Stabilisation frontière du système élastodynamique dans un polygone plan

In this paper, we study the boundary stabilization of the elastodynamic system in a plane polygonal domain. Here, we take in account singularities which appear when changing boundary conditions. To cite this article: R. Brossard, J.-P. Lohéac, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A level-set formulation of immersed boundary methods for fluid–structure interaction problems

We propose a level set formulation of the immersed boundary method for fluid–structure problems in two and three dimensions. We prove that the resulting model verifies an energy estimate. To cite this article: G.-H. Cottet, E. Maitre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Construction du préconditionneur Neumann–Neumann de décomposition de domaine de niveau 2 pour des problèmes élastodynamiques en grandes déformations

In this Note, we present the construction of a Neumann–Neumann domain decomposition preconditioner with scalability properties to solve nonlinear elastostatic and elastodynamic problems. By using the results given in Alart et al. [C. R. Acad. Sci. Paris, Ser. I 331 (2000) 399–404] and according to the work realized in Farhat et al. [Int. J. Numer. Methods Engrg. 38 (1995) 3831–3853] and Fragakis and Papadrakakis [Comput. Methods Appl. Mech. Engrg. 193 (2004) 4611–4662], we give the construction steps of the two level preconditioner which are based on the introduction of a coarse space especially adapted to finite deformations problems with static and dynamic process. To cite this article: M. Barboteu, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Une méthode par éléments finis mixte préservant la positivité pour le problème de contact en élasticité

We consider a mixed finite element method for the contact problem in elasticity that furnishes approximated fields (displacements and constraints) satisfying the sign conditions of the continuous problem. We prove that the method verifies similar a priori error estimates as the standard method. To cite this article: P. Hild, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Approximation par la méthode des éléments finis de la formulation en domaine régulier de problèmes de fissures

The discretization by various mixed finite element methods of a new variational formulation of crack problems is considered. The new formulation, called the smooth domain method, is derived for crack problems in the case of a simplified model of an elastic membrane. Inequality type boundary conditions are prescribed at the crack faces. The resulting model takes the form of an unilateral contact problem on the crack. The mathematical analysis for the method leads to optimal convergence rates, as given in this Note. To cite this article: Z. Belhachmi et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

The dynamics of functioning investigating societal transitions with partial differential equations

In this article a mathematical framework is introduced and explored for the study of processes in societal transitions. A transition is conceptualised as a fundamental shift in the functioning of a societal system. The framework views functioning as a real-valued field defined upon a real variable. The initial status quo prior to a transition is captured in a field called the regime and the alternative that possibly takes over is represented in a field called a niche. Think for example of a transition in an energy supply system, where the regime could be centrally produced, fossil fuel based energy supply and a niche decentralised renewable energy production. The article then proceeds to translate theoretical notions on the interactions and dynamics of regimes and niches from transition literature into the language of this framework. This is subsequently elaborated in some simple models and studied analytically or by means of computer simulation.

Films courbés minces ferromagnétiques

We consider a thin curved ferromagnetic film not submitted to an external magnetic field. The behavior of the film is described by an energy depending on the magnetization of the film verifying the saturation constraint. The energy is composed of an induced magnetostatic energy and an energy term with density including the exchange energy and the anisotropic energy. We study the behavior of this energy when the thickness of the curved film goes to zero. We show with Γ-convergence arguments that the minimizers of the free energy converge to the minimizers of a local energy depending on a two-dimensional magnetization. To cite this article: H. Zorgati, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Some extensions of Carlitz on cyclic sums and generating functions

Recently, the author (Proc. Amer. Math. Soc., 57 1976, 271–275) derived two theorems involving double series, which gave as a consequence new and known generating functions for the Jacobi polynomial. The method of proof differed from that of previous workers. Using an extension of this procedure, we present in this paper two theorems for double and m-dimensional series which generalize our previous work. These formulas also yield new generating functions for the Jacobi polynomial and extend some formulas of Carlitz (Boll. U.M.I. (3), 16 1961, 150–155) and others. A feature of this work is the inclusion of the Jacobi polynomial within the framework of m-dimensional cyclic sums, thus generalizing a main result of Carlitz (SIAM Rev., 6 1964, 20–30).     

On the contact boundaries of normal surface singularities

The abstract boundary M of a normal complex-analytic surface singularity is canonically equipped with a contact structure. We show that if M is a rational homology sphere, then this contact structure is uniquely determined by the topological type of M. An essential tool is the notion of open book carrying a contact structure, defined by E. Giroux. To cite this article: C. Caubel, P. Popescu-Pampu, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

An example of nonuniqueness for the continuous static unilateral contact model with Coulomb friction

The aim of this Note is to propose an example of nonuniqueness for the continuous static unilateral contact model with Coulomb friction in linear elasticity. To cite this article: P. Hild, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Heterogeneous wires made of martensitic materials

In this paper we present a direct derivation of a theory of heterogeneous wires starting from three-dimensional nonlinear hyperelasticity augmented by an interfacial energy term. The derivation involves no a priori choice of asymptotic expansion or ansatz. It yields a wire theory with two Cosserat vector fields. The theory is applied to multiwell energy functions appropriate for martensitic materials. A formal derivation of higher theories of homogeneous wires is given, which yields three additional Cosserat vector fields and an explicit form for the bending and torsion energy. To cite this article: H. Le Dret, N. Meunier, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Generation and Enumeration of Some Classes of Interval Orders

In this paper we consider the class of interval orders, recently considered by several authors from both an algebraic and an enumerative point of view. According to Fishburn’s Theorem (Fishburn J Math Psychol 7:144–149, 1970), these objects can be characterized as posets avoiding the poset 2?+?2. We provide a recursive method for the unique generation of interval orders of size n?+?1 from those of size n, extending the technique presented by El-Zahar (1989) and then re-obtain the enumeration of this class, as done in Bousquet-Melou et al. (2010). As a consequence we provide a method for the enumeration of several subclasses of interval orders, namely AV(2?+?2, N), AV(2?+?2, 3?+?1), AV(2?+?2, N, 3?+?1). In particular, we prove that the first two classes are enumerated by the sequence of Catalan numbers, and we establish a bijection between the two classes, based on the cardinalities of the principal ideals of the posets.     

