×

A simple infinite element. (English) Zbl 0619.65097

Infinite elements provide one of the most attractive alternatives for dealing with differential equations in unbounded domains. The region where loads, sources, inhomogeneities and anisotropies exist is modelled by finite elements and the far, uniform region is represented by infinite elements.
We propose a new infinite element which can represent any type of decay towards infinity. The element is so simple that explicit expressions can be obtained for the element matrix in many cases, yet large improvements in the accuracy of the solution are obtained as compared with the truncated mesh. Explicit expressions are in fact given for the Laplace equation and \(1/r^ n\) decay. The element is conforming with linear triangles and bilinear quadrilaterals in two dimensions.
The element can be used with any standard finite-element program without having to modify the shape function library or the numerical quadrature library of the program. The structure or bandwidth of the stiffness matrix of the finite portion of the mesh is not modified when the infinite elements are used. An example problem is solved and the solution found to be better than several other methods in common usage. The proposed method is thus highly recommended.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] McDonald B.H., Finite element solution of unbounded field problems (1972)
[2] DOI: 10.1137/0715030 · Zbl 0402.65060 · doi:10.1137/0715030
[3] McDonald B.H., Finite Elements in Electrical pp 161– (1980)
[4] Zienkiewicz O.C., Proc. ASME Winter Meeting, Washington D.C. (1981)
[5] Ungless R.F., M. A. Sc. Thesis
[6] Bettess P., Engng. 11 pp 53– (1977)
[7] Bettess P., Engng. 11 pp 1271– (1977)
[8] DOI: 10.1002/eqe.4290060404 · doi:10.1002/eqe.4290060404
[9] Bettess P., Engng. 15 pp 1613– (1980)
[10] Lynn P.P., Engng. 17 pp 347– (1981)
[11] Chow Y.K., Engng. 17 pp 503– (1981)
[12] Beer G., Engng. 17 pp 43– (1981)
[13] Medina F., Engng. 17 pp 1177– (1981)
[14] Emson C., Proc. ASME Winter Meeting, Washington D.C. (1981)
[15] Pissanetzky S., Engng. 19 pp 913– (1983)
[16] S. Pissanetzky, MAGNUS: Computer-aided design of electromagnets. Implementation of the twoscalar-potentials method at the Central Scientific Computing facility of the Brookhaven National Laboratory, BNL Report 28416 (1980).
[17] Pissanetzky S., Solution of three-dimensional, anisotropic, nonlinear problems of magnetostatics using two scalar potentials, finite and infinite multipolar elements and automatic mesh generation (1982)
[18] Kagawa Y., J. Engng. Design 1 pp 1– (1983)
[19] DOI: 10.1108/eb009982 · Zbl 0619.65108 · doi:10.1108/eb009982
[20] Zienkiewicz O.C., Engng. 11 pp 355– (1977)
[21] DOI: 10.1108/eb009972 · doi:10.1108/eb009972
[22] Pissanetzky S., Engng. 17 pp 255– (1981)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.