Grégory Vial

Interactions between moderately close inclusions for the Laplace equation

The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain Ω0. This question has been studied extensively for a single inclusion or well-separated inclusions. In two-dimensional situations, we investigate the case where the distance between the holes tends to zero but remains large with respect to their characteristic size. We first consider two perfectly insulated inclusions. In this configuration we give a complete multiscale asymptotic expansion of the solution to the Laplace equation. We also address the situation of a single inclusion close to a singular perturbation of the boundary ∂Ω0. We also present numerical experiments implementing a multiscale superposition method based on our first order expansion.

On Generalized Ventcel's Type Boundary Conditions for Laplace Operator in a Bounded Domain

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to well-posed variational problems under a sign condition of a coefficient. Nevertheless, situations where this condition is violated appeared in several works. The well-posedness of such problems was still open. This manuscript establishes, in the generic case, the existence and uniqueness of the solution for the Ventcel boundary value problem without the sign condition. Then we consider perforated geometries and give conditions to remove the genericity restriction.

High Order Finite Element Calculations for the Cahn-Hilliard Equation

In this work, we propose a numerical method based on high degree continuous nodal elements for the Cahn-Hilliard evolution. The use of the p-version of the finite element method proves to be very efficient avoiding difficult computations or strategies like

Effect of surface defects on structure failure: A two-scale approach

\mathcal{C}^{1}

Small defects in mechanics

elements, adaptive mesh refinement, multi-grid resolution or isogeometric analysis. Beyond the classical benchmarks and comparisons with other existing methods, a numerical study has been carried out to investigate the influence of a polynomial approximation of the logarithmic free energy.

Artificial conditions for the linear elasticity equations

This work aims to take into account the influence of boundary defects on the behaviour till rupture of structures without any fine geometrical description of the domain. This is achieved by appealing to two approaches: an asymptotic analysis of Navier equations and strong discontinuity models. We present in this work a strategy to couple the two approaches in order to provide the analysis till rupture of the structure behavior. The approach is validated on an academic example.

Effect of micro-defects on structure failure Coupling asymptotic analysis and strong discontinuity

In this paper, we present a method to compute rapidly the solution of the Navier equation in domains with small inclusions close to each other. The main feature of our method is the use of a coarse description of the geometry. The computation relies on asymptotic expansion and computation of profiles, which are solution of a problem posed in unbounded domain. We propose and compare several artificial boundary conditions to compute these profiles efficiently.

Multi-scale asymptotic expansion for a singular problem of a free plate with thin stiffener

In this paper, we consider the equations of linear elasticity in an exterior domain. We exhibit artificial boundary conditions on a circle, which lead to a non-coercive second order boundary value problem. In the particular case of an axisymmetric geometry, explicit computations can be performed in Fourier series proving the well-posedness except for a countable set of parameters. A perturbation argument allows to consider near-circular domains. We complete the analysis by some numerical simulations.

Artificial boundary conditions to compute correctors in linear elasticity

This work aims at taking into account the influence of geometrical defects on the behavior till complete failure of structures. This is achieved without any fine description of the exact geometry of the perturbations. The proposed strategy is based on two approaches: asymptotic analysis of Navier equations and strong discontinuity approach.

Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer

In this paper, we consider a partially clamped plate which is stiffened on a portion of its free boundary. Our aim is to build an asymptotic expansion of the displacement, solution of the Kirchhoff-Love model, with respect to the thickness of the stiffener. Due to the mixed boundary conditions, singularities appear, obstructing the construction of the terms of the asymptotic expansion in the same way as if the plate was surrounded by the stiffener on its whole boundary. Using a splitting into regular and singular parts, we are able to formulate an asymptotic expansion involving profiles which allow to take into account the singularities.