Vladimir Salnikov

Dina Razafindralandy, Vladimir Salnikov, Aziz Hamdouni et al.

This article reviews some integrators particularly suitable for the numerical resolution of differential equations on a large time interval. Symplectic integrators are presented. Their stability on exponentially large time is shown through numerical examples. Next, Dirac integrators for constrained systems are exposed. An application on chaotic dynamics is presented. Lastly, for systems having no exploitable geometric structure, the Borel-Laplace integrator is presented. Numerical experiments on Hamiltonian and non-Hamiltonian systems are carried out, as well as on a partial differential equation. Keywords: Symplectic integrators, Dirac integrators, long-time stability, Borel summation, divergent series.

Vladimir Salnikov, Daniel Choi, Philippe Karamian-Surville

In this paper we describe efficient methods of generation of representative volume elements (RVEs) suitable for producing the samples for analysis of effective properties of composite materials via and for stochastic homogenization. We are interested in composites reinforced by a mixture of spherical and cylindrical inclusions. For these geometries we give explicit conditions of intersection in a convenient form for verification. Based on those conditions we present two methods to generate RVEs: one is based on the Random Sequential Adsorption scheme, the other one on the time driven Molecular Dynamics. We test the efficiency of these methods and show that the first one is extremely powerful for low volume fraction of inclusions, while the second one allows us to construct denser configurations. All the algorithms are given explicitly so they can be implemented directly.