Y. Yan and D. Keyes
Journal of Computational Physics, 281, pp. 759-786, (2015)
Optimal control, Regularization, Numerical smoothing, Numerical continuation, Multiphysics, Conjugate heat transfer, NavierStokes
We study a new optimization scheme that generates smooth
and robust solutions for Dirichlet velocity boundary control (DVBC) of
conjugate heat transfer (CHT) processes. The solutions to the DVBC of
the incompressible Navier–Stokes equations are typically nonsmooth, due
to the regularity degradation of the boundary stress in the adjoint
Navier–Stokes equations. This nonsmoothness is inherited by the
solutions to the DVBC of CHT processes, since the CHT process couples
the Navier–Stokes equations of fluid motion with the
convection–diffusion equations of fluid–solid thermal interaction. Our
objective in the CHT boundary control problem is to select optimally the
fluid inflow profile that minimizes an objective function that involves
the sum of the mismatch between the temperature distribution in the
fluid system and a prescribed temperature profile and the cost of the
control.
Our strategy to resolve the nonsmoothness of
the boundary control solution is based on two features, namely, the
objective function with a regularization term on the gradient of the
control profile on both the continuous and the discrete levels, and the
optimization scheme with either explicit or implicit smoothing effects,
such as the smoothed Steepest Descent and the Limited-memory
Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) methods. Our strategy to
achieve the robustness of the solution process is based on combining the
smoothed optimization scheme with the numerical continuation technique
on the regularization parameters in the objective function. In the
section of numerical studies, we present two suites of experiments. In
the first one, we demonstrate the feasibility and effectiveness of our
numerical schemes in recovering the boundary control profile of the
standard case of a Poiseuille flow. In the second one, we illustrate the
robustness of our optimization schemes via solving more challenging
DVBC problems for both the channel flow and the flow past a square
cylinder, which use initial control profiles far from optimal and
require the numerical continuation technique applied on regularization
parameters. We believe our solution strategy is general and can be
applied to other large-scale optimal control problems which involve
multiphysics processes and require smooth approximations to the optimal
control profile.