Sebastian Münstermann

Shuwei Zhou, Christian Häffner, Sophie Stebner et al.

Physics-informed neural networks provide a mesh-free framework for solving partial differential equation-governed problems in solid mechanics. However, most existing formulations in linear elasticity still learn the displacement field directly, which does not explicitly exploit the analytic structure of two-dimensional elasticity and becomes restrictive for fracture problems with crack face discontinuities and crack tip singularities. Moreover, existing Kolosov--Muskhelishvili informed neural network formulations still rely on residual-based loss functions with multiple boundary and interface terms, whereas a variational concept has not yet been established. To address these issues, a variational Kolosov--Muskhelishvili informed neural network framework for two-dimensional linear elastic problems with and without cracks is proposed in this work. The solution is represented by two holomorphic Kolosov--Muskhelishvili potentials and trained through an energy-based loss function derived from the principle of minimum total potential energy. For crack problems, a discontinuous stress potential representation is further introduced to embed the crack face condition and crack tip singularity directly into the solution ansatz. The proposed framework is validated on a series of benchmark problems with or without crack problems. The results show that variational Kolosov--Muskhelishvili informed neural network can accurately predict stress and displacement field as well as stress intensity factors. Compared with traditional neural network models, it achieves higher accuracy, simpler loss construction, and faster convergence in the considered cases. Overall, the proposed variational Kolosov--Muskhelishvili informed neural network provides an effective and physically consistent variational framework for two-dimensional linear elastic fracture analysis.

Johannes Rosenberger, Johannes Tlatlik, Sebastian Münstermann

To this date the safety assessment of materials, used for example in the nuclear power sector, commonly relies on a fracture mechanical analysis utilizing macroscopic concepts, where a global load quantity K or J is compared to the materials fracture toughness curve. Part of the experimental effort involved in these concepts is dedicated to the quantitative analysis of fracture surfaces. Within the scope of this study a methodology for the semi-supervised training of deep learning models for fracture surface segmentation on a macroscopic level was established. Therefore, three distinct and unique datasets were created to analyze the influence of structural similarity on the segmentation capability. The structural similarity differs due to the assessed materials and specimen, as well as imaging-induced variance due to fluctuations in image acquisition in different laboratories. The datasets correspond to typical isolated laboratory conditions, complex real-world circumstances, and a curated subset of the two. We implemented a weak-to-strong consistency regularization for semi-supervised learning. On the heterogeneous dataset we were able to train robust and well-generalizing models that learned feature representations from images across different domains without observing a significant drop in prediction quality. Furthermore, our approach reduced the number of labeled images required for training by a factor of 6. To demonstrate the success of our method and the benefit of our approach for the fracture mechanics assessment, we utilized the models for initial crack size measurements with the area average method. For the laboratory setting, the deep learning assisted measurements proved to have the same quality as manual measurements. For models trained on the heterogeneous dataset, very good measurement accuracies with mean deviations smaller than 1 % could be achieved...