Optimal experimental design and modeling for parameter identification for inhomogeneous problems

The reliable prediction of numerical simulations requires not only physically based mathematical modeling but also determination of the associated model constants based on experimental data. However, deficiencies of experimental data as well as model deficiency may have an impact on the stability of the identified parameters. This application is based on the following working hypothesis: Stable material parameters are a prerequisite for a reliable prediction and in this way for the validation of a model. The overall aim of this proposal is therefore an optimal experimental design as well as an optimal model design for stable identification of material parameters for models with partial differential equations for hyperelasticity and plasticity.

The assessment of stability and thus reliability of the identified parameters is based on a confidence matrix and an associated design function. The design function is intended to give the goodness of the estimation in terms of a real number, which allows to compare different estimations and thus provides the opportunity for optimization. In this research project, three design functions known from the literature on optimal experimental design are used. Initially, control variables will be used as design variables to control load, geometry as well as position of the measuring points. Regarding the confidence matrix, three options are considered.

For confidence matrices calculated by aid of statistics, the required experiments are usually time consuming and cost intensive. One way to artificially increase the number of experiments is to generate synthetic data using a stochastic model. It has to be noted, that only aleatory but no epistemic uncertainties are taken into account. To represent the synthetic data, an existing method based on B-splines for spatial dependencies is extended to time dependencies.

Another aim of the research project is to estimate the reliability of stable parameter identification with respect to the model structure, whereby material parameters are now also variables of the design function.

The main result of the project will be material parameters for models of hyperelasticity and plasticity with optimized confidence ranges with respect to experimental and model design. Final investigations are intended to verify the above working hypothesis.