Mixed-dimensional modeling of vascular tissues with reduced Lagrange multipliers
This paper presents a numerical method for the simulation of multiscale materials composed of an elastic matrix and slender active inclusions. The setting is motivated by the modeling of vascularized tissues and by problems arising in the context of medical imaging techniques, where the estimation of effective (i.e., macroscale) material properties is affected by the presence of microscale structures and microscale dynamics, such as fluid flow in the vasculature. We propose a method where the background solid material and the active slender inclusions are discretized independently, imposing the required interface conditions via non-matching Lagrange multipliers. The intrinsic geometrical complexity of the resulting computational model is simplified by relying on a reduced Lagrange multiplier framework, where the functional space of the Lagrange multiplier is replaced by the tensor product between an infinite dimensional Sobolev space defined on a lower-dimensional characteristic set of co-dimension two, and a finite dimensional space defined on the cross-sections of the inclusions. In view of the coupling with one-dimensional blood flow models, we derive a non-standard boundary condition that enforces a local deformation on the solid-fluid boundary, and we present the details of its stability analysis in the continuous elasticity setting. The method is validated with different numerical examples in two and three dimensions, assessing its convergence properties and its potential for the in silico characterization of tissues samples.