Optimal design of midsurface of shells: Differentiability proof and sensitivity computation

We suppose that a shell submitted to a given load (self-weight or wind, for instance), has to resist as well as possible towards given criteria. We aim at the following problem: Is it possible to find an optimal design of the midsurface of the shell with respect to this criteria? This problem can be worked using gradient-type algorithms. In this paper we work on the differentiability proof and numerical computation of the gradient. For a given shape of the midsurface, we consider that the shell works in linear elastic conditions. We use the Budiansky-Sanders model for elastic shells, from which we get the displacement field in the shell. The criteria to be minimized are supposed to depend on the shape directly, and also through the displacement field. In this paper, we prove that the displacement field depends on the shape in a Fréchet-differentiable manner (for an appropriate topology on the set of admissible shapes). Then we give a way to compute the gradient of a given criteria from a theoretical point of view and from a numerical point of view. This allows us to use descent-type methods of optimization. They will lead to shapes which react better and better. Notice that we know nothing about convergence of these methods, the existence and unicity of a theoretical optimal solution. But from a practical point of view, it is quite interesting to be able to modify a given shape to obtain a better one.     

Approximation par éléments finis de problèmes elliptiques d'optimisation de forme

We consider a problem of elliptic optimal design. The control is the shape of the domain on which the Dirichlet problem for the Laplace equation is posed. In dimension n=2, S?veràk proved that there exists an optimal domain in the class of all open subsets of a given bounded open set, whose complements have a uniformly bounded number of connected components. The proof (J. Math. Pures Appl. 72 (1993) 537–551) is based on the compactness of this class of domains with respect to the complementary-Hausdorff topology and the continuous dependence of the solutions of the Dirichlet Laplacian in H¹ with respect to it. In this Note we consider a finite-element discrete version of this problem and prove that the discrete optimal domains converge in that topology towards the continuous one as the mesh-size tends to zero. The key point of the proof is that finite-element approximations of the solution of the Dirichlet Laplacian converge in H¹ whenever the polygonal domains converge in the sense of that topology. To cite this article: D. Chenais, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Homéomorphisme entre ouverts lipschitziens

Résumé Dans cet article, on démontre un résultat concernant l'ensemble π(θ, h, R) des partie uniformément bornées deR ⁿ qui sont des ouverts uniformément lipschitziens de constantes données θ, h et R. On démontre que si la différence symétrique entre2 tels ouverts lipschitziens éléments de π(θ, h, R), est de mesure assez petite, alors ces2 ouverts sont homéomorphes. De plus, l'homéomorphisme est lipschitzien, de constante ne dépendant que de θ, h et R. Entrata in Redazione il 5 aprile 1977.     

Continuité et différentiabilité d'Éléments propres: Application à l'optimisation de structures

The buckling load of a structure may usually be computed with an eigenvalue problem: it is the eigenvalue of smallest absolute value. In optimizing structures with a constraint on the buckling load, repeated eigenvalues are likely to occur. We prove continuity and differentiability results of eigenelements with respect to design variables using the variational characterization of eigenvalues. We illustrate these results with a classical problem: buckling of a beam. Application to arch buckling is presented in another article.     

Deterioration of a finite element method for arch structures when the thickness goes to zero

Summary In this paper, we establish that a popular finite element method for arch structures degenerates when the thickness tends to zero. This is due to the fact that, for null thickness, the energy functional looses the ellipticity property. We show then how to link the step size to the thickness in order to get required precision. Numerical results finally illustrate the theoretical analysis.     

On the locking phenomenon for a class of elliptic problems

Summary. In this paper we study the numerical behaviour of elliptic problems in which a small parameter is involved and an example concerning the computation of elastic arches is analyzed using this mathematical framework. At first, the statements of the problem and its Galerkin approximations are defined and an asymptotic analysis is performed. Then we give general conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter. Finally we study an example in computation of arches working in linear elasticity conditions. We build one finite element scheme giving a locking behaviour, and another one which does not. Revised version received October 25, 1993     

Design sensitivity for arch structures with respect to midsurface shape under static loading

We consider in this paper an example of structural optimization in which the structure is a loaded arch and the design variable is the shape of the arch. We concentrate on differentiability of static response with respect to shape changes. After recalling the arch equation with its functional spaces and the optimization problem, we state a differentiability theorem and provide a detailed proof. Numerical use of this result is finally discussed.     

High-performance liquid chromatographic analysis of the unusual pathway of oxidation of L-arginine to citrulline and nitric oxide in mammalian cells

A very unusual pathway of the oxidation of L-arginine to citrulline and nitric oxide has been discovered recently in cytotoxic macrophages. In an attempt to detect molecules generated through this metabolic pathway, a fast radio high-performance liquid chromatographic method was developed to analyse the whole set of radiolabelled L-arginine-derived metabolites produced by mammalian cells after appropriate induction. A new intermediate which might be NG-hydroxy-L-arginine was found.     

Shape Optimal Design Problem with Convective and Radiative Heat Transfer: Analysis and Implementation

We present a study of an optimal design problem for a coupled system, governed by a steady-state potential flow equation and a thermal equation taking into account radiative phenomena with multiple reflections. The state equation is a nonlinear integro-differential system. We seek to minimize a cost function, depending on the temperature, with respect to the domain of the equations. First, we consider an optimal design problem in an abstract framework and, with the help of the classical adjoint state method, give an expression of the cost function differential. Then, we apply this result in the two-dimensional case to the nonlinear integro-differential system considered. We prove the differentiability of the cost function, introduce the adjoint state equation, and give an expression of its exact differential. Then, we discretize the equations by a finite-element method and use a gradient-type algorithm to decrease the cost function. We present numerical results as applied to the automotive industry.