×

The multigrid method in solid mechanics. I: Algorithm description and behaviour. (English) Zbl 0724.73269

Summary: A multigrid algorithm is described that can be used to obtain the finite element solution of linear elastic solid mechanics problems. The method is applied to some two-dimensional problems to evaluate its strenths and weaknesses. Extensive studies are made to determine the convergence behaviour of the method. In general, this depends on many factors: the number of degrees-of-freedom in the discretization, characteristics of the algorithm, Poisson’s ratio when it is closed to \(0\cdot 5\), the amount of bending deformation in the problem under consideration, and the degree of nonuniformity in the mesh. Only certain values of the multigrid parameters allow a converged solution to be obtained with a computational effort proportional to the number of degrees-of-freedom. These values include the optimum ones, i.e. those that lead to convergence with the least computational effort. The constant of proportionality is only independent of the number of degrees-of-freedom and still depends on the other factors listed above.

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
74S05 Finite element methods applied to problems in solid mechanics
65F10 Iterative numerical methods for linear systems

Citations:

Zbl 0724.73270
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] and , Finite Element Solution of Boundary Value Problems, Academic Press, New York, 1984.
[2] Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1982.
[3] Braess, Comp. Mech. 3 pp 321– (1988)
[4] Brandt, Math. Comp. 31 pp 333– (1977)
[5] Multi-Grid Methods and Applications, Springer-Verlag, Berlin, 1985. · Zbl 0595.65106
[6] and (eds.), Multigrid Methods, Springer-Verlag, Berlin, 1981.
[7] and (eds.) Multigrid Methods II, Springer-Verlag, Berlin, 1985.
[8] and , Applied Iterative Methods, Academic Press, New York, 1981. · Zbl 0459.65014
[9] The Finite Element Method, Prentice-Hall, Englewood Cliffs, N.J., 1987.
[10] Parsons, Int. j. numer. methods eng. 29 pp 739– (1990)
[11] and , ’Multigrid methods: Fundamental algorithms, model problem analysis and applications’, in and (eds.), Multigrid Methods, Springer-Verlag, Berlin, 1982, pp. 1-176.
[12] The Finite Element Method, McGraw-Hill, New York, 1983.
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.