Simultaneous Single-Step One-Shot Optimization with Unsteady PDEs

TL;DR

This paper extends the single-step one-shot optimization method to unsteady PDEs solved by time-marching schemes, enabling efficient PDE-constrained optimization for complex unsteady problems.

Contribution

It introduces a framework that adapts the one-shot method for unsteady PDEs, including Navier-Stokes and advection-diffusion equations, with adaptive time scales.

Findings

Successful application to Navier-Stokes optimal control

Improved simulation scheme with adaptive time scales

Numerical validation on advection-diffusion problem

Abstract

The single-step one-shot method has proven to be very efficient for PDE-constrained optimization where the partial differential equation (PDE) is solved by an iterative fixed point solver. In this approach, the simulation and optimization tasks are performed simultaneously in a single iteration. If the PDE is unsteady, finding an appropriate fixed point iteration is non-trivial. In this paper, we provide a framework that makes the single-step one-shot method applicable for unsteady PDEs that are solved by classical time-marching schemes. The One-shot method is applied to an optimal control problem with unsteady incompressible Navier-Stokes equations that are solved by an industry standard simulation code. With the Van-der-Pol oscillator as a generic model problem, the modified simulation scheme is further improved using adaptive time scales. Finally, numerical results for the advection-diffusion equation are presented.