Identification of Material Parameters for Constitutive Equations

Abstract

This contribution addresses various topics on parameter identification for constitutive equations on the basis of experimental data. Starting from the basic characteristics of inverse problems illustrated by simple examples, four different identification methods are introduced. Then, particular aspects of the least–squares approach are Outlined, such as direct and adjoint state differentiation methods, optimization, discretization error of FEM (finite element method), model error quantification of hierarchical modeling, consequences of instabilities, stochastic simulation, and statistical analysis. Uniform small strain problems as well as non‐uniform large strain problems are considered, where, for the latter, finite element results are incorporated into the optimization process. The examples are concerned with goal‐oriented adaptive refinement and statistical analysis for hierarchical modeling on the basis of an optical method for generation of experimental data in large strain elasticity.

The Good, the Bad and the Ugly: Augmenting a Black-Box Model with Expert Knowledge

“…1) The bad: The first model,Ê bad , is a fully-connected MLP as shown in Figure 1a with a conventional least squares loss function. Such kind of models have already been applied successfully to parameter identification problems with injective backward mappings [2]. However, we expect them to fail for our non-injective problem since the model will in fact be trained to predict the mean of the ambiguities in the data as proven in Ref.…”

“…The hardening stress R depends on the accumulated plastic strain ε and the material model parameters p ≡ (γ 1 , γ 2 , β 1 , β 2 ). The so-called parameter identification problem [2] entails finding material parameters p for a given stress-strain curve C = {R 1 (ε 1 ), . .…”

“…from the domain of stress-strain curves C, to the domain of material parameters P. A summary of typical solution strategies can be found in Ref. [2] and references therein. However, all of these approaches only consider injective backward mappings.…”

Mechanical Properties of Arteries : An In Vivo Parameter Identification Method

“…The inverse problem (IP) does not have a solution since measurement errors exist in general and in this particular case the response function cannot account for the hysteresis present in the pressure-radius data, see Figure 3b, because it is not a function of time. By rewriting the inverse problem as an optimization problem, however, in which the differences between the measurements and the model predictions are minimized, the problem can be solved (Mahnken, 2004). The optimization problem is typically formulated as a least-squares minimization problem according to minimize…”

“…there are many different model parameter combinations which solve (MP) equally well. It is, therefore, not possible to select one set of model parameters over another (Mahnken, 2004).…”