Speaker
Description
Design optimization of frame structures is crucial for numerous civil engineering applications requiring lightweight yet dynamically robust designs. The minimum weight design of such structures under dynamic loading presents significant computational challenges due to the non-convexity of the feasible domain, polynomial dependence of stiffness on design variables, and disconnected feasible set with singularities. Traditional nonlinear programming approaches often fail to provide reliable solutions to these problems, necessitating more sophisticated mathematical tools.
We present a novel topology optimization approach based on the moment-sum-of-squares (mSOS) hierarchy of semidefinite programming relaxations to address these challenges. This marks the first application of mSOS techniques to dynamic topology optimization. Our method builds upon our previous research in weight minimization under fundamental free-vibration eigenvalue constraints, extending it to encompass both standard and robust dynamic compliance constraints, as well as peak power constraints.
Our contribution establishes theoretical connections between these formulations, demonstrating that for loads excited below the lowest resonance frequency, both dynamic compliance and peak power optimization constraints can be viewed as special cases of the free-vibration eigenvalue constraints. We also establish a direct relationship between the worst-case state fields in robust optimization scenarios and the free-vibration eigenmodes associated with the lowest non-zero eigenvalue.
The efficacy of our approach is validated through numerical examples, demonstrating: (1) finite convergence of the mSOS hierarchy, and (2) the ability to generate high-quality feasible points even at low relaxation degrees. These results enable minimum weight design of frame structures under dynamic loads with guaranteed global ε-optimality certificates.
