Design Sensitivity Analysis (Frontiers in Applied Mathematics) (No. 25)

Design Sensitivity Analysis

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F R O N T I E R S IN APPLIED MATHEMATICS The SIAM series on Frontiers in Applied Mathematics publishes monographs dealing with creative work in a substantive field involving applied mathematics or scientific computation. All works focus on emerging or rapidly developing research areas that report on new techniques to solve mainstream problems in science or engineering. The goal of the series is to promote, through short, inexpensive, expertly written monographs, cutting edge research poised to have a substantial impact on the solutions of problems that advance science and technology. The volumes encompass a broad spectrum of topics important to the applied mathematical areas of education, government, and industry.

EDITORIAL BOARD H.T. Banks, Editor-in-Chief, North Carolina State University Richard Albanese, U.S. Air Force Research Laboratory, Brooks AFB Carlos Castillo Chavez, Cornell University Doina Cioranescu, Universite Pierre et Marie Curie (Paris VI) Lisa Fauci,Tulane University Pat Hagan, Bear Stearns and Co., Inc. Belinda King,Virginia Polytechnic Institute and State University Jeffrey Sachs, Merck Research Laboratories, Merck and Co., Inc. Ralph Smith, North Carolina State University AnnaTsao, Institute for Defense Analyses, Center for Computing Sciences

BOOKS PUBLISHED IN FRONTIERS IN A P P L I E D MATHEMATICS Stanley, Lisa G. and Stewart, Dawn L, Design Sensitivity Analysis: Computational Issues of Sensitivity Equation Methods Vogel, Curtis R., Computational Methods for Inverse Problems Lewis, F. L; Campos,].; and Selmic, R., Neuro-Fuzzy Control of Industrial Systems with Actuator Nonlinearities Bao, Gang; Cowsar, Lawrence; and Masters, Wen, editors, Mathemot/co/ Modeling in Optical Science Banks, H.T.; Buksas, M.W.; and Lin,T., Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts Oostveen, Job, Strongly Stabilizable Distributed Parameter Systems Griewank, Andreas, Evaluating Derivatives: Principles andTechniques of Algorithmic Differentiation Kelley, C.T., Iterative Methods for Optimization Greenbaum,Anne, Iterative Methods for Solving Linear Systems Kelley, C.T., Iterative Methods for Linear and Nonlinear Equations Bank, Randolph E., PLTMG:A Software Package for Solving Elliptic Partial Differential Equations. Users'Guide 7.0 More, Jorge J. and Wright, Stephen J., Optimization Software Guide Riide, Ulrich, Mathematical and Computational Techniques for Multilevel Adaptive Methods Cook, L. Pamela, Transonic Aerodynamics: Problems in Asymptotic Theory Banks, H.T., Control and Estimation in Distributed Parameter Systems Van Loan, Charles, Computational Frameworks for the Fast Fourier Transform Van Huffel, Sabine and Vandewalle, Joos, The Total Least Squares Problem: Computational Aspects and Analysis Castillo, Jose E., Mothematical Aspects of Numerical Grid Generation Bank, R. E., PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. Users' Guide 6.0 McCormick, Stephen F., Multilevel Adaptive Methods for Partial Differential Equations Grossman, Robert, Symbolic Computation: Applications to Scientific Computing Coleman,Thomas F. and Van Loan, Charles, Handbook for Matrix Computations McCormick, Stephen F., Multigrid Methods Buckmasterjohn D., The Mathematics of Combustion Ewing, Richard E., The Mothematics of Reservoir Simulation

Design Sensitivity Analysis Computational Issues of Sensitivity Equation Methods

Lisa G. Stanley Montana State University Bozeman, Montana

Dawn L. Stewart Air Force Institute of Technology Wright Patterson AFB, Ohio

siam Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2002 by the Society for Industrial and Applied Mathematics. 1098765432 I All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Stanley, Lisa G. Design sensitivity analysis : computational issues of sensitivity equation methods / Lisa G. Stanley, Dawn L. Stewart. p. cm.-- (Frontiers in applied mathematics) Includes bibliographical references and index. ISBN 0-89871-524-5 I. Engineering design—Mathematical models. 2. Mathematical optimization. I. Stewart, Dawn L. II.Title. III. Series.

TA174.S7622002 620'.0042-dc2l 2002075757

is a registered trademark.

Contents List of Figuresh

xi

List of Tables

hxv

Foreword

xvii

Preface

xix

1

Introduction

1

2

Mathematical Framework for Linear Elliptic Problems 2.1 Preliminaries 2.1.1 Notation 2.1.2 Function Space Tools 2.1.3 Bilinear Forms and Variational Tools 2.2 Regularity of Elliptic Boundary Value Problems 2.2.1 Strong Solutions 2.2.2 Weak Solutions 2.3 Mappings and Implications to Sensitivity Analysis 2.3.1 Implicit Function Theorem 2.3.2 Operator Formulations of Sensitivity Equations

5 5 6 8 9 11 12 12 14 14 15

3

Model Problems 3.1 Heat in a Thin Rod with a Parameter-Dependent Forcing Term . . . . 3.2 Heat in a Thin Rod with a Parameter-Dependent Boundary 3.2.1 State Equation 3.2.2 Sensitivity Equation 3.3 Nonlinear Model 3.3.1 State Equation 3.3.2 Sensitivity Equation

17 17 19 20 20 22 22 23

4

Computational Algorithms 4.1 The Method of Mappings 4.1.1 Transformation Techniques 4.1.2 An Algebraic Transformation

25 25 25 27

vii

viii

Contents

4.2

4.3

SEMs 4.2.1 Hybrid SEM 4.2.2 Abstract Version of the Semianalytic Method 4.2.3 Applying the H-SEM to the Nonlinear Model Approximation Framework 4.3.1 Variational Formulations 4.3.2 Piecewise Linear Finite Elements

28 28 31 33 35 35 37

5

Numerical Results 5.1 Linear Model 5.1.1 State Approximations 5.1.2 H-SEM 5.1.3 A-SAM 5.2 Nonlinear Model 5.2.1 Convergence of the State and the Sensitivity 5.2.2 Sensitivities in Optimal Design 5.3 State Gradient Approximations 5.3.1 A Global Projection Scheme 5.3.2 A Local Projection Scheme 5.3.3 Numerical Results

43 43 43 44 47 49 49 52 60 61 62 62

6

Mathematical Framework for Navier-Stokes Equations 6.1 The Homogeneous Dirichlet Problem 6.1.1 Function Spaces and Notation 6.1.2 Existence and Uniqueness of Solutions to the Variational Form 6.2 The Nonhomogeneous Dirichlet Problem 6.3 An Abstract Framework for Navier-Stokes 6.3.1 The Framework 6.3.2 Using the Framework 6.3.3 Continuity of Solutions with Respect to Data 6.4 Analysis of the Sensitivity Equations 6.4.1 A General Formulation of the Sensitivity Equations ... 6.4.2 Existence and Uniqueness of Solutions to the Sensitivity Equations 6.5 Differentiability of Solutions with Respect to q

71 71 71

7

Two-Dimensional Flow Problems 7.1 Flow around a Cylinder 7.2 Flow over a Bump 7.3 A Finite Element Formulation 7.3.1 Adaptive Methodology 7.4 Some Numerical Results 7.4.1 Flow around a Cylinder 7.4.2 Flow over a Bump

72 73 75 75 76 77 78 78 79 80 81 81 82 84 84 87 87 88

Contents

8

Adaptive Mesh Refinement Strategies 8.1 A Local Projection for Higher Dimensions 8.2 Numerical Results for Two-Dimensional Problems 8.2.1 Flow around a Cylinder 8.2.2 Flow over a Bump

ix

95 95 98 98 119

Bibliography

135

Index

139

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List of Figures 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27

Finite element approximations to w( , 1.5) Approximations to w(x, 1.5) Hl error of z N ( x , q) for q ranging from 1.1 to 1.9 Finite element approximations tos(x, 1.5) H-SEM approximations to s(x, 1.5) Approximation of 1.5) with N = 3 Finite element approximations to p( , 1.5) A-SAM approximations to s(x, 1.5) Hl errors for sensitivity calculations Numerical approximations to the solution of the nonlinear model at q = 2 and q = 1.2 Numerical approximations to the solution of the sensitivity equation at q — 2 using PWC derivatives L2 error of the state and sensitivity approximations at q = 2 with N = M. Numerical approximations to the solution of the sensitivity equation at q = 2 using PWC derivatives with N = 2 and mesh refinement in M. . . Numerical approximations to the solution of the sensitivity equation at q = 2 using PWC derivatives with N = 4 and mesh refinement in M. . . Numerical approximations to the solution of the sensitivity equation at q = 1.4 using PWC derivatives Numerical approximations to the solution of the sensitivity equation at q = 1.2 using PWC derivatives L2 error of the solution and sensitivity approximations at q = 1.2 L2 error of sensitivity approximations using PWC derivatives Gauss-Newton algorithm Data generated at q = 2 Data generated at q = 1.4 The cost function and its approximations for p = 16 and q* ~ 2 The cost function and its approximations for p = 16 and q* ~ 1.4. . . . Finite element derivatives with projections at N = 4 and q = 2 Finite element derivatives with projections at N = 8 and q = 1.2. . . . . L2 error on each element for N = 4 and q = 2 L2 error on each element for N = 8 and q = 1.2 xi

44 44 45 45 46 46 47 48 48 50

50 51 52 53 53 54 54 55 55 56 56 58 58 63 64 64 65

xii

List of Figures

5.28 5.29 5.30 5.31 5.32

Sensitivity approximations at q = 2 Sensitivity approximations at q = 1.4 Sensitivity approximations at q = 1.2 Model problem—L2 error of sensitivity approximations (PWC and local). Model problem—L2 error of sensitivity approximations (global and local).

