Optimal experimental design and optimal modeling for parameter identification of inhomogeneousproblems

数值模拟的可靠预测不仅需要基于物理的数学建模，还需要基于实验数据确定相关的模型常数。然而，实验数据的不足以及模型的不足可能会影响识别参数的稳定性。该应用程序基于以下工作假设：稳定的材料参数是可靠预测的先决条件，从而验证模型。因此，本提案的总体目标是为具有超弹性和塑性偏微分方程的模型的材料参数的稳定识别提供最佳实验设计和最佳模型设计。对确定参数的稳定性和可靠性的评估是基于置信度矩阵和相关的设计函数。设计函数旨在根据实数给出估计的优点，这允许比较不同的估计，从而提供优化的机会。在本研究项目中，使用了从文献中已知的三个优化实验设计函数。最初，控制变量将作为设计变量来控制载荷、几何形状以及测点的位置。对于置信矩阵，考虑了三种选择。对于借助统计计算的置信矩阵，所需要的实验通常是耗时且成本高的。人为增加实验次数的一种方法是使用随机模型生成合成数据。必须指出的是，只有偶然性而没有认识性的不确定性被考虑在内。为了表示合成数据，将现有的基于b样条的空间依赖关系方法扩展到时间依赖关系。研究项目的另一个目的是估计相对于模型结构的稳定参数识别的可靠性，其中材料参数现在也是设计函数的变量。该项目的主要结果将是超弹性和塑性模型的材料参数，在实验和模型设计方面具有优化的置信范围。最后的调查旨在验证上述工作假设。

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.