Tailored derivative-based optimization for the characterization of thermal and piezoelectric material behaviour

时至今日，压电陶瓷材料参数的可靠表征仍然是一个重大的挑战。由于过渡到无铅材料的变体，这仍然是一个重要的问题，特别是在实际应用中。从数学的角度来看，有必要解决一个必须明确处理非线性材料行为的逆问题。子项目OPT的目标是对所得到的优化问题进行理论分析，以发展该问题的伴随微积分。由于状态方程的性质非常重要，因此将与子项目MESS和ANA进行密切合作。基于这些理论见解，将开发和分析用于高功率超声应用的压电陶瓷材料参数表征的适应性优化策略。由于基于梯度的优化方法将应用于表征，因此特别关注在算法微分的帮助下一致导数信息的有效计算，作为“离散然后优化”方法的一部分。为了解决逆问题固有的不适定性，本文还特别强调了优化初始猜测的确定以及定制化正则化策略的推导。作为子项目SIM背景下的密集合作的一部分，开发的方法将集成到HighPerMeshes框架中。这也将为所获得的研究成果在所考虑的应用之外的利用铺平道路。在整个项目生命周期中，研究小组将开发模型问题的层次结构。这将允许在不同难度的问题上验证优化策略。此外，在与子项目SIM和MESS的合作下，所开发的方法将被应用于表征尽可能真实的模型。在此基础上，用较少的简化量实现对非线性行为的描述。

To this day the reliable characterization of material parameters of piezoceramics still poses a significant challenge. Due to the transition to lead-free material variants this remains an important issue especially for practical applications. From a mathematical point of view it is necessary to solve an inverse problem where the nonlinear material behaviour must be handled explicitly. The goal of the subproject OPT is a theoretical analysis of the resulting optimization problem in order to develop adjoint-calculus for this problem. As the properties of the state equation are fundamentally important there will be an intensive cooperation with subprojects MESS and ANA. Based on these theoretical insights an adapted optimization strategy for the characterization of the material parameters of piezoceramic materials for high-power ultrasonic applications shall be developed and analyzed. As gradient-based optimization methods will be applied for the characterization there is a special focus on the efficient computation of consistent derivative information with the help of Algorithmic Differentiation as part of a 'discretize-then-optimize' approach. In order to deal with the inherent ill-posedness of inverse problems special emphasis is also put on the determination of appropriate initial guesses for the optimization as well as the derivation of tailored regularization strategies. As part of an intensive cooperation in the context of subproject SIM the developed approaches will be integrated into the HighPerMeshes framework. This will also pave the way for utilization of obtained research results beyond the applications considered. During the whole project life span the research group will develop a hierarchy of model problems. This will allow for validation of the optimization strategies with problems of varying difficulty. Furthermore, in cooperation with subprojects SIM and MESS the methods developed will be applied to characterize models that are as realistic as possible. Based on the results obtained, it is intended to achieve a description of the nonlinear behaviour with few simplified quantities.