Cognitive trajectory of proof by contradiction for transition-to-proof students

History and research on proof by contradiction suggests proof by contradiction is difficult for students in a number of ways. Students’ comprehension of already-written proofs by contradiction is one such aspect that has received relatively little attention. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, we constructed and tested a cognitive model that describes how a student might construct the concept ‘proof by contradiction’ in an introduction to proof course. Data for this study was collected from students in a series of five teaching interventions focused on proof by contradiction. This paper will report on two participants as case studies to illustrate that our cognitive trajectory for proof by contradiction is a useful model for describing how students may come to understand the proof method.     

Le problème de Signorini dans la théorie des plaques minces de Kirchhoff–LoveSignorini's problem in the Kirchhoff–Love theory of plates

In the framework of the Kirchhoff–Love asymptotic theory of elastic thin plates we consider the unilateral contact problem with friction for a plate on a rigid foundation (Signorini problem with friction). First, we notice, when the thickness vanishes, that the order of the friction force must be lower than that of the contact pressure. These two measures are connected by Coulomb law. Consequently, at least formally, the friction force must be vanishing when the thickness goes to zero. We actually prove that any sequence of solution of the sequence of three-dimensional scaled Signorini problems with friction strongly converges to the unique solution of a two-dimensional Signorini plate problem without friction. To cite this article: J.-C. Paumier, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 567–570.     

Existence of solutions for a dynamic Signorini's contact problem

The purpose of this work is to present an existence result for the dynamic frictionless contact problem between an elastic body and a rigid foundation. The proof is based on five fundamental steps: a discretization in time which leads to a discretized problem with unique solution; the construction of functions approximating a solution of the problem; the treatment of the contact condition by means of a Lagrange multiplier whose orthogonality properties allow us to get a priori estimates; the convergence of said functions and, finally, the pass to the limit obtaining a weak solution of the continuous problem. To cite this article: M.T. Cao, P. Quintela, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

An optimization problem related to the modeling of atmospheric inorganic aerosols

A mathematical model for the computation of chemical equilibrium of atmospheric inorganic aerosols is proposed. The equilibrium is given by the minimum of the Gibbs free energy for a system involving an aqueous phase, a gas phase and solid salts. A primal-dual method solving the Karush–Kuhn–Tucker conditions is detailed. An active set/Newton method permits the computation of the minimum and track solid salts at the equilibrium. To cite this article: N.R. Amundson et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Méthode d'éléments finis avec hybridisation frontière pour les problèmes de contact avec frottementFinite element method with Lagrange multipliers for contact problems with friction

In this Note, we propose a finite element method with Lagrange multipliers in order to approximate contact problems with friction. The discretized normal and tangential constraints at the candidate contact interface are expressed by using continuous piecewise linear Lagrange multipliers in the saddle-point formulation. An optimal error estimate is established and several numerical studies corresponding to this choice of the discretized normal and tangential constraints are achieved. To cite this article: L. Baillet, T. Sassi, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 917–922.     

Existence of a solution to a dynamic unilateral contact problem for a cracked viscoelastic body

In this paper we study a dynamic unilateral contact problem with friction for a cracked viscoelastic body. The viscoelastic model is characterized by Kelvin–Voigt's law and a nonlocal friction law is investigated here. The existence of a solution to the problem is obtained by using a penalty method. Several estimates are obtained on the solution to the penalized problem, which enable us to pass to the limit by using compactness results. To cite this article: M. Cocou, G. Scarella, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Generalized ridge regression,least squares with stochastic prior information,and Bayesian estimators

The ridge estimator of the usual linear model is generalized by the introduction of an a priori vector r and an associated positive semidefinite matrix S. It is then shown that the generalized ridge estimator can be justified in two ways: (a) by the minimization of the residual sum of squares subject to a constraint on the length, in the metric S, of the vector of differences between r and the estimated linear model coefficients, (b) by incorporating prior knowledge, r playing the role of the vector of means and S proportional to the precision matrix. Both a Bayesian and an Aitken generalized least squares frameworks are used for the latter. The properties of the new estimator are derived and compared to the ordinary least squares estimator. The new method is illustrated with different assumptions on the form of the S matrix.     

MV-closures of Wajsberg hoops and applications

In this paper we construct, given a Wajsberg hoop A, an MV-algebra MV(A) such that the underlying set A of A is a maximal filter of MV(A) and the quotient MV(A)/A is the two element chain. As an application we provide a topological duality for locally finite Wajsberg hoops based on a previously known duality for locally finite MV-algebras. We also give another duality for k-valued Wajsberg hoops based on a different representation of k-valued MV-algebras and show the relation to the first duality. We also apply this construction to give a topological representation for free k-valued Wajsberg hoops.