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

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Thanks to the advancement of additive manufacturing technologies, mechanical metamaterials have attracted a great deal of attention in recent years. With the employment of such technologies, materials with exceptional or tailored mechanical properties can be easily manufactured mainly by 3D printing of different microstructures rather than by changing the material composition. These lattice materials can provide remarkable material properties in spite of being significantly lighter than typical bulk materials. Due to the large number of degrees of freedom for engineering structures, single-scale numerical simulation of such cellular materials is computationally demanding. Therefore, two-scale computational homogenization approaches, such as FE 2 and FE-FFT, can perform a key role in the cost-effective numerical modeling of metamaterials. Twoscale computational homogenization methods rely on solving a boundary value problem (BVP) for each of the macroscopic and microscopic scales in a nested procedure. Although representative homogenization techniques have been widely used to study materials with heterogeneous microstructures, there still exist some challenges in their employment for lattice materials. This study addresses main challenges in two-scale-based computational homogenization methods for numerical modeling of mechanical metamaterials. High dependence of convergence rate and accuracy on phase contrast for fast Fourier transform (FFT) solvers and comparable macro and micro characteristic lengths in metamaterials (i.e. the applicability of the principle of scale separation) are some examples of such challenges.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Challenges in two‐scale computational homogenization of mechanical metamaterials

Danesh

Brepols

Reese

2023

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“…Concerning efficient two-scale simulation approaches utilizing the FE-FFT-based approach, different methods may be used: For example, using a coarse discretized microstructure for the two-scale simulation followed by a post-processing step for the generation of highly resolved microstructural results for macroscopic integration points of interest [5,14] or using a database base instead of the FE-FFT-based approach, while this FE-FFT-based approach is only used in macroscopic critical areas [15]. A general overview on FE-FFT-based two-scale methods is given in [6].…”

Section: Introductionmentioning

confidence: 99%

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

Gierden

Waimann

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.

“…Recent developments regarding two‐scale FE–FFT‐based simulations are summarized in the paper by Gierden et al. [14]. In this context, the pioneering work of Spahn et al.…”

Section: Introductionmentioning

confidence: 99%

“…The review papers by Schneider [12] and Lucarini et al [13] provide a good overview of the state of the art in FFT-based microstructure simulation and the available solution algorithms. Recent developments regarding two-scale FE-FFT-based simulations are summarized in the paper by Gierden et al [14]. In this context, the pioneering work of Spahn et al [15] who performed the first FE-FFT-based simulation and the work of Kochmann et al [16] who applied this method to polycrystalline materials should be mentioned.…”

Section: Introductionmentioning

confidence: 99%

Modeling the material behavior of heterogeneous materials considering a two‐scale FE–FFT‐based simulation framework

Schmidt,

Gierden,

Waimann

et al. 2023

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In various engineering applications, components subjected to high mechanical or multi‐physical loads such as thermal, thermo‐mechanical, or electro‐thermal loads are predominantly made of metals or composites. These materials are characterized by polycrystalline or multi‐phase microstructures which determine the overall material response. However, since the microscopic material behavior is directly related to the distribution, size, and morphology of the individual grains or phase constituents, precise predictions of the macroscopic material response require simulations of the microstructural behavior. Hence, this work considers a two‐scale finite element (FE) and fast Fourier transform (FFT)‐based simulation framework, which is an efficient alternative to the classical method for the simulation of periodic unit cells. Assuming scale separation, the macroscopic and microscopic boundary value problems are first solved individually and subsequently linked by performing a scale transition in terms of averaging the macroscopic quantities over the corresponding local fields. While the macroscopic solution is computed by means of the finite element method, the microscopic boundary value problem, which is embedded as a periodic unit cell in each macroscopic integration point, is solved using an FFT‐based simulation method. In general, these microscopic simulations allow to model grain‐scale phenomena such as martensitic phase transformation or plastic deformation caused by dislocation glide which can be observed in polycrystalline materials. In this work, we present a two‐scale FE–FFT‐based model to capture the martensitic phase transformations in shape memory alloys. In order to demonstrate the applicability of the model, a numerical example is given.