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

直到今天，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变体，这仍然是一个重要的问题，特别是对于实际应用。从数学的角度来看，有必要解决一个反问题，其中的非线性材料行为必须明确处理。子项目OPT的目标是对由此产生的优化问题进行理论分析，以便为该问题开发伴随演算。由于状态方程的性质至关重要，因此将与MESS和ANA子项目进行密切合作。基于这些理论见解，应开发和分析用于表征高功率超声应用的压电陶瓷材料的材料参数的自适应优化策略。由于基于梯度的优化方法将被应用于表征，因此特别关注在作为“离散化然后优化”方法的一部分的数学微分的帮助下对一致的衍生信息的有效计算。为了处理反问题的固有不适定性，特别强调的是还把适当的初始猜测的优化以及定制的正则化策略的推导确定。作为在SIM子项目背景下的密集合作的一部分，所开发的方法将被纳入HighPerMeshes框架。这也将为在所考虑的应用之外利用所获得的研究成果铺平道路。在整个项目生命周期中，研究小组将开发一个模型问题的层次结构。这将允许验证的优化策略与不同难度的问题。此外，在SIM和MESS次级项目的合作下，将应用所开发的方法来描述尽可能现实的模型。基于所获得的结果，它是为了实现与几个简化量的非线性行为的描述。

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.