Topological derivative
Encyclopedia
In the fields of shape optimization
and topology optimization
, a topological derivative is, conceptually, a derivative
of a function of a region 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 often used, it comes from the first-order term of the topological asymptotic expansion, dealing only with infinitesimal perturbations.
Shape optimization concerns itself with finding an optimal shape. That is, find
to minimize some scalar-valued objective function, 
. Neglecting changes in topology, an initial guess can be improved by perturbing the shape of 
by methods of calculus of variations
and functional analysis
.
Shape optimization
Shape optimization is part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given constraints...
and topology optimization
Topology optimization
- Introduction :Topology optimisation is a mathematical approach that optimises material layout within a given design space, for a given set of loads and boundary conditions such that the resulting layout meets a prescribed set of performance targets...
, a topological derivative is, conceptually, a derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of a function of a region 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 often used, it comes from the first-order term of the topological asymptotic expansion, dealing only with infinitesimal perturbations.
Shape optimization concerns itself with finding an optimal shape. That is, find
to minimize some scalar-valued objective function, . Neglecting changes in topology, an initial guess can be improved by perturbing the shape of by methods of calculus of variationsCalculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
and functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
.
External links
- Allaire and al. Structural optimization using topological and shape sensitivity via a level set method
- S. Amtutz, I. Horchani and M. Masmoudi Crack detection by the topological gradient method
- J. Sokołowski and A. Zochowski On topological derivative in shape optimization