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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.

Definition

Let $\Omega$ be an open bounded domain of $\mathbb {R} ^{d}$, with $d\leq 2$, which is subject to a nonsmooth perturbation confined in a small region $\omega _{\varepsilon }({\tilde {x}})={\tilde {x}}+\varepsilon \omega$ of size $\varepsilon$ with ${\tilde {x}}$ an arbitrary point of $\Omega$ and $\omega$ a fixed domain of $\mathbb {R} ^{d}$. Let $\Psi$ be a characteristic function associated to the unperturbed domain and $\Psi _{\varepsilon }$ be a characteristic function associated to the perforated domain $\Omega _{\varepsilon }=\Omega \backslash {\overline {\omega _{\varepsilon }}}$. A given shape functional $\Phi (\Psi _{\varepsilon }({\tilde {x}}))$ associated to the topologically perturbed domain, admits the following topological asymptotic expansion:

$\Phi (\Psi _{\varepsilon }({\tilde {x}}))=\Phi (\Psi )+f(\varepsilon )g({\tilde {x}})+o(f(\varepsilon ))$

where $\Phi (\Psi )$ is the shape functional associated to the reference domain, $f(\varepsilon )$ is a positive first order correction function of $\Phi (\Psi )$ and $o(f(\varepsilon ))$ is the remainder. The function $g({\tilde {x}})$ is called the topological derivative of $\Phi$ at ${\tilde {x}}$.

Applications

Shape and topology optimization

The topological derivative can be applied to shape optimization problems in structural mechanics. Shape optimization concerns itself with finding an optimal shape. That is, find $\Omega$ to minimize some scalar-valued objective function, $J(\Omega )$. This technique can be coupled with level set method.

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.

Image processing

In the field of image processing, in 2006, the topological derivatives 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 $O(Nlog(N))$ operations are needed to detect edges, where $N$ is the number of pixels.

During the following years, other problems have been considered: classification, segmentation, inpainting and super-resolution. This approach can be applied to gray-level or color images. Until 2010, isotropic diffusion was used for image reconstructions. The topological gradient is also able to provide edge orientation and this information can be used to perform anisotropic diffusion.

Inverse problems

In 2005, the topological asymptotic expansion for the Laplace equation with respect to the insertion of a short crack inside a plane domain has 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.

In 2009, the topological gradient method has been applied to tomographic reconstruction.

References

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Books

A. A. Novotny and J. Sokolowski, Topological derivatives in shape optimization, Springer, 2013.

External links

• Allaire and al. Structural optimization using topological and shape sensitivity via a level set method