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A finite element method for large domains. (English) Zbl 0687.73065
Summary: A combined analytical and numerical method is devised to solve elliptic boundary value problems in large or infinite domains. First the domain is divided by an artificial boundary \({\mathcal B}\) into a small computational domain \(\Omega\) and a large or infinite residual domain D. In D the problem is solved analytically, and from the solution an exact nonlocal relation between the solution and its derivatives on \({\mathcal B}\) is deduced. This relation is used as a boundary condition to complete the formulation of a problem in \(\Omega\) which has exactly the same solution there as the original problem. Then a finite element formulation of this new problem in \(\Omega\) is presented. The exact nonlocal boundary condition is given explicitly for Laplace’s equation, for the equations of plane stress and plane strain in linear elastostatics, and for certain equations governing beams and axisymmetric cylindrical shells. The artificial boundary is chosen to be a sphere or a circle. The properties and computational implications of the boundary condition are discussed. Some numerical examples are presented, and the results are compared with those obtained by the standard finite element method using different approximate local boundary conditions on the artificial boundary.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Software:
DLEARN
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References:
[1] Bathe, K.J.; Wilson, E.L., Numerical methods in finite element analysis, (1976), Prentice-Hall Englewood Cliffs, NJ · Zbl 0528.65053
[2] Bettess, P., Infinite elements, Internat. J. numer. methods engrg., 11, 53-64, (1977) · Zbl 0362.65093
[3] Zienkiewicz, O.C., The finite element method, (1977), McGraw-Hill Maidenhead, UK · Zbl 0435.73072
[4] Cruse, T.; Rizzo, F., Boundary integral equation method: computational applications in applied mechanics, Asme amdii, (1975)
[5] Brebbia, C.A., The boundary element method for engineers, (1978), Pentech Press London · Zbl 0414.65060
[6] Zienkiewicz, O.C.; Kelly, D.W.; Bettess, P., The coupling of the finite element method and boundary solution procedures, Internat. J. numer. methods engrg., 11, 355-375, (1977) · Zbl 0347.65048
[7] Shaw, R.P.; Falby, W., FEBIE—A combined finite element boundary integral equations method, Comput. & fluids, 6, 153-160, (1978) · Zbl 0385.76027
[8] Margulies, M., Progress in boundary element methods, (), 258-288, Chapter 8
[9] Johnson, C.; Nedelec, J.C., On the coupling of boundary integrals and finite element methods, Math. comp., 35, 1063-1079, (1980) · Zbl 0451.65083
[10] Greenspan, D.; Werner, P., A numerical method for the exterior Dirichlet problem for the reduced wave equation, Arch. rational mech. anal., 23, 288-316, (1967) · Zbl 0161.12701
[11] Keller, J.B.; Givoli, D., An exact nonreflecting boundary condition, J. comput. phys., 82, 172-192, (1989) · Zbl 0671.65094
[12] Fix, G.J.; Marin, S.P., Variational methods for underwater acoustic problems, J. comput. phys., 28, 253, (1978) · Zbl 0384.76048
[13] MacCamy, R.C.; Marin, S.P., A finite element method for exterior interface problems, Internat. J. math. math. sci., 3, 311-350, (1980) · Zbl 0429.65108
[14] Marin, S.P., Computing scattering amplitudes for arbitrary cylinders under incident plane waves, IEEE trans. antennas and propagation AP-30, 1045, (1982)
[15] Marin, S.P., A finite element method for problems involving the Helmholtz equation on two dimensional exterior regions, ()
[16] Goldstein, C.I., A finite element method for solving Helmholtz type equations in wave guides and other unbounded domains, Math. comp., 39, 309-324, (1972)
[17] Givoli, D., A finite element method for large domain problems, () · Zbl 0825.73732
[18] Morse, P.M.; Feshbach, H., Methods of theoretical physics, (1953), McGraw-Hill New York · Zbl 0051.40603
[19] Hughes, T.J.R.; Ferencz, R.M.; Raefsky, A.M., (), Chapter 11
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