Project 07: Navier-Stokes Control with Mixed Boundary Conditions - NMOPT

Goal¶

Move from Stokes to stationary Navier-Stokes control with mixed Dirichlet and Neumann boundary controls.

This project combines nonlinearity, fluid constraints, boundary control, and Newton linearization.

Let the state be velocity-pressure $(y,\pi)$ and the control be

u=(u_D,u_N),

where $u_D$ is imposed on an inflow boundary $\Gamma_D$ and $u_N$ is a traction-like control on $\Gamma_N$ .

Minimize

J(y,u_D,u_N) = \frac12\|y-y_d\|_{L^2(\Omega)^d}^2 + \frac{\alpha_D}{2}\|u_D-u_{D,\mathrm{ref}}\|_{L^2(\Gamma_D)^d}^2 + \frac{\alpha_N}{2}\|u_N-u_{N,\mathrm{ref}}\|_{L^2(\Gamma_N)^d}^2

subject to

\begin{cases} -\nu\Delta y+(y\cdot\nabla)y+\nabla\pi=f & \text{in }\Omega,\\ \nabla\cdot y=0 & \text{in }\Omega,\\ y=u_D & \text{on }\Gamma_D,\\ (-\pi I+\nu\nabla y)n=u_N & \text{on }\Gamma_N,\\ y=0 & \text{on }\Gamma_0. \end{cases}

The adjoint equation is the formal transpose of the linearized Navier-Stokes operator:

\begin{cases} -\nu\Delta p-(y\cdot\nabla)p+(\nabla y)^T p+\nabla\lambda=y-y_d & \text{in }\Omega,\\ \nabla\cdot p=0 & \text{in }\Omega, \end{cases}

with boundary conditions depending on the chosen weak formulation.

The unconstrained boundary optimality conditions are formally

\alpha_D(u_D-u_{D,\mathrm{ref}})-\mathcal T_D(p,\lambda)=0,

\alpha_N(u_N-u_{N,\mathrm{ref}})-p|_{\Gamma_N}=0,

where $\mathcal T_D(p,\lambda)$ denotes the adjoint boundary traction associated with Dirichlet control.

Course material: lecture14.md for boundary controls, lecture15.md for nonlinear KKT/Newton ideas.
deal.II: step-22 for Stokes and step-57 or other Navier-Stokes tutorials for nonlinear flow solvers.