Effective transmission conditions for thin-layer transmission problems in elastodynamics. The case of a planar layer model

Bonnet, Marc; Burel, Aliénor; Duruflé, Marc; Joly, Patrick

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 43-75.

This article is concerned with the design, analysis, numerical approximation and implementation of effective transmission conditions (ETCs) for the propagation of elastic waves through a thin planar elastic layer with small uniform thickness $\eta$ which is embedded in a reference elastic medium, under transient conditions, with both materials assumed to have isotropic properties. A family of ETCs of order $k$ (i.e. whose approximation error is of expected order $O\left(\eta {}^{k+1}\right)$) is formulated by deriving and exploiting a formal asymptotic expansion in powers of $\eta$ of the transmission solution inside the layer. The second-order ETCs are then retained as the main focus for the remainder of the article, and given a full justification in terms of both the stability of the resulting transient elastodynamic problem and the error analysis. The latter is performed by establishing and justifying asymptotic expansions for the solutions of both the exact transmission problem and its approximation based on the second-order ETCs. As a result, the error (in energy norm) between those two solutions is shown to be, as expected, of order $O\left(\eta {}^3\right)$. Finally, the numerical approximation of the proposed second-order ETC within the framework of spectral element methods is studied, with special attention devoted to the selection of a robust time-stepping scheme that is mostly explicit (and conditionally stable). Among these, a scheme that is implicit only for the interfacial degrees of freedom, termed semi-implicit, is shown to be stable under the same stability condition as for the layer-less configuration. The main theoretical results of this work are illustrated and validated by 2D and 3D numerical experiments under transient elastodynamic conditions.

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