66 66 67 67 68

6.1

Sample domain with boundaries

74

7.1 Geometry for two-dimensional flow around a cylinder. 7.2 Geometry for flow over a bump 7.3 Crouzier-Raviart element 7.4 Initial and adapted meshes for a cylinder problem 7.5 w-velocity contours for flow around a cylinder. 7.6 u-velocity contours for flow around a cylinder. 7.7 w-velocity sensitivity contours for flow around a cylinder. 7.8 v-velocity sensitivity contours for flow around a cylinder. 7.9 Initial and adapted meshes for a bump problem 7.10 u, v-velocity contours for flow over a bump 7.11 u, v-velocity sensitivity contours for flow over a bump 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 8.20 8.21

Typical subdomain of an element vertex Element with three quadratic expressions for g* Meshes for cylinder problem at Re = 100 u-velocity sensitivities on initial mesh for Re = 100 v-velocity sensitivities on initial mesh for Re = 100 Error of sensitivity approximations on initial mesh for Re = 100 w-velocity sensitivities on first adapted mesh for Re = 100 y-velocity sensitivities on first adapted mesh for Re = 100 Error of w-velocity sensitivity approximations on first adapted mesh for Re = 100 Error of v-velocity sensitivity approximations on first adapted mesh for Re = 100 Initial meshes and u, v-velocity contours for L = 6, 15 and Re = 350. w-velocity sensitivity contours for L = 6, 15 and Re = 350 v-velocity sensitivity contours for L = 6, 15 and Re = 350 Meshes for cylinder problem at Re = 350 u-velocity sensitivities on initial mesh for Re = 350 v-velocity sensitivities on initial mesh for Re = 350 Error of sensitivity approximations on initial mesh for Re = 350 u-velocity sensitivities on first adapted mesh for Re = 350 y-velocity sensitivities on first adapted mesh for Re = 350 Error of w-velocity sensitivity approximations on first adapted mesh for Re = 350 Error of u-velocity sensitivity approximations on first adapted mesh for Re = 350

82 83 85 88 89 90 90 91 91 92 93 96 97 99 100 101 102 103 104 106 107 . 108 109 110 1ll 112 113 114 115 116 117 118

List of Figures

8.22 8.23 8.24 8.25 8.26 8.27 8.28 8.29 8.30 8.31 8.32 8.33 8.34 8.35

xiii

Flow about a cylinder—L2 error of u -sensitivity approximations on initial mesh 120 Row about a cylinder—L2 error of v-sensitivity approximations on initial mesh 120 Sensitivity vectors, L = 10, flow over a bump 121 u-velocity contours and sensitivity vectors, Re = 500, flow over a bump. 122 Flow over a bump, initial and adapted meshes for Re = 1000, L = 20. . 123 u-velocity contours for flow over a bump 124 v-velocity contours for flow over a bump 125 u-velocity sensitivity contours for flow over a bump 126 v-velocity sensitivity contours for flow over a bump 127 Sensitivity vector plots on initial and adapted meshes 128 Values ofJ h /(q(S))along a line 132 Values of along a line 132 Values of Jh(q(S)) along a line 133 Values of along a line 133

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List of Tables 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Matching data for the optimization Optimization results for p = 16, qopt ~ 2, qinit = 1.2, PWC derivatives. Optimization results for p = 16, qopt ~ 1.4, qinit — 2.0, PWC derivatives. L2 errors for the derivative approximations Optimization results for Case 1, global projection scheme Optimization results for Case 1, local projection scheme Optimization results for Case 2, global projection scheme Optimizations results for Case 2, local projection scheme

8.1 8.2 8.3

L2 errors for flow and sensitivities at Re = 100 L2 errors for flow and sensitivities at Re = 350 Values of Jh along a line, Re = 1, and A = 0.5

xv

57 59 59 65 68 69 69 70 108 115 131

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Foreword Sensitivity analysis consists of a set of tools that can be utilized in the context of optimization, optimal design, or simply system analysis to assess the influence of parameters on the state of the system. Continuous sensitivity methods construct a sensitivity equation whose solution provides the sensitivity of the state variable with respect to a parameter. To utilize such information, computational methods are typically applied. This book discusses certain pitfalls that can arise when sensitivities are computed for systems whose sensitivities exist in a function space. In addition, it carefully presents the theory and motivating examples for avoiding these pitfalls. As computing power has progressed, computational methods have opened the way to making sensitivity analysis a tractable design tool. It is this aspect of sensitivity analysis that this book addresses. A first (and perhaps obvious) step to computing sensitivities might be to approximate the state equation and then to use this approximation to compute sensitivities. Although this seems like a benign order of steps (and one often applied), the authors show that errors can arise that are sometimes catastrophic with respect to computational accuracy. The book presents illuminating examples of what can go wrong in computing sensitivities when the domain is a function space. Although the examples are for spatial variables in one and two dimensions, they give insight that should lead to further research in more complicated domains. This work is application unspecific; the authors provide concrete example problems to point out some of the problems that can arise and what improvements can be obtained. There are some simple one-dimensional partial differential equation problems and some more sophisticated two-dimensional steady state flow problems. The authors consider how to obtain accurate state gradient calculations, and why accuracy may be compromised if one computes sensitivities of computational models without considering the underlying distributed parameter system. The authors present examples in which computing the sensitivity by approximating the continuous sensitivity equation and solving can be different from computing the sensitivity of the approximation to the state equation. A major advantage of continuous sensitivity methods as presented in this book is the great increase in computational speed. The intended audience for this work is engineers, scientists, and applied mathematicians who want to use continuous sensitivity equations in a computational setting. An undergraduate knowledge of analysis and numerical analysis is assumed along with a familiarity with finite element methods. Necessary concepts and results for Hilbert spaces are provided or referenced for the reader who is unfamiliar with these ideas. The editorial board of this book series believes this edition is timely and will prove most helpful to the reader who wants to use sensitivity analysis in a computational context. xvii

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Preface This book provides an introduction to the computational aspects of continuous sensitivity equation methods (CSEMs) for partial differential equations. It is intended for advanced undergraduate and graduate students in the areas of numerical analysis and applied and computational mathematics as well as other scientists and engineers who have a particular interest in modeling, design, control, and optimization of physical systems. With the significant advances in computer technology during the twentieth century, industries that design and manufacture high-performance products are increasingly interested in exploiting the advantages of computer-aided design, numerical analysis, and optimal design methods. For example, a detailed analysis of aerodynamic systems, automated manufacturing processes, and casting or molding processes can be performed using a software package prior to time-consuming, expensive, and labor-intensive physical experiments. During the last decade, considerable effort has been devoted to the development of computational tools that allow designers to mathematically analyze such physical processes. Sensitivity analysis plays an important role in this effort. Many simulation tools can be enhanced by using local information provided by state sensitivities to obtain nearby solutions. Hence, one is able to develop a qualitative picture of how small changes in design parameters might affect the performance of a physical system. This book is intended to give the reader an overview of applications and computational issues regarding sensitivity calculations performed using CSEMs. The focus is on the construction and analysis of algorithms for computing sensitivities. Consequently, adjoint methods and other methods used primarily in the context of optimization applications are omitted from this text. Myriad algorithms have been developed for the computation of sensitivities. In Chapter 1, a number of these methods are briefly discussed. The methods are addressed in sufficient detail to illustrate their respective advantages and disadvantages. A number of schemes fall under the heading of CSEMs, and these methods have been applied to a variety of problems. The term sensitivity equation methods refers to a broad class of computational methods that compute a sensitivity by deriving and solving an equation known as the sensitivity equation. This text focuses on the use of CSEMs, which refers to the formal mathematical differentiation of governing equations (usually partial differential equations) to derive the sensitivity equations (which also take the form of partial differential equations). Chapter 1 gives a general overview of this approach and Chapter 2 provides the appropriate definitions to answer the question of what exactly is meant by the term sensitivity. Essentially, the sensitivity is interpreted as a partial derivative with respect to a design parameter. Chapters 2 and 6 present the mathematical framework developed by the authors for the purpose of rigorous mathematical justification of the formal differentiation xix

xx

Preface

techniques that form the base of the CSEM concept. These chapters are very mathematical and are intended for readers with a background in real analysis and function analysis. A general framework is given for a class of linear elliptic boundary value problems in Chapter 2, while Chapter 6 provides an abstract framework for Navier-Stokes equations. Chapter 3 introduces the one-dimensional model problems used throughout the book, to provide the reader with a relatively simple platform for absorbing the concepts presented. Furthermore, these model problems allow the authors to demonstrate numerical techniques in a simple format and to illustrate some subtle computational issues in subsequent chapters. The models are constructed in detail and are related to the real-world applications on which they are based. In the course of outlining these problems, many of the fundamental characteristics of CSEMs are discussed. For example, an inherent by-product of CSEMs is the coupling of the sensitivity equation (partial differential equation) to the original state equation (partial differential equation). This coupling can take the form of sensitivity boundary conditions that require state gradient information, or, in the case of nonlinear state equations, the sensitivity equation may include terms that involve the state variable or the gradient of the state variable. Chapters 4 and 5 focus on the computational algorithms used for sensitivity calculations throughout the book. Computational schemes are developed for both linear and nonlinear elliptic problems. The method of mappings is incorporated into the CSEM as an approach to computing shape sensitivities for a class of heat transfer applications. Numerical results for such algorithms are explored and discussed in significant detail in Chapter 5. To develop a framework for fluid flow applications for both high and low Reynolds number situations and for applications involving boundary layers, considerable effort is concentrated on the analysis of a nonlinear model problem. In the case of nonlinear state equations, terms involving state gradients often appear as coefficients in the sensitivity equations. This phenomenon produces a special set of obstacles when the goal is to quickly and cheaply obtain accurate sensitivity approximations. Indeed, one of the model problems illustrates that obtaining accurate state gradient approximations is of fundamental importance for sensitivity computations in the presence of a boundary layer. Hence, an entire section of Chapter 5 is devoted to the topic of extracting and improving state gradient approximations. Projection techniques first developed for a posteriori error estimators in finite element codes are used to improve the accuracy of these state gradient approximations. In addition, Chapter 5 contains a brief section that explores the usefulness of sensitivity approximations within the framework of an optimal design problem. Finally, Chapters 7 and 8 investigate the validity of the CSEM approach for some examples of two-dimensional fluid flow problems. In Chapter 7, the continuous sensitivity equation method is shown to be numerically efficient for computing sensitivities for flow around a cylinder and for flow over a bump. However, Chapter 7 points out that, in certain instances, the accuracy of the sensitivity calculations can be greatly improved by taking advantage of adaptive refinement strategies that may already be available within the flow solver producing the state approximations. Chapter 8 presents an example that combines adaptive refinement with projection techniques in order to provide exceptionally accurate sensitivity approximations for nominal extra computational expense. The authors would like to take this opportunity to thank the many people and agencies who have made this book possible. Lisa Stanley would like to express her appreciation to the Air Force Office of Scientific Research for the AASERT fellowship that provided

Preface

xxi

funding for research and travel during 1997-1999. Major Dawn Stewart is also grateful to the Air Force for granting its permission for this book to be written. The Interdisciplinary Center for Applied Mathematics provided both authors with the computer facilities, the office facilities, and, most important, the mentoring necessary to carry out the research component of their dissertations. These mentors include the thesis advisor Dr. John A. Burns and other friends and colleagues, such as Dr. Jeff T. Borggaard, Dr. Gene M. Cliff, and Dr. Belinda B. King. A special note of thanks to Dr. Dominique Pelletier for allowing the authors the opportunity to use and modify the code used to generate steady state solutions and sensitivities for Navier-Stokes equations. The authors are also extremely grateful to everyone at SLAM, especially Dr. Tom Banks, Editor-in-Chief of the Frontiers series, and Marianne Will, Acquisitions Editor, for providing the opportunity to publish work in the well-respected Frontiers series. Lisa Stanley would also like to acknowledge the Burns Telecommunication Center, in particular Ritchie Boyd, on the campus of Montana State University for the computer facilities needed for some graphics editing. Finally, this section would not be complete without the recognition of our families for their constant encouragement, support, patience, and never-ending love and understanding throughout the years. Lisa G. Stanley Dawn L. Stewart

