The topological derivative is, conceptually, a derivative of a shape functional with respect to infinitesimal changes in its topology, such as adding an infinitesimal hole or crack. When used in higher dimensions than one, the term topological gradient is also used to name the first-order term of the topological asymptotic expansion, dealing only with infinitesimal singular domain perturbations. It has applications in shape optimization, topology optimization, image processing and mechanical modeling.
Contents
Definition
Let
where
Structural mechanics
The topological derivative can be applied to shape optimization problems in structural mechanics. The topological derivative can be considered as the singular limit of the shape derivative. It is a generalization of this classical tool in shape optimization. Shape optimization concerns itself with finding an optimal shape. That is, find
In 2005, the topological asymptotic expansion for the Laplace equation with respect to the insertion of a short crack inside a plane domain had been found. It allows to detect and locate cracks for a simple model problem: the steady-state heat equation with the heat flux imposed and the temperature measured on the boundary. The topological derivative had been fully developed for a wide range of second-order differential operators and in 2011, it had been applied to Kirchhoff plate bending problem with a fourth-order operator.
Image processing
In the field of image processing, in 2006, the topological derivative has been used to perform edge detection and image restoration. The impact of an insulating crack in the domain is studied. The topological sensitivity gives information on the image edges. The presented algorithm is non-iterative and thanks to the use of spectral methods has a short computing time. Only
In 2012, a general framework is presented to reconstruct an image
where
In this framework, the asymptotic expansion of the cost function
Thanks to the topological gradient, it is possible to detect the edges and their orientation and to define an appropriate
In image processing, the topological derivatives have also been studied in the case of a multiplicative noise of gamma law or in presence of Poissonian statistics.
Inverse problems
In 2009, the topological gradient method has been applied to tomographic reconstruction. The coupling between the topological derivative and the level set has also been investigated in this application.
Books
A. A. Novotny and J. Sokolowski, Topological derivatives in shape optimization, Springer, 2013.