A comparison between different numerical methods for the modeling of polycrystalline materials with an elastic–viscoplastic behavior

Robert, Camille; Mareau, Charles

doi:10.1016/j.commatsci.2015.03.028

Cited by 22 publications

(17 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are several papers [26,27,28,29,30,31] comparing FFT-based methods with FEM and other discretisation methods. As a limitation, the accuracy of local fields cannot be deduced from the homogenised properties when a non-conforming method is used.…”

Section: Introductionmentioning

confidence: 99%

Energy-based comparison between the Fourier–Galerkin method and the finite element method

Vondřejc

Geus

2020

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

The Fourier-Galerkin method (in short FFTH) has gained popularity in numerical homogenisation because it can treat problems with a huge number of degrees of freedom. Because the method incorporates the fast Fourier transform (FFT) in the linear solver, it is believed to provide an improvement in computational and memory requirements compared to the conventional finite element method (FEM). Here, we compare the two methods using the energetic norm of local fields, which has the clear physical interpretation as being the error in the homogenized properties. This enables the comparison of memory and computational requirements at the same level of approximation accuracy. We show that the methods' effectiveness rely on the smoothness (regularity) of the solution and thus on the material coefficients. Thanks to its approximation properties, FEM outperforms FFTH for problems with jumps in material coefficients, while ambivalent results are observed for the case that the material coefficients vary continuously in space. FFTH profits from a good conditioning of linear system, independent of the number of degrees of freedom, but generally needs more degrees of freedom. More studies are needed for other FFT-based schemes, non-linear problems, and dual problems (which require special treatment in FEM). keywords:energy-based comparison, finite element method, Fourier-Galerkin method, fast Fourier transform, numerical homogenisation, computational effectiveness arXiv:1709.08477v1 [math.NA]

show abstract

Section: Introductionmentioning

confidence: 99%

Energy-based comparison between the Fourier–Galerkin method and the finite element method

Vondřejc

Geus

2020

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…This form has been extended to (geometrically) non-linear problems [7][8][9][10][11][12], however still relying on a reference medium, which may impact the convergence and obscures consistent linearization. It has nevertheless been successfully applied to large-scale computations [13][14][15][16]. Currently, even a mature open-source implementation is available featuring different solution techniques [17].…”

mentioning

confidence: 99%

Finite strain FFT-based non-linear solvers made simple

Geus

Vondřejc

Zeman

et al. 2017

Computer Methods in Applied Mechanics and Engineering

117

122

View full text Add to dashboard Cite

Computational micromechanics and homogenization require the solution of the mechanical equilibrium of a periodic cell that comprises a (generally complex) microstructure. Techniques that apply the Fast Fourier Transform have attracted much attention as they outperform other methods in terms of speed and memory footprint. Moreover, the Fast Fourier Transform is a natural companion of pixel-based digital images which often serve as input. In its original form, one of the biggest challenges for the method is the treatment of (geometrically) non-linear problems, partially due to the need for a uniform linear reference problem. In a geometrically linear setting, the problem has recently been treated in a variational form resulting in an unconditionally stable scheme that combines Newton iterations with an iterative linear solver, and therefore exhibits robust and quadratic convergence behavior. Through this approach, well-known key ingredients were recovered in terms of discretization, numerical quadrature, consistent linearization of the material model, and the iterative solution of the resulting linear system. As a result, the extension to finite strains, using arbitrary constitutive models, is at hand. Because of the application of the Fast Fourier Transform, the implementation is substantially easier than that of other (Finite Element) methods. Both claims are demonstrated in this paper and substantiated with a simple code in Python of just 59 lines (without comments). The aim is to render the method transparent and accessible, whereby researchers that are new to this method should be able to implement it efficiently. The potential of this method is demonstrated using two examples, each with a different material model

show abstract

“…Please note that such approach is close to the well‐established frameworks of multilevel FE method (FEM) (see, for example, Smit et al, Feyel and Chaboche, Miehe et al,() and Schröder for a recent review). A comparison between the FFT‐ and the FE‐method can be found, inter alia, in Robert and Mareau …”

Section: Introductionmentioning

confidence: 99%

“…A comparison between the FFT-and the FE-method can be found, inter alia, in Robert and Mareau. 44 Irrespective of the numerical scheme used, the computation of an algorithmically consistent macroscopic tangent is an important point. Only when this tangent is known, quadratic convergence of a Newton-Raphson scheme at macroscopic scale can be obtained.…”

Section: Introductionmentioning

confidence: 99%

An algorithmically consistent macroscopic tangent operator for FFT‐based computational homogenization

Göküzüm

Keip

2017

Numerical Meth Engineering

View full text Add to dashboard Cite

Summary The present work addresses a multiscale framework for fast‐Fourier‐transform–based computational homogenization. The framework considers the scale bridging between microscopic and macroscopic scales. While the macroscopic problem is discretized with finite elements, the microscopic problems are solved by means of fast‐Fourier‐transforms (FFTs) on periodic representative volume elements (RVEs). In such multiscale scenario, the computation of the effective properties of the microstructure is crucial. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of the RVE, the computation of the associated moduli is not straightforward. The key contribution of the present paper is the derivation and implementation of an algorithmically consistent macroscopic tangent operator which directly resembles the effective moduli of the microstructure. The macroscopic tangent is derived by means of the classical Lippmann‐Schwinger equation and can be computed from a simple system of linear equations. This is performed through an efficient FFT‐based approach along with a conjugate gradient solver. The viability and efficiency of the method is demonstrated for a number of two‐ and three‐dimensional boundary value problems incorporating linear and nonlinear elasticity as well as viscoelastic material response.

show abstract

A comparison between different numerical methods for the modeling of polycrystalline materials with an elastic–viscoplastic behavior

Cited by 22 publications

References 46 publications

Energy-based comparison between the Fourier–Galerkin method and the finite element method

Energy-based comparison between the Fourier–Galerkin method and the finite element method

Finite strain FFT-based non-linear solvers made simple

An algorithmically consistent macroscopic tangent operator for FFT‐based computational homogenization

Contact Info

Product

Resources

About