Upscaling and reliable two-scale Fourier/finite element-based simulations

• Applicant: Professor Dr. Hermann Georg Matthies
• Subject Area: Applied Mechanics, Statics and Dynamics ; Mechanics ; Mechanical Properties of Metallic Materials and their Microstructural Origins
• Term: from 2017 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 324231889

In this project, we focus on the two-scale modelling of deterministic and also stochastic elastic and inelastic problems in the small strain regime, such as elasto-plasticity with isotropic hardening. On the micro-scale we consider Fast Fourier Transform (FFT)-based solvers, enabling very efficient three-dimensional image-based studies. Building on our previous results on variational FFT-based techniques in the Fourier-Galerkin setting, we consider a scheme equivalent to the Moulinec-Suquet algorithm and an improved scheme based on exact integration. For a macro-scale problem, we consider the standard finite element method with FE2-like coupling procedures and also the mesh-in-element (MIEL) method, which allows to treat problems without the assumption of scale separation.The project will further increase the computational efficiency of FFT-based solvers (e.g. preconditioning, acceleration by low-rank techniques) for micro-scale problems, which will be modelled with random material fields. In the stochastic setting, discretisation and solution procedures within a variational setting will be developed, providing a probabilistic description of macro material properties. Alternative boundary conditions will be considered for FFT-based solvers to enable its coupling not only in the FE2-framework, but also in the MIEL method; the transfer of randomness from micro- to macro-scale will be of particular interest. For those coupled stochastic problems, two-scale quasi-Newton methods with emphasis on line-search and trust-region algorithms will be developed.As a result, the discretisation and solution procedures will be developed for two-scale nonlinear problems with FFT-based solvers on the micro-scale and FEM solvers on the macro-scale. We expect that the collaboration with the research group in the Czech Republic will lead to a significant increase in the efficiency of two-scale simulations with realistic microstructural representations, making them accessible on conventional computer platforms.

• DFG Programme: Research Grants
• International Connection: Czech Republic
• Partner Organisation: Czech Science Foundation
• Cooperation Partner: Dr.-Ing. Jan Zeman, Ph.D.