Patrick Kurzeja

Gian-Luca Geuken, Jörn Mosler, Patrick Kurzeja

The concept of Rao-Blackwellization is employed to improve predictions of artificial neural networks by physical information. The error norm and the proof of improvement are transferred from the original statistical concept to a deterministic one, using sufficient information on physics-based conditions. The proposed strategy is applied to material modeling and illustrated by examples of the identification of a yield function, elasto-plastic steel simulations, the identification of driving forces for quasi-brittle damage and rubber experiments. Sufficient physical information is employed, e.g., in the form of invariants, parameters of a minimization problem, dimensional analysis, isotropy and differentiability. It is proven how intuitive accretion of information can yield improvement if it is physically sufficient, but also how insufficient or superfluous information can cause impairment. Opportunities for the improvement of artificial neural networks are explored in terms of the training data set, the networks' structure and output filters. Even crude initial predictions are remarkably improved by reducing noise, overfitting and data requirements.

Gian-Luca Geuken, Patrick Kurzeja, David Wiedemann et al.

This work investigates different sufficient and necessary criteria for hyperelastic, isotropic polyconvex material models, focusing on neural network implementations for incompressible materials. Furthermore, the expressiveness, accuracy, simplicity as well as the efficiency of those models is analyzed. This also enables an assessment of the practical applicability of the models. Convex Signed Singular Value Neural Networks (CSSV-NNs) are applied to compressible materials and tailored to incompressibility (inc-CSSV-NNs), resulting in a universal approximation for frame-indifferent, isotropic polyconvex energies for the compressible as well as incompressible case. While other existing approaches also guarantee frame-indifference, isotropy and polyconvexity, they impose too restrictive constraints and thus limit the expressiveness of the model since they are only based on sufficient but not necessary criteria. This is further substantiated by numerical examples of several, well-established classical models (Neo--Hooke, Mooney--Rivlin, Gent and Arruda--Boyce) and Treloar's experimental data. Moreover, the numerical examples include an explicitly constructed energy function that cannot be approximated by neural networks constrained by Ball's criterion for polyconvexity. This substantiates that Ball's criterion, though sufficient, is not necessary for polyconvexity.

Gian-Luca Geuken, Patrick Kurzeja, David Wiedemann et al.

This paper presents a novel framework of neural networks for isotropic hyperelasticity that enforces necessary physical and mathematical constraints while simultaneously satisfying the universal approximation theorem. The two key ingredients are an input convex network architecture and a formulation in the elementary polynomials of the signed singular values of the deformation gradient. In line with previously published networks, it can rigorously capture frame-indifference and polyconvexity - as well as further constraints like balance of angular momentum and growth conditions. However and in contrast to previous networks, a universal approximation theorem for the proposed approach is proven. To be more explicit, the proposed network can approximate any frame-indifferent, isotropic polyconvex energy (provided the network is large enough). This is possible by working with a sufficient and necessary criterion for frame-indifferent, isotropic polyconvex functions. Comparative studies with existing approaches identify the advantages of the proposed method, particularly in approximating non-polyconvex energies as well as computing polyconvex hulls.