In this paper, we derive models for the interaction of a linearized three-dimensional elastic structure with a thin elastic layer of possibly different material attached to it. Rigorous derivation is performed by considering a thin three-dimensional layer and the asymptotics of the solution of the full remaining three-dimensional problem when the thickness [Formula: see text] of the thin layer tends to zero. Furthermore, the attached thin material is assumed to have the elasticity coefficients which are of order [Formula: see text], for [Formula: see text] with respect to the coefficients of the three-dimensional body. In the limit, five different models are obtained with respect to different choices of p, namely [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. Furthermore a three-dimensional–two-dimensional model is proposed that has the same asymptotics as the original three-dimensional problem. This is convenient for applications because one does not have to decide in advance which limit model to use.

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We formulate a new free-boundary type mathematical model describing the interac-4 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. 10In particular, this work provides the first dimension reduction-based fully coupled model of mesh-11 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 com-15 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 imple-18 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 Key words. TO DO 22 AMS subject classifications. TO DO 231. Introduction. In this paper we formulate a free-boundary type mathematical 24 model of the interaction between shells and mesh-like structures consisting of thin 25 rods. Composite structures of this type arise in many engineering and biological 26 applications where an elastic mesh is used to reinforce the underlying shell structure. 27The main motivation for this work comes from the study of the interaction between 28 vascular devices called stents, and vascular walls. See Figure 1. Coronary stents 29 have been used to reinforce coronary arteries that suffer from coronary artery disease, 30 which is characterized by occlusion or narrowing of coronary arteries due to plaque 31 deposits. Stents, which are metallic mesh-like tubes, are implanted into coronary 32 arteries to prop the arteries open and to recover normal blood supply to the heart 33 muscle. Understanding the interaction between vascular walls and stents is important 34 in determining which stents produce less complications such as in-stent re-stenosis 35 [7]. Mathematical modeling of stents and other elastic mesh-like structures has been 36primarily based on using 3D approaches: the entire structure is assumed to be a 37 single 3D structure, and 3D finite elements are used for the numerical approximati...

In this article, we explore the possibility of modeling the interaction of a 2d elastic body with a thin 2d elastic body of possibly higher thickness using a 1d model for the thin body. We use the asymptotic analysis with respect to the small thickness of the 2d interaction model and formulate five different limit models depending on the order of stiffness of the thin body with respect to the thickness. Then we formulate a 2d–1d model which has the same asymptotics as the 2d–thin 2d model with respect to thickness. Finally, we numerically test the approximation of the 2d–thin 2d model by the 2d–1d model on two problems, one with an analytical solution and one more realistic problem.

The standard engineer's model for heat transfer between the fluid flowing through the pipe and the exterior medium neglects the effects of the pipe's wall. The goal of this paper is to prove that they are not always negligible. Comparing the ratio between diffusivities of the fluid and the wall with the wall's thickness, using rigorous asymptotic analysis, we find five different models for effective description of the heat exchange process.