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Mathematical Problems in Engineering
Volume 2018, Article ID 1895208, 11 pages
https://doi.org/10.1155/2018/1895208
Research Article

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

1Saarland University, 66041 Saarbrücken, Germany
2Department of Mathematics, Saarland University, 66041 Saarbrücken, Germany
3Mathematical Image Analysis Group, Saarland University, 66041 Saarbrücken, Germany

Correspondence should be addressed to Thomas Schuster; ed.bs-inu.mun@retsuhcs.samoht

Received 20 July 2017; Revised 8 December 2017; Accepted 10 January 2018; Published 18 February 2018

Academic Editor: Mohsen Asle Zaeem

Copyright © 2018 Stefan Diebels et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.