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Jacobi-accelerated FFT-based solver for smooth high-contrast data

Martin Ladecký, Ivana Pultarová, François Bignonnet, Indre Jödicke, Jan Zeman, Lars Pastewka

TL;DR

This work tackles the slow convergence of FFT-based solvers for smooth, high-contrast microstructures by introducing a Green-Jacobi preconditioner, forming the Jacobi-accelerated FFT (J-FFT) method that preserves the $O(N_ ext{N} \log N_ ext{N})$ FFT complexity. By combining a global Green operator with a local Jacobi scaling in a matrix-free, FFT-accelerated framework, the authors demonstrate substantial reductions in CG iterations for problems with smooth data and high phase contrast, particularly in phase-field topology optimization scenarios and grid-adapted settings. The results show a clear trade-off: Green excels for sharp interfaces, while Green-Jacobi markedly improves convergence for smooth variations, with nonlinear elasticity experiments illustrating robustness to spatially varying tangents. The proposed method offers a practical pathway to accelerate FFT-based solvers in multiscale materials simulations, including topology optimization and phase-field fracture, where smooth microstructures are common and grid adaptation is beneficial.

Abstract

The computational efficiency and rapid convergence of fast Fourier transform (FFT)-based solvers render them a powerful numerical tool for periodic cell problems in multiscale modeling. On regular grids, they tend to outperform traditional numerical methods. However, we show that their convergence slows down significantly when applied to microstructures with smooth, highly-contrasted coefficients. To address this loss of performance, we introduce a Green-Jacobi preconditioner, an enhanced successor to the standard discrete Green preconditioner that preserves the quasilinear complexity, $\mathcal{O}(N \log N)$, of conventional FFT-based solvers. Through numerical experiments, we demonstrate the effectiveness of the Jacobi-accelerated FFT (J-FFT) solver within a linear elastic framework. For problems characterized by smooth data and high material contrast, J-FFT significantly reduces the iteration count of the conjugate gradient method compared to the standard Green preconditioner. These findings are particularly relevant for phase-field fracture simulations, density-based topology optimization, and solvers that use adaption of the grid, which all introduce smooth variations in the material properties that challenge conventional FFT-based solvers.

Jacobi-accelerated FFT-based solver for smooth high-contrast data

TL;DR

This work tackles the slow convergence of FFT-based solvers for smooth, high-contrast microstructures by introducing a Green-Jacobi preconditioner, forming the Jacobi-accelerated FFT (J-FFT) method that preserves the FFT complexity. By combining a global Green operator with a local Jacobi scaling in a matrix-free, FFT-accelerated framework, the authors demonstrate substantial reductions in CG iterations for problems with smooth data and high phase contrast, particularly in phase-field topology optimization scenarios and grid-adapted settings. The results show a clear trade-off: Green excels for sharp interfaces, while Green-Jacobi markedly improves convergence for smooth variations, with nonlinear elasticity experiments illustrating robustness to spatially varying tangents. The proposed method offers a practical pathway to accelerate FFT-based solvers in multiscale materials simulations, including topology optimization and phase-field fracture, where smooth microstructures are common and grid adaptation is beneficial.

Abstract

The computational efficiency and rapid convergence of fast Fourier transform (FFT)-based solvers render them a powerful numerical tool for periodic cell problems in multiscale modeling. On regular grids, they tend to outperform traditional numerical methods. However, we show that their convergence slows down significantly when applied to microstructures with smooth, highly-contrasted coefficients. To address this loss of performance, we introduce a Green-Jacobi preconditioner, an enhanced successor to the standard discrete Green preconditioner that preserves the quasilinear complexity, , of conventional FFT-based solvers. Through numerical experiments, we demonstrate the effectiveness of the Jacobi-accelerated FFT (J-FFT) solver within a linear elastic framework. For problems characterized by smooth data and high material contrast, J-FFT significantly reduces the iteration count of the conjugate gradient method compared to the standard Green preconditioner. These findings are particularly relevant for phase-field fracture simulations, density-based topology optimization, and solvers that use adaption of the grid, which all introduce smooth variations in the material properties that challenge conventional FFT-based solvers.

Paper Structure

This paper contains 22 sections, 1 theorem, 42 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction and Motivation
  2. Problem setup: Small-strain elasticity
  3. Discretization - finite element method (FEM) on regular grid
  4. Linearization - Newton's method
  5. Linear solver - conjugate gradient (CG) method
  6. Preconditioning strategies
  7. Green preconditioner
  8. Jacobi preconditioner
  9. Green-Jacobi preconditioner
  10. Experiments and results
  11. Laminate
  12. Cosine geometry with voids (infinite contrast).
  13. Phase field topology optimization
  14. Smooth vs sharp interphases
  15. Nonlinear elasticity
  16. ...and 7 more sections

Key Result

Lemma 1

Let $c$, $\boldsymbol{\mathsfit{K}}$, $\boldsymbol{\mathsfit{J}}$, $\boldsymbol{\mathsfit{K}}_0$ and $\boldsymbol{\mathsfit{K}}_{\boldsymbol{\mathsfit{J}}}$ be defined as in the previous paragraph and let $\alpha\in(0,1)$. Then for every $n\in\mathbb{N}$, there exists a subspace $V_{n}\subset\mathbb The coefficient of the second term does not depend on $\alpha$.

