Title: The one-shot method for a pressure correction Navier-Stokes solver: an example where optimisation can be faster than simulation

Authors: Johannes Rückelt; Günter Bärwolff; Thomas Slawig

Addresses: Christian-Albrechts-University Kiel, Department of Computer Science, Christian-Albrechts-Platz 4, 24118 Kiel, Germany ' TU Berlin, Department of Mathematics, Straße des 17. Juni 136, 10623 Berlin, Germany ' Christian-Albrechts-University Kiel, Department of Computer Science, Christian-Albrechts-Platz 4, 24118 Kiel, Germany

Abstract: We show an exemplary optimisation problem for a fluid flow where a numerical optimisation run can be as fast as and even faster than a pure simulation. In this example, the optimisation parameter is the upper boundary value in a two-dimensional driven cavity flow, mathematically described by the stationary incompressible Navier-Stokes equations. The stationary solution of the equations is computed by pseudo-time stepping, i.e., by running a transient solver into a steady state, using additional inner pressure correction steps. We compare optimisation methods of Newton and quasi-Newton type that use the fully converged state and derivative with an alternating algorithm that performs a parameter update already after a few steps of the state and derivative iterations. All derivatives are computed via automatic differentiation. The alternating algorithm is able to produce the mentioned fast optimisation results. A second example configuration shows that this phenomenon is problem-dependent and cannot be generalised.

Keywords: automatic differentiation; one-shot optimisation; fixed point solver; pressure-correction solver; Navier-Stokes equations; fluid flow.

DOI: 10.1504/IJMMNO.2013.055201

International Journal of Mathematical Modelling and Numerical Optimisation, 2013 Vol.4 No.2, pp.114 - 135

Received: 27 Jul 2012
Accepted: 01 Feb 2013
Published online: 26 Jul 2014 *

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