RÉSUMÉ

Multimodel and multiscale methods are critical for the analysis of defects in the form of distributed cracking (damage) or discrete cracks as well as for simulation of multiphysics problems and the design of complex engineering systems. Development of multiscale methods generated significant interest in the computational engineering community in the past decade. Among the barriers that need to be overcome are the development of rigorous interscale transfer operators and supporting numerical methodologies. The five articles comprising this special issue address various elements of the multimodel, multiphysics, and multiscale approaches.
The article by Ammar, Chinesta, and Joyot addresses various aspects of the numerical modeling and computation of electronic structures. It provides an overview of the computational challenges and development efforts in computational physics and mechanics of materials at nanoscale. The article addresses the dimensionality issue. The development of methods bridging different models and scales is essential to make the multiscale models computationally tractable. The Arlequin method provides a framework for communication between various models and scales; in the article by Ben Dhia, the Arlequin modeling paradigm is presented and analyzed. The parameters of this methodology are investigated through the mathematical analysis of the representative Arlequin problem. Nearly optimal Arlequin parameters (like the partition of energies functions, the coupling operators, etc.) are identified. The Arlequin method provides the framework for coupling continuum models as well for linking atomistic and continuum models. The article by Dureisseix and Neron provides the domain decomposition framework for simulation of steady state porohermoelasticity problems in which the spatial scales are different. A variant of the periodic homogenization method and a multiscale surface transfer operator based on the mortar-like transmission technique are developed. The issue of an effective solution of the discrete system of equations arising from the nonlinear multimodel is addressed within the framework of the FETI dual domain decomposition method for nonlinear problems by Pebrel, Rey, and Gosselet. The effectiveness of this methodology is verified by several numerical examples. The temporal multiscale problem is considered in the article by Rodrigues, in which the space-time local mesh refinement technique for elastodynamics is developed. Different time steps are used in various spatial regions. The accuracy and stability of the proposed formulation are studied. Of particular interest is a post-processing strategy that removes the so-called aliasing phenomena while enhancing the accuracy of the time-stepping scheme.
We hope that the special issue will be of interest to the computational engineering community concerned with the modeling and simulation of multimodel, multiphysics, and multiscale problems.