Finite strain FFT-based non-linear solvers made simple

Abstract

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

FFT-based interface decohesion modelling by a nonlocal interphase

“…A comparison of different FFT formulations and solution approaches in a crystal plasticity constitutive framework [4] was presented in [5]. Recently, an FEM perspective on an FFT based spectral formulation for small strain non-linear material behaviour was given in [6] and its extension to a finite strain setting was presented in [7]. Alongside such improvements, much effort has gone into making the method suitable for various applications.…”

An algorithm for stress and mixed control in Galerkin‐based FFT homogenization

“…The RVE contains four particles occupying a volume fraction of 0.2, and the domain is discretized using 64 3 voxels (Figure A). The matrix and particle mechanical behavior follows a Saint Venant‐Kirchhoff hyperelastic model, following the works of Kabel et al and de Geus et al The matrix properties are μ = 28 Pa and k = 46.67 Pa (shear and bulk modulus), and the particles are one order of magnitude stiffer (both moduli are 10 times larger than the matrix ones). The uniaxial tensile test reaches an elongation of , with L and L 0 being the target and the initial cell length in the loading direction and the other two directions being unconstrained.…”

“…The test functions are virtual deformation gradients premultiplied by a projection operator to enforce their compatibility. After some algebraic work, the discrete form of the linear momentum balance can be written as where P is the first Piola‐Kirchhoff stress, F is the deformation gradient field, α are a set of internal variables defining the history, and is a linear map that acts on a voxel tensor field P defined in to create a new voxel tensor field defined in the same space. The linear map corresponds to a convolution in the real space that is computed as where and are the discrete Fourier transform and its inverse and is the projection operator in the Fourier space (fourth‐order tensor field).…”

“…All the corrections δ F are fluctuation terms (average free), and the applied deformation gradient enters through . The linear equations have to be solved by a projection‐based linear solver such as the conjugate gradient method . If the constitutive equations are rate dependent, the stress depends on both the deformation gradient, F , and the velocity gradient, L .…”

Coupled Crystal Plasticity–Phase Field Fracture Simulation Study on Damage Evolution Around a Void: Pore Shape Versus Crystallographic Orientation

“…23,27,28 The collocation-based discretization approach typically used for spectral solvers leads to oscillations in the vicinity of strain or stress jumps that are caused by Gibbs phenomenon. Since these oscillations might introduce significant numerical artifacts in the case of brittle damage modeling and a finite element-based spectral discretization 29,30 is not available within DAMASK, a hybrid finite element-spectral approach is used for the presented study.…”