A Dimension-Reduction Based Coupled Model of Mesh-Reinforced Shells

Abstract

We formulate a new free-boundary type mathematical model describing the interaction between a shell and a mesh-like structure consisting of thin rods. Composite structures of this type arise in many applications. One example is the interaction between vascular walls treated with vascular devices called stents. The new model embodies two-way coupling between a two-dimensional (2D) Naghdi type shell model and a 1D network model of curved rods, describing no-slip and balance of contact forces and couples (moments) at the contact interface. The work presented here provides a unified framework within which 3D deformation of various composite shell-mesh structures can be studied. In particular, this work provides the first dimension reduction-based fully coupled model of mesh-reinforced shells. Using rigorous mathematical analysis based on variational formulation and energy methods, the existence of a unique weak solution to the coupled shell-mesh model is obtained. The existence result shows that weaker solution spaces than the classical shell spaces can be used to obtain existence, making this model particularly attractive to finite element method based computational solvers, where Lagrangian elements can be used to simulate the solution. An example of such a solver was developed within Freefem++ and applied to study mechanical properties of four commercially available coronary stents as they interact with the vascular wall. The simple implementation, low computational costs, and low memory requirements make this newly proposed model particularly suitable for fast algorithm design and for the coupling with fluid flow in fluid-composite structure interactions problems.