Spectral methods for spherical representative volume elements

• Applicant: Professor Dr.-Ing. Martin Diehl, since 4/2021
• Subject Area: Mechanics ; Mechanical Properties of Metallic Materials and their Microstructural Origins
• Term: from 2018 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 411252165

Multiscale simulations allow in principle to track and predict the microstructure evolution during forming process simulations, which can be used to obtain more precise simulations. Unfortunately, the numerical costs for such simulations are enormous, since on every integration- or collocation point on the macro scale a simulation of a representative volume element (RVE) is carried out. One can think of an RVE as a small virtual material sample on which the effective mechanical properties are measured, which enter the simulation at the macroscale material point level.Since all RVE simulations are on the same domain, one does not need the flexibility of the finite element method (FEM), which allows to develop solvers adapted to the special properties of the RVE boundary value problem. Thus, for RVE that tesselate the 3D space, like cubical RVE with periodic boundary conditions, spectral solvers offer advantages over FE solvers.On the other hand, the numerical costs can be reduced by reducing the RVE simulation domain directly. One way to do so is to maximize the representativity of the microstructure. Another approach is to use spherical instead of hexahedral RVE, which allow for smaller domain volumes at equal solution quality by reducing the boundary influence via the surface to volume ration. Further, spherical RVE do not induce an artificial anisotropy.For thermomechanical problems on spherical domains, an optimized solution technique is not at hand. Unfortunately, the spectral method is not applicable directly. Thus, the aim of this project is to develop a fast and reliable solver for thermomechanical boundary- and initial value problems that is adapted to and optimized for spherical domains.In the first phase of the project, proposals based on radial basis approaches, spectral and pseudospectral approaches and combinations of these are be developed. The candidates are to be implemented in a high level programming language like python, which allows for a fast implementation and testing. This is followed by an evaluation of the candidates.In the second phase, the most promising candidate is to be ported into the Düsseldorfer Advanced Material Simulation Toolkit (DAMASK), which is an free and open multiscale simulation tool. This requires the solver to be implemented in a low-level but high performance programming language.The solver is then compared to presently included finite element solvers. The project concludes with engineering part simulations that use the new multiscale scheme, in which the approach can be tested and evaluated.

• DFG Programme: Research Grants
• Ehemaliger Antragsteller: Privatdozent Dr.-Ing. Rainer Glüge, until 3/2021