Geometrical Criteria#

This page presents the geometrical criteria currently supported by OpenPisco and their corresponding mathematical formulations.

Overview#

Geometrical criteria only depend on the current geometry represented by the level-set function.

Most geometrical criteria do not require a physical solver.

The corresponding classes are collected in:

Supported Criteria#

  • Volume

  • Surface

  • Mass

  • Minimal thickness

  • Maximal thickness

  • Center of mass deviation

Volume#

\[J(\Omega) = \int_{\Omega}{ dx}\]

Surface#

\[J(\Omega) = \int_{\partial\Omega}{ dx}\]

Mass#

\[J(\Omega) = \rho \int_{\Omega}{ dx}\]

where \(\rho\) is the constant density.

Minimal Thickness#

\[J(\Omega) = \int_{ \partial \Omega}{ \int_{0}^{d_{min}}{ \left[ ( d_{\Omega}(s - z n(s)))^+ \right]^2 } \:dz \:dx},\]

where \(d_{\Omega}\) is the signed distance function, \(d_{min}\) is the desired minimal thickness and \(n\) is the normal vector.

Maximal Thickness#

\[J(\Omega) = \int_{\Omega}{\left[ \left( d_{\Omega} (x) + \frac{d_{max}}{2} \right)^{-}\right]^2 \:dx}.\]

where \(d_{\Omega}\) is the signed distance function and \(d_{max}\) is the desired maximal thickness.

Center of Mass Deviation#

\[J(\Omega) = \| (\int_\Omega x dx) / |\Omega| - c\|^2\]

where \(c\) is the reference center of mass.

Associated Implementation#

3

  1. Allaire, F. Jouve, G. Michailidis, Thickness control in structural optimization via a level set method, SMO 53, 1349-1382 (2016)