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Stability and convergence of a class of enhanced strain methods. (English) Zbl 0855.73073

The authors are interested in providing a foundation for enhanced strain methods including ones introduced in an earlier paper of the second author and M. S. Rifai [Int. J. Numer. Methods Eng. 29, No. 8, 1595-1638 (1990; Zbl 0724.73222)]. In that paper, the enhanced strains were assumed to belong to a space orthogonal to the space of stresses, and other assumptions were made as well. In the current paper, these assumptions are revisited and generalized, and well-posedness of the resulting discrete problem is shown to be their consequence. In addition, the authors show that, in the incompressible limit, the enhanced strain formulation can stabilize the unstable “bilinear displacement and discontinuous linear pressure” element, giving its behavior similar to the behavior of “bilinear displacement and discontinuous constant pressure” element.
The authors consider a boundary value problem of linear elasticity in a domain in two- or three-dimensional space. This domain is polygonal and can be described as a union of affine images of a single square or cubic finite element. To the usual spaces of stresses and strains the authors add the so-called “enhanced strains”. While the additional variables do not greatly change the continuous problem, their presence changes the discrete problem in a fundamental manner. In the discrete case, the enhanced strains are drawn from a space which is orthogonal to the usual space of strains, are not images of admissible displacements, and are orthogonal in a norm weighted by polynomials of low enough order in the elastic modulus. These conditions are sufficient to prove the error estimates from which the well-posedness can be concluded.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Citations:

Zbl 0724.73222
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