This talk showcases applications of semidefinite programming (SDP) techniques to two problems of structural topology optimization.

First, we consider the problem of designing optimum truss reinforcement of a thin-walled composite laminate to withstand manufacturing and operational loads and to suppress elastic wall instabilities. For this problem, the instabilities can be conveniently captured with free-vibration eigenmodes, allowing thus for a convex SDP representation. To accelerate the solution of these types of problems, we utilize the static generalized Schur complement lemma, which additionally provides us with an upper bound on the maximum admissible fundamental eigenfrequency, and a lower bound on the minimum admissible compliance of the manufacturing load case. Finally, we report on manufacturing the composite beam prototype with a 3D-printed internal structure and its experimental validation to conclude that the structural response agrees well with the model predictions.

In the second part of the talk, we consider the topology optimization of frame structures. In this case, the SDP formulation is no longer convex in general. However, because the SDP constraints are polynomial matrix inequalities, we adopt the moment-sums-of-squares hierarchy for their solution. It turns out that each relaxation of this hierarchy generates both lower and upper bounds on the optimal design, which provides us not only with a measure of the solution quality but also an inexpensive sufficient condition of global optimality. Besides, we observed finite convergence for all the tested problems.

This a joint work between Faculty of Civil Engineering, CTU in Prague, and Compo Tech PLUS, SuĊĦice.