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

Abstract. We formulate a new free-boundary type mathematical model describing the interac4 tion between a shell and a mesh-like structure consisting of thin rods. Composite structures of this 5 type arise in many applications. One example is the interaction between vascular walls treated with 6 vascular devices called stents. The new model embodies two-way coupling between a 2D Naghdi 7 type shell model, and a 1D network model of curved rods, describing no-slip and balance of contact 8 forces and couples (moments) at the contact interface. The work presented here provides a unified 9 framework within which 3D deformation of various composite shell-mesh structures can be studied. 10 In particular, this work provides the first dimension reduction-based fully coupled model of mesh11 reinforced shells. Using rigorous mathematical analysis based on variational formulation and energy 12 methods, the existence of a unique weak solution to the coupled shell-mesh model is obtained. The 13 existence result shows that weaker solution spaces than the classical shell spaces can be used to 14 obtain existence, making this model particularly attractive to Finite Element Method-based com15 putational solvers, where Lagrangian elements can be used to simulate the solution. An example 16 of such a solver was developed within Freefem++, and applied to study mechanical properties of 17 four commercially available coronary stents as they interact with vascular wall. The simple imple18 mentation, low computational costs, and low memory requirements make this newly proposed model 19 particularly suitable for fast algorithm design and for the coupling with fluid flow in fluid-composite 20 structure interactions problems. 21