Topology Optimisation of Multiple Scales in Thermoelasticity

The design of multifunctional materials and structures that combine tailored mechanical performance with thermal behaviour is a central challenge in engineering, particularly in the context of the rapid development of additive manufacturing technologies. While topology optimisation provides a powerful computational framework to address this challenge, its extension to heterogeneous materials and multiscale systems remains non-trivial, especially when thermal and mechanical effects interact. In this work, a multiscale topology optimisation framework is presented for mechanical, thermal, and thermoelastic problems. The approach integrates asymptotic expansion homogenisation within a hierarchical formulation. Both fully coupled thermoelastic formulations and decoupled multiobjective strategies are investigated, allowing the relative influence of mechanical stiffness and thermal conductivity to be controlled through weighting factors. All problems are solved using an in-house developed numerical platform. The results demonstrate that the proposed framework provides accurate and robust solutions for single-scale, multiscale, and inverse homogenisation problems. While purely mechanical and thermal cases converge reliably, fully coupled thermoelastic formulations exhibit pronounced instabilities associated with the non-monotonic nature of their sensitivities. These findings motivate the adoption of decoupled multiobjective formulations as a stable and physically meaningful alternative for multiscale design. Overall, the proposed methodology establishes a computationally efficient and versatile basis for the multiscale optimisation of heterogeneous materials, with particular relevance to the design of additively manufactured multifunctional structures operating under combined thermal and mechanical conditions.