Astrid S. Pechstein

Astrid S. Pechstein

Recently, "Tangential Displacement Normal Normal Stress" (TDNNS) elements were introduced for small-deformation piezoelectric structures. Benefits of these ele- ments are that they are free from shear locking in thin structures and volume locking for nearly incompressible materials. We extend these elements to the large defor- mation case for electro-active polymers in the framework of an updated Lagrangian method. We observe that convergence does not deteriorate as the material becomes nearly incompressible with growing Lamé parameter $λ$, and that the discretization of slender structures by flat volume elements is feasible. The elements are freely available in the open source software package Netgen/NGSolve.

Astrid S. Pechstein, Joachim Schöberl

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 H(curl) for the displacement. We define the space H(div 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 H(curl) and a discrete H(div div) norm. A-priori error estimates of optimal order with respect to these norms are obtained.

Dietrich Braess, Astrid S. Pechstein, J. Schöberl

We develop an a posteriori error estimator for the Interior Penalty Discontinuous Galerkin approximation of the biharmonic equation with continuous finite elements. The error bound is based on the two-energies principle and requires the computation of an equilibrated moment tensor. The natural space for the moment tensor consists of symmetric tensor fields with continuous normal-normal components. It is known from the Hellan-Herrmann-Johnson (HHJ) mixed formulation. We propose a construction that is totally local. The procedure can also be applied to the original HHJ formulation, which directly provides an equilibrated moment tensor.