Abstract
The tangential-displacement normal-normal-stress (TDNNS) method is a finite element method for mixed elasticity. As the name suggests, the tangential component of the displacement vector as well as the normal-normal component of the stress are the degrees of freedom of the finite elements. The TDNNS method was shown to converge of optimal order, and to be robust with respect to shear and volume locking. However, the method is slightly nonconforming, and an analysis with respect to the natural norms of the arising spaces was still missing. We present a sound mathematical theory of the infinite dimensional problem using the space {{mathbf {H}}}(mathbf {curl}) for the displacement. We define the space {underline{{mathbf {H}}}}({text {div}},mathbf {{div}}) for the stresses and provide trace operators for the normal-normal stress. Moreover, the finite element problem is shown to be stable with respect to the {{mathbf {H}}}(mathbf {curl}) and a discrete {underline{{mathbf {H}}}}({text {div}},mathbf {{div}}) norm. A-priori error estimates of optimal order with respect to these norms are obtained. Beside providing a new analysis for the elasticity equation, the numerical techniques developed in this paper are a foundation for more complex models from structural mechanics such as Reissner Mindlin plate equations, see Pechstein and Schöberl (Numerische Mathematik 137(3):713–740, 2017).
Highlights
In [19], we introduced the tangential-displacement normal-normal-stress (TDNNS) method for treating the problem of linear elasticity
In [19], we introduced the TDNNS method for treating the problem of linear elasticity
We showed that the TDNNS method is capable of overcoming shear locking [20] and volume locking [22]
Summary
In [19], we introduced the TDNNS method for treating the problem of linear elasticity. The TDNNS method is slightly nonconforming, as the stress finite elements are not in the infinite-dimensional distributional space H(div div), which was introduced in [19]. The analysis of TDNNS finite elements provided in our former work [19,20] is based on discrete, broken norms rather than the natural norms of the infinite-dimensional spaces H(curl) and H(div div). We want to provide an analysis based on the natural norms of the Sobolev spaces This analysis takes the fact that the stress space is nonconforming into account, and leads to optimal order a-priori error estimates. The necessity of this new framework becomes evident in the analysis of problems in structural mechanics. Compared to the TDNNS formulation, the rotation takes the role of the displacement, while the symmetric stress tensor in H (div div) is replaced by the tensor of bending and twisting moments
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