Denise Chenais

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.

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.

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.

SHAPE OPTIMIZATION IN SHELL THEORY: Design Sensitivity of the Continuous Problem

Abstract We consider a non-shallow shell made of an isotropic homogeneous material, working in linear elastic conditions, subjected to a given load. Our aim is to change the shape of the shell so that it resists better towards a given criterion. By shape, we mean essentially the midsurface of the shell. The thickness could be added without any difficulty. The important aspect that we study here is the midsurface. This problem is worked by gradient type methods. We prove that if the criterion depends on the displacement field through a differentiable function, then it depends on the shape in a differentiable manner, because the displacement field is a differentiable function of the shape. Then we present an analytical formula giving the exact gradient of the criteria before any discretization. After that, we explain how to compute numerically an approximation to this exact gradient. Then we give numerical results.

Discrete Gradient and Discretized Continuum Gradient Methods for Shape Optimization of Shells

ABSTRACT This paper is concerned with a family of problems of optimization of systems governed by partial differential equations (PDE). One example is considered here: the shape optimization of a shell working in linear elasticity. The displacement field is the solution of an elliptic PDE that has a design variable in its coefficients. The criterion to be minimized depends on the design variable through the solution of the PDE. It will be solved by a descent method, which requires computation of the gradient. Since such a problem must be solved numerically, it is necessary to go through discretizations. It is possible to discretize before or after differentiating. The specific subject discussed here is the comparison between two dif ferent methods: (1rpar; Discrete Gradient (DG): discretize first, then differentiate; lpar;2rpar;Discretized Continuum Gradient (DCG): differentiate first, then discretize.

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

SummaryIn 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.

Necessary conditions to avoid numerical locking in singular perturbations. Application to arch equations

We study the problem of numerical computation of the solution of equations posed in infinite dimensional vector spaces (for example, partial differential equations). We focus specially on the case of linear equations depending on a real parameter t ϵ [0, 1]. The solution is supposed to exist and to be unique for any given t ϵ [0, 1], but for t = 0, there are several solutions: the kernel of the operator defining the equation degenerates, for t = 0 it is not reduced to {0}. Several specific equations of this type have been investigated in the literature. The authors usually try to build approximation schemes which converge to the solution of the equation uniformly with respect to the parameter when it tends to zero. In this paper, we study, first in a general setting, the existence of any approximation scheme which might converge uniformly with respect to t ϵ [0, 1]. This is done using the notion of nth which of a subset of a Banach space. Then the result is applied to the equation governing the linear static equilibrium of an arch. The small parameter is the thickness of the arch. The result says that the best possible convergence is uniform in t. Furthermore one finite element method which is very commonly used in industrial problems, degenerates when t becomes small, but another finite element method which has been proposed more recently does converge uniformly with respect to t. Our result says that it is optimal.

Why does it seem difficult to avoid homogenization type methods in order to change topology in shape optimization

This paper shows that in a large class of situations, moving continuously a domain in order to optimize its shape prevents from introducing new holes.This gives an explanation about the introduction of homogenization in shape optimization when the topology is not known a priori.

Finite Element Approximation of Wild Optimal Shapes

We consider a problem of optimal design in which the functional to minimize depends on the shape of the unknown domain through the solution of a Laplace-Dirichlet equation which is posed on this domain. We are interested in optimal controls whose topology and regularity are not known a priori. In the continuous case, in dimension n = 2, ˘ak [?] proved that there exists an optimal domain in the class of all open subsets of a given bounded open set which have a uniformly bounded number of holes. Here, we consider a finite-element discrete version of this problem and prove that sequences of discrete optimal domains converge in the complementary