Novel insights into Pareto fronts in multiobjective topology optimization and a comparative study of scalarization strategies
Topology optimization seeks structures that minimize an objective function subject to constraints. While extensive research has focused on single-objective formulations, design is usually a trade-off between conflicting criteria. This naturally leads topology optimization to the multiobjective optimization field. Unfortunately, despite their long coexistence, interaction between these fields has remained limited. This work aims to bridge this gap. First, we provide a theoretical comparison of two common scalarization techniques, the weighted-sum and $\varepsilon$-constraint method, to the established Pascoletti-Serafini scalarization. Then, we perform extensive bi-objective numerical experiments, including minimization of volume, compliance, maximum stress and dynamic compliance. For each experiment, the four scalarization methods are aggregated, often alongside design space sampling and continuation, to better resolve the Pareto front. The major contribution of this work is the observation that the Pareto front consists of segments belonging to local fronts. Contrary to the smooth, convex fronts often reported in the literature, we show that this property can lead to discontinuous and nonconvex fronts. Finally, we numerically compare the approximation quality of the scalarization methods. The numerical examples demonstrate the robustness of the Pascoletti-Serafini scalarization against fronts with pronounced discontinuities, nonconvexities and high-curvature regions. In contrast, the weighted-sum and $\varepsilon$-constraint methods can produce clustering and gaps, depending on the shape of the front.