Keywords: finite element, thin layer modeling, thin layers, interface, interphase.

Several theoretical works have been published concerning the modeling of very thin layers with material properties which differ significantly from those of the surrounding media. Traditionally, there have been two extreme ways to handle the physical modeling of these layers. The first consists of a full finite element analysis model and the other consists of ignoring the layer altogether. Both have significant disadvantages, the first with respect to the computational time needed, and the other one with respect to accuracy. The model here proposed is a healthy compromise between the two.

Theoretical models of thin layers based on asymptotic analysis were proposed recently by several authors, among which Benveniste et al. [1,2,3] In this new approach, the layer is being replaced by an interface which has no thickness; the effect of the layer is then modeled by imposing special jump conditions on the interface. While the theoretical models are available for heat conduction and elasticity, with different orders of accuracy, this paper focuses on two-dimensional heat conduction.

The main goal of this paper is to transition from these theoretical models to construct new finite element formulations allowing the use of this model in an effective manner. In order to demonstrate the satisfactory nature of the proposed model, we focus on a heat conduction problem in a circular domain composed of two concentric rings, on which Dirichlet boundary conditions have been imposed along the two exterior boundaries. This problem is axi-symmetric, and an analytical solution is available for this model. The comparison between the analytical model and the computational model based on jump conditions proves to be very satisfactory. An additional investigation shows that the solution converges to the exact solution with an optimal rate of convergence.

This paper is a first introduction to the computational implementation of thin layer modeling using finite elements. In future work, models will be developed for non- axisymmetric problems, as well as for elasticity problems in various configurations.

1

Y. Benveniste, "A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media", J. Mech. Phys. Solids, 54, 708-734, 2006. doi:10.1016/j.jmps.2005.10.009

2

Y. Benveniste, G. Baum, "An interface model of a graded three-dimensional anisotropic curved interphase", Proc. Royal Society A-Mathematical Physical and Engineering Sciences, 463, 419-434, 2007. doi:10.1098/rspa.2006.1777

3

Y. Benveniste, "An Interface Model for a Three-Dimensional Curved Thin Piezoelectric Interphase between Two Piezoelectric Media", Math. and Mech. of Solids, 14, 102-122, 2009. doi:10.1177/1081286508092605

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