Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 1895208, 11 pages

https://doi.org/10.1155/2018/1895208

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

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

Correspondence should be addressed to Thomas Schuster

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.

#### 1. Introduction

A material modeling process cannot be seen as a simple and single procedure, since it consists of multiple tasks. Usually, the beginning is the determination of the material behavior out of experimental data. In this step of a material modeling process, the data basis for the further work is established. It has to be decided if elastic, plastic, or viscose effects or combinations of these have to be included. The underlying structure of the model is established on this choice. The mathematical description has to follow physical principles, so that in a second step, the theoretical background from continuum mechanics has to be considered, typically leading to a model which consists of a coupled set of nonlinear differential equations. Subsequently, the outcoming material model has to be realized numerically. Based on these results, it is now possible to quantitatively compare the simulation to the experiments, which yields a parameter identification. In general, an inverse method is proposed, which determines the model parameters from the minimization of the error between model and experiment.

This last point, the parameter identification, is mainly examined in this contribution. Most of the attention is usually dedicated to the three firstly mentioned steps: experiment, theory, and numerics, whereby the final point of the identification process is often disregarded. Especially in some recent works, it comes up that the importance of the parameter finding increases more and more since the models try to include more material characteristics often resulting in a high number of model parameters. Moreover, some previous contributions have shown that a mechanical characterization only based on uniaxial data is not sufficient for a realistic description of three-dimensional, inhomogeneous problems. For details concerning the necessity of a multiaxial approach the reader is referred to the contributions of, for example, Baaser et al. [1–3], Johlitz and Diebels [4], and Seibert et al. [5]. In a model based only on uniaxial data the simulations for a parameter identification can be executed on ordinary geometries, such as simple cubes, whereby in a multiaxial description the complete specimen geometry and the resulting inhomogeneities have to be considered. This results in inverse calculations, where the detailed experimental conditions are reproduced in the numerics and subsequently compared to the simulation.

A similar effort is necessary when a geometrical structure has to be represented. This is the case when a complete geometry of an assembly is investigated [6, 7] or if a microstructure is examined, for example, for composites [8–10] or foams [11].

Both the increasing model complexity and the necessity of multiaxial approaches result in a more complex simulation which needs high effort to be solved. Finally, the computational costs are very high and lead to long-lasting computations. Regarding the parameter identification, many simulations have to be executed to find a matching parameter set. Hence, it is definitely recommended to treat the identification process much more attentively. An efficient parameter identification can reduce the duration for a finished modeling process with matching parameters a lot.

In a material modeling process it is usual to invest a high effort in the data acquisition in the experiments and the model description, whereby it is a common method to use only very simple optimisation algorithms for a parameter finding. Stochastic methods such as evolution strategies [12, 13] or the* pattern search*-algorithm [14, 15] as well as the method* fminsearch* based on the Nelder-Mead simplex algorithm [16] are often applied. The advantage of all of these methods is that no gradient information of the model is needed so that the effort in realizing and starting the parameter identification process is very low, whereby the performance of the algorithms is normally not satisfying. Concerning the realization it is only necessary to run the simulation and to define a fitness function which is able to compare the experimental and numerical results adequately. Due to the ill-posedness of the inverse problem, small errors in the input data of the identification may lead to large errors in the model parameters. This problem can be overcome by the introduction of appropriate regularization strategies. Regularization methods are stable solution strategies for inverse problems. They assure that the computed solution fits the given mathematical model and at the same time attenuate the noise that is contained in the measured data. Theory and application of regularization methods in Hilbert spaces are well-founded; we refer to the standard textbooks [17–19]. In the last decade the theory was extended to Banach spaces; see [20]. Amongst the most popular regularization tools are Tikhonov functionals [21]. These consist of two parts, a data fitting term and a penalty term. The latter is responsible for the stability and at the same time for the incorporation of any a priori information. That is why Tikhonov functionals might be an interesting tool to tackle parameter identification problems connected to constitutive equations of elastic and viscoelastic materials. This is the research hypothesis to be validated in the present article. We construct Tikhonov functionals that are adapted to our models and demonstrate the performance of Tikhonov regularization in different settings and condition numbers of the underlying system.

The applied methods and their advantages are shown with respect to a material model according to the work of Scheffer [22]. In order to focus on the process itself the data bases for the identification are not the original experimental values, where signal noise and statistics play an important role. Moreover, it is not finally sure whether the resulting parameters in the work of Scheffer [22] represent the optimal set since the model is of high complexity in order to describe a nonlinear viscoelastic material behavior depending on the deformation velocity. Hence, for this article the model with the already identified parameters is taken as the reference and the parameters are reidentified. As the main advantage the exact values of the parameters are known and it can be proven if the proposed methods find this exact solution of the problem.

*Organization of the Article*. In Section 2 the theoretical aspects concerning the material description and the resulting constitutive equations are discussed. In Section 3 the problem of parameter identification as an inverse problem is addressed. Section 4 verifies the usage of the proposed methods by showing theoretical results for simplified material models as well as by reconstructing given material parameters. The paper is concluded with a summary and an outline of future research in Section 5.

#### 2. Mathematical Model

A new approach for identifying material parameters is discussed here. There are many viscoelastic materials, and different models exist to characterize them fully. To test the approach, the following model is used: It is inspired by real life experiments and is based on a rheological model, where a spring is positioned in parallel to Maxwell elements. Each of the Maxwell elements consists of a spring connected in series to a dashpot; compare, Figure 1 (Figure 1 is reprinted by permission from Springer Customer Service Centre GmbH: [23]).