1991), 121-145
MSC: Primary 73D25; Secondary 65N30, 65N38, 73C99, 73V05
DOI: https://doi.org/10.1090/qam/1096236
MathSciNet review: MR1096236
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The problem under consideration is the scattering of elastic waves by inhomogeneous obstacles. The main goal is to obtain approximation techniques which are amenable to numerical implementation. For time-periodic problems a coupling procedure involving finite elements and boundary integral equations is described. For general time-dependent problems, artificial boundary methods are studied. In both cases the concept of generalized stress, as originated by Kupradze, plays a central role. The analysis is restricted to planar two-dimensional problems since these illustrate the essential ideas.

Garth A. Baker, Error estimates for finite element methods for second order hyperbolic equations, SIAM J. Numer. Anal. 13 (1976), no. 4, 564–576. MR 423836, DOI https://doi.org/10.1137/0713048
A. Barry, J. Bielak, and R. C. MacCamy, On absorbing boundary conditions for wave propagation, J. Comput. Phys. 79 (1988), no. 2, 449–468. MR 973337, DOI https://doi.org/10.1016/0021-9991%2888%2990025-3
Jacobo Bielak and R. C. MacCamy, An exterior interface problem in two-dimensional elastodynamics, Quart. Appl. Math. 41 (1983/84), no. 1, 143–159. MR 700668, DOI https://doi.org/10.1090/S0033-569X-1983-0700668-X

R. Clayton and B. Engquist, Absorbing boundary conditions for acoustic and elastic wave equations, Bull. Seismol. Soc. Amer. 67, 1529–1540 (1977)

M. Costabel and E. P. Stephan, Integral equations for transmission problems in linear elasticity, J. Integral Equations Appl. 2 (1990), no. 2, 211–223. MR 1045769, DOI https://doi.org/10.1216/JIE-1990-2-2-211
Martin Costabel and Ernst P. Stephan, On the convergence of collocation methods for boundary integral equations on polygons, Math. Comp. 49 (1987), no. 180, 461–478. MR 906182, DOI https://doi.org/10.1090/S0025-5718-1987-0906182-9
R. Courant and D. Hilbert, Shu-hsüeh wu-li fang fa. Chüan I, Science Press, Peking, 1965 (Chinese). Translated from the English by Ch’ien Min and Kuo Tun-jen. MR 0195654
Todd Dupont, $L^{2}$-estimates for Galerkin methods for second order hyperbolic equations, SIAM J. Numer. Anal. 10 (1973), 880–889. MR 349045, DOI https://doi.org/10.1137/0710073
G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin-New York, 1976. Translated from the French by C. W. John; Grundlehren der Mathematischen Wissenschaften, 219. MR 0521262

S. Emerman and R. A. Stephan, Comment on “Absorbing boundary conditions for acoustic and elastic wave equations” by R. Clayton and B. Engquist, Bull. Seismol. Soc. Amer. 73, 661–665 (1983)

Bjorn Engquist and Andrew Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977), no. 139, 629–651. MR 436612, DOI https://doi.org/10.1090/S0025-5718-1977-0436612-4
Björn Engquist and Andrew Majda, Radiation boundary conditions for acoustic and elastic wave calculations, Comm. Pure Appl. Math. 32 (1979), no. 3, 314–358. MR 517938, DOI https://doi.org/10.1002/cpa.3160320303
American Institute of Physics Handbook, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1957. MR 0095765

M. E. Gurtin, The linear theory of elasticity, Handbuch der Physik v. VIa/2, Springer-Verlag, New York-Heidelberg-Berlin, 1972

G. Hellwig, Partial Differential Equations, Blaisdell Press, 1964

Robert L. Higdon, Numerical absorbing boundary conditions for the wave equation, Math. Comp. 49 (1987), no. 179, 65–90. MR 890254, DOI https://doi.org/10.1090/S0025-5718-1987-0890254-1
Stefan Hildebrandt and Ernst Wienholtz, Constructive proofs of representation theorems in separable Hilbert space, Comm. Pure Appl. Math. 17 (1964), 369–373. MR 166608, DOI https://doi.org/10.1002/cpa.3160170309
George C. Hsiao and Wolfgang L. Wendland, A finite element method for some integral equations of the first kind, J. Math. Anal. Appl. 58 (1977), no. 3, 449–481. MR 461963, DOI https://doi.org/10.1016/0022-247X%2877%2990186-X
Michihiro Kitahara, Boundary integral equation methods in eigenvalue problems of elastodynamics and thin plates, Studies in Applied Mechanics, vol. 10, Elsevier Scientific Publishing Co., Amsterdam, 1985. MR 798471
V. D. Kupradze, Potential methods in the theory of elasticity, Israel Program for Scientific Translations, Jerusalem; Daniel Davey & Co., Inc., New York, 1965. Translated from the Russian by H. Gutfreund; Translation edited by I. Meroz. MR 0223128
Douglas Bradley Meade, Interface problems in elastodynamics, ProQuest LLC, Ann Arbor, MI, 1989. Thesis (Ph.D.)–Carnegie Mellon University. MR 2637397
Eli Sternberg, On the integration of the equations of motion in the classical theory of elasticity, Arch. Rational Mech. Anal. 6 (1960), 34–50 (1960). MR 119547, DOI https://doi.org/10.1007/BF00276152