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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