Figures (8)

  • Figure 1: Number of iterations of the preconditioned conjugate gradient (PCG) method required to solve mechanical equilibrium on a regular grid of $256^2$ nodal points as a function of the number of Gaussian filter applications, $i$. The green dashed line indicates the results for Green PCG, while black solid line indicates results for Green-Jacobi PCG. Figure (a.1) shows the initial two-phase material density $\rho_{\mathrm{0}}$, while $\rho_{\mathrm{I}}$ in (a.2) is the density for which the number of iterations of Green PCG attains its maximum. The last density $\rho_{\mathrm{II}}$ in (a.3) has the smallest total phase contrast $\chi_{\mathrm{II}} = 10^{2}$. Panel (b) shows cross sections of material densities $\rho_{0}$, $\rho_{\mathrm{I}}$, and $\rho_{\mathrm{II}}$, at the middle row of nodal points shown by the dotted, dashed-dotted, and dashed lines in panels (a.1) to (a.3), respectively. Figure (c) shows maximum gradient of density field $\nabla \rho_{i}$, and Figure (d) shows total phase contrast $\max(\rho_{i})/\min(\rho_{i})$.
  • Figure 2: Example of data samplings and discretization grids of two density functions. In (a), we see the linear density function $\rho_{\text{laminate}}(\boldsymbol{x})$ used in the first experiment from Section \ref{['sec:n_laminate']}. In (b), we see the cosine density function $\rho_{\text{cos}}(\boldsymbol{x})$ used in the second experiment from Section \ref{['sec:periodic']}. Both material density functions are sampled with resolutions $4^2$, $8^2$, and $16^2$ pixels denoted as $\mathcal{G}_4$, $\mathcal{G}_8$, and $\mathcal{G}_{16}$, respectively. Finite element discretization grids $\mathcal{T}_4$, $\mathcal{T}_8$, and $\mathcal{T}_{16}$ consist of $4^2,8^2$, and $16^2$ nodal points, respectively. The parameter $\chi^{\rm tot}$ controls the total phase contrast
  • Figure 3: Number of iterations of PCG method needed to solve mechanical equilibrium \ref{['eq:lin_system_exp']}, for the laminate geometry from Section \ref{['sec:n_laminate']}. Panels (a._) show results for the Green preconditioner \ref{['eq:prec_lin_systemG']}, panels (b._) show results for the Jacobi preconditioner \ref{['eq:prec_lin_systemJ']}, and panels (c._) show results for the Green-Jacobi preconditioner \ref{['eq:prec_lin_systemJG']}. The first row (_.1) shows results for total phase contrast $\chi^{\rm tot}=10^1$, and second row panels (_.2) shows results for total phase contrast $\chi^{\rm tot}=10^4$. Each panel shows: i) on the horizontal axis the number of nodal points in $x_1$-direction $n$ of $\mathcal{T}_{n}$, ii) on the vertical axis the number of data sampling points in $x_1$-direction $p$ of $\mathcal{G}_{p}$. The upper limit for the number of iterations is $999$. The color coding, with color bars on the right, represents the number of iterations to highlight trends rather than exact iteration counts.
  • Figure 4: Number of iteration of PCG method needed to solve mechanical equilibrium \ref{['eq:lin_system_exp']}, for the cosine geometry from Section \ref{['sec:periodic']}. Panels (a._) show results for the Green preconditioner \ref{['eq:prec_lin_systemG']}, panels (b._) show results for the Jacobi preconditioner \ref{['eq:prec_lin_systemJ']}, and panels (c._) show results for the Green-Jacobi preconditioner \ref{['eq:prec_lin_systemJG']}. First row panels (_.1) show results for total phase contrast $\chi^{\rm tot}=\infty$, and second row panels (_.2) show results for total phase contrast $\chi^{\rm tot}=10^4$. Each panel has: i) on the horizontal axis number of discretization points in $x_1$-direction $n$ of $\mathcal{T}_{n}$, ii) on the vertical axis the number of data sampling points in $x_1$ direction $p$ of $\mathcal{G}_{p}$. The upper limit for the number of iterations is $999$. The color coding, with color bars on the right, represents the number of iterations to highlight trends rather than exact iteration counts.
  • Figure 5: Number of iteration of the PCG method needed to solve mechanical equilibrium \ref{['eq:lin_system_topopt']}, with respect to the geometry obtained during the topology optimization process (using L-BFGS optimizer). In the top row we see: initial geometry (random noise) (a.1), near uniform geometry (a.2), initial two-phase geometry (a.3), two-phase geometry (a.4), and converged geometry (a.5). Blue color indicates void (very soft material), while red indicates the bulk material. In the graph (b), the green lines correspond to the Green preconditioner \ref{['eq:prec_lin_systemG']}, the blue lines to the Jacobi preconditioner \ref{['eq:prec_lin_systemJ']}, and and the black lines to the Green-Jacobi preconditioner \ref{['eq:prec_lin_systemJG']}. Graph (c) displays the norm of the density gradient, while graph (d) shows the total phase contrast of the material. All three graphs shows results for 2 meshes $\mathcal{T}_{512}$, and $\mathcal{T}_{1024}$, using dash-dotted, and solid lines, respectively.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Lemma 1