Etherogeneous Dimensional Coupling of Vascularized Tissues: A Reduced Lagrange Multiplier Framework for Bidirectional Elastic Matrix-Fluid Inclusion Interactions
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We present a novel numerical framework for the simulation of vascularized tissues, modeled as biphasic multiscale materials consisting of an elastic matrix and fluid-filled inclusions. The accurate prediction of effective macroscale material properties in such tissues is influenced by microscale structures and fluid dynamics, such as the flow within vascular networks. At the same time, the flow in the vascular networks is influenced by the strain and stresses in the surrounding elastic matrix. To capture the complex interaction of these systems, we propose a Reduced Lagrange Multiplier approach, where the interface between solid and fluid domains is not directly resolved within the computational mesh of the elastic material but is instead discretized independently, with interface conditions enforced using non-matching Lagrange multipliers. This approach exploits the multiscale and etherogeneous dimensional nature of the problem by reducing the Lagrange multipliers space of the background elasticity problem to a lower-dimensional set of co-dimension two, coupled bi-directionally with one-dimensional blood flow models. We present the principle behind the coupling method, its implementation, and its application to the simulation of realistic vascularized tissues.