Topological Sensitivity Analysis for the Anisotropic Laplace Problem

This paper is concerned with the reconstruction of objects immersed in anisotropic media from boundary measurements. The aim of this paper is to propose an alternative approach based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. The idea is to formulate the reconstruction problem as a topology optimization one minimizing an energy-like function. We derive a topological asymptotic expansion for the anisotropic Laplace operator. The unknown object is reconstructed using level-set curve of the topological gradient. We make finally some numerical examples proving the efficiency and accuracy of the proposed algorithm.

Topological sensitivity analysis for a three-dimensional parabolic type problem

This work focuses on the topological sensitivity analysis of a three-dimensional parabolic type problem. The considered application model is described by the heat equation. We derive a new topological asymptotic expansion valid for various shape functions and geometric perturbations of arbitrary form. The used approach is based on a rigorous mathematical framework describing and analyzing the asymptotic behavior of the perturbed temperature field.

Topological asymptotic formula for the 3D non-stationary Stokes problem and application

International audience This paper is concerned with a topological asymptotic expansion for a parabolic operator. We consider the three dimensional non-stationary Stokes system as a model problem and we derive a sensitivity analysis with respect to the creation of a small Dirich-let geometric perturbation. The established asymptotic expansion valid for a large class of shape functions. The proposed analysis is based on a preliminary estimate describing the velocity field perturbation caused by the presence of a small obstacle in the fluid flow domain. The obtained theoretical results are used to built a fast and accurate detection algorithm. Some numerical examples issued from a lake oxygenation problem show the efficiency of the proposed approach. Ce papier porte sur l'analyse de sensibilité topologique pour un opérateur parabolique. On considère le problème de Stokes instationnaire comme un exemple de modèle et on donne une étude de sensibilité décrivant le comportement asymptotique de l'opérateur relativement à une petite perturbation géométrique du domaine. L'analyse présentée est basée sur une estimation du champ de vitesse calculée dans le domaine perturbé. Les résultats de cette étude ont servi de base pour développer un algorithme d'identification géométrique. Pour la validation de notre approche, on donne une étude numérique pour un problème d'optimisation d'emplacement des injecteurs dans un lac eutrophe. Des exemples numériques montrent l'efficacité de la méthode proposée

Topological asymptotic analysis of a diffusive–convective–reactive problem

Purpose The purpose of this study is sensitivity analysis of the L2-norm and H1-seminorm of the solution of a diffusive–convective–reactive problem to topological changes of the underlying material. Design/methodology/approach The topological derivative method is used to devise a simple and efficient topology design algorithm based on a level-set domain representation method. Findings Remarkably simple analytical expressions for the sensitivities are derived, which are useful for practical applications including heat exchange topology design and membrane eigenvalue maximization. Originality/value The topological asymptotic expansion associated with a diffusive–convective–reactive equation is rigorously derived, which is not available in the literature yet.

An Overview of the Topological Gradient Approach in Image Processing: Advantages and Inconveniences

Image analysis by topological gradient approach is a technique based upon the historic application of the topological asymptotic expansion to crack localization problem from boundary measurements. This paper aims at reviewing this methodology through various applications in image processing; in particular image restoration with edge detection, classification and segmentation problems for both grey level and color images is presented in this work. The numerical experiments show the efficiency of the topological gradient approach for modelling and solving different image analysis problems. However, the topological gradient approach presents a major drawback: the identified edges are not connected and then the results obtained particularly for the segmentation problem can be degraded. To overcome this inconvenience, we propose an alternative solution by combining the topological gradient approach with the watershed technique. The numerical results obtained using the coupled method are very interesting.

An introduction to the topological asymptotic expansion with examples

International audience To find an optimal domain is equivalent to look for Its characteristic function. At first sight this problem seems to be nondifferentiable. But it is possible to derive the variation of a cost function when we switch the characteristic function from 0 to 1 or from 1 to 0 a small area. Classical and two generalized adjoint approaches are considered in this paper. Their domain of validity is given and Illustrated by several examples. Using this gradient type Information, It is possible to build fast algorithms. Generally, only one Iteration Is needed to find the optimal shape. Trouver un domaine optimal est équivalent à la recherche de sa fonction caractéristique. A première vue, ce problème semble non différentiable, mais Il est possible de calculer la variation de la fonction coût lorsque la fonction caractéristique passe de 1 à 0 ou de 0 à 1 dans une région de petite taille. On s’appuiera sur une approche adjointe classique et deux généralisations de cette méthode. Le domaine de validité de ces différentes approches est donné et illustré par différents exemples. Cette Information de type gradient permet de construire des algorithmes très efficaces: en général, une seule Itération suffit pour trouver le domaine optimal.