Coupling methods for non-matching meshes through distributed Lagrange multipliers

Nature and engineering commonly present multi-physics problems, i.e., complex systems involving a number of mutually interacting subsystems that can be modelled by (non linearly coupled) Partial Differential Equations (PDEs). As a tool to numerically model such problems, we study the fictitious domain method with Lagrange multipliers. After introducing an example model problem, a constrained Poisson equation, where the constraint is applied either to a codimension one domain or to a codimension zero domain, we deal with the technical problems related to the implementation of the coupled system, with a focus on the computation of coupling matrices, and the numerical coupling between arbitrarily distributed non-matching meshes. To conclude, we use the fictitious domain method to study composites materials, in particular fiber reinforced materials. After introducing the necessary tools of continuum mechanics and differential geometry, we develop a full three-dimensional model, where the effect of the fibers is imposed through a distributed Lagrange multiplier approach. We study our model using inf-sup conditions from Mixed Finite Elements, and derive a one-dimensional model where the coupling is achieved introducing some additional modellistic hypotheses.