Novel LOD Methods for the Nonlinear Simulation of Heterogeneous Materials with Application to Cryogels

• Applicants: Professor Dr.-Ing. Daniel Balzani; Professor Dr. Patrick Henning
• Subject Area: Mechanics ; Applied Mechanics, Statics and Dynamics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 574914916

This project aims at simulating the mechanical properties of geometrically nonlinear, hyperelastic materials with highly fluctuating properties. As a prototype problem for our methodological developments, we focus on so-called cryogels, i.e., special hydrogels with a porous polymeric meso-structure that are particularly used in biomedical applications, e.g. as scaffolds to regenerate damaged tissues in the human body. The simulation of such materials comes with severe numerical challenges which are caused by the nonlinearity on the one hand, and fine structural variations on the other hand. Especially the latter aspect brings considerable practical issues as the resolution of the heterogeneous structures requires very fine computational meshes, which in turn results in a tremendous computational complexity. In this project we address this issue by the so-called Localized Orthogonal Decomposition (LOD), a numerical multiscale method that is applicable to heterogenous material structures beyond assumptions of scale-separation or periodicity. It is based on the construction of (low dimensional) generalized finite element spaces with particular shape functions that incorporate information about the fine material structure. Though this approach proved to be very successful for linear problems, its generalization to nonlinear problems is still in the developing phase, where we plan to take major steps forward in this project. In particular, the project is concerned with questions regarding the update of LOD spaces within the iterations of nonlinear solvers, such as Newton's method, as well as the derivation of a posteriori error estimators that help us to adaptively steer the computation of LOD shape functions. In particular, we plan to support the efficient LOD realization for nonlinear mechanical problems by a rigorous mathematical analysis and by detailed numerical benchmarking. Furthermore, we investigate extensions of the methodology to models for incompressible materials as often encountered in practical applications. Finally, in the last part of the project we address the modeling of cryogels based on real data and the application of the novel LOD algorithms for realistic simulations of their mechanical properties.

• DFG Programme: Research Grants