Analysis of an Averaging Operator for Atomic-to-Continuum Coupling Methods by the Arlequin Approach

摘要

A new coupling term for blending panicle and continuum models with the Arlequin framework is investigated in this work. The coupling term is based on an integral operator defined on the overlap region that matches the continuum and particle solutions in an average sense. The present exposition is essentially the continuation of a previous work (Bauman ct al.. On the application of the Arlequin method to the coupling of particle and continuum models. Computational Mechanics, 42, 511-530, 2008) in which coupling was performed in terms of an H'-type norm. In that case, it was shown that the solution of the coupled problem was mesh-dependent or, said in another way, that the solution of the continuous coupled problem was not the intended solution. This new formulation is now consistent with the problem of interest and is virtually mesh-independent when considering a particle model consisting of a distribution of heterogeneous bonds. The mathematical properties of the formulation are studied for a one-dimensional model of harmonic springs, with varying stiffness parameters, coupled with a linear elastic bar, whose modulus is determined by classical homogeni/.ation. Numerical examples are presented for one-dimensional and two-dimensional model problems that illustrate the approximation properties of the new coupling term and the effect of mesh size.