Séminaire Martin Ladecký
mercredi 21 octobre à 14 h 00 – 15 h 00
Jacobi-Accelerated FFT Solver for Topology Optimization of Metamaterials
Martin Ladecký
martin.ladecky@imtek.uni-freiburg.de
The exotic properties of mechanical metamaterials arise from the geometry of their microstructure. Topology optimization of microstructures [1] is nontrivial and computationally expensive, as it requires repeatedly solving homogenization problems. Solvers based on the fast Fourier transform (FFT) are a standard tool for homogenization problems [2,3], offering quasilinear O(N log N) complexity and excellent performance on regular grids. However, their convergence deteriorates markedly when the material coefficients are smooth and highly contrasted [4], a regime frequently encountered in density-based topology optimization.
In my talk, I will present the Jacobi-accelerated FFT (J-FFT) solver [4], designed specifically for this setting. The method augments the classical discrete Green operator [5] with a diagonal Jacobi scaling that captures local material contrast while preserving the O(N log N) complexity per iteration. Although the strict grid-size independence of the standard Green preconditioner is relaxed, the J-FFT solver yields substantially faster convergence for smooth, high-contrast coefficient fields. In phase-field topology optimization of mechanical metamaterials, where the density field becomes smooth and highly contrasted, J-FFT consistently outperforms the standard FFT-based solver.
References
[1] O. Sigmund, Materials with prescribed constitutive parameters: An inverse homogenization problem, Int. J. Solids Struct. 31(17) (1994) 2313–2329.
[2] H. Moulinec and P. Suquet, A fast numerical method for computing the linear and nonlinear mechanical properties of composites, C. R. Acad. Sci. Paris, Série II 318(11) (1994) 1417–1423.
[3] M. Schneider, A review of nonlinear FFT-based computational homogenization methods, Acta Mechanica 232 (2021) 2051–2100.
[4] M. Ladecký, I. Pultarová, F. Bignonnet, I. Jödicke, J. Zeman, and L. Pastewka, Jacobi-accelerated FFT-based solver for smooth high-contrast data, Comput. Methods Appl. Mech. Engrg. 461 (2026) 119269.
[5] M. Ladecký, R. J. Leute, A. Falsafi, I. Pultarová, L. Pastewka, T. Junge, and J. Zeman, An optimal preconditioned FFT-accelerated finite element solver for homogenization, Appl. Math. Comput. 446 (2023) 127835.