# Tangential Continuous Displacement and Normal-Normal Continuous Stress Mixed Finite Elements for Linear Elasticity

## Dipl.-Ing.in Dr.in Astrid Pechstein

Jan. 9, 2007, 3:30 p.m. T 1010

Abstract. We consider the mixed formulation of linear elasticity, which contains the displacement $u$ as well as the stress tensor $\sigma$ as unknowns,

$\begin{eqnarray} −div(\sigma) & = & f \\ A\sigma & = & \epsilon(u) = \frac 1 2 \left(\nabla u + \nabla u^T \right) \end{eqnarray}$

We derive a variational formulation of the problem, choosing $u \in H(curl)$. For the stress tensor $\sigma$ we do not only need $\sigma \in L^2$, but further

$div \sigma \in H(curl)^∗ = H^{−1}(div).$

We refer to this space as $H(div div)$. We see that the variational problem is well posed. We use the Finite Element Method to discretize the problem. For the displacement, we use tangential-continuous Nédélec ﬁnite elements, whereas for the stresses we propose a new family of ﬁnite elements. These elements are symmetric, tensor-valued, and normal-normal continuous. This formulation is suitable for nearly incompressible materials, where the Poisson ratio $ν$ approaches 1/2. Also, the elements do not suﬀer from shear locking when anisotropic elements are used.

We present shape functions of arbitrary order for the ﬁnite element spaces. To map the reference element onto the physical element, we use transformations that keep the tangential trace for the displacement and the normal-normal trace for the stresses. We see that the discrete system is stable without further stabilization.

In order to obtain a positive deﬁnite system matrix, we hybridize the stress space. This way we ﬁnd a preconditioner that works for nearly incompressible materials.

The solution satisﬁes optimal order error estimates. We discuss the implementation of the new elements, and give numerical results.