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1 Introduction

As modern computing capabilities increase, considerable effort is devoted to the development of computational tools for the analysis, design, control, and optimization of complex physical systems [4], [19], [29]. A critical, and sometimes expensive, first step in the process of computer-aided analysis and design is the formulation of a detailed and accurate model of the system. The physical performance of the design is modeled by a system of mathematical equations, usually ordinary or partial differential equations. For many engineering applications, analysis of the design requires efficient construction and manipulation of complex geometries along with fast and accurate numerical methods for solving partial differential equations. Generally, there are a number of physical parameters (or design variables) that designers can adjust (either manually or within a computer simulation) to improve the design. In applications such as aerodynamic design, growth and control of thin films, or design of smart materials, some design variables may influence the shape (geometry) of the design. Consequently, designers then become interested in how sensitive the state variables are to small changes in the design variables. For example, when analyzing a composite material, one may be interested in the sensitivity of the heat flow through the material to small changes in the thickness of the film. Sensitivity analysis is a mathematical tool that provides a methodology for investigating such questions. This book has two main goals. The first is to describe and illustrate the application of continuous sensitivity equation methods (CSEMs) to several examples. The examples were chosen to demonstrate that there are many computational schemes available within the context of CSEMs. Furthermore, the choice of a specific algorithm should be guided by the overall objectives of the problem. The second goal is to provide the foundation of a mathematical framework that can be used to derive sensitivity equations. We provide mathematical constructions for elliptic boundary value problems and Navier-Stokes equations. Within the context of elliptic boundary value problems where shape is the design parameter, the method of mappings is explored as a tool for mathematically justifying and rigorously analyzing sensitivity equations. Once a sufficient amount of background is constructed, we point out the mathematical as well as the numerical issues that are inherent to CSEMs. To make effective use of sensitivities, accurate and efficient computational algorithms are essential. In addition, the natural trade-off between accuracy and efficiency can often

1

2

Chapter 1. Introduction

be exploited to produce the best overall computational tool for a specific application. For example, if sensitivity calculations are used to enhance a simulation tool, accuracy may be more important than speed. However, when used for gradient computations within an optimization algorithm, speed may be more crucial than high accuracy. Hence, the scientific motivation behind the sensitivity calculation can often influence the choice of computational algorithm. Many numerical methods have been developed for the purpose of sensitivity computation. The first, and maybe the best known, is the finite difference technique. The method is very simple to implement and has a background that is fundamentally established in mathematics. However, for complex fluid flow problems where each state approximation requires considerable computational resources, this method requires two flow solves, each at different parameter values, to form a difference quotient. In large aerospace flow problems, this is sometimes an extremely severe requirement, since mesh generation itself can take weeks to months. In addition, a step-size sufficiently small to ensure accurate approximations is not generally known a priori. Such obstacles can make finite difference techniques impractical in many situations and difficult to incorporate into optimization schemes. A procedure for estimating an optimal step-size is described in [36]. The optimal step-size attempts to balance truncation error and round-off error. Unfortunately, since the discretization is generally nonuniform, the truncation error varies from point to point so that the results must be used in combination with some other decision algorithm. Another early numerical technique was the "discretize-then-differentiate" approach. This scheme approximates sensitivities by first employing some discretization scheme to approximate the solution to a partial differential equation and then implicitly differentiating this result to obtain a sensitivity approximation scheme. Recent advances in this area include the development of automatic differentiation packages (see [25], [47]), such as ADIFOR. Given source code for state calculations, this package generates source code for sensitivity calculations. Once a computer code for generating a finite-dimensional state approximation has been constructed, the code is viewed as a sequence of compositions of elementary functions, and differentiation is achieved by clever and repeated use of the chain rule. Automatic differentiation has been applied to shape optimization problems and optimal design problems (see [16] and [34]), and the ideas have been considered and refined for a variety of other applications; see [26], [27], [33]. For more references and a general discussion comparing this method with other shape optimization algorithms, see [22]. A disadvantage of the discretize-then-differentiate technique is that in cases in which the mesh is parameter dependent, as is the case in shape-optimization problems, then differentiation of the discrete PDF leads to mesh sensitivities on the right-hand side. Although much work has been done in recent years to get a handle on these quantities (see [53]), calculating mesh derivatives is still not well understood, particularly in cases in which the meshes are prescribed adaptively. Several new techniques have been developed in an attempt to alleviate some of the disadvantages of the two approaches mentioned above. The idea is to employ a "differentiatethen-discretize" scheme, or a so-called CSEM, which consists of implicitly differentiating a partial differential equation to obtain a sensitivity equation and choosing an appropriate discretization scheme based on that equation's structure. Then both the partial differential equation and the sensitivity equation are discretized to obtain finite-dimensional equations for numerical approximations to the partial differential equation and the sensitivity equation.

Chapter 1. Introduction

3

As demonstrated in this work, a number of schemes fall under the heading of CSEMs. Early use of the sensitivity equation for optimal design purposes is seen in [38], and theoretical framework describing sensitivity functions can be found, e.g., [29] and [42]. Variants of sensitivity equation methods have been applied to a variety of problems. In [23], Godfrey investigates several uses of sensitivity equation methods (SEMS) for the analysis of chemically reacting flows in aerodynamic applications. SEMs are used to compute aerodynamic stability derivatives in [24] and [45]. Optimal control of fluid flows and shape optimization algorithms using these methods can be found in [7], [8], [9], [10], [11], [12], [14], [28]. The references given here are by no means comprehensive or exhaustive, but they serve as a starting point for the interested reader. As is the case with most new research, the CSEM approach provides its own set of advantages and disadvantages. Although the CSEM has a very mathematical foundation, derivation of the sensitivity equation can be difficult at times. Parameters can enter a model in a variety of ways, and much of the analysis still must be done on an individual problem basis for many cases. Identifying the correct boundary conditions for the sensitivity equation can be tricky. The mathematical constructions in Chapters 2 and 6 certainly provide a platform for further analysis. However, developing a comprehensive mathematical theory for CSEMs will require much more sophisticated tools than those presented here. Even for a relatively small class of elliptic partial differential equations, there is no blanket approach to sharply characterizing the smoothness of the sensitivity. During the course of our research, we have seen that smoothness may be gained, lost, or maintained when one compares the smoothness of the state to that of the sensitivity. Once a firm mathematical foundation is developed, the advantage of the CSEM approach is that it will provide insight into situations where the sensitivity equation must be interpreted in a weak sense. This may be the case with discontinuous coefficients or forcing functions when the discontinuities are somehow dependent on the parameter of interest. Another area where CSEMs may be useful is in optimal design. CSEMs can provide cheap and accurate sensitivity approximations for gradient calculations. However, the issue of consistent gradients is far from resolved; see [7]. In addition, optimal shape design is an area in which the authors would like to make a contribution. This topic has long been linked to variational problems through study in the area of the calculus of variations; see [39], [40], and [41] for applications focusing on structural design optimization. Optimal shape design encompasses a mathematically challenging set of problems as the domains of differential operators become parameter dependent. In this setting, the concept of differentiation of an operator becomes quite difficult to characterize in certain instances. Clearly, the advantages, disadvantages, and potential applications of CSEMs are still revealing themselves as the research continues. This book is a compilation of the authors' dissertation work while attending Virginia Polytechnic Institute and State University. Some of the numerical results in this book were generated using MATLAB on various UNIX platforms, including SUN workstations and Silicon Graphics Indigo and Indigo 2 workstations. The two-dimensional fluid flow results were generated using an adaptive finite element code developed by Dominique Pelletier and other researchers at Ecole Polytechnique de Montreal. The code was modified to calculate sensitivities at Virginia Polytechnic Institute and State University by Dr. Jeff Borggaard and Dr. Dawn Stewart.

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2 Mathematical Framework for Linear Elliptic Problems

This chapter surveys the mathematical background needed to analyze the regularity of a particular class of linear elliptic partial differential equations, which includes some of the model problems introduced in Chapter 3. In particular, the state equation takes the form of a linear elliptic boundary value problem defined on a parameter-dependent domain. First, we give the general formulation of the class of state equations to be considered. We then present the necessary notation and framework to provide the existence and uniqueness of both strong and weak solutions for the general form. Finally, we discuss the implications this framework has for the derivation of sensitivity equations. Readers who are interested in an overview of the general approach of SEMs may choose to skip this chapter on a first reading; it contains a number of technical mathematical details and is not essential reading for those who seek a general exposure to SEMs. However, readers who are already familiar with the topic may find that the details of this chapter help to solidify their understanding of the mathematical concepts used in the derivation of sensitivity equations. Let Q c Rm denote an open domain of design parameters. For each q Q, assume that C Rn, n = 1,2, or 3, is a domain satisfying the cone condition with the boundary (q) such that locally, (q) lies on one side of (q). Consider the elliptic boundary value problem

where each of the data K(x , q), f ( x , q), and g(x, q) may depend on the design parameter q. Throughout this chapter, we assume that the coefficient K ( x , q) is a real-valued function satisfying 0 < m1 K ( x , q) m2 for all x (q), q Q. The long-term research goal is to mathematically characterize the dependence of the state w(x, q) on small changes in the design described by q.

2.1

Preliminaries

This section introduces some preliminary concepts and results that are used throughout the following three chapters. The trace theorems for Sobolev spaces are presented in the 5

6

Chapter 2. Mathematical Framework for Linear Elliptic Problems

context needed for this work. Variational tools such as bilinear forms, Gelfand triples, and their related properties are reviewed, and the results from the theory of partial differential equations relevant to this work are surveyed. The authors concede that the material in this chapter gives only an abbreviated introduction to the relevant mathematical concepts, and the reader should be cautioned that some of the material relies on knowledge of basic concepts from functional analysis, such as linear operators, inner-product spaces, and duality pairings. Additional reading for the related functional analysis topics includes [43], [52], and [55].

2.1.1

Notation

Most of the notation used throughout the book is consistent with that of Wloka [54]. We begin with some general geometric assumptions. Sobolev spaces as well as their inner products and related norms are described. Let Q represent a bounded, connected, open subset of R", n = 1, 2, or 3, satisfying the cone condition. Denote the boundary of by We assume that is piecewise smooth and that, locally, lies on one side of Unless otherwise stated, the variable x denotes the spatial variable or vector of spatial variables. The state and sensitivity variables are denoted by boldface letters such as u, w, and s to indicate that this variable may represent a vector of scalar variables. We assume a finite number of design parameters, and the notation q denotes the vector of design parameters. The design space is denoted by Q. Partial derivatives with respect to either the spatial variable or to a parameter are denoted as

Similarly, higher-order derivatives are expressed as

The derivatives referred to in this work are Frechet derivatives unless otherwise specified. The following definition is given as a reminder and to preface the mathematical framework constructed in later sections: Let (X, \\ • \\ X ), (Y, || • ||y), and (Q, \\ • \\Q) be normed linear spaces, and denote the set of all bounded linear operators mapping X to Y by B(X, Y). Let G B(X, Y). Then G is said to be Frechet differentiable at x0 X if there exists a continuous linear operator D x G(x 0 ) B(X, 7) such that for every h X with XQ + h X,

Frechet derivatives along with other notions of differentiation are discussed in [46]. If X, Y, and Z are normed linear spaces, then the product space X x Q has norm defined by

As before, we assume q Z and G: X x Q —> Y. Given qQ int(Q), define G1: X —> Y byG 1 (x) = G(x,qo). If G 1 ( x ) has a Frechet derivative at x = XQ , then G (x, q0) has a partial Frechet derivative with respect to x at (XQ, qo), and this derivative is denoted

2.1. Preliminaries

by

7

As is usual, we denote the space of (Lebesgue) square integrable functions defined on and with the standard inner product

is a Hilbert space. Let Hm whose denote the subspace of functions in L2 generalized partial derivatives up to order m also belong to L2 In this chapter, we will make particular use of the Sobolev spaces Hl and HQ defined by

and

with inner product,

norm,

and seminorm,

defined by

respectively. Let the space of infinitely differentiable functions defined on be denoted by C Likewise, the subset of these functions that have compact support in is denoted by It is well known that the set C Hm is dense in Hm and the completion of C with respect to the norm \\ • \\m is defined to be the set Hm0 . The dual space of Hm0 , that is, the set of all bounded linear functionals defined on H0 m is denoted as H-m . For a general Hilbert space, V, we denote its dual space as V*. Recall that the norm on the dual space is the operator norm given by

where

When there is no chance of confusion, we use the notation

to depict the duality pairing. In general, we use the notation {•, •} for both inner products and duality pairings. The subscripts are used only when this notation could be confused with that of an L2 inner product, and we remark on the specific meaning of the notation whenever the usage is ambiguous.

