# zbMATH — the first resource for mathematics

Non-reflecting boundary conditions for elastic waves. (English) Zbl 0708.73012
Summary: An exact non-reflecting boundary condition is devised for time-harmonic two-dimensional elastodynamics in infinite domains. The domain is made finite by the introduction of a circular artificial boundary on which this exact condition is imposed. In the finite computational domain a finite element method is employed. Numerical examples are presented in which the accuracy and efficiency of the method using the exact non-local boundary condition are compared with those of methods based on approximate local boundary conditions. The method is also used to solve problems in large finite domains by reducing them to smaller domains. In addition, local boundary conditions are derived which are exact for waves with a limited number of angular Fourier components.

##### MSC:
 74J10 Bulk waves in solid mechanics 74S05 Finite element methods applied to problems in solid mechanics 74J15 Surface waves in solid mechanics 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text:
##### References:
 [1] Engquist, B.; Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Math. comp., 31, 629-652, (1977) · Zbl 0367.65051 [2] Bayliss, A.; Turkel, E., Radiation boundary conditions for wave-like equations, Comm. pure appl. math., 33, 707-725, (1980) · Zbl 0438.35043 [3] Feng, K., Finite element method and natural boundary reduction, (), 1439-1453 · Zbl 1003.65085 [4] Ting, L.; Miksis, M.J., Exact boundary conditions for scattering problems, J. acoust. soc. am., 80, 1825-1827, (1986) [5] Higdon, R.L., Numerical absorbing boundary conditions for the wave equation, Math. comp., 49, 65-90, (1987) · Zbl 0654.65083 [6] MacCamy, R.C.; Marin, S.P., A finite element method for exterior interface problems, Int. J. math. math. sci., 3, 311-350, (1980) · Zbl 0429.65108 [7] Goldstein, C.I., A finite element method for solving Helmholtz type equations in wave guides and other unbounded domains, Math. comp., 39, 309-324, (1982) · Zbl 0493.65046 [8] Keller, J.B.; Givoli, D., Exact non-reflecting boundary conditions, J. comp. phys., 82, 172-192, (1989) · Zbl 0671.65094 [9] Lysmer, J.; Kuhlemeyer, R.L., Finite dynamic model for infinite media, J. eng. mech. div. ASCE, 95, 859-877, (1969) [10] Smith, W.D., A nonreflecting plane boundary for wave propagation problems, J. comp. phys., 15, 492-503, (1974) · Zbl 0287.73024 [11] Cerjan, C.; Kosloff, D.; Kosloff, R.; Reshef, M., A nonreflecting boundary condition for discrete acoustic and elastic wave equations, Geophysics, 50, 705-708, (1985) [12] Sochacki, J.; Kubicheck, R.; George, J.; Fletcher, W.R.; Smithson, S., Absorbing boundary conditions and surface waves, Geophysics, 52, 60-71, (1987) [13] Engquist, B.; Majda, A., Radiation boundary conditions for acoustic and elastic wave calculations, Comm. pure app. math., 32, 313-357, (1979) · Zbl 0387.76070 [14] Clayton, R.W.; Engquist, B., Absorbing boundary conditions for acoustic and elastic wave equations, Bull. seismol. soc. amer., 67, 1529-1540, (1977) [15] Scandrett, C.L.; Kriegsmann, G.A.; Achenbach, J.D., Application of the limiting amplitude principle to elastodynamic scattering problems, SIAM J. sci. stat. comp., 7, 571-590, (1986) · Zbl 0585.73044 [16] R.L. Higdon, “Absorbing boundary conditions for elastic waves,” SIAM J. Num. Anal., to appear. · Zbl 0955.65061 [17] Givoli, D.; Keller, J.B., A finite element method for large domains, Comp. math. appl. mech. eng., 76, 41-66, (1989) · Zbl 0687.73065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.