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

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.

A method for determining the parameters in a rheological model for viscoelastic materials by minimizing Tikhonov functionals

“…In particular, the Maxwell element with the smallest relaxation time (here: (µ 1 , τ 1 )) is most affected. This confirms stability investigations obtained in [8].…”

“…However, it is obvious that the reconstructions of (µ 1 , τ 1 ) show a tremendous error, while the other parameters are computed rather stably. This confirms considerations of the authors in [8] where the authors have proven that the reconstruction of small relaxation times always is severely ill-conditioned and the condition We repeat this experiment with 100 different perturbed data sets with the same noise level and plot the values of (µ 1 , τ 1 ) in Figure 6.1. We see that there is a very large variance in values spread over 100 different experiments.…”

“…Therefore, the parameter identification of viscoelastic materials is in great demand and is used in many fields. Results for the identification of material parameters of viscoelastic structures with a linear model can also be found in [8,11]. High-order discontinuous Galerkin methods for anisotropic viscoelastic wave equations are found in [38].…”

Parameter Identification and Validation of Shape-Memory Polymers within the Framework of Finite Strain Viscoelasticity

“…Park and Schapery [ 23 ] used the Prony series to describe the relaxation and creep behavior of a viscoelastic material. Diebels et al [ 24 ] identified the elastic and viscoelastic material parameters from constitutive equations by means of a Tikhonov regularization and inspired an extra penalty term from the stress-strain relationships to expect better results. Haupt et al [ 25 ] used a relatively simple identification method based on the concept of fractional calculus and obtain the model-inherent material parameters.…”