27 research outputs found
Shape optimization for quadratic functionals and states with random right-hand sides
In this work, we investigate a particular class of shape optimization
problems under uncertainties on the input parameters. More precisely, we are
interested in the minimization of the expectation of a quadratic objective in a
situation where the state function depends linearly on a random input
parameter. This framework covers important objectives such as tracking-type
functionals for elliptic second order partial differential equations and the
compliance in linear elasticity. We show that the robust objective and its
gradient are completely and explicitly determined by low-order moments of the
random input. We then derive a cheap, deterministic algorithm to minimize this
objective and present model cases in structural optimization
A consistent relaxation of optimal design problems for coupling shape and topological derivatives
In this article, we introduce and analyze a general procedure for approximating a ‘black and white’ shape and topology optimization problem with a density optimization problem, allowing for the presence of ‘grayscale’ regions. Our construction relies on a regularizing operator for smearing the characteristic functions involved in the exact optimization problem, and on an interpolation scheme, which endows the intermediate density regions with fictitious material properties. Under mild hypotheses on the smoothing operator and on the interpolation scheme, we prove that the features of the approximate density optimization problem (material properties, objective function, etc.) converge to their exact counterparts as the smoothing parameter vanishes. In particular, the gradient of the approximate objective functional with respect to the density function converges to either the shape or the topological derivative of the exact objective. These results shed new light on the connections between these two different notions of sensitivities for functions of the domain, and they give rise to different numerical algorithms which are illustrated by several experiment
A Mesh Adaptation Strategy to Predict Pressure Losses in LES of Swirled Flows
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Program (FP/2007-2013) / ERC Grant Agreement ERC-AdG 319067-INTECOCIS
Multi-phase structural optimization
We consider the optimal distribution of several elastic materials in a fixed working
domain. In order to optimize both the geometry and topology of the mixture we rely on the
level set method for the description of the interfaces between the different phases. We
discuss various approaches, based on Hadamard method of boundary variations, for computing
shape derivatives which are the key ingredients for a steepest descent algorithm. The
shape gradient obtained for a sharp interface involves jump of discontinuous quantities at
the interface which are difficult to numerically evaluate. Therefore we suggest an
alternative smoothed interface approach which yields more convenient shape derivatives. We
rely on the signed distance function and we enforce a fixed width of the transition layer
around the interface (a crucial property in order to avoid increasing “grey” regions of
fictitious materials). It turns out that the optimization of a diffuse interface has its
own interest in material science, for example to optimize functionally graded materials.
Several 2-d examples of compliance minimization are numerically tested which allow us to
compare the shape derivatives obtained in the sharp or smoothed interface cases
A LINEARIZED APPROACH TO WORST-CASE DESIGN IN PARAMETRIC AND GEOMETRIC SHAPE OPTIMIZATION
The purpose of this article is to propose a deterministic method for optimizing a structure with respect to its worst possible behavior when a ‘small ’ uncertainty exists over some of its features. The main idea of the method is to linearize the considered cost function with respect to the uncertain parameters, then to consider the supremum function of the obtained linear approximation, which can be rewritten as a more ‘classical ’ function of the design, owing to standard adjoint techniques from optimal control theory. The resulting ‘linearized worst-case ’ objective function turns out to be the sum of the initial cost function and of a norm of an adjoint state function, which is dual with respect to the considered norm over perturbations. This formal approach is very general, and can be justified in some special cases. In particular, it allows to address several problems of considerable importance in both parametric and shape optimization of elastic structures, in a unified framework. Contents Abstract
Scalarization of Set-Valued Optimization Problems in Normed Spaces
This work focuses on scalarization processes for nonconvex set-valued optimization problems whose solutions are defined by the socalled l-type less order relation, the final space is normed and the ordering cone is not necessarily solid. A scalarization mapping is introduced, which generalizes the well-known oriented distance, and its main properties are stated. In particular, by choosing a suitable norm it is shown that it coincides with the generalization of the so-called Tammer-Weidner nonlinear separation mapping to this kind of optimization problems. After that, two concepts of solution are characterized in terms of solutions of associated scalar optimization problems defined through the new scalarization mapping