Editing Topology optimization (section)

== Problem statement ==

A topology optimization problem can be written in the general form of an [[optimization problem]] as:

: <math>\begin{align}
&\underset{\rho}{\operatorname{minimize}} & &F = F(\mathbf{u(\rho), \rho}) =  \int_{\Omega}  f(\mathbf{u(\rho), \rho}) \mathrm{d}V \\
&\operatorname{subject\;to} 
& &G_0(\rho) = \int_{\Omega} \rho \mathrm{d}V - V_0 \leq 0 \\
&&&G_j(\mathbf{u}(\rho), \rho) \leq 0 \text{ with } j = 1, ..., m
\end{align}</math>

The problem statement includes the following:

* An [[objective function]] <math>F(\mathbf{u(\rho), \rho})</math>. This function represents the quantity that is being minimized for best performance. The most common objective function is compliance, where minimizing compliance leads to maximizing the stiffness of a structure.
* The material distribution as a problem variable. This is described by the density of the material at each location <math> \rho(\mathbf{x}) </math>. Material is either present, indicated by a 1, or absent, indicated by a 0. <math> \mathbf{u}=\mathbf{u}(\mathbf{\rho})</math> is a state field that satisfies a linear or nonlinear state equation depending on <math> \rho </math>.
* The design space <math> (\Omega)</math>. This indicates the allowable volume within which the design can exist.  Assembly and packaging requirements, human and tool accessibility are some of the factors that need to be considered in identifying this space . With the definition of the design space, regions or components in the model that cannot be modified during the course of the optimization are considered as non-design regions.
* <math>\scriptstyle m </math> [[constraint (mathematics)|constraint]]s <math> G_j(\mathbf{u}(\rho), \rho) \leq 0 </math> a characteristic that the solution must satisfy. Examples are the maximum amount of material to be distributed (volume constraint) or maximum stress values.

Evaluating <math> \mathbf{u(\rho)} </math> often includes solving a differential equation. This is most commonly done using the [[finite element method]] since these equations do not have a known analytical solution.