8

Chapter 2. Mathematical Framework for Linear Elliptic Problems

2.1.2

Function Space Tools

Let represent a bounded, connected, open subset of Rn, n = 1,2, or 3. We begin by defining the geometric properties needed for the assumptions on the domain . The following definitions along with a more general presentation of these concepts can be found in Wloka's book [54]. A function belongs to the class C if and all its partial derivatives up to order k are continuous, differentiable, and bounded on and the kth partial derivatives of 0 are A-Holder continuous. With this groundwork, the notion of a (k, )-diffeomorphism can be introduced. Consider two domains and Definition 2.1. LetT: be an isomorphism. The mapping T is a morphism, or T if the following conditions hold:

-diffeo-

1. The coordinate functions of the map T denoted by each belong to the class 2. The coordinate functions of the inverse map T-l denoted by xi = Mi ( y 1 , . . . , y n ), i = 1 , . . . ,n, each belong to the class < 3. Ifk > 1, we require that the Jacobian determinant ofT

satisfies

Here, the constants c and C are independent of the spatial variables x. The domain is said to be of class C if the region is (k, )-smooth. That is, for each point x € , there exists a neighborhood Ux of x that can be transformed by a (k, )-diffeomorphism to the unit ball in Rn in such a way that the boundary is in oneto-one correspondence with the central plane xn = 0 and so that Ux lies on one side of the central plane while Ux lies on the other side of the central plane. The notation denotes the complement of the closure of . One can think of this diffeomorphism as a mapping that maintains an orientation with respect to the boundary . Now that the groundwork has been laid, we assume that the domain belongs to the class C . The appropriate values of k and are assigned as we consider each class of solutions in the following sections. We summarize the trace theorems taken from section 8 of [54] in the following results. Theorem 2.2 (trace theorem). Suppose that is (k, )-smooth, and let l — m > , with m a nonnegative integer and l + 1 < k + . Then there exists a continuous linear trace operator

with the property that

Here

denotes differentiation in the direction of the outward normal.

9

2.1. Preliminaries

For the case that we are only interested in the mapping T0 ,we can relax the restriction l + 1 < k + to l < k + X. One can also characterize the kernel of the trace operator introduced above. We condense this theorem to concentrate only on the particular result needed for the analysis in the following sections, and we also need the inverse theorem found in [54]. Theorem 2.3. Suppose that operator given above, then

is (0, 1)-smooth. IfT0: Hl(ti) ->

H1/2(

is the trace

Theorem 2.4 (inverse theorem). Let bea(k, ) -smooth domain, and let Then there exists a continuous, linear extension operator

with the property that

More detailed results concerning Sobolev spaces, trace operators, and related subjects can be found in [1]. 2.1.3

Bilinear Forms and Variational Tools

To obtain theoretical results concerning weak solutions to (2.1)-(2.2), variational formulations rely on the concept of a Gelfand triple. We also need some elementary definitions and results concerning bilinear forms and their properties. Definition 2.5. Let V and H be Hilbert spaces. Suppose that the mapping i: V —> H is a continuous, injective embedding with Im(i') dense in H. Then the embedding i': H —> V* described by is continuous and injective and Im(i') is dense in V*. The preceding construction

with continuous, injective, dense embeddings is called a Gelfand triple. The assumptions on V can be relaxed to a reflexive Banach space, but for the following sections, the restriction to Hilbert spaces is sufficient. Since Im(i') = i'(H) is dense in V*, we can uniformly approximate each linear functional (on the unit ball) in V* by the inner product on H. That is, for each / V*, we can write

The completeness of H and the continuity of i' imply the existence of h1 H such that

Chapter 2. Mathematical Framework for Linear Elliptic Problems

10

This provides a continuous extension of the inner product on H to a continuous linear functional on V* x V, and it gives a representation of the linear functionals in V* by means of the inner product in H. Definition 2.6. Let V be a Hilbert space; then a mapping a: V x V —> R is called a bilinear form if it is linear in each variable. That is, a(•, •) must satisfy

and

for allx,y,z

V, where a,

€ R.

Suppose V <—> H <-+ V* is a Gelfand triple. Let a(•, •) be a bilinear form on V. Definition 2.7. The bilinear form, a(•, •), is V-elliptic if the following conditions hold:

and

where the constants c 1 ,c 2 are independent of the functions x,y

V.

If a: V x V —> R is a continuous bilinear form, that is, if (2.7) holds, then a uniquely determines a continuous linear mapping L: V —> V by

and || a || = ||L||. The proof of this statement is a consequence of the Riesz representation theorem, and a more general version can be found in Theorem 17.8 of [54]. Moreover, let (h 1 , i(v))H denote the representation (2.4) of an arbitrary linear functional l V*. Then by the Riesz representation theorem, we have the relation

where R is the Riesz isomorphism R: V* obtain

V. Using this relation along with (2.9), we

Hence, there exists a continuous representation operator A: V —> V* given by A = R - l o L such that

And A is said to be V-elliptic if the bilinear form a(•, •) is. The construction given above leads us to the Lax-Milgram theorem, which provides the existence and uniqueness of weak solutions to elliptic boundary value problems in the following sections.

2.2. Regularity of Elliptic Boundary Value Problems

11

Theorem 2.8 (Lax-Milgram theorem). If the bilinear form a(x, y) is V-elliptic, then the corresponding operator A: V —> V* given in (2.11) is a linear topological isomorphism between the spaces V and V*. The norms are bounded in the following way:

where c\ and c2 are the constants given in Definition 2.7. Or, equivalently, for each f V*, the variational equation Ax = / has a unique solution in V, and this solution depends continuously on f. Many forms of this theorem are given in the literature; see [5] and [21]. This particular version is found in [54]. Once these fundamental results have been introduced, we are ready to move to the regularity results for the general form of the state equation given in (2. l)-(2.2).

2.2

Regularity of Elliptic Boundary Value Problems

We note that for a fixed q, the problem (2.1)-(2.2) defines an elliptic boundary value problem on a fixed domain, = (q). Also, as we show in Chapter 4, the method of mappings is often used to transform the problem (2. l)-(2.2) on (q) to a problem on a fixed computational domain In either case, without loss of generality, we may study the regularity of solutions by fixing the domain . This section contains a survey of theoretical results characterizing the regularity of elliptic boundary value problems of the general form

As noted above, we assume for the moment that the domain is independent of the parameter q. In section 2.3 we describe how these results can be used, in conjunction with mapping techniques, to treat differential equations defined on parameter-dependent domains. Specific assumptions on the coefficient k(k, q) and on the domain and the boundary are given in each section. To simplify the discussion further, we first transform (2.13)-(2.14) into a problem with homogeneous boundary conditions. Assume that wg(•) H1 satisfies

then the trace where T0 is the trace operator from section 2.1.2. Observe that if g theorem implies the existence of such that (2.15) holds. If satisfying (2.15). In the then the trace theorem implies that there exists then the precise meaning of (2.15) requires some more general setting, if discussion that is beyond the scope of this book. In the case of sections 2.2.1 and 2.2.2, the preceding justification is sufficient for the formulation of the homogeneous problem. Let z = w — wg; then z satisfies (at least formally)

12

Chapter 2. Mathematical Framework for Linear Elliptic Problems

In the following sections, we focus on the homogeneous problem of the form

2.2.1

Strong Solutions

This section presents a result describing the data that yield strong solutions to the elliptic equation (2.16)-(2.17). First, we specify the meaning of the term strong solution. Definition 2.9. A strong (or classical) solution to (2.16)—(2.17) is a function such that and

Ifz is a strong solution to (2.16)-(2.17) and if both and then we define to be a strong solution to (2.13)-(2.14). Let the operator

be defined as

where the domain of. is given by Note that A0 is a strongly elliptic, self-adjoint differential operator; see [54]. The following theorem characterizes solutions of the operator equation

To say that z is a solution to (2.20) is equivalent to asserting that (2.16)-(2.17) has a strong solution. Theorem 2.10. Assume that , and let If the data and then there exists a unique solution z D(A 0 ) to (2.20), andz(x) is a strong solution of (2.16)-(2.17). This result is a consequence of the theory presented in Chapter 2 of [54]. 2.2.2

Weak Solutions

In this section, we outline the variational techniques used to obtain the existence and uniqueness of weak solutions to a state equation of the form given in (2.16)-(2.17). The variational form of the state equation is expressed, and Theorem 2.8 can be used to assert the existence of the weak solution to the state equation. Note that

forms a Gelfand triple, with the identity (inclusion) being the map from For a fixed value of the parameter q in (2.1), define the bilinear form

13

2.2. Regularity of Elliptic Boundary Value Problems

R by the following:

The Cauchy-Schwarz and Poincare inequalities can be used to show that this bilinear form is continuous and -elliptic. In particular, there exist constants m2 and c such that

and

Here, m2 is the upper bound placed on the coefficient k(x, q) for all x and for all q Q. The constant c involves the lower bound on the coefficient k(k, q) as well as the constant prescribed by the Poincare inequality. See [1], [13], and [54] for details regarding these results as well as many other theoretical results related to variational formulations of partial differential equations. We are now ready to define the term weak solution. Definition 2.11. A weak solution to (2.16)-(2.17) is a Junction z

for all If z is a weak solution to (2.16)-(2.17) and if define w = z + wg to be a weak solution to (2.13)-(2.14).

such that

then we

Using the bilinear form defined above, one can construct the operator A: D(A) where and

Note that the variational equation given in (2.22) is equivalent to the operator equation (in

The following result characterizes weak solutions to (2.16)-(2.17) and is a direct consequence of the construction given above and the Lax-Milgram theorem. Theorem 2.12. Let the domain ) and the boundary data to the operator equation then there exists a unique solution (2.24), and z(•) is a weak solution to the state equation (2.16)-(2.17). This work focuses on state equations with Dirichlet boundary conditions; however, section 21 of Chapter 3 in [54] contains several examples of elliptic equations with various types of boundary conditions treated using variational techniques. The reader may also

Chapter 2. Mathematical Framework for Linear Elliptic Problems

14

choose to consult several other classical texts, such as [2], [20], and [37], for general expositions on the theory of partial differential equations. In the previous sections, we constructed a mathematical framework for analyzing the regularity of elliptic boundary value problems on a fixed domain. However, we want to apply these concepts to equations defined on parameter-dependent domains in order to derive sensitivity equations for shape parameters. With this goal in mind, we now transition to the method of mappings that provides a mechanism for transforming equations on parameterdependent domains to equations defined on fixed domains, thereby allowing us to use the results of this chapter to analyze sensitivity equations.

