# A New Family of Mixed Finite Elements for Elasticity

## Dipl.-Ing.^{in} Dr.^{in} Astrid Pechstein

**Jan. 20, 2009, 3 p.m. MZ 005A**

In our thesis, we introduce a new ﬁnite element method to discretize the equations of elasticity. It is analyzed thoroughly both in the inﬁnite-dimensional and in the discrete setting, where ﬁnite element schemes of arbitrary order are presented. As the main result of this work, we prove that the new method is locking-free with respect to volume and shear locking, i.e that it is applicable for both nearly incompressible materials and the discretization of the thin structures using ﬂat elements.

To date, several well-known methods for the discretization and subsequent solution of the equations of elasticity have been introduced: the primal method using continuous ﬁnite element functions for the displacement, the mixed method due to Hellinger and Reissner, and mixed methods with weak symmetry. However, each of these methods has its drawbacks, which motivates the need for yet another formulation. The primal method suﬀers from volume and shear locking phenomena. Mixed methods can be shown to be stable with respect to volume locking; however, the construction of ﬁnite elements can only be done at high polynomial orders, and therefore high computational cost. Mixed methods with weak symmetry are easier to construct, but the exact symmetry of the stress tensor is lost.

The new method lies “in between” the primal method and the mixed Hellinger-Reissner method. The vector-valued displacement function is chosen in the space $H$(curl), which ensures continuity of the tangential component across interfaces. This choice implies to search for the stresses in the newly introduced space $H$(div div) consisting of symmetric tensor valued functions, whose divergence again allows for a distributional divergence lying in $H^{−1}$ . It is shown that such tensor ﬁelds have their normal-normal component continuous. The resulting mixed formulation is referred to as “Tangential-Displacement-Normal-Normal-Stress (TD-NNS) formulation”. Its stability in the inﬁnite-dimensional setting is analyzed.

To discretize the TD-NNS formulation, a stable pair of ﬁnite elements is introduced. For the displacement space, Nédélec elements as standard choice for an $H$(curl) conforming discretization are used. For the stress space, a new family of symmetric tensor-valued ﬁnite elements of arbitrary order is constructed. Stability and optimal order approximation properties of the resulting ﬁnite element method are provided.

A major drawback of mixed methods is the indeﬁniteness of the resulting system matrix. In order to obtain a positive deﬁnite matrix, we apply hybridization. The normal-normal continuity of the stresses is torn, and re-enforced by Lagrangian multipliers resembling the normal displacement on interfaces. After local elimination of the stresses, one obtains a positive deﬁnite matrix. In this setup, stability with respect to volume locking is provided. An additive Schwarz block preconditioner is shown to be an optimal choice.

Finally, the TD-NNS method is used on thin structures, where ﬂat prismatic elements are used. Stability and error estimates which are independent of the aspect ratio of the elements are derived. All results are conﬁrmed by computational examples.