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.

Application of the topological gradient method to tomography

International audience A new method for parallel beam tomography is proposed. This method is based on the topological gradient approach. The use of the topological asymptotic analysis for detecting the main edges of the data allows us to filter the noise while inverting the Radon transform. Experimental results obtained on noisy data illustrate the efficiency of this promising approach in the case of Magnetic Resonance Imaging. We also study the sensitivity of the algorithm with respect to several regularization and weight parameters. Une nouvelle méthode de reconstruction pour la tomographie par faisceaux parallèles est proposée. Cette méthode est basée sur l’approche du gradient topologique. En détectant les contours sur les données grâce à l’analyse asymptotique topologique, il est possible de filtrer le bruit dans le processus d’inversion de la transformée de Radon. Des résultats expérimentaux obtenus sur des données bruitées illustrent les possibilités de cette approche prometteuse dans le domaine de traitement d’images IRM. Nous étudions également la sensibilité de l’algorithme par rapport aux différents paramètres de régularisation et pondération.

Sensitivity of the macroscopic response of elastic microstructures to the insertion of inclusions

This paper proposes an exact analytical formula for the topological sensitivity of the macroscopic response of elastic microstructures to the insertion of circular inclusions. The macroscopic response is assumed to be predicted by a well-established multi-scale constitutive theory where the macroscopic strain and stress tensors are defined as volume averages of their microscopic counterpart fields over a representative volume element (RVE) of material. The proposed formula—a symmetric fourth-order tensor field over the RVE domain—is a topological derivative which measures how the macroscopic elasticity tensor changes when an infinitesimal circular elastic inclusion is introduced within the RVE. In the limits, when the inclusion/matrix phase contrast ratio tends to zero and infinity, the sensitivities to the insertion of a hole and a rigid inclusion, respectively, are rigorously obtained. The derivation relies on the topological asymptotic analysis of the predicted macroscopic elasticity and is presented in detail. The derived fundamental formula is of interest to many areas of applied and computational mechanics. To illustrate its potential applicability, a simple finite element-based example is presented where the topological derivative information is used to automatically generate a bi-material microstructure to meet pre-specified macroscopic properties.

OPTIMAL EDGE DETECTION BY TOPOLOGICAL ASYMPTOTIC ANALYSIS

In this paper, we consider a variational approach to the problem of edge detection without using a priori information. To begin with, we derive an asymptotic expansion of a functional inspired by the Mumford–Shah model at its global minimum. Then, we show that, according to our model, the optimal set of image edges is indicated by the set of points for which the dominant term of this expansion is minimal and the topological derivative associated with the considered functional is equal to zero. These two conditions form the basis for the introduced method to edge detection, which does not require prior setting of parameters. The analytical results presented in this paper are additionally validated by numerical experiments.