A Hierarchical Flow Solver for Optimisation with PDE Constraints
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2011-12-21
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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.
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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