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

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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.