  The design of minimum-compliance bending-resistant structures with continuous
cross-section parameters is a challenging task because of its inherent
non-convexity. Our contribution develops a strategy that facilitates computing
all guaranteed globally optimal solutions for frame and shell structures under
multiple load cases and self-weight. To this purpose, we exploit the fact that
the stiffness matrix is usually a polynomial function of design variables,
allowing us to build an equivalent non-linear semidefinite programming
formulation over a semi-algebraic feasible set. This formulation is
subsequently solved using the Lasserre moment-sum-of-squares hierarchy,
generating a sequence of outer convex approximations that monotonically
converges from below to the optimum of the original problem. Globally optimal
solutions can subsequently be extracted using the Curto-Fialkow flat extension
theorem. Furthermore, we show that a simple correction to the solutions of the
relaxed problems establishes a feasible upper bound, thereby deriving a simple
sufficient condition of global $\varepsilon$-optimality. When the original
problem possesses a unique minimum, we show that this solution is found with a
zero optimality gap in the limit. These theoretical findings are illustrated on
several examples of topology optimization of frames and shells, for which we
observe that the hierarchy converges in a finite (rather small) number of
steps.