2.3

Mappings and Implications to Sensitivity Analysis

In Chapter 4, we discuss a technique for transforming an elliptic boundary value problem defined on a parameter-dependent domain to a "transformed" boundary value problem that is also elliptic but is now defined on a domain independent of the design parameter. The process of transforming can introduce an explicit dependence of the transformed equation on the design variable. The notation used in the following sections reflects this situation. In this setting, the sensitivity of the transformed state can be defined precisely, and the sensitivity equation for the transformed state can be derived in a mathematically rigorous fashion. In this section, we introduce the necessary mathematical framework and derive the (operator) sensitivity equation. The relationship between the operator equation and the differential equation is addressed, and some examples are presented in Chapter 3. 2.3.1

Implicit Function Theorem

Theorem 2.13 (implicit function theorem). Assume Z, Y, and Q are Hilbert spaces, and let G: D(G) C Z x Q —> Y be continuous on a neighborhood U of the point ( Z 0 , q0) int[D(G)]. If 1. G(z0,q0) = 0,

2. G has a strong partial Frechet derivative G(z 0 , q 0 ), and 3. [ z G(z 0 , q 0 ) ] - l exists and belongs to B(Y, Z), then there exist open neighborhoods U, W with Z0 U C Z and q0 W for any q W, the equation G(z, q) = 0

Q such that

has a unique solution z = u(q) and the mapping u: W —> U is continuous. Thus, u(q) satisfies the equation G(u(q), q) = 0 forq W. Moreover, if q G(z 0 , q0) exists, then u(q) is Frechet differentiable at q0 and

A more general version of this theorem can be found in Zeidler's book [56].

2.3. Mappings and Implications to Sensitivity Analysis

2.3.2

15

Operator Formulations of Sensitivity Equations

In this section, we outline an approach for deriving sensitivity equations that relies on the implicit function theorem. For the analysis of elliptic boundary value problems in section 2.2, each of the constructions ultimately takes the form of an operator equation such that for each q Q

where C: Q (Z, Y) is a continuous linear operator, Z and Y are Banach spaces, and F: Q —> Y. One can further generalize this equation by defining the operator G: Zx Q Y as follows:

With each value of the design parameter q, we associate the corresponding state that solves (2.26) to form the pair (z(q), q) Z x Q such that

That is, for a fixed q Q, we characterize the state z(•, q) as the solution to the operator equation (2.28). This induces a natural mapping z: Q Z described by

To characterize the sensitivity at q, we are actually interested in determining the operator D q z(q 0 ): Q Z. Hence, each q Q determines a sensitivity (an operator) whose range, in turn, belongs to the function space Z. Under certain conditions, one can use Theorem 2.13 to derive an operator sensitivity equation for D q z(q 0 ). In the following paragraphs, we provide the construction and conditions necessary to apply the theorem. We then move to some examples to illustrate the process. Let(z0, q0) = (z0(q0), q0) Zx Q satisfy (2.29). The partial derivative zG(z0, q0): Z Y exists and is given by

Moreover, if the operator £ is Frechet differentiable at q0, then the derivative [ D q L ( q o ) ] is a bounded linear operator [Dq£(qo)]: Q B(Z, Y). If we define the operator [L(qo)]: Q Y by

then it is straightforward to show that L (q0) is a bounded linear operator from Q to Y. Further suppose that D q F(q 0 ): Q Y exists in the Frechet sense. Then the partial derivative of G with respect to q at the point (z0, q0) exists, and [ G(z0, q 0 ) ] : Q Y is given by

With this construction, the following theorem can be established.

16

Chapter 2. Mathematical Framework for Linear Elliptic Problems

Theorem 2.14. Let q0 int( Q) be fixed. Suppose that there exists a unique Z 0 (q 0 ) int(Z) so that G(z0(g0), q0) = 0. Further suppose that [L(q 0 )] -1 exists in B(Y, Z). If the Frechet derivatives Dq£(q0) and D q F(q 0 ) exist in B(Q, B(Z, F)) and B(Q, 7), respectively, then the sensitivity p = Dqz(+qo): Q —> Z exzste and satisfies the operator equation

Proof. Equation (2.33) follows from applying the operator equation given in (2.25) of Theorem 2.13 to equations (2.30) and (2.32) given above. Theorem 2.14 implies that for a given parameter q0, the sensitivity p(., q0) belongs to the function space Z, and it is the solution to the operator equation (2.33). At the beginning of the following chapter, we present an example to demonstrate the derivation of the sensitivity equation. It is important to note that the choice of function spaces Z and Y is nontrivial. Moreover, one can construct relatively simple state equations for which the preceding theorem does not apply. In the next chapter, we present some examples and, where appropriate, relate the examples with the framework presented here.

3 Model Problems

This chapter describes three examples of state equations used throughout the book to explore the concept of a sensitivity and to illustrate several of the mathematical issues involved in its computation. For clarity of presentation, we have chosen state equations whose spatial domains are one-dimensional. In addition to their simplicity, these examples are convenient for comparing numerical results since both the state and the sensitivity equations have analytical solutions. They also provide the reader insight into several aspects of sensitivity computation that are inherent to our approach. The first example is used to illustrate the use of the mathematical framework outlined in the previous chapter. A state equation is introduced, the operator formulation is given, and an operator sensitivity equation is derived. The remaining model problems—one is linear and one is a nonlinear boundary value problem—provide us with one-dimensional examples of state equations with parameterdependent domains. The state and sensitivity equations are outlined along with their respective closed-form solutions. In subsequent chapters, several aspects pertaining to sensitivity analysis and the implementation of CSEMs are explored for each of these problems. Computational methods for numerically approximating the solutions to the sensitivity equations are described, and numerical results are given.

3.1

Heat in a Thin Rod with a Parameter-Dependent Forcing Term

We begin with a state equation that is examined in greater detail in section 4.2.1. The parameter of interest is denoted by q. The domain of the state equation is independent of the parameter; however, the q appears explicitly in the forcing term.

with boundary conditions

17

Chapter 3. Model Problems

18

The forcing function f(x, q) is given by

Using results from section 2.2.2, it can be shown that the boundary value problem (3. l)-(3.2) possesses a strong solution in H 2 (0, 1) for each q Q = (1,2). Define the sensitivity

The mathematical framework described in the previous chapter can be used to derive the corresponding sensitivity equation for p(., q). Here, we illustrate the importance of choosing the function spaces Z and Y correctly, and we note that a formal derivation can often provide the insight necessary for this choice. The first approach might be to let Z == H2(0, 1) H10(0, 1) and Y = L2(0, 1). In this case, the operator £: Q -> B(H2(0, 1) H10(0, 1), L2(0, 1)) is defined by

for each q Q, where A0 is defined in (2.19) in section 2.2.1. In this case, £ = £(q 0 ) = A0 is constant with respect to the parameter q. Furthermore, the operator F: Q L2(0, 1) is defined by Using this framework, the operator G: Z x Q L2(0, 1), and the operator equation given by G(w,q) = £(q)w-F(q) = 0 is equivalent to (3.1)-(3.2). Clearly, the operator £ ( q ) - l = [A0]-1 exists in (L2(0, 1), H 2 (0, 1) H10(0, 1)) as it is a special case of the differential operator A0 given in section 2.2.1. The structure of L implies that its Frechet derivative is the zero operator in B(Q, B(Z, Y)). However, the derivative DqF(q) does not exist in the L2-sense, and Theorem 2.14 is not applicable for this construction. However, if one chooses the spaces Z = H 1 0 (0, 1)and Y = H - l ( ), then the operator £(q): Q B(H 1 0 (0,1), H-l is defined by

The A operator is defined in (2.23). It follows from the framework presented in section 2.2.2 that A-l = [£(q) - 1 exists in B ( H - 1 H 1 0 (0, 1)), and the Frechet derivative D q L(q 0 ) 1 -l is the zero operator in B(H 0 (0, 1), H Furthermore, one now defines the operator F: by Then the operator G :

and the operator equation given by G(w,q) = £(q)w-F(q) = 0

3.2. Heat in a Thin Rod with a Parameter-Dependent Boundary

19

is equivalent to (3.1)-(3.2). Using this construction, one can now represent the Frechet derivative of F with respect to q in the appropriate fashion. In particular, for a given qo, DqF(qo): Q H - l ( ) exists and is given by

is the Dirac delta function for all u(•) H10(0, 1) and for each h Q. Here, with mass at x = The hypotheses of Theorem 2.14 are now satisfied. Therefore, the sensitivity p exists (in H 1 0 (0, 1)) and satisfies the operator equation (in -l H (0,1)) given by That is, for any qo equation

Q, the sensitivity equation is well defined and is given by the variational

Furthermore, the sensitivity p(•, q0) H10(0, 1) is the unique solution to (3.5). From the derivation of the operator sensitivity equation, one observes that the differential operator describing the state equation, A in the above paragraphs, is also the differential operator associated with the sensitivity equation. Hence, the operator sensitivity equation is derived with the appropriate mathematical rigor, and a corresponding differential equation, our sensitivity equation, is constructed using the operator expression. In particular, p( , q) satisfies the following differential equation

where over,

belongs to H- l (0,1) for each

(1,2). More-

where

The boundary conditions for is the Dirac delta function with mass at 1 are clear since p(•, q) H 0(0,1). In the following sections, we consider other examples of state equations and their associated sensitivity equations. Once the examples are introduced, we examine specific mathematical and computational issues related to sensitivity analysis for these types of problems. Although they are somewhat simple in nature, these model problems illustrate many of the fundamental issues addressed in this book.

3.2

Heat in a Thin Rod with a Parameter-Dependent Boundary

The model given in this section describes the steady state temperature distribution in a thin rod. The shape parameter determines the length of the rod, and a heat source is applied to only one section of the rod.

