A model order reduction method for finite strain FFT solvers using a compressed sensing technique

Proc Appl Math and Mech

Schmidt

et al. 2021

Self Cite

To capture all the individual microstructural effects of complex and heterogeneous materials in structural finite element simulations, a two‐scale simulation approach is necessary. Since the computational effort of such two‐scale simulations is extremely high, different methods exist to overcome this problem. In terms of a FFT‐based microscale simulation, one possibility is to use a reduced set of frequencies leading to a reduced numerical solution of the Lippmann‐Schwinger equation [?]. In a post‐processing step, highly resolved microstructural fields may then be reconstructed by using the compressed sensing technique [?]. Since the stress evaluation of this method is in real space and therefore not reduced, it is most beneficial in terms of linear elastic material behavior. Another very recent method to reduce the computational effort of a microscale simulation is the self‐consistent clustering analysis [?,?]. Such a self‐consistent clustering analysis is split into an offline and an online stage. Within the offline stage, the material points of the high‐fidelity representation of the unit cell are grouped into clusters with similar material behavior. Thereafter, in the online stage, a self‐consistent clustering analysis is used to solve the boundary value problem by a clustered Lippmann‐Schwinger equation. Since the generation of clusters may be based on linear elastic simulations, we propose to use a reduced set of frequencies for these simulations to improve the efficiency of the total algorithm. Elastic and elasto‐plastic composites are investigated in a small strain setting as representative simulation examples.

Section: Clustering Approachmentioning

confidence: 99%

Microstructure simulation using self‐consistent clustering analysis

Proc Appl Math and Mech

Schmidt

et al. 2021

Self Cite

“…The FFT-based simulation using a reduced set of Fourier modes was introduced utilizing a fixed set of modes [4,12]. In order to increase the accuracy of this MOR technique, a geometrically adapted selection of Fourier modes was developed in [8].…”

Section: Introductionmentioning

confidence: 99%

FFT‐based simulation of evolving microstructures utilizing an adapting reduced set of Fourier modes

Proc Appl Math and Mech

Svendsen

et al. 2023

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The mechanical behavior of a periodic heterogeneous microstructure may be predicted by using a fast Fourier transform (FFT) based simulation approach. To reduce the computational effort of this method, we introduced a model order reduction (MOR) technique utilizing a reduced set of Fourier modes for the computations in Fourier space. To increase the accuracy of this MOR technique we developed a geometrically adapted sampling pattern for choosing the considered Fourier modes based on the representation of phases within the microstructure. Since the phase distribution of, for example, martensite and austenite in a polycrystalline microstructure evolves with increasing mechanical or thermal loads, the set of considered Fourier modes should also evolve according to the underlying micromechanical fields. We present the accuracy and the adaptability of this adaptive reduced set of Fourier modes by investigating the micromechanical fields of a polycrystal considering such phase transformations.

A Review of FE-FFT-Based Two-Scale Methods for Computational Modeling of Microstructure Evolution and Macroscopic Material Behavior

Kochmann

Arch Computat Methods Eng

et al. 2022

The overall, macroscopic constitutive behavior of most materials of technological importance such as fiber-reinforced composites or polycrystals is very much influenced by the underlying microstructure. The latter is usually complex and heterogeneous in nature, where each phase constituent is governed by non-linear constitutive relations. In order to capture such micro-structural characteristics, numerical two-scale methods are often used. The purpose of the current work is to provide an overview of state-of-the-art finite element (FE) and FFT-based two-scale computational modeling of microstructure evolution and macroscopic material behavior. Spahn et al. (Comput Methods Appl Mech Eng 268:871–883, 2014) were the first to introduce this kind of FE-FFT-based methodology, which has emerged as an efficient and accurate tool to model complex materials across the scales in the recent years.