TY - JOUR

T1 - The numerical solution of the Helmholtz Equation for wave propagation problems in underwater acoustics

AU - Bayliss, A.

AU - Goldstein, C. I.

AU - Turkel, E.

N1 - Funding Information:
Acknowledgements--This work was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract No. DE-AC02-76CH00016. One of the authors was partially supported by the National Aeronautics and Space Administrationu nder NASA Contract NAS 1-17130 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23665.

PY - 1985

Y1 - 1985

N2 - The Helmholtz Equation (- Δ - K2n2)u = 0 with a variable index of refraction, n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. A numerical algorithm has been developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range. The equation is discretized by using the finite element method, thus allowing for the modeling of complicated geometries (including interfaces) and complicated boundary conditions. A global radiation boundary condition is imposed at the far-field boundary that is exact for an arbitrary number of propagating modes. The resulting large, nonselfadjoint system of linear equations with indefinite symmetric part is solved by using the preconditioned conjugate-gradient method applied to the normal equations. A new preconditioner based on the multigrid method is developed. This preconditioner is vectorizable and is extremely effective over a wide range of frequencies, provided the number of grid levels is reduced for large frequencies. A heuristic argument is given that indicates the superior convergence properties of this preconditioner. The relevant limit to analyze convergence is for K increasing and a fixed prescribed accuracy level. The efficiency and robustness of the numerical algorithm are confirmed for large acoustic models, including interfaces with strong velocity contrasts.

AB - The Helmholtz Equation (- Δ - K2n2)u = 0 with a variable index of refraction, n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. A numerical algorithm has been developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range. The equation is discretized by using the finite element method, thus allowing for the modeling of complicated geometries (including interfaces) and complicated boundary conditions. A global radiation boundary condition is imposed at the far-field boundary that is exact for an arbitrary number of propagating modes. The resulting large, nonselfadjoint system of linear equations with indefinite symmetric part is solved by using the preconditioned conjugate-gradient method applied to the normal equations. A new preconditioner based on the multigrid method is developed. This preconditioner is vectorizable and is extremely effective over a wide range of frequencies, provided the number of grid levels is reduced for large frequencies. A heuristic argument is given that indicates the superior convergence properties of this preconditioner. The relevant limit to analyze convergence is for K increasing and a fixed prescribed accuracy level. The efficiency and robustness of the numerical algorithm are confirmed for large acoustic models, including interfaces with strong velocity contrasts.

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U2 - 10.1016/0898-1221(85)90162-2

DO - 10.1016/0898-1221(85)90162-2

M3 - Article

AN - SCOPUS:0021462701

SN - 0898-1221

VL - 11

SP - 655

EP - 665

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

IS - 7-8

ER -