Chapter 3. Model Problems

20

3.2.1

State Equation

Let the design space be the interval Q = (1,2). For a given q equation described by the elliptic boundary value problem

Q, consider the state

with homogeneous Dirichlet boundary conditions

Here f : (0, +00)

R is the piecewise continuous function given by

The goals are to solve (3.7)-(3.8) for the state, w(x, q), for a given value of q and to determine the sensitivity of the state to small changes in the parameter. The forcing function, f(•), is discontinuous on (0, 2) and belongs to the space L2(0, 2). For a fixed q (1, 2), the state w(•, q) is a function belonging to H2(0, q) DH10(0,q). In particular, one can verify that the analytical solution to (3.7)-(3.8) is

The sensitivity equation for this problem is discussed in the following section. 3.2.2

Sensitivity Equation

To mathematically describe how small perturbations in q affect w(x, q), we first define the sensitivity

From (3.10), we see that for each q and has the following form:

Q = (1, 2), the sensitivity is a linear function of x

Note that for a fixed q Q, s(x, q) is a C function on (0, q). We now derive a differential equation for which the function given in (3.12) is a solution. Beginning with the state equation defined on the interval (q), one can implicitly differentiate (3.7)-(3.8) to obtain a sensitivity equation. At this stage, it is important to note that this differentiation is rather formal. In general, the partial derivatives w(x, q) and w(x, q) need to be continuous to interchange the order of differentiation. In this case w(x, q) is discontinuous, but one can verify by hand that the sensitivity s(x, q) = w ( x , q ) satisfies the differential equation

3.2. Heat in a Thin Rod with a Parameter-Dependent Boundary

21

with boundary conditions

The sensitivity equation (3.13)-(3.14) is a linear elliptic boundary value problem with Dirichlet boundary conditions. The boundary conditions should be derived with care as the right endpoint of the domain (0, q) depends explicitly on the parameter q. Taking the total derivative of the right boundary condition in (3.8) with respect to the parameter q leads to the correct boundary condition for the sensitivity

Remark. Observe that the condition at x = q in (3.14) requires boundary information from the spatial derivative of the state. This coupling arises as a result of the domain of the state equation, [0, q] in this case, depending explicitly on the parameter q. Even for very simple linear elliptic state equations of this type, the corresponding sensitivity equations are coupled to the state through the boundary conditions. This feature is characteristic of sensitivity equations when the parameter of interest defines the shape of the domain. The appearance of the gradient in the boundary conditions (3.14) raises certain mathematical as well as computational issues. In the case of an elliptic sensitivity equation with Dirichlet boundary conditions, the function determining the parameter-dependent boundary condition w(q) will be required to belong to H ( (q)) (for each value of q) to ensure that the sensitivity equation possesses a unique, classical solution. To guarantee that a weak solution exists, the function w(q) should belong to H ( (q)). Further details regarding the regularity of solutions to elliptic partial differential equations are presented in Chapter 2 and references therein. Also noteworthy is the fact that the data on the right side of the sensitivity equation are very smooth. That is, the sensitivity equation is homogeneous since the source term f(x) in (3.7) does not explicitly depend on the parameter q. As shown in the previous example, when the source term, or forcing function, depends explicitly on the parameter, derivation of the sensitivity equation needs to be done with care but can be rigorously achieved using variational techniques. Recall that the state w(x, q) in (3.10) has a piecewise linear, discontinuous second derivative. For this particular problem, the sensitivity is very smooth as compared to the state. This fact is somewhat counterintuitive, since one might expect to lose smoothness when examining derivatives of the state. Over the course of our research, we have found that smoothness may be gained, lost, or even preserved when moving from the state equation to the sensitivity equation. Indeed, a general rule for this phenomena is yet unclear. This is partly because there are so many different ways in which a parameter can appear in a mathematical model of a physical system. Furthermore, the study of linear elliptic systems is only a first step since many of the most complicated physical systems are modeled using hyperbolic or parabolic partial differential equations. For our work, we continue to answer the question of smoothness case by case. In the next section, we examine a nonlinear example that allows us to gain significant insight into computation of sensitivities for Navier-Stokes equations presented in later chapters.

Chapter 3. Model Problems

22

3.3

Nonlinear Model

In this section, we consider a simple one-dimensional nonlinear boundary value problem with spatial domain in one dimension. As in the case of the linear model, the design parameter is denoted by q, and the design parameter determines the length of the interval over which the differential equation is defined. The corresponding sensitivity equation is given, and we note that the sensitivity equation for this model problem is linear although the original state equation is a nonlinear boundary value problem. 3.3.1

State Equation

We concentrate on the boundary value problem defined by the second-order, nonlinear differential equation

with boundary conditions

For each q > 1, the solution to this boundary value problem is given by

At this stage, we construct a simple inverse design problem, addressed in Chapter 5, in order to comment on the use of continuous sensitivity equation methods in conjunction with optimization algorithms. Let 0 < x1 < x2 < • • • < xp < 1 be fixed locations and assume that Wj is a real number representing a desired value of w(x) at Xj for j = 1,2,... , p. Consider the inverse design problem: Find q* > 1 such that

where w(x, q) is the solution of (3.15)-(3.16) and the integer p represents the number of data points. In gradient-based optimization one needs the derivative

Again, we use the notation

to define the sensitivity. One approach to the evaluation of this gradient at a given q is to and the sensitivity, and to form the computation compute the state, (3.19). This involves first solving (3.15)-(3.16) for w(x, q) and then computing s(x, q).

3.3. Nonlinear Model

3.3.2

23

Sensitivity Equation

One benefit of using the model problem is that we can calculate the sensitivity by direct differentiation of (3.17). In particular,

On the other hand, we can implicitly differentiate the boundary value problem (3.15)-(3.16) and obtain a boundary value problem for the sensitivity s(x, q). It follows that s(x) satisfies the linear differential equation

with boundary conditions

As in the previous example, the boundary conditions in (3.22) require the application of the chain rule and should be treated with care.

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4 Computational Algorithms

This chapter describes two computational methods for solving sensitivity equations. Note that the model problems given in Chapter 3 are defined on parameter-dependent domains. We address this issue here, and suggestions for dealing with this situation in the context of numerical sensitivity approximations are given. In particular, mapping techniques are discussed, and we show how these techniques can be blended with SEMs to numerically approximate sensitivities. We also show that there are some pitfalls associated with combining these techniques.

4.1

The Method of Mappings

The examples introduced in Chapter 3 belong to a class of elliptic boundary value problems defined on domains that depend on a parameter. For many engineering applications, a typical approach to such problems is to begin by transforming the problem to a fixed computational domain that is not parameter dependent. This computational domain is often more regular in shape, which simplifies grid generation and can improve the accuracy of numerical calculations. This mapping technique is very common for problems involving complex geometries that occur in fluid dynamics. The book [53] is an excellent source of information about these topics, and [32] provides some examples with computational details. 4.1.1

Transformation Techniques

We begin with some comments concerning the theoretical aspects of the method of mappings. The technique of mapping a given domain to one with a different coordinate system is used extensively in the theory of partial differential equations. We briefly summarize the theoretical results that provide the mathematical foundation for the use of this technique. The transformation theorem on page 80 of [54] gives the conditions under which Sobolev spaces defined on the physical domain, are equivalent to those defined on the computational space Let T +: be a Ck diffeomorphism between the domains. Here, we are implicitly assuming that the domain is smooth enough to construct the transformation 25

Chapter 4. Computational Algorithms

26

T. Then the Sobolev spaces

and are equivalent for any an integer). Furthermore, the trace operator, T0, commutes with the operation of transforming; see page 131 in [54]. Finally, the theory of elliptic differential operators from [54] also gives the important result that under the appropriately smooth diffeomorphism, (strongly) elliptic operators defined on transform to (strongly) elliptic operators defined on . This is also true for hyperbolic and parabolic operators as noted in [32]. A general form for the relationship between the original operator and the transformed operator is given on page 143 of [54]. We now address some of the practical aspects of using transformations. From a computational standpoint, transforming can be a complex process for partial differential equations with spatial domains in R2 or R3. In addition to the issue of regularity, one is concerned with loss of accuracy. The mappings are implemented by defining a new coordinate system that maps to a fixed computational space, . Transformations that produce coordinate systems that are orthogonal at boundaries are preferred. Otherwise, accuracy decreases as orthogonality declines (see [53, pp. 3-5]). A two-dimensional coordinate system on takes the form

where (x, y) represents the coordinate system in given by

The transformation

is

Aside from the issue of orthogonality, T must also satisfy other mathematical properties. In particular, T is required to be one-to-one, and the mapping should maintain some smoothness in the distribution of grid points. Ideally, T should also provide some mechanism for clustering grid points once a discretization has been constructed. For problems involving complex geometries, determining the mappings can be a complicated process. The transformations can be constructed using conformal mapping methods, partial differential equation methods, or algebraic methods. Conformal mapping methods are based on the theory of complex variables and are used only when working with a physical domain given in two spatial dimensions. Partial differential equation methods are also widely used. With this approach, the computational domain is rectangular with a uniform grid, and the location of corresponding points in the physical domain is determined through the solution of a system of linear, or nonlinear, partial differential equations. Elliptic grid generators are the most commonly used. However, there are methods that use hyperbolic or parabolic equations to determine the mappings. To determine a mapping, these equations are usually solved numerically. Consequently, this method may introduce numerical errors into the grid-generation aspect of the overall computational scheme. As we will see later, the spatial derivatives of the transformations are needed to construct the transformed equations. Hence, questions of invertibility and differentiability become more tedious to resolve when the mappings are computed numerically. For this reason, computing the transformations

27

4.1. The Method of Mappings

numerically can be a drawback. However, for very complicated geometries, this approach may be the only one suitable for numerical computation. Finally, algebraic methods determine an algebraic expression which relates points in with those in . Interpolation schemes are used to represent the interior points of the domain in terms of the points along the boundaries. Implementation of these methods is often simple and fast. Another advantage of algebraic methods is that the derivatives of the transformations can be computed analytically, thereby reducing computational time and avoiding the introduction of additional numerical inaccuracies into the computational method. However, these methods may be difficult to implement for very complex geometries. An algebraic approach is used to construct the transformations presented in this chapter. Once the transformation is determined, the process of transforming the differential equation begins. We remark here that the crucial step is to derive the relationship between the differential operator defined on and the corresponding operator defined on £2. This requires the computation of the derivatives of the mappings with respect to the spatial variables. An example of this process is given in the following section, and the details of this relationship will be presented in the context of the example. For equations posed on parameter-dependent domains, the mappings are also parameter dependent. In order to construct the transformed equations on the computational domain, derivatives of the mapping with respect to the spatial variables are computed, and these derivatives will also involve the parameter. Hence, the design parameter often appears explicitly in the transformed differential equation although the domain of definition no longer depends on the parameter. This can be seen in the example given in the following sections. One important issue for sensitivity calculations is that the choice of the computational method can result in the need to also compute the partial derivative of the mapping with respect to the parameter. Usually referred to as a "mesh sensitivity," this derivative is often difficult or even impossible to compute in two-dimensional and three-dimensional problems. In particular, this derivative is difficult to obtain when the transformations are computed numerically or when adaptive algorithms are used. The mesh sensitivity appears in the example presented in the following section. We comment further on this issue as we examine the following example. 4.1.2

An Algebraic Transformation

In this section, a specific example of an algebraic transformation technique is presented in the context of the model problems given in sections 3.2 and 3.3. We now define the transformations used to move between the physical and the computational domains. For the model problems, the physical domain is the interval (0, q), where q is a parameter taking on values from the interval (1,2). The computational domain is the unit interval = (0,1). For (1,2) define the transformation let and for each fixed by Note that the function M

defined by

28

Chapter 4. Computational Algorithms

is the inverse of T and is commonly referred to as the mesh map. As noted earlier, transforming can be a complex process for two-dimensional and three-dimensional problems. For the one-dimensional examples given here, transforming is straightforward. Some of the difficulties that occur in two-dimensional and three-dimensional problems, such as holes and corners in the domain, are not present in this case.

4.2

SEMs

This section presents two examples of SEMs used for the numerical approximation of sensitivities. We give a detailed description of the schemes used to obtain the numerical calculations presented in subsequent chapters. The transformation techniques described in section 4.1.1 are used to develop two computational methods for sensitivity approximation. The computational methods presented in sections 4.2.1 and 4.2.2 use the linear model problem given in (3.7)-(3.8) as a platform for investigating the implementation of the mapping techniques, in combination with SEMs, for the numerical approximation of sensitivities. In section 4.2.3, we demonstrate how one of the SEMs can be applied to the nonlinear model givenin(3.15)-(3.16). As illustrated in this chapter, there are several ways to implement SEMs, and these variations can yield algorithms with different convergence properties. We consider two specific SEMs used in conjunction with the transformation techniques described in section 4.1. The first is based on transforming both the state and the sensitivity equations to the computational domain, solving the transformed equations, and mapping these solutions back to the physical domain. The second approach transforms the state equation and then derives its sensitivity equation. Once the state and sensitivity systems are solved, the solutions are mapped back to the physical domain. There are benefits and drawbacks to each method. Indeed, it is not always obvious which scheme is best for a given problem. Many questions need to be answered before a complete theory can be developed, and we address some of these issues with the computational schemes and numerical results presented here. We now describe each of the SEMs in general and in the context of the linear model problem.

