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1. Köster, Michael. A Hierarchical Flow Solver for Optimisation with PDE Constraints.

Degree: 2011, Technische Universität Dortmund

URL: http://hdl.handle.net/2003/29239

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.
*Advisors/Committee Members: Turek, Stefan.*

Subjects/Keywords: 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; 510

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Köster, M. (2011). A Hierarchical Flow Solver for Optimisation with PDE Constraints. (Thesis). Technische Universität Dortmund. Retrieved from http://hdl.handle.net/2003/29239

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Köster, Michael. “A Hierarchical Flow Solver for Optimisation with PDE Constraints.” 2011. Thesis, Technische Universität Dortmund. Accessed March 02, 2021. http://hdl.handle.net/2003/29239.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Köster, Michael. “A Hierarchical Flow Solver for Optimisation with PDE Constraints.” 2011. Web. 02 Mar 2021.

Vancouver:

Köster M. A Hierarchical Flow Solver for Optimisation with PDE Constraints. [Internet] [Thesis]. Technische Universität Dortmund; 2011. [cited 2021 Mar 02]. Available from: http://hdl.handle.net/2003/29239.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Köster M. A Hierarchical Flow Solver for Optimisation with PDE Constraints. [Thesis]. Technische Universität Dortmund; 2011. Available from: http://hdl.handle.net/2003/29239

Not specified: Masters Thesis or Doctoral Dissertation

2. Köster, Michael. A Hierarchical Flow Solver for Optimisation with PDE Constraints.

Degree: 2011, Technische Universität Dortmund

URL: http://dx.doi.org/10.17877/DE290R-6950

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.
*Advisors/Committee Members: Turek, Stefan (advisor), Meyer, Christian (referee).*

Subjects/Keywords: Block-Glätter; Czochralski; Elliptisch; EOJ Stabilisierung; Finite Elemente; Hierarchisch; Hierarchisches Lösungskonzept; Inexaktes Newton-Verfahren; Instationär; Inverse Probleme; Kantenbasierte Stabilisierung; Kristallwachstum; Krylov; lineare Komplexität; Mehrgitter; Mehrgitter-Krylov; Monolithisch; Navier-Stokes; Nichtparametrische Finite Elemente; Optimierung; Raum-Zeit; Sattelpunkt; Schurkomplement-Vorkonditionierer; Stokes; Unstrukturierte Gitter; Vanka; Verteilte Kontrolle; Wärmeleitung; Wärmeleitungsgleichung; Block smoother; CFD; Crank-Nicolson; Crystal growth; Distributed Control; Edge-oriented stabilisation; Elliptic; EOJ stabilisation; FEAT; FEATFLOW; 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; Inexact Newton; Inverse Problems; KKT system; Large-Scale; linear complexity; Monolithic; Multigrid; Multigrid-Krylov; Multilevel; Nonparametric finite elements; Nonstationary; OPTFLOW; Optimisation; Optimization; PDE Constraints; saddle point; SAND; Schur complement preconditioning; Space-time; SQP; Theta schema; Theta scheme; Time-dependent; Transient; Unstructured Grids; 510

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Köster, M. (2011). A Hierarchical Flow Solver for Optimisation with PDE Constraints. (Doctoral Dissertation). Technische Universität Dortmund. Retrieved from http://dx.doi.org/10.17877/DE290R-6950

Chicago Manual of Style (16^{th} Edition):

Köster, Michael. “A Hierarchical Flow Solver for Optimisation with PDE Constraints.” 2011. Doctoral Dissertation, Technische Universität Dortmund. Accessed March 02, 2021. http://dx.doi.org/10.17877/DE290R-6950.

MLA Handbook (7^{th} Edition):

Köster, Michael. “A Hierarchical Flow Solver for Optimisation with PDE Constraints.” 2011. Web. 02 Mar 2021.

Vancouver:

Köster M. A Hierarchical Flow Solver for Optimisation with PDE Constraints. [Internet] [Doctoral dissertation]. Technische Universität Dortmund; 2011. [cited 2021 Mar 02]. Available from: http://dx.doi.org/10.17877/DE290R-6950.

Council of Science Editors:

Köster M. A Hierarchical Flow Solver for Optimisation with PDE Constraints. [Doctoral Dissertation]. Technische Universität Dortmund; 2011. Available from: http://dx.doi.org/10.17877/DE290R-6950