Tobias Scheffer

Lagrangian Strain Tensor Computation with Higher Order Variational Models.

The reliable estimation of the Lagrangian stress tensor from an image sequence is a challenging problem in mechanical engineering. Since this tensor involves first order motion derivatives, it appears tempting to estimate the optical flow field with a highly accurate variational model and compute its derivatives afterwards. In this paper we explain why this idea is inappropriate due to lower order smoothness assumptions and the ill-posedness of differentiation. As a remedy, we propose a variational framework that performs higher order regularisation of the optical flow field and directly computes the Lagrangian stress tensor from the image measurements. Due to its recursive structure, this framework is very generic. It can incorporate smoothness assumptions of arbitrary high order and allows to compute derivatives of any desired order in a stable way. With a biaxial tensile experiment with an elastomer we demonstrate that our novel approach gives substantially better results for the Lagrangian stress tensor than computing derivatives of the optical flow field. Moreover, it also outperforms a frequently used commercial software that marks the state-of-the-art for Lagrangian stress tensor computation.

Identifying Elastic and Viscoelastic Material Parameters by Means of a Tikhonov Regularization

For studying the interaction of displacements, stresses, and acting forces for elastic and viscoelastic materials, it is of utmost importance to have a decent mathematical model available. Usually such a model consists of a coupled set of nonlinear differential equations together with appropriate boundary conditions. However, since the different material classes vary significantly with respect to their physical and mechanical behavior, the parameters which appear in these equations are unknown and therefore have to be determined before the equations can be used for further investigations or simulations. It is this very step which is addressed in this article where we consider elastic as well as viscoelastic material behavior. The idea is to compute the parameters as solutions of a minimization problem for Tikhonov functionals. Tikhonov regularization is a well-established solution technique for tackling inverse problems. On the one hand, it assures a computation that is stable with respect to noisy input data, and on the other hand, it involves desired a priori information on the solution. In this article we develop problem adapted Tikhonov functionals and prove that a Tikhonov regularization improves the accuracy especially when the underlying system is ill-conditioned.