4.2.1

Hybrid SEM

In essence, the hybrid SEM (H-SEM) falls into the category of a differentiate-then-discretize method. Given the state equation defined on the physical domain, the corresponding sensitivity equation is first derived. For some problems the derivation may only be done formally. As discussed in Chapter 2, rigorous mathematical derivation of a sensitivity equation required the use of differential operators and subsequent differentiation of those operators. However, in many instances, the formal implicit differentiation of the boundary value problem and boundary conditions is acceptable. Once the state and sensitivity equations are established, the transformation techniques are used to derive the corresponding transformed state and sensitivity equations posed on the computational domain. The discretization is then applied to these transformed systems, and numerical calculations are performed. The preceding discussion outlines the general structure of the H-SEM, and in the following paragraphs we construct a particular example by applying the H-SEM approach to the model problem given in section 3.2.1. The first step with this approach is to derive

4.2. SEMs

29

a sensitivity equation that is also posed on the physical domain. This step is completed in section 3.2.2. Once the state and sensitivity equations are described, the method of mappings is applied. For the sake of clarity, the equations are given here. The state equation is given by with

The corresponding sensitivity equation defined on the physical domain is given by

The transformations defined in section 4.1.2 are used to construct the transformed functions. The transformed state is denoted by (-, •); likewise, the notation (-, •) refers to the transformed sensitivity. For and q (1,2), define

and

To determine the transformed boundary value problems on , one must also determine the action of the forcing function (•) under the mapping. Once transformed, the original forcing function becomes

which now depends explicitly on the parameter q. Using the above definitions and the chain rule, the spatial derivatives of the original functions and those of the transformed functions are related by

and

30

Chapter 4. Computational Algorithms

As noted in the previous section, the differential operators acting on the respective domains are related through algebraic expressions involving the spatial derivatives of the mapping. As a result, the transformed equations can be algebraically more complicated than their original counterparts. The identities developed above are used to derive transformed boundary value problems for both the transformed state and the transformed sensitivity defined in (4.2). The transformed state equation is given by the differential equation

with boundary conditions

Likewise, the transformed sensitivity satisfies the differential equation

with boundary conditions

Note that equations (4.7)-(4.8) and (4.9)-(4.10) remain weakly coupled through the boundary conditions of the transformed sensitivity equation, (4.10). Once the transformations are constructed and the transformed equations are derived, one can proceed to numerically approximate the solutions to these equations. The discretization for this particular example is discussed in section 4.3. The H-SEM is summarized below to illustrate the key components of the approach. H-SEM Step 1. Solve the transformed state equation (4.7)-(4.8) for Step 2. Solve the transformed sensitivity equation (4.9)-(4.10) for Step 3. Map

back to the physical domain to obtain the sensitivity given by

Observe that by using (3.10), (3.12), and the definitions of the transformed functions given in (4.2), the closed-form solutions to the transformed state and transformed sensitivity equations are given by

and

4.2. SEMs

31

respectively. We take a moment here to address the issue of regularity for the transformed equations. Some of the notation and language used in this paragraph is clarified in Chapter 2. For a fixed value of (1, 2), it follows from Theorem 2.10 that the transformed state equation given Theorem 2.2 implies that the in (4.7)-(4.8) has a unique strong solution can only be interpreted as a function trace of the function at the boundary Hence, the boundary data (4.10) of the transformed belonging to the function space sensitivity equation only satisfy the requirements of Theorem 2.12, and the transformed for a sensitivity equation is only guaranteed to possess a weak solution fixed value of the parameter. In this one-dimensional example, the transformed sensitivity is actually a strong solution to the differential equation. In particular, However, in higher dimensions, the appearance of the trace of the gradient of the state within the boundary conditions of the sensitivity equation is an issue that presents not only theoretical but also computational dilemmas. The computational issues are addressed in this as well as in later chapters. First, we describe an alternative approach for obtaining a sensitivity equation. 4.2.2

Abstract Version of the Semianalytic Method

The preceding section developed the transformed sensitivity and its relationship to the original sensitivity under the transformation M. One can also investigate the sensitivity of the transformed state. An advantage of this approach is that the derivation of a sensitivity equation can be mathematically justified, and the details of this justification are presented in Chapter 2. Here, we give the formal derivation of the sensitivity equation. In Chapter 3, we saw that the formal derivation can help guide the construction of the operator framework which mathematically justifies the formal argument given below. This section describes an abstract version of the semianalytical method. The process outlined here differs from the hybrid method in the order in which the sensitivity equation is derived and the transformations are performed. This approach to the computation of the sensitivity is similar in spirit to the semianalytical method (SAM), a technique often used in the engineering community [6]. Roughly speaking, the SAM begins by first transforming the state equation to the computational domain. The second step is to discretize the state equation, thereby producing an algebraic system. This discrete equation is then differentiated to obtain a discrete sensitivity equation, which is solved using special techniques. An abstract version of this method (A-SAM) may be constructed by deriving a sensitivity equation after transforming but before discretizing the state equation. We focus on this approach in order to compare results with the Hybrid method of the previous section. In particular, this approach is applied to the model problem given in section 3.2.1. The first step applies the method of mappings to derive the transformed state equation as given in (4.7)-(4.8). The infinite-dimensional transformed state equation is then differentiated to obtain an equation for the sensitivity of the transformed state. The framework for the rigorous mathematical derivation of the sensitivity equation is given in Chapter 2, and the details for this particular example can be found in section 3.1. In the following, the mechanics of the approach are described and an overview of the method is given. define the sensitivity of the transformed state by On the computational domain, is a The goal is to derive a differential equation for which

32

Chapter 4. Computational Algorithms

solution. Beginning with the transformed state equation

and boundary conditions

we apply the framework given in Chapter 2, and the sensitivity, p( , q), satisfies the differential equation

where is the Dirac delta function with mass at The reader is referred to section 3.1 for the details. can be calculated directly from (4.11). In particular, For this example,

Clearly, the transformed sensitivity is not equal to the sensitivity of the transformed state; that is, Observe that the sensitivity of the transformed state, is less smooth than the transformed sensitivity in (4.12). Furthermore, the relationship between the transformed sensitivity and the sensitivity of the transformed state can be derived using (4.5)-(4.6) and the chain rule. Beginning with the definition of the sensitivity of the transformed state, it follows that

The relationship between s(x, q} and can be obtained by using (4.2) and the definition of the transformation T. Direct computation yields

and the spatial derivative of Observe the appearance of the mesh sensitivitv. in this equation. For this example, can be calculated analytically using algebraic techniques. However, for two-dimensional and three-dimensional problems, the

4.2. SEMs

33

transformations are often constructed using numerical algorithms such as the partial differential equation method discussed earlier. Obtaining derivatives of these maps can be very difficult, and as shown in the following sections, the computation of the spatial derivative can also affect the quality of the sensitivity approximation. Below is an outline of the key steps for this method. A-SAM Step 1. Solve the transformed state equation (4.7)-(4.8) for Step 2. Solve (4.15)-(4.16) for the sensitivity of the transformed state Step 3. Map

back to the physical domain and obtain the sensitivity s(x, q) using

Observe that in order to calculate s(x, q) using the A-SAM approach, one generally needs to compute not only the sensitivity of the transformed state but also the spatial derivative of the transformed state and the mesh derivative As mentioned, the mesh derivative is calculated analytically for this example. Turning to the mathematical issues of existence and regularity of these sensitivities, we note that (4.18) provides an avenue with which to pursue theoretical issues regarding the original sensitivity s(-, •) by means of the sensitivity of the transformed state p(-, •)• That is, we may analyze the existence and regularity of s(-, •) by first examining these issues for p(-, •) and then applying those results to s(-, •) through the equation (4.18). In some situations, such as the current example, this can be advantageous. Recall that the existence of and the rigorous derivation of the corresponding sensitivity equation are discussed in section 3.1. Before moving to the numerical solution of the state and sensitivity equations for each of the methods previously detailed, we first illustrate how the H-SEM approach can be applied to the nonlinear model problem given in section 3.3.1.

4.2.3

Applying the H-SEM to the Nonlinear Model

The discussion in this section provides some insight into applying the H-SEM to more realistic nonlinear problems, such as the Navier-Stokes equations discussed in Chapter 7. Recall that the physical domain for the nonlinear problem given in (3.15)-(3.16) is hence, the transformations defined in section 4.1.2 are used to construct For this problem, we also include an intermediate step for homogenizing the boundary conditions of the transformed state equation. Hence, we define z( ), the function we refer to as the transformed state for this problem, by After applying this transformation, we obtain the transformed state equation defined on the computational domain and given by the Dirichlet problem

34

Chapter 4. Computational Algorithms

with homogeneous Dirichlet boundary conditions

In a similar manner, we transform the sensitivity equation (3.21)-(3.22) to the computational domain. In particular, we let and define by

It follows that the transformed sensitivity

satisfies the equation

with Dirichlet boundary conditions and

As seen in previous sections, the definition in (4.22) requires boundary information from the gradient of the transformed state, and the transformed sensitivity equation (4.23) is weakly coupled to the transformed state equation through the appearance of the terms involving In practice, one must use some numerical scheme to solve the boundary value problem (3.15)-(3.16), and the computation of s(x, q) must be accomplished by using this approximate solution to obtain state gradient approximations. As shown in section 5.3, there are many natural numerical schemes that one can employ in this approach. Although we discuss several schemes, we concentrate on a projection method approach in later chapters. The basic idea can be extended to complex aerodynamic flow problems. However, many theoretical and technical issues are not yet settled. Comment. As noted, the construction of the transformed state equation (4.20)-(4.21) and the transformed sensitivity equation (4.23)-(4.24) requires the derivative (in space) In particular, one needs of the transformation . or else a numencal approximation of This issue is addressed in many computational fluid dynamics (CFD) codes, and there are good methods for dealing with this problem (e.g., see [531). However, there is no need to compute the mesh sensitivity, with this approach. On the other hand, if one follows the approach described by the abstract version of the SAM from section 4.2.2, then the mesh sensitivity, or an approximation, is required to recover the original sensitivity; see (4.18). The numerical approximation of a mesh sensitivity is the source of considerable computational complexity for many realistic CFD applications. Consequently, one advantage of mapping both the state and the sensitivity equation is that the computation of this gradient can be eliminated. are computed on Finally, we note that once and the state and sensitivity can be recovered on through the expressions

4.3. Approximation Framework

35

and

respectively. When applying the inverse transforms to the numerical solutions, numerical errors can be induced. In particular, the map T(x, q) is often constructed numerically for practical CFD problems. Moreover, in (4.26) the presence of the derivative z(l, q) at the boundary can introduce additional errors. These are practical issues that are important to address in more complex problems. Later in this chapter, some of these practical issues are presented, and suggestions are given in section 5.3 for dealing with the issue of numerically approximating state gradient information.

4.3

Approximation Framework

In this section, we outline the finite element methods used for the numerical implementation of the algorithms discussed in the previous sections. Variational formulations for each of the equations are given. Brief descriptions of the finite element spaces, discretization, and grid construction are also included.

4.3.1

Variational Formulations

For the following discussion, note that the underlying function space, V, is defined to be the Sobolev space See Chapter 2 and the references therein for a precise definition and further discussion. Much of the same notation is used for both the linear and nonlinear models. Linear Model We begin by considering the transformed state equation in (4.7)-(4.8). Multiplying by an arbitrary function n V and integrating by parts, we have the following integral equation:

This equation, along with the bilinear form given in section 2.1.3 (with K ( x , q) = 1) and the L2 inner product defined in section 2.1.1, produces the weak form of the transformed state equation. In particular, (4.7)-(4.8) is equivalent to the following variational equation. Find such that

for all n(-) V. See [5] and [54] for further details regarding the equivalence of the variational formulation to the boundary value problem. The H-SEM method requires us to solve the transformed sensitivity equation given in (4.9)-(4.10). Note that the boundary conditions (4.10) are nonhomogeneous. To simplify

36

Chapter 4. Computational Algorithms

notation, we denote the right boundary condition of (4.10) by defined by the function

then

and It follows that differential equation

If

is

satisfies the boundary conditions s*(0) = 0 and belongs to V and solves the

with homogeneous Dirichlet boundary conditions

Now that s*(-, q) has been chosen, the corresponding variational equation is defined in V. Find v(.) V such that

for all n(.) V. Once is computed, the transformed sensitivity is recovered using the relationship Details regarding this technique for nonhomogeneous boundary conditions can be found in [5]. In this particular case of the one-dimensional problem, the solution to (4.32) is the trivial solution, and the preceding construction of s* and v is somewhat unnecessary. However, the definition of allows us to clearly point out a numerical issue inherent to continuous SEMs. With the reader's indulgence, we discuss the issue in some detail in the following sections. For the implementation of the A-SAM method, one must numerically approximate the sensitivity of the transformed state, p(., q). The derivation of the sensitivity equation is discussed in mathematical detail as an example in section 3.1. The sensitivity equation (4.15)-(4.16) is guaranteed to have a weak solution p( , q) (0, 1) for each q (1, 2). That is, (4.15)-(4.16) should be interpreted in the weak sense (see section 2.2.2), and the sensitivity p(., q) is the unique solution to the variational equation

a(., •) is the bilinear form defined in section 2.2.2 with K(X, q) = 1. The bounded linear functional is given by

Nonlinear Model Turning our attention to the nonlinear problem in section 4.2.3, recall that one seeks to approximate the weakly coupled system defined by the transformed state equation

4.3. Approximation Framework

37

and the transformed sensitivity equation

with homogeneous boundary conditions and

respectively. Observe that we have homogenized boundary conditions prior to any variational formulation for this example. To numerically approximate the solutions to (4.35)(4.38), we first obtain the weak formulations using an approach similar to that outlined and exist and belong to in the sections above. Note that for each Then for each V the transformed state z satisfies the variational equation

Likewise, the transformed sensitivity equation satisfies

for all V. We now move to a brief description of the finite element algorithm that is used to obtain numerical results.

4.3.2

Piecewise Linear Finite Elements

In the following sections, we briefly describe the discretization and numerical implementations used to compute state and sensitivity approximations. Linear Model

For the finite element approximation of the variational equations, we begin by constructing the grid. Recall the variational form of the transformed state equation given in (4.28). Note in the computational domain. that the function (., q) is discontinuous at the point Hence, a grid point of the mesh is placed at that point. We partition the domain into Let subintervals where for We the width of the ith subinterval be denoted by choose the finite-dimensional subspace of (0, 1) to be the space spanned by N piecewise linear basis functions denoted

38

Chapter 4. Computational Algorithms

where is the standard continuous piecewise linear basis function. For the computations presented for the linear model, we use the partition of developed above for both state and sensitivity approximations. Hence, the mesh for 7 = 0, 1,..., N is used to calculate approximations to ( , q), v( ), and p( , q). The function v( ) is approximated in the same manner as the transformed state. In particular, we let

and

denote the finite element approximations of ( , q), v( ), and p( , q), respectively. Combining (4.28), (4.32), and (4.33) with the approximations (4.42), (4.43), and (4.44) produces the N x N linear systems of equations

and

respectively. The vectors of unknowns are defined in the logical manner and The matrix K is given by

for i, j = 1,2,... , N, and it is sometimes referred to as the stiffness matrix. The vectors on the right-hand side of the equations are given as follows:

and

4.3. Approximation Framework

39

Once vN( ) has been obtained, the approximation of the transformed sensitivity is computed using

where

is the approximation to

given by

Note that the subscript notation used on refers to the use of the H-SEM approach for the sensitivity calculation. The approximation to the original sensitivity is obtained using the mapping T and the relation Likewise, we use the notation to denote the sensitivity approximation obtained using the abstract version of the SAM method. For this approach, the approximation and the relationship

are used to calculate the sensitivity approximation. Note that the spatial derivative and the sensitivity of the mapping M are hardwired into the relation. Therefore, no error is incurred from the mapping terms when we recover using the expression above. Comment. The use of for introduces an error into that is independent of the error in vN( , q). This error can significantly affect the accuracy of and, eventually, that of . It is also noteworthy that although the weak form in (4.33) does not require spatial information about ( ), the spatial derivative is required to reconstruct through the expression above. These issues play an important role in the numerical results presented in section 5.1.3. Nonlinear Model

Although there are several possible choices for finite element spaces, we limit our discussion to the simplest (convergent) scheme. Recall that the nonlinear model motivates the algorithm we apply to two-dimensional flow problems in later chapters; hence, we note that in more complex problems one must choose these spaces with care to ensure the algorithm satisfies the appropriate convergence criteria (inf-sup conditions, etc.). For this example, we construct a finite element scheme that allows for more flexibility than in the previous linear model. In particular, the weak coupling of the transformed state and transformed sensitivity equations allows one to choose different mesh sizes for each of these numerical calculations. We remark that such an option is certainly viable for the linear model; however, it is not explored in this work. In this section, N denotes the number of basis functions used for the finite element solution of the transformed state equation, and M defines the number of basis functions used for the transformed sensitivity calculation. For each of the numerical calculations, the grid points are equally spaced within the domain

40

Chapter 4. Computational Algorithms

Recall the formulations given in (4.35)-(4.38) and let

and

denote Galerkin approximations of the pair z(.) and u(.). Observe that respectively. This leads us to the variational equations

and

for i = 1, 2,... , N and k = 1, 2,... , M. The equation in (4.53) leads to an N x N system of nonlinear equations that must be solved to determine the vector of unknowns (Z 1 ,Z 2 , . . . , Z N ) T , and we denote this set of equations as

Also note that the coefficients of the unknowns involve the computation of several forms of inner products dictated by (4.53). With a certain amount of algebra and patience, these coefficients can be determined analytically. This was our approach; however, these coefficients can also be generated numerically if one prefers. The nonlinear system given above can be solved using a black box nonlinear solver, or one can generate one's own code as an exercise in numerical methods. The approximations given later in the book were generated using Newton's method with a line search. Although it might seem a bit impervious, the equation in (4.54) yields an N x N linear system of equations. Once ZN is known, then the terms involving can be used to determine the matrix and right-hand-side vectors for this equation. Note that UM (•) depends on ZN (•) and its spatial derivative (•) on To emphasize the dependence, we let u N,M (.) denote the solution of (4.54), given that z N (.) obtained from (4.53) is used in (4.54). At this point, two important observations play a key role in the construction of accurate numerical sensitivities: • The freedom to choose separate finite element spaces for the transformed state z( , q) and the transformed sensitivity u( , q) allows for the development of schemes that

4.3. Approximation Framework

41

simultaneously converge to the state and the sensitivity. In addition, both h-refinement (mesh refinement) and p-refinement (the selection of higher-order elements) can be combined to construct numerical solutions of ZN ( , q) and uN,M ( , q) so that the error in u N,M ( , q) is sufficiently small to ensure convergence of optimal design algorithms based on the SEM (see [7], [9], and [10]). • The solution U N , M ( ) depends not only on zN( ) but also on its derivative . Moreover, since zN( ) is piecewise linear, is a piecewise constant (discontinuous) function. However, the actual transformed sensitivity u( ) is smooth, and one might expect to lose at least one order of accuracy in u N,M ( ). In fact, things can be much worse unless special care is exercised. There are two obvious fixes with which to address these issues. One could use higherorder splines for the sensitivity variable u( ). However, this method will be more expensive, and it is not reasonable to expect great improvements unless higher-order schemes are also used for the state equation. The other obvious fix is to use mesh refinement in M (assuming accuracy in N). A third approach makes use of smoothing projections. The idea is similar to the method used to obtain a posteriori error estimators for adaptive mesh generation (see [11], [30], [57], and [58]). This approach is outlined in section 5.3 and is applied to the model problem and to a two-dimensional fluid flow problem.

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5 Numerical Results

This chapter contains numerical calculations associated with the computational algorithms outlined in Chapter 4. Numerical results are presented for the two SEMs applied to the linear model. This is followed by a section containing numerical approximations for the nonlinear model using the Hybrid method.

5.1

Linear Model

Here we present numerical approximations to w(x, q) and s(x, q) obtained by using the computational methods described in sections 4.2.1 and 4.2.2. All computations presented for the linear model use the same grids for both state and sensitivity approximations. Since the transformed state must be used in the application of both H-SEM and A-S AM, we begin with a brief section reporting state approximations including error calculations. Sensitivity approximations using each of the SEMs are then presented.

5.1.1

State Approximations

It is important to recall that a node is placed at Figure 5.1 shows the finite element approximations to for various values of N; these approximations converge rapidly with grid refinement. Similar behavior is observed over a range of parameter values. The corresponding approximations to , obtained by transforming the finite element approximation, ( , q), back to the physical domain, are shown in Figure 5.2. Comparing Figures 5.1 and 5.2, one can see that convergence of the approximations is preserved under the domain transformation. Figure 5.3 shows the Hl error in WN(X, q) for values of q between 1.1 to 1.9. The values of N range from 3 to 33 and are indicated in the legend. Note that the rate of convergence is better for q 1 as the quadratic term (see (4.11)) in the transformed state becomes less dominant. 43

44

Chapter 5. Numerical Results

Figure 5.1. Finite element approximations to

Figure 5.2. Approximations to w(x, 1.5). 5.1.2

H-SEM

In this section, we present sensitivity calculations obtained by applying the H-SEM algorithm. Figure 5.4 shows the convergence of the finite element approximations to ( , 1.5). Observe that the entire error results from the approximation of the boundary condition Since the transformation M is smooth, the

5.1. Linear Model

45

Figure 5.3. Hl error of WN(X, q)forq ranging from 1.1 to 1.9.

Figure 5.4. Finite element approximations to (x, 1.5). only error in (x, 1.5) is due to this approximation. Figure 5.5 shows the slow convergence of (x, 1.5) to (x, 1.5). Although numerical results are given only for q = 1.5, the error in the approximate boundary condition and the qualitative behavior of the convergence are similar over the entire parameter range. We take a moment to clarify this issue in the following paragraph. Recall that ( ) is obtained using a piecewise linear approximation. Thus, the

46

Chapter 5. Numerical Results

Figure 5.5. H-