A 5D concept for space-time optimal control problems with application to simplified Carreau flow

TL;DR

The paper addresses optimal control of non-Newtonian Carreau-flow through a 5D space-time framework that couples 3D spatial dynamics, 1D time, and 1D optimization. It solves the parameter-dependent space-time KKT system using fully space-time finite elements, discretizing with $P_1$ elements on a pentatope mesh and employing a tracking-type cost $J_\rho(y,u)$ that includes a regularization $\rho$. A key demonstration shows how the approach yields large-scale systems (e.g., $\sim 31\times 10^6$ dofs) and highlights the influence of $\rho$ on the optimal control magnitude, illustrating the computational and modeling complexities of space-time optimization for non-Newtonian flows. The work points to parallel computation, adaptive mesh refinement, and more realistic nonlinear models as crucial future directions, aiming to make this 5D framework practical for engineering and scientific applications in complex fluids.

Abstract

This work presents a 5D concept to optimizing non-Newtonian fluid flows through a simplified Carreau flow model. We solve the optimization problem by approximating the solution of the KKT System with fully space-time finite element methods instead of the more traditional time-stepping technique combined with spatial finite element discretization. Therein, the finite element method is formulated in 3D in space, 1D in time, and 1D in the optimization loop, yielding a 5D overall framework.

Figures (1)

• Figure 1: Optimal control $u_h$ for different values of the regularization parameter $\rho$: $\rho = 0.01$ (left); $\rho = 0.001$ (middle); $\rho = 0.0001$ (right).