46 research outputs found
A new approach for approximating linear elasticity problems
No full textInternational audienceIn this Note, we present and analyze a new method for approximating linear elasticity problems in dimension two or three. This approach directly provides approximate strains, i.e., without simultaneously approximating the displacements, in finite element spaces where the Saint Venant compatibility conditions are exactly satisfied in a weak form. To cite this article: P.G. Ciarlet, P. Ciarlet, Jr., C. R. Acad. Sci. Paris, Ser. I 346 (2008). © 2008 Académie des sciences
Finite Element Heterogeneous Multiscale Method for the Helmholtz Equation
No full textAbstract We show that standard Finite Element Heterogeneous Multiscale Method (FE-HMM) can be used to approximate the effective behavior of solutions of the classical Helmholtz equation in highly oscillatory media. Using a novel combination of well known results about FE-HMM and the notion of T -coercivity we derive an a priori error bound. Numerical experiments corroborate the analytical findings. To cite this article: P. Ciarlet Jr., C. Stohrer, C. R. Acad. Sci. Paris, Ser
Tools for solving the div-curl problem with mixed boundary conditions in a polygonal domain
No full textFollowing [Assous, Ciarlet, Jr., Sonnendrücker, Resolution of the Maxwell equations in a domain with reentrant corners (1998)], we continue to study the resolution of two-dimensional problems in non convex domains. In this previous paper, we considered several methods for solving Maxwell’s equations (stationary or instationary) with a perfectly conducting boundary condition
A justification of Peek's empirical law in electrostatics [Justification de la loi de Peek en électrostatique]
No full textInternational audienceWe consider the computation of the electrostatic charge density at the tip of a rounded corner. The relation between the curvature radius and the electrostatic field is given by Peek's empirical law which is valid only for thin, cylindrical or spherical, geometries. In this Note, we justify mathematically this law and extend it to other geometries. With the help of multiscaled asymptotic expansions, we derive an expression for the charge density for geometries which coincide at infinity with a cone. A numerical illustration is provided. To cite this article: P. Ciarlet Jr., S. Kaddouri, C. R. Acad. Sci. Paris, Ser. I 343 (2006). © 2006 Académie des sciences
On Saint Venant's compatibility conditions and Poincaré's lemma
No full textSaint Venant's theorem constitutes a classical characterization of smooth matrix fields as linearized strain tensor fields. This theorem has been extended to matrix fields with components in L2 by the second author and P. Ciarlet, Jr. in 2005. One objective of this Note is to further extend this characterization to matrix fields whose components are only in H-1. Another objective is to demonstrate that Saint Venant's theorem is in fact nothing but the matrix analog of Poincaré's lemma
Characterization of the kernel of the operator CURL CURL
No full textIn a simply-connected domain Ω in R3, the kernel of the operator CURLCURL acting on symmetric matrix fields from L2s (Ω) to H−2 s (Ω) coincides with the space of linearized strain tensor fields. For not simply-connected domains, Volterra has characterized this kernel for smooth fields. Here we extend this result for domains with a Lipschitz-continuous boundary for fields in L2s (Ω). To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007). © 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved
Système de Stokes avec flux de vitesse et pression imposés
No full textInternational audienceDans cette Note, nous étudions le système de Stokes avec flux de vitesse et pression imposés, dans un domaine borné, à bord régulier par morceaux
Electrowetting of a 3D drop : numerical modelling with electrostatic vector fields
No full textInternational audienceThe electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable for the full 3D case. © EDP Sciences, SMAI, 2010
Augmented formulations for solving Maxwell equations
No full textInternational audienceWe consider augmented variational formulations for solving the static or time-harmonic Maxwell equations. For that, a term is added to the usual H (curl) conforming formulations. It consists of a (weighted) L2 scalar product between the divergence of the EM and the divergence of test fields. In this respect, the methods we present are H (curl, div) conforming. We also build mixed, augmented variational formulations, with either one or two Lagrange multipliers, to dualize the equation on the divergence and, when applicable, the relation on the tangential or normal trace of the field. It is proven that one can derive formulations, which are equivalent to the original static or time-harmonic Maxwell equations. In the latter case, spurious modes are automatically excluded. Numerical analysis and experiments will be presented in the forthcoming paper [Augmented formulations for solving Maxwell equations: numerical analysis and experiments, in preparation]
ELECTROWETTING OF A 3D DROP: NUMERICAL MODELLING WITH ELECTROSTATIC FIELDS
No full textThe electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable for the full 3D case
