On reduced models in nonlinear solid mechanics

International audienceThe capabilities and limits of current model reduction methods are examined in the case of solid mechanics problems involving significant nonlinearities—such as (visco)plasticity, damage, contact with friction, …—and parameters. Particular emphasis will be put on the PGD method (Proper Generalized Decomposition) and its last developments. These reduced models are the key to the introduction of materials physics in simulation-driven structural design, a domain in which quasi real-time simulations are mandatory

The variational theory of complex rays for the calculation of medium-frequency vibrations

A new approach called the ``variational theory of complex rays’’ (VTCR) is developed for calculating the vibrations of weakly damped elastic structures in the medium-frequency range. Here, the emphasis is put on the most fundamental aspects. The effective quantities (elastic energy, vibration intensity, etc.) are evaluated after solving a small system of equations which does not derive from a finite element discretization of the structure. Numerical examples related to plates show the appeal and the possibilities of the VTCR

Construction of a micromechanics-based intralaminar mesomodel, and illustrations in ABAQUS/Standard

International audienceThe recent advances in the modeling of degradations in stratified composites have led to improved models on all scales. In particular, today, micromechanics derived in a generic framework enables one to define a reference virtual material which integrates most of the knowledge of a material. Thus, a model using damage mechanics on the mesoscale and usable for structural analysis can be built as a homogenized version of this reference model through previously-developed bridges. The objective is to derive a refined model worthy of micromechanics confidence, but transposable into a commercial code (here, ABAQUS/Standard)

A new numerical strategy for the resolution of high-Péclet advection-diffusion problems

International audienceThis paper introduces a discontinuous method for the efficient determination of an approximate numerical solution of the two-dimensional advection-diffusion equation. Using the VTCR methodology, this method involves free-space solutions of the governing partial differential equation. For the advection-diffusion equation with constant coefficients, the free-space solutions are exponential functions with a sharp gradient. The continuity of the solution across element boundaries is enforced weakly through a dedicated variational formulation. Preliminary results for a certain type of benchmark problem suggest that this approach is a promising numerical tool for handling such problems

On Trefftz and weak Trefftz discontinuous Galerkin approaches for medium-frequency acoustics

International audienceIn this paper, the wave approach called the Variational Theory of Complex Rays (VTCR), which was developed for medium-frequency acoustics and vibrations, is revisited as a discontinuous Galerkin method. Extensions leading to a weak Trefftz constraint are introduced. This weak Trefftz discontinuous Galerkin approach enables hybrid FEM/VTCR strategies to be developed easily, and paves the way for new computational techniques for the resolution of engineering problems. This paper presents some of the fundamental properties of the approach, which is illustrated by several numerical examples

On a Computational Strategy with Time-Space Homogenization for Heterogeneous Materials

International audienceA new multiscale computational strategy was recently proposed for the analysis of structures described both on a fine space scale and a fine time scale. This strategy, which involves homogenization in space as well as in time, could replace in several domains of application the standard homogenization techniques, which are generally limited to the space domain and present some drawbacks. It is an iterative strategy which calls for the resolution of problems on both a micro (fine) scale and a macro (homogenized) scale. Here, we review the bases of this approach and present improved approximation techniques to solve the micro and macro problems