Geometric Finite Element Discretization of Maxwell Equations in Primal and Dual Spaces

Based on a geometric discretization scheme for Maxwell equations, we unveil a mathematical\textit{\}transformation between the electric field intensity $E$ and the magnetic field intensity $H$, denoted as Galerkin duality. Using Galerkin duality and discrete Hodge operators, we construct two system matrices, $[ X_{E}]$ (primal formulation) and $[ X_{H}$ (dual formulation) respectively, that discretize the second-order vector wave equations. We show that the primal formulation recovers the conventional (edge-element) finite element method (FEM) and suggests a geometric foundation for it. On the other hand, the dual formulation suggests a new (dual) type of FEM. Although both formulations give identical dynamical physical solutions, the dimensions of the null spaces are different.Comment: 22 pages and 4 figure

Complementarity bilateral bounds on forces in magnet systems

It is well known that complementarity can provide bilateral bounds on energy, in numerical approximations of nonlinear magnetostatics. Force is, up to sign, the derivative of energy, so can such bounds also apply to forces, torques, etc.? On the face of it, no, since inequalities are not preserved by differentiation. Yet, useful bounds can be given in some cases, as shown here, following an idea by E. Matagne (Eur. J. Appl. Phys., 25, 107 (2004))

Hierarchic finite element bases on unstructured tetrahedral meshes

The problem of constructing hierarchic bases for finite element discretization of the spaces H1, H(curl), H(div) and L2 on tetrahedral elements is addressed. A simple and efficient approach to ensuring conformity of the approximations across element interfaces is described. Hierarchic bases of arbitrary polynomial order are presented. It is shown how these may be used to construct finite element approximations of arbitrary, non-uniform, local order approximation on unstructured meshes of curvilinear tetrahedral elements

Application of Discrete Differential Forms to Spherically Symmetric Systems in General Relativity

In this article we describe applications of Discrete Differential Forms in computational GR. In particular we consider the initial value problem in vacuum space-times that are spherically symmetric. The motivation to investigate this method is mainly its manifest coordinate independence. Three numerical schemes are introduced, the results of which are compared with the corresponding analytic solutions. The error of two schemes converges quadratically to zero. For one scheme the errors depend strongly on the initial data.Comment: 22 pages, 6 figures, accepted by Class. Quant. Gra

New insights in gill/buccal rhythm spiking activity and CO2 sensitivity in pre- and post-metamorphic tadpoles (Pelophylax ridibundus)

Central CO2chemosensitivity is crucial for all air-breathing vertebrates and raises the question of itsrole in ventilatory rhythmogenesis. In this study, neurograms of ventilatory motor outputs recorded infacial nerve of premetamorphic and postmetamorphic tadpole isolated brainstems, under normo- andhypercapnia, are investigated using Continuous Wavelet Transform spectral analysis for buccal activityand computation of number and amplitude of spikes during buccal and lung activities. Buccal burstsexhibit fast oscillations (20-30 Hz) that are prominent in premetamorphic tadpoles: they result from thepresence in periodic time windows of high amplitude spikes. Hypercapnia systematically decreases thefrequency of buccal rhythm in both pre- and postmetamorphic tadpoles, by a lengthening of the interburstduration. In postmetamorphic tadpoles, hypercapnia reduces buccal burst amplitude and unmasks smallfast oscillations. Our results suggest a common effect of the hypercapnia on the buccal part of the CentralPattern Generator in all tadpoles and a possible effect at the level of the motoneuron recruitment inpostmetamorphic tadpoles

Electromagnetic wormholes and virtual magnetic monopoles

We describe new configurations of electromagnetic (EM) material parameters, the electric permittivity $\epsilon$ and magnetic permeability $\mu$, that allow one to construct from metamaterials objects that function as invisible tunnels. These allow EM wave propagation between two points, but the tunnels and the regions they enclose are not detectable to EM observations. Such devices function as wormholes with respect to Maxwell's equations and effectively change the topology of space vis-a-vis EM wave propagation. We suggest several applications, including devices behaving as virtual magnetic monopoles.Comment: 4 pages, 3 figure

Commentaire de : " Transformation thermodynamics: cloaking and concentrating heat flux " - Opt. Express 20, 8207 (mars 2012).

Optics Express a publié en mars 2012 un article concernant la transposition au domaine de la thermique (équation de la chaleur) du principe de la cape d'invisibilité optique. Les auteurs y présentent en particulier la théorie bidimensionnelle d'un métamatériau en forme de disque, aux propriétés thermiques remarquables en conduction pure (équation de la chaleur), dont ils proposent ensuite une réalisation approchée, formée de dix couches d'isolant thermique séparées par dix couches de conducteurs de conductivités thermiques décroissant avec le rayon. Nous donnons ici les résultats d'une expérience numérique complémentaire, consistant à comparer le comportement thermique de la réalisation proposée de ce métamatériau à celui d'une configuration tout à fait banale à deux couches. Cette expérience montre clairement que les auteurs sont allés trop loin dans l'interprétation pratique de leurs résultats théoriques. En particulier, et conformément aux résultats habituels de la thermodynamique, la réalisation approchée qu'ils proposent pour leur matériau théorique (tout comme plus généralement toute autre réalisation, aussi soignée soit-elle), ne permet en aucun cas de protéger un objet de la chaleur mieux que ne le fait un simple isolant d'épaisseur équivalente. L'isolation qu'ils obtiennent est même moins bonne, ce qui enlève tout intérêt pratique à leur travail, qui contient par ailleurs d'autres erreurs

On the degrees of freedom of lattice electrodynamics

Using Euler's formula for a network of polygons for 2D case (or polyhedra for 3D case), we show that the number of dynamic\textit{\}degrees of freedom of the electric field equals the number of dynamic degrees of freedom of the magnetic field for electrodynamics formulated on a lattice. Instrumental to this identity is the use (at least implicitly) of a dual lattice and of a (spatial) geometric discretization scheme based on discrete differential forms. As a by-product, this analysis also unveils a physical interpretation for Euler's formula and a geometric interpretation for the Hodge decomposition.Comment: 14 pages, 6 figure

The Simplicial Ricci Tensor

The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton to define a non-linear, diffusive Ricci flow (RF) that was fundamental to Perelman's proof of the Poincare conjecture. Analytic applications of RF can be found in many fields including general relativity and mathematics. Numerically it has been applied broadly to communication networks, medical physics, computer design and more. In this paper, we use Regge calculus (RC) to provide the first geometric discretization of the Ric. This result is fundamental for higher-dimensional generalizations of discrete RF. We construct this tensor on both the simplicial lattice and its dual and prove their equivalence. We show that the Ric is an edge-based weighted average of deficit divided by an edge-based weighted average of dual area -- an expression similar to the vertex-based weighted average of the scalar curvature reported recently. We use this Ric in a third and independent geometric derivation of the RC Einstein tensor in arbitrary dimension.Comment: 19 pages, 2 figure

Variational Inequalities in Critical-State Problems

Similar evolutionary variational inequalities appear as convenient formulations for continuous quasistationary models for sandpile growth, formation of a network of lakes and rivers, magnetization of type-II superconductors, and elastoplastic deformations. We outline the main steps of such models derivation and try to clarify the origin of this similarity. New dual variational formulations, analogous to mixed variational inequalities in plasticity, are derived for sandpiles and superconductors.Comment: Submitted for publicatio