Polynomial optimization in the design of globally optimal frame structures under dynamic loads

Designing frame and shell structures for globally optimal mechanical performance is challenging because stiffness and inertia depend non-linearly on sizing variables, rendering an ensuing optimization problem highly non-convex. For static minimum compliance problems, we have recently demonstrated how to overcome this kind of difficulty by constructing a convergent Lasserre hierarchy of convexified problems. This solves the polynomial optimization problem to certified global optimality, under mild assumptions. In this project, we aim to extend these results to structures under mechanical vibration loads by developing (i) efficient techniques for free-vibration problems, (ii) formulation and solution techniques for steady-state damped harmonic vibrations, (iii) problem-specific methods to accelerate optimization algorithms, and (iv) proof-of-concept validation experiments. We aim to achieve these goals by extending and adapting contemporary techniques of polynomial optimization to the dynamics of structures.