A Hierarchical Flow Solver for Optimisation with PDE Constraints

Abstract

Active flow control plays a central role in many industrial applications such as e.g. control of crystal growth processes, where the flow in the melt has a significant impact on the quality of the crystal. Optimal control of the flow by electro-magnetic fields and/or boundary temperatures leads to optimisation problems with PDE constraints, which are frequently governed by the time-dependent Navier-Stokes equations.

The mathematical formulation is a minimisation problem with PDE constraints. By exploiting the special structure of the first order necessary optimality conditions, the so called Karush-Kuhn-Tucker (KKT)-system, this thesis develops a special hierarchical solution approach for such equations, based on the distributed control of the Stokes-- and Navier--Stokes. The numerical costs for solving the optimisation problem are only about 20-50 times higher than a pure forward simulation, independent of the refinement level.

Utilising modern multigrid techniques in space, it is possible to solve a forward simulation with optimal complexity, i.e., an appropriate solver for a forward simulation needs O(N) operations, N denoting the total number of unknowns for a given computational mesh in space and time. Such solvers typically apply appropriate multigrid techniques for the linear subproblems in space. As a consequence, the developed solution approach for the optimal control problem has complexity O(N) as well. This is achieved by a combination of a space-time Newton approach for the nonlinearity and a monolithic space-time multigrid approach for 'global' linear subproblems. A second inner monolithic multigrid method is applied for subproblems in space, utilising local Pressure-Schur complement techniques to treat the saddle-point structure. The numerical complexity of this algorithm distinguishes this approach from adjoint-based steepest descent methods used to solve optimisation problems in many practical applications, which in general do not satisfy this complexity requirement.