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A posteriori error estimators related to equilibrium defaults of finite element solutions for elastostatic problems. (English) Zbl 0917.73074

Summary: This paper presents a study of two types of a posteriori error estimator based on equilibrium defaults in finite element solutions of planar linear elastic problems. The \(\widetilde{G}\)-type estimator, developed at LTAS-Infographie, uses explicit relations between the equilibrium defaults and the energy of the error. This type is represented here with a new physical interpretation, and with the reference tractions on the sides of elements redefined so as to satisfy equilibrium. The \(\widehat\sigma\)-type estimator uses a reference solution for stress \(\widehat\sigma\), which has no equilibrium defaults, i.e. \(\widehat\sigma\) is a statically admissible stress field. The reference solutions considered in this paper are based on the formulation first proposed by Ladèveze. The present study considers several alternative formulations of equilibrating tractions suitable for both types of estimator, and it is restricted to errors in models composed of 4-noded bilinear Lagrange elements. Numerical results are presented for several examples with smooth or singular stress fields. It is observed that significant improvements to the effectiveness of the \(\widetilde{G}\)-estimator can be obtained, particularly for coarser meshes. With judicious choice of equilibrating tractions it is concluded that this estimator can be more effective than the \(\widehat\sigma\)-estimator.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74B10 Linear elasticity with initial stresses
65N15 Error bounds for boundary value problems involving PDEs
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