Trajectory-Aware Flow Matching for Topology Optimisation

2026-07-16

AI Summary

This work proposes a flow-matching-based topology optimization framework (FMTO) to address the high computational cost of traditional methods, which rely on repeated finite element analyses and sensitivity calculations, as well as the difficulty existing generative approaches face in simultaneously ensuring structural feasibility and physical consistency. FMTO innovatively embeds intermediate optimization trajectories generated by the BESO algorithm into a conditional generative process, constructing a volume-fraction-indexed probability path and a target velocity field to enable physics-guided, efficient generation. Without increasing inference overhead, the method significantly enhances generation quality and stability. Compared to diffusion model baselines, FMTO drastically reduces the number of sampling steps while producing diverse, high-fidelity topologies that better satisfy compliance performance and volume fraction constraints, demonstrating effectiveness in both two- and three-dimensional problems.

Abstract

Topology optimisation (TO) often requires repeated finite element analysis and sensitivity-based material updates, which can be costly when multiple candidate designs are needed under varying physical and design conditions. Generative TO offers a route to rapid design exploration, but existing models may rely on adversarial training, long reverse-diffusion sampling, or external guidance to maintain structural feasibility and physical consistency. This study develops a flow matching-based topology optimisation (FMTO) framework for conditional topology generation. Linear FMTO is first formulated as an endpoint-based baseline by interpolating between a Gaussian source field and the BESO reference topology. To introduce mechanically meaningful intermediate states, a trajectory-aware FMTO formulation is proposed, where volume-fraction-indexed BESO states are used to construct the probability path and target velocity field. This incorporates physics-guided optimisation history into generative flow learning without adding inference-time optimisation. A path--velocity mismatch analysis explains why moderate trajectory weighting can improve generation stability, whereas excessive guidance may over-constrain the learned transport. Numerical examples show that FMTO generates diverse topology candidates with improved compliance-related performance, volume-fraction satisfaction, topology fidelity, and substantially fewer sampling steps than a diffusion-based baseline. Under limited training data, trajectory-aware FMTO achieves the best overall performance with a moderate trajectory weight. Studies on trajectory-anchor density and three-dimensional topology generation further demonstrate the influence of path design and the applicability of the proposed framework beyond two-dimensional problems.

Problem

• topology optimisation
• generative modeling
• structural feasibility
• physical consistency
• computational cost

Innovation

• flow matching
• topology optimisation
• trajectory-aware generation
• physics-guided generative modeling
• BESO