Abstract
The local and global behavior of materials with internal microstructures is often investigated on a (representative) volume element. Typically, periodic boundary conditions are applied on such “virtual specimens” to mimic the situation in the bulk of the material. While, in general, different types of boundary value solvers can be used to solve for mechanical equilibrium, spectral methods have been established as a powerful numerical tool especially suited for this task [for application examples see 1–7]. Starting from the pioneering work of Moulinec and Suquet [8], several improvements in performance and stability have been achieved for solving mechanical boundary value problems [9–13]. Recent advancements of using the spectral approach to solve coupled field equations enable the modeling of multiphysical phenomena such as fracture propagation, temperature evolution, chemical diffusion, and phase transformation in conjunction with the mechanical boundary value problem. The fundamentals of such a multi-physics framework, which is implemented in the Düsseldorf Advanced Materials Simulation Kit (DAMASK) [14, 57] are presented in the following together with implementation details and illustrative examples.
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Notes
- 1.
To simplify the notation, in the following, the argument x is dropped whenever it is possible, i.e., F(x) is denoted as F only
- 2.
Quantities in real space and Fourier space are distinguished by notation Q(x) and Q(k), respectively, with x the position in real space, k the frequency vector in Fourier space, and i2 = − 1. ℱ−1[⋅] denotes the inverse Fourier transform.
- 3.
The solution for the deformation gradient field, i.e., the actual spectral method procedure, is performed in parallel to these iterations.
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Acknowledgments
PS and FR acknowledge funding through SFB 761 Steel – ab initio by the Deutsche Forschungsgemeinschaft (DFG). MD acknowledges the funding of the TCMPrecipSteel project in the framework of the SPP 1713 Strong coupling of thermo-chemical and thermo-mechanical states in applied materials by the DFG.
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Shanthraj, P., Diehl, M., Eisenlohr, P., Roters, F., Raabe, D. (2018). Spectral Solvers for Crystal Plasticity and Multi-physics Simulations. In: Schmauder, S., Chen, CS., Chawla, K., Chawla, N., Chen, W., Kagawa, Y. (eds) Handbook of Mechanics of Materials. Springer, Singapore. https://doi.org/10.1007/978-981-10-6855-3_80-1
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DOI: https://doi.org/10.1007/978-981-10-6855-3_80-1