Toplogical derivative for nonlinear magnetostatic problem

The topological derivative represents the sensitivity of a domain-dependent functional with respect to a local perturbation of the domain and is a valuable tool in topology optimization. Motivated by an application from electrical engineering, we derive the topological derivative for an optimization problem which is constrained by the quasilinear equation of two-dimensional magnetostatics. Here, the main ingredient is to establish a sufficiently fast decay of the variation of the direct state at scale 1 as $|x|\rightarrow \infty$. In order to apply the method in a bi-directional topology optimization algorithm, we derive both the sensitivity for introducing air inside ferromagnetic material and the sensitivity for introducing material inside an air region. We explicitly compute the arising polarization matrices and introduce a way to efficiently evaluate the obtained formulas. Finally, we employ the derived formulas in a level-set based topology optimization algorithm and apply it to the design optimization of an electric motor.Comment: 54 pages, 9 figure

Devoirs à domicile différenciés :: comment les instaurer et quelle en est la perception d’enseignants jurassiens ?

Ce travail de mémoire est basé sur la thématique des devoirs à domicile. En effet, il s’agit d’un thème très important, qui fait débat depuis plusieurs années maintenant, car beaucoup de personnes s’interrogent sur leur finalité. À cet effet, je me suis intéressée à une pratique très peu connue, qui est celle des devoirs à domicile différenciés. En effet, même si la différenciation est une pratique très actuelle dans les écoles jurassiennes, elle n’est pas présente lors de la distribution des devoirs à domicile. Afin de répondre aux diverses interrogations ayant trait à cette thématique, j’ai introduit cette pratique dans une classe de 4ème année, afin d’en observer les effets. Par conséquent, il a fallu réfléchir à toutes les modalités à mettre en place pour que cette manière de fonctionner puisse être avantageuse pour les élèves. Au terme de cette expérimentation, j’ai comparé mes résultats avec les avis de deux enseignantes, travaillant dans le même degré

TOPOLOGICAL ASYMPTOTIC ANALYSIS FOR A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS

International audienceTopological asymptotic expansions for quasilinear elliptic equations have not been studied yet. Such questions arise from the need to apply topological asymptotic methods in shape optimization to nonlinear elasticity equations as in imaging to detect sets with codimensions ≥ 2 (e.g. points in 2D or segments in 3D). Our main contribution is to provide topological asymptotic expansions for a class of quasilinear elliptic equations, perturbed in non-empty subdomains. The obtained topological gradient can be split into a classical linear term and a new term which accounts for the non linearity of the equation. With respect to topological asymptotic analysis, moving from linear equations onto quasilinear ones requires to heavily revise the implemented methods and tools. By comparison with the steps carried out to obtain such expansions with the Laplace equation, the core issue for a quasilinear equation lies in the ability to defi ne the variation of the direct state at scale 1 in R^N . Accordingly we build dedicated weighted quotient Sobolev spaces, which semi-norms encompass both the L^p norm and the L^2 norm of the gradient in R^N. Then we consider an appropriate class of quasilinear elliptic equations, to ensure that the problem de ning the direct state at scale 1 enjoys a combined p and 2 ellipticity property. The needed asymptotic behavior of the solution of the nonlinear interface problem in R^N is then proven. An appropriate duality scheme is set up between the direct and adjoint states at each stage of approximation

Calculation of tapered monoplane wings

The tapered wing shape increases the lift in the middle of the wing and thus reduces the bending moment of the lifting forces in the plane of symmetry. Since this portion of the wing is the thickest, the stresses of the wing material are reduced and desirable space is provided for stowing the loads in the wing. This statically excellent form of construction, however, has aerodynamic disadvantages which must be carefully weighed, if failures are to be avoided. This treatise is devoted to the consideration of these problems