Variational principles in nonlinear mechanics using convex optimization and automated numerical tools

Habilitation manuscript (Habilitation à Diriger des Recherches - HDR) This manuscript explores advanced approaches to solving non-linear mechanics problems using convex analysis and optimization. It introduces key concepts in convex optimization, focusing on algorithms and solvers relevant to mechanical applications. Variational principles in solid and structural mechanics are analyzed, with an emphasis on dissipative materials. The work advances limit analysis theory with versatile numerical tools applied to structures like plates, shells, and reinforced concrete. It extends these tools to shape and topology optimization, maximizing load-bearing capacity and addressing multi-material design challenges. Strategies for modeling uncertainty are proposed, incorporating robust and stochastic optimization techniques. Finally, innovative regularization approaches for softening behaviors, including dynamic fracture and plasticity, are presented, linking non-convex problems to related convex formulations for efficient resolution. This research bridges theoretical insights and practical engineering challenges, offering novel tools and methodologies.