Efficient localization indicators in computational inelasticity - from enhanced tensor decompositions to full field visualization

• Applicants: Professor Dr.-Ing. Andreas Menzel; Professor Dr. Gerik Scheuermann
• Subject Area: Image and Language Processing, Computer Graphics and Visualisation, Human Computer Interaction, Ubiquitous and Wearable Computing ; Mechanics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 582038168

The Finite Element Method (FEM) is nowadays established as a key computational approach to solve advanced boundary value problems, as most relevant for, e.g., the design and property prediction of engineering devices, processes, and systems. FE solutions of challenging problems in areas and disciplines such as production engineering, geophysics or biomedical applications may show localization phenomena, i.e. significantly higher deformation in particular zones in comparison to their vicinity, which results in changes of the type of the underlying partial differential equation to be solved (loss of ellipticity) and thereby in mesh dependent simulation results. Such results lack physical plausibility and essentially are useless -- the detection of such states and loading levels at which mesh dependencies occur is of cardinal importance for reliable computational design in engineering applications. One criterion for the identification of localization is based on wave propagation-type analysis, respectively on the investigation of the so-called acoustic tensor - the related computational costs of such analysis are, however, extremely high so that a comprehensive analysis based on the acoustic tensor is, in view of computational resources, currently not feasible for FE simulations of relevant engineering application. In general, tensor visualization is a central tool to study tensor fields in applications including mechanics. Different tensor decompositions have been used to support tensor (field) visualization. Few works use decompositions into traceless symmetric tensors, also called deviators, to visualize e.g. the stiffness tensor. Since the acoustic tensor is related to tensors of the type of stiffness tensors, this deviatoric decomposition can be transferred to the decomposition of the acoustic tensor as well. The particular properties of this decomposition are particularly attractive for the visualization of related properties and, moreover, are computationally more efficient than other already established approaches. The successful implementation of such localization related visualization approaches and visualization tools significantly improves the interpretability of such tensors on the one hand, and, on the other, contributes to a significant improvement of efficiently identifying admissible limits of applicability in the context of finite element simulations in engineering practice. In consequence, the main goal of this proposal is to establish the efficient computation and visualization of localization indicators to significantly speed up related simulations and to provide these new indicators to the computational engineering community.

• DFG Programme: Research Grants