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This study introduces a flow matching-based topology optimization (FMTO) framework that enhances conditional topology generation by incorporating trajectory-aware formulations. By utilizing volume-fraction-indexed BESO states to create a probability path and target velocity field, the method effectively integrates physics-guided optimization history without the need for inference-time optimization. The results show that FMTO not only generates diverse topology candidates with improved performance metrics but also significantly reduces the number of sampling steps compared to traditional diffusion-based methods.
Moderate trajectory weighting in FMTO leads to superior topology generation stability and performance, challenging conventional approaches that rely heavily on adversarial training or extensive sampling.
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.
