Theoretical and numerical study of some interaction problems in continuum mechanics

Abstract

This thesis investigates elastic–elastic and fluid–structure interaction (FSI) problems under generalized interface conditions that extend beyond classical continuity and balance laws. The work develops a rigorous functional framework, establishing weak formulations, existence, and stability results for systems where coupling laws incorporate impedance, stiffness, or damping effects through positive definite operators. These additional terms modify the natural energy balance and lead to new conditions for well-posedness. On the computational side, finite volume schemes are designed on admissible meshes, with particular attention to conservative and stable treatment of interface terms. Penalization techniques are introduced to enforce generalized interface laws, and convergence to the original model is demonstrated. Theoretical findings are validated through a set of numerical experiments, which confirm convergence, illustrate the influence of interface parameters, and highlight the role of generalized coupling in dynamic responses. Overall, the thesis bridges rigorous mathematical analysis with efficient numerical strategies, offering a unified framework for analyzing and simulating complex coupled systems.