Large-Scale PDE-Constrained Optimization in Applications : Lecture Notes in Applied and Computational Mechanics - Subhendu Bikash Hazra

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Large-Scale PDE-Constrained Optimization in Applications

By: Subhendu Bikash Hazra

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Hardcover | 24 June 2010

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224 Pages

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This book results from the authors work done on simulation based optimization problems at the Department of Mathematics, University of Trier, and reported in his postdoctoral thesis ("Habilitationsschrift") accepted by the Faculty-IV of this University in 2008. The focus of the work has been to develop mathematical methods and algorithms which lead to efficient and high performance computational techniques to solve such optimization problems in real-life applications. Systematic development of the methods and algorithms are presented here. Practical aspects of implementations are discussed at each level as the complexity of the problems increase, supporting with enough number of computational examples. It consists of two parts: first part deals with time dependent optimization problems with applications in environmental engineering and the second part deals with steady state optimization problems, in which the PDEs are solved using semi-iterative or pseudo-time-stepping techniques, with applications in aerodynamics.

This book will be useful for scientists and engineers who are looking for efficient numerical methods for PDE-constrained optimization problems. It will be helpful for graduate and Ph.D. students in applied mathematics, aerospace engineering, mechanical engineering, civil engineering and computational engineering during their training and research. This also will provide exciting research and development areas involving realistic applications.

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"This book is based on the habilitation thesis ... of the author. It documents the author's extensive experience with two classes of optimization algorithms applied to two particular PDE constrained optimization problems. ... The book offers a wealth of practical information about the design and numerical performance of specific optimization algorithms in two specific application settings. Engineers ad practitioners who are experienced in computational design and who seek practical numerical recipes to solve optimization problems in these two application areas will find the monograph useful." (Pavel B. Bochev, Mathematical Reviews, Issue 2012 b)

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Introduction	p. 1
Partial Differential Equations in Mathematical Modeling of Fluid Flow Problems	p. 5
Introduction	p. 5
The Navier-Stokes Equations for Compressible Viscous Flow	p. 12
The Euler Equations for Compressible Inviscid Flow	p. 12
Vector Form of the Navier-Stokes Equations for Compressible Viscous Flow	p. 13
Non-dimensionalization	p. 14
Turbulence and Its Modeling	p. 14
Turbulent Averaged Quantities	p. 15
The Reynolds Averaged Navier-Stokes Equations	p. 16
Analytic Aspects of the PDEs	p. 17
PDE-Constrained Optimization Methods	p. 19
Unconstrained Optimization Problem	p. 19
Constrained Optimization Problem	p. 21
Nested Analysis and Design (NAND)	p. 21
Simultaneous Analysis and Design (SAND)	p. 24
Full Newton SAND	p. 24
Applications in Environmental Engineering
Mathematical Model of Multiphase Flow through Porous Media	p. 29
Introduction	p. 29
General form of the Multiphase Flow Equations	p. 30
Isothermal Water-Gas System (Two-Phase Flow)	p. 31
Nonisothermal Water-Gas Systems (Two-Phase Two-Component Flow)	p. 32
Constitutive Relationships	p. 34
The Forward Simulation Problem	p. 36
Governing Equations	p. 36
Discretization	p. 38
Implicit Time Discretization	p. 40
The Software System MUFTE_UG	p. 41
Parameter Identification in Multiphase Flow through Porous Media	p. 43
Introduction	p. 43
Least-Squares Formulation	p. 44
The Multiple Shooting Parameter Estimation Approach	p. 44
A Reduced Generalized Gauss-Newton Method	p. 45
Computation of (Inexact) Derivatives	p. 47
Numerical Results and Discussion	p. 50
Isothermal Case (Two-Phase flow)	p. 50
Non-isothermal Case (Two Phase Two-Component Flow)	p. 52
Conclusions	p. 61
Applications in Aerodynamics
Simultaneous Pseudo-Time Stepping for PDE-Model Based Optimization Problems	p. 65
Introduction	p. 65
The Optimization Problem and Pseudo-unsteady Formulation of the KKT Conditions	p. 67
Reduced SQP Methods	p. 69
Pseudo-Time-Stepping for Optimization Problems	p. 71
Application to a Model Problem	p. 72
Analysis of the Hessian	p. 73
Numerical Implementation	p. 75
Results and Discussion	p. 76
Conclusions	p. 80
Aerodynamic Shape Optimization Using Simultaneous Pseudo-Time-Stepping	p. 81
Introduction	p. 81
Pseudo-Time-Stepping for Optimization Problems	p. 83
Detailed Equations of the Aerodynamic Shape Optimization Problem in 2D	p. 83
Discretization	p. 86
Reduced Hessian Updates	p. 92
Numerical Results and Discussion	p. 95
Drag Reduction with Geometric Constraint for an RAE2822 Airfoil	p. 95
Drag Reduction with Geometric Constraints for Supersonic Cruise Transport (SCT) Wing	p. 102
Conclusions	p. 104
Indirect Treatment of State Constraints Aerodynamic Shape Optimization Using Simultaneous Pseudo-Time-Stepping	p. 105
Introduction	p. 105
Pseudo-Time-Stepping for the Constrained Optimization Problem	p. 105
Numerical Results and Discussion	p. 109
Conclusions	p. 116
Direct Treatment of State Constraints in Aerodynamic Shape Optimization Using Simultaneous Pseudo-Time-Stepping	p. 177
Introduction	p. 177
Scalar State Constraints	p. 118
Partial Reduction of the Problem	p. 119
Solution Strategy of the Constrained Problem	p. 120
Back Projection	p. 121
Numerical Results and Discussion	p. 122
Applications in 2D	p. 123
Application in 3D	p. 127
Conclusions	p. 132
Multigrid One-Shot Pseudo-Time-Stepping Method for Aerodynamic Shape Optimization	p. 135
Introduction	p. 135
The Multigrid Algorithm	p. 136
Numerical Results and Discussion	p. 137
Drag Reduction with Constant Thickness for RAE2822 Airfoil	p. 137
Drag Reduction with Geometric Constraints for SCT Wing	p. 147
Conclusions	p. 152
Multigrid One-Shot Pseudo-Time Stepping Method for State Constrained Aerodynamic Shape Optimization	p. 155
Introduction	p. 155
The Multigrid Algorithm	p. 156
Numerical Results and Discussion	p. 157
Drag Reduction with Constant Lift on (193×33) Grid	p. 158
Drag Reduction with Constant Lift on (321×57) Grid	p. 164
Conclusions	p. 173
One-Shot Pseudo-Time-Stepping Method for Aerodynamic Shape Optimization Using the Navier-Stokes Equations	p. 175
Introduction	p. 175
Detailed Equations of the Aerodynamic Shape Optimization Problem	p. 176
Numerical Results and Discussion	p. 183
Conclusions	p. 188
References	p. 189
Index	p. 199
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Large-Scale PDE-Constrained Optimization in Applications

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