Topology optimization of thermoelastic structures with single and functionally graded materials exploring energy and stress-based formulations

Topology optimization problem formulations have lately included stresses, besides compliance, to ensure mechanical strength feasibility, which is of utmost importance in structural engineering practice. A mechanically induced stress field has often been considered in optimal structural design. However, one realizes that thermal stresses can also greatly influence efficient designs, especially when addressing highly constrained structures. Moreover, stress mitigation has been achieved by enlarging the design domain to multi-material solutions. This motivates to pursue stress-based topology optimization of thermoelastic structures and the extension of the multi-material setting to Functionally Graded Materials (FGMs), with greater potential in stress mitigation. Two optimization problems are investigated: (1) elastic strain energy minimization and (2) maximum von Mises stress minimization. In the former, the single-material problem is revisited, but in the frame of a multi-objective formulation, weighting mechanical and thermal strain energy terms, as they can be decoupled. Insights into thermal stresses allow to propose a well-posed stress-based formulation for the topology optimization thermoelastic problem. In the latter, stress mitigation is sought on account of optimizing the spatial mixture (composition) of two solids amidst prescribed or predicted voids. It is assumed that the RAMP interpolation scheme has the physical meaning of rendering the thermoelastic properties for the continuous variation of composition. Linear thermoelasticity and plane stress benchmarks are used. In the multi-objective energy-based problem, the trade-offs between the conflicting design objectives, in the Pareto sense, are highlighted. Regarding the stress-based problem, lower stress peaks are obtained in FGM solutions, as stresses are more evenly distributed.