Hamamuki N.,Hokkaido University |
Ntovoris E.,CERMICS ENPC
Interfaces and Free Boundaries | Year: 2016
In this paper we set up a rigorous justification for the reinitialization algorithm. Using the theory of viscosity solutions, we propose a well-posed Hamilton-Jacobi equation with a parameter, which is derived from homogenization for a Hamiltonian discontinuous in time which appears in the reinitialization. We prove that, as the parameter tends to infinity, the solution of the initial value problem converges to a signed distance function to the evolving interfaces. A locally uniform convergence is shown when the distance function is continuous, whereas a weaker notion of convergence is introduced to establish a convergence result to a possibly discontinuous distance function. In terms of the geometry of the interfaces, we give a necessary and sufficient condition for the continuity of the distance function.We also propose another simpler equation whose solution has a gradient bound away from zero. © 2016 European Mathematical Society.
Dobson M.,CERMICS ENPC |
Luskin M.,University of Minnesota |
Ortner C.,Mathematical Institute
Computer Methods in Applied Mechanics and Engineering | Year: 2011
Force-based atomistic-continuum hybrid methods are the only known pointwise consistent methods for coupling a general atomistic model to a finite-element continuum model. For this reason, and due to their algorithmic simplicity, force-based coupling methods have become a popular class of atomistic-continuum hybrid models as well as other types of multiphysics models. However, the recently discovered unusual stability properties of the linearized force-based quasicontinuum (QCF) approximation, especially its indefiniteness, present a challenge to the development of efficient and reliable iterative methods.We present analytic and computational results for the generalized minimal residual (GMRES) solution of the linearized QCF equilibrium equations. We show that the GMRES method accurately reproduces the stability of the force-based approximation and conclude that an appropriately preconditioned GMRES method results in a reliable and efficient solution method. © 2010 Elsevier B.V.
Noumir Y.,Ecole Normale Superieure de Cachan |
Le Guilcher A.,CERMICS ENPC |
Lardjane N.,CEA DAM Ile-de-France |
Monneau R.,CERMICS ENPC |
Sarrazin A.,CERMICS ENPC
Journal of Computational Physics | Year: 2015
We develop a new algorithm for the computation of the Geometrical Shock Dynamics (GSD) model. The method relies on the fast-marching paradigm and enables the discrete evaluation of the first arrival time of a shock wave and its local velocity on a Cartesian grid. The proposed algorithm is based on a first order upwind finite difference scheme and reduces to a local nonlinear system of two equations solved by an iterative procedure. Reference solutions are built for a smooth radial configuration and for the 2D Riemann problem. The link between the GSD model and p-systems is given. Numerical experiments demonstrate the efficiency of the scheme and its ability to handle singularities. © 2014 Elsevier Inc..
Al Haj M.,CERMICS ENPC |
Forcadel N.,University of Paris Dauphine |
Monneau R.,CERMICS ENPC
Archive for Rational Mechanics and Analysis | Year: 2013
In this article, we study the existence and the uniqueness of traveling waves for a discrete reaction-diffusion equation with bistable nonlinearity, namely a generalization of the fully overdamped Frenkel-Kontorova model. This model consists of a system of ODEs which describes the dynamics of crystal defects in lattice solids. Under very weak assumptions, we prove the existence of a traveling wave solution and the uniqueness of the velocity of propagation of this traveling wave. The question of the uniqueness of the profile is also studied by proving Strong Maximum Principle or some weak asymptotics on the profile at infinity. © 2013 Springer-Verlag Berlin Heidelberg.