Filtering of the design variable¶

In some applications such as density based topology optimization, it can be useful to apply some transformation to the design variable. The decorator filtered_optimizable() allows to automate this process, it converts an optimization problem

\[\begin{aligned} \min_{x\in \R^n}& \quad J(x)\\ \textrm{s.t.} & \left\{\begin{aligned} g_i(x)&=0, \text{ for all } 1\leqslant i\leqslant p,\\ h_j(x) &\leqslant 0, \text{ for all }1\leqslant j \leqslant q,\\ \end{aligned}\right. \end{aligned}\]

into the problem

\[\begin{aligned} \min_{x\in \R^n}& \quad J(\rho(x))\\ \textrm{s.t.} & \left\{\begin{aligned} g_i(\rho(x))&=0, \text{ for all } 1\leqslant i\leqslant p,\\ h_j(\rho(x)) &\leqslant 0, \text{ for all }1\leqslant j \leqslant q,\\ \end{aligned}\right. \end{aligned}\]

where \(\rho\,:\,\R^n\to\R^n\) is a transformation of the design variable.

The decorator is used as follows:

python

from nullspace_optimizer import EuclideanOptimizable, filtered_optimizable

# Filter the design variable
# The optimizer will call J(filter(x)), G(filter(x)), etc.
@filtered_optimizable(filter, diff_filter)
class Problem(EuclideanOptimizable):
     def x0(self):
         #x0
     #...

     def J(self, x):
     # J

     # etc...

The decorator takes two arguments:

filter: the transformation map
diff_filter: the derivative of the transformation map. It should be implemented such that diff_filter(x,v)=v @ dfilter(x) where dfilter(x) is the Jacobian matrix of the filter. This is required so that the decorator filtered_optimizable() can update consistently the sensitivities DJ, DG and DH according to the chain rule

\[\newcommand{\D}{\mathrm{D}} \frac{\D}{\D \texttt{x}}[\texttt{J}(\rho(\texttt{x}))]\cdot \texttt{dx} = \frac{\D \texttt{J}}{\D \texttt{x}}(\rho(\texttt{x})) \cdot \frac{\D \rho(\texttt{x})}{\D \texttt{x}}\cdot \texttt{dx}.\]