# A domain decomposition method for geometrically nonlinear elasticity

## Klemens Reindl

Feb. 1, 2013, 2:30 p.m. S2 416

The goal of this diploma thesis is the numerical solution of the geometrically nonlinear elasticity equation by finite elements. The well-known simplification of linearized elasticity is only valid for small deformations (or small forces). For large deformations, the displacement is large, the quadratic term in the strain tensor cannot be anymore neglected, and so the problem becomes nonlinear. Usually, one solves the corresponding discrete system with the Newton method, where in each step, a new stiffness matrix has to be assembled.

In the scope of the diploma thesis, we want to exploit situations where the displacement is small under a local point of view (i.e. relative to its neighborhood) and reduce the degree of nonlinearity. We use the established Floating Frame of Reference Formulation (FFRF): by partitioning the computational domain into subdomains (frames), we can divide the total displacement into a rigid displacement of the corresponding frame and a (small) local displacement relative to the frame. For the latter, we are then allowed to linearize the strain tensor. The overall system is still nonlinear, but the nonlinearity is restricted to the (few) rigid displacements of the frames.

To solve the system efficiently, we use a combination of FFRF and the finite element tearing and interconnecting (FETI) method. The advantage of this approach is, that the stiffness matrix of each frame stays constant in the whole scheme, and thus needs to be assembled and factorised only once. In each Newton step, only a small system corresponding to the rigid frame displacements has to be assembled and solved. Finally, we show numerical results for a 2D model problem.