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Theorie

FFTjax solves variational partial differential equations on voxelized domains using a spectral Fourier-based

Spectral solution of PDEs

Consider a generic equilibrium problem in small-strain elasticity:

\[ \nabla \cdot \sigma(\mathbf{x}) = 0 \]

with constitutive relation

\[ \sigma = \mathbb{C} : \varepsilon, \qquad \varepsilon = \frac{1}{2}(\nabla u + \nabla u^\top) \]

In the spectral approach, fields are transformed into Fourier space:

\[ \hat{f}(\mathbf{k}) = \mathcal{F}[f(\mathbf{x})] \]

Spatial derivatives become multiplications:

\[ \nabla \rightarrow i\mathbf{k} \]

which converts differential operators into algebraic expressions. The equilibrium equation is then solved iteratively in Fourier space using a Green operator formulation, enabling efficient convolution-based updates.

The computational complexity scales as \( \mathcal{O}(N \log N) \) due to the use of Fast Fourier Transforms.

Variational formulation

The solver is based on an energy minimization principle. For elasticity and phase-field fracture, the total energy functional reads

\[ \Pi(u, d) = \int_\Omega g(d)\,\psi_e(\varepsilon(u)) \, d\Omega + \int_\Omega G_c \left( \frac{d^2}{2\ell} + \frac{\ell}{2} |\nabla d|^2 \right) d\Omega \]

where

  • \( u \) is the displacement field
  • \( d \) is the phase-field variable
  • \( g(d) \) is the degradation function
  • \( \mathcal{G}_c \) is the fracture toughness
  • \( \ell \) is the length scale parameter

Staggered solution scheme

A variational staggered scheme is employed:

  1. Elastic step
    Minimize \( \Pi(u, d^{n}) \) with respect to \(u\) → Linear/nonlinear equilibrium solved via spectral operator.
\[ A_u \, u = b_u , \]

where \( A_u \) represents the spectral stiffness operator. The system is solved iteratively using the Conjugate Gradient (CG) method, exploiting the matrix-free application of \( A_u \) in Fourier space.

  1. Phase-field step
    Minimize \( \Pi(u^{n+1}, d) \) with respect to \(d\)
    → Helmholtz-type equation solved in Fourier space. [ A_d \, d = b_d , ] which is likewise solved using CG with spectral evaluation of the differential operators.

The two fields are updated alternately until convergence of the coupled system.

Image-based discretization

The computational domain is defined on a regular voxel grid derived from segmented experimental data. Material heterogeneity is directly assigned per voxel, avoiding geometric idealization and enabling direct microstructure-to-simulation coupling.

Periodic boundary conditions are naturally satisfied within the spectral framework.

This formulation enables:

  • Efficient voxel-scale simulations
  • Direct use of image-based microstructures
  • Differentiability for inverse parameter identification
  • GPU acceleration via JAX