Reduced-order topology optimisation of high-resolution microstructures with internal contact

Project Information

microFFTTO

Grant agreement ID: 101106585

DOI

10.3030/101106585

Project closed

EC signature date 7 July 2023

Start date 1 March 2024

End date 28 February 2026

Funded under

• Marie Skłodowska-Curie Actions (MSCA)

Total cost

No data

EU contribution

€ 189 687,36

Coordinated by

ALBERT-LUDWIGS-UNIVERSITAET FREIBURG
Germany

Periodic Reporting for period 1 - microFFTTO (Reduced-order topology optimisation of high-resolution microstructures with internal contact)

Reporting period: 2024-03-01 to 2026-02-28

Summary of the context and overall objectives of the project

Additive manufacturing (3D printing) technologies are progressively advancing toward the microscale, enabling the fabrication of so-called architectured (meta)materials — materials whose mechanical properties arise not from their chemical composition but from their carefully engineered internal geometric structure. Such materials can be lighter, stronger, and more multifunctional than conventional bulk materials, and hold considerable promise for applications in aerospace, biomedical, and sustainable engineering.

Realising this potential requires moving beyond trial-and-error design toward automated computational methods. Topology optimisation — a technique that automatically determines the optimal material arrangement within a given design domain — is well suited to this task. However, applying it at the microscale is computationally very demanding: a single cubic millimetre of material discretised at micrometre resolution yields billions of computational elements, which exceeds the capacity of even state-of-the-art solvers. Algorithms based on the fast Fourier transform (FFT) offer a practical way forward, as they are well suited to the regular grid structures that arise naturally in additive manufacturing — but further development was needed before they could be applied routinely at the resolutions modern manufacturing now enables.

The µFFTTO project, conducted at the University of Freiburg within the livMatS Cluster of Excellence, set out to address this gap by developing faster and more memory-efficient FFT-based algorithms for the topology optimisation of high-resolution microstructures.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The project began with a systematic assessment of low-rank tensor approximations as a strategy for reducing the computational cost of FFT-based computational methods. A database of representative mechanical metamaterial microstructures was assembled and used to evaluate how well different tensor formats could represent complex geometries. This study showed that the assessed low-rank tensor formats are poorly suited to microstructures with features that are not aligned with the main grid directions. The tensor rank strongly depends on the geometry of the microstructure and therefore increases the computational cost of the solution for complex structures. Similarly, organically shaped geometries typical of topology-optimised designs are hard to represent by these types of tensors. The development of this approach was therefore not pursued further.

The project proceeded to develop µFFTTO — an open-source library for FFT-based microstructure topology optimization. Within this library, a microstructure topology optimization algorithm was implemented using a parallelised finite element FFT-based solver. Testing on high-resolution metamaterial microstructures confirmed the viability of the approach. However, these tests also revealed significant performance limitations for certain classes of problems — specifically, microstructures with smooth material distributions and high phase contrast between material and void regions — motivating further development. The results of this work will be reported in a forthcoming publication titled 'Topology optimization of metamaterials with FFT-accelerated micromechanical solvers’.

A significant challenge emerged when the standard Green's operator preconditioned FFT-based solver proved unexpectedly slow for this class of problems. This motivated the development of a novel diagonal scaling strategy — the Green-Jacobi preconditioner — which forms the basis of the J-FFT solver. The resulting method reduced computation time by an order of magnitude compared to the standard approach, representing the principal algorithmic contribution of the project. The results of this work have been reported in a paper titled 'Jacobi-accelerated FFT-based solver for smooth high-contrast data', currently available as a preprint on arXiv.

Beyond the J-FFT solver, the project yielded deeper insight into the convergence behaviour of Green's operator preconditioned FFT-based solvers, establishing a clearer link between solver performance and the spatial distribution of material properties in the microstructure. This understanding provides a firmer theoretical foundation for the further development of efficient FFT-based topology optimisation methods.

In parallel with the J-FFT development, local grid adaptation techniques were also investigated as a complementary cost-reduction strategy. While keeping regular data structures throughout, these techniques relocate points of the computational grid locally based on solution complexity and microstructural geometry — substantially reducing discretisation error while keeping computational costs manageable. This approach is compatible with both the standard FFT solver and the J-FFT solver developed within the project. The results of this work have been published in Comptes Rendus. Mécanique.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The principal scientific result of µFFTTO is the J-FFT solver — a Jacobi-accelerated FFT-based solver that addresses a fundamental performance bottleneck in microstructure topology optimization. Standard FFT-based solvers perform well for microstructures with sharp material interfaces, but become inefficient for smooth material distributions with high contrast between material and void regions. The J-FFT solver resolves this issue, delivering an order-of-magnitude speedup over conventional approaches for this class of problems.

The project also contributed to the development of local grid adaptation techniques as a complementary cost-reduction strategy, compatible with both the standard FFT-based solver and the J-FFT solver. Additionally, a systematic assessment of low-rank tensor approximations established that these techniques are not well suited to the general microstructure geometries arising in topology optimization — providing useful guidance for future work in the field.

Together, these results advance the computational tools available for the automated, high-resolution design of mechanical metamaterials, contributing to the broader goal of making such methods practically accessible for engineering applications.