prof. Ing. Didier Henrion, Ph.D.

This PhD thesis is co-supervised with Prof. Martin Kružík and Dr. Milan Korda. It is dedicated to exploring polynomial optimization techniques in applied calculus of variations, in particular in continuum mechanics of solids. Variational methods represent a powerful tool for analysis and numerics of problems in elasticity, plasticity, or viscoelasticity where the notion of energy plays a key role. Their analysis often leads to the global minimization of nonconvex objective functions, a challenging task. The PhD student should develop a novel, mathematically rigorous approach to modeling and analysis in nonlinear mechanics of solids. Potential avenues of exploration include: dynamical/time-dependent problems in the mechanics of solids, minimizing movements and spatial/temporal discretization. Numerical approaches to minimizing movements play a significant role in these topics because the existence-of-solutions proofs are usually done via semidiscretization in time. This, together with spatial discretization, leads to large-scale but highly structured polynomial optimization problems. Alternative, mesh-free numerical methods based on semidefinite optimization and the moment–sums of squares hierarchy, are also to be investigated and developed. We expect close cooperation with the Polynomial optimization group at LAAS-CNRS in Toulouse on theoretical and numerical aspects of polynomial optimization.