Unsteady rugby ball

This example is an unsteady extension of the classic "rugby ball" example by Borrvall & Petersson 2003.

The objective is to minimize the time integral of the dissipation

\[ \mathcal{F} = \int_0^T \frac{1}{|\Omega_\text{obj}|}\int \frac{1}{2} \left[ \nabla \mathbf{u} \cdot \left(\nabla \mathbf{u} + (\nabla \mathbf{u})^T \right) + \chi \mathbf{u} \cdot \mathbf{u} \right] d\Omega, dt \]

where the velocity boundary condition progressively increase from 0 to 1 over the course of 0.5 time units.

It is also subject to a volume constraint

\[ \mathcal{C} = \frac{1}{|\Omega_\text{opt}|}\int_{\Omega_\text{opt}} \rho d\Omega. \]

More information regarding objectives and constraints can be found in Objectives and constraints.