Mechanical conditions for stability and optimal convergence of mixed finite elements for linear plane elasticity. (English) Zbl 0729.73208

Stein, Erwin; Rolfes, R.

Comput. Methods Appl. Mech. Eng. 84, No. 1, 77-95 (1990).

Summary: In order to develop an efficient and manageable tool for checking stability and optimal convergence of mixed finite elements (LBB- and ‘equilibrium’ condition), three mechanical conditions are stated. The first requires the continuity of the normal component of the stress tensor across interelement boundaries, the second forbids spurious modes on a two element patch, and the third is to avoid zero-energy-stresses on an element. The mathematical proof shows that the conditions are necessary and sufficient. Finally, the hybrid implementation of two plane mixed elements is carried out, and comparisons are made with two standard displacement elements. In particular, the mixed element with constant displacement shape functions (MMC) surpasses the linear displacement element by far and also the quadratic displacement element if the computational effort is compared.

MSC:

74S05

Finite element methods applied to problems in solid mechanics

Keywords:

LBB- and ‘equilibrium’ condition; continuity of the normal component of the stress tensor; interelement boundaries; forbids spurious modes on a two element patch; avoid zero-energy-stresses; hybrid implementation of two plane mixed elements

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