Authors: Köster, Michael
Title: A Hierarchical Flow Solver for Optimisation with PDE Constraints
Language (ISO): en
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.
Subject Headings: Block-Glätter
Block smoother
CFD
Crank-Nicolson
Crystal growth
Czochralski
Distributed Control
Edge-oriented stabilisation
Elliptic
Elliptisch
EOJ stabilisation
EOJ Stabilisierung
FEAT
FEATFLOW
Finite Elemente
Finite Elements
First discretise then optimise
First discretize then optimize
First optimise then discretise
First optimize then discretize
Flow-Around-Cylinder
Full Newton-SAND
Heat equation
Hierarchical
Hierarchical solution concept
Hierarchisch
Hierarchisches Lösungskonzept
Inexact Newton
Inexaktes Newton-Verfahren
Instationär
Inverse Probleme
Inverse Problems
Kantenbasierte Stabilisierung
KKT system
Kristallwachstum
Krylov
Large-Scale
linear complexity
lineare Komplexität
Mehrgitter
Mehrgitter-Krylov
Monolithic
Monolithisch
Multigrid
Multigrid-Krylov
Multilevel
Navier-Stokes
Nichtparametrische Finite Elemente
Nonparametric finite elements
Nonstationary
OPTFLOW
Optimierung
Optimisation
Optimization
PDE Constraints
Raum-Zeit
saddle point
SAND
Sattelpunkt
Schur complement preconditioning
Schurkomplement-Vorkonditionierer
Space-time
SQP
Stokes
Theta schema
Theta scheme
Time-dependent
Transient
Unstructured Grids
Unstrukturierte Gitter
Vanka
Verteilte Kontrolle
Wärmeleitung
Wärmeleitungsgleichung
URI: http://hdl.handle.net/2003/29239
http://dx.doi.org/10.17877/DE290R-6950
Issue Date: 2011-12-21
Appears in Collections:Lehrstuhl III: Angewandte Mathematik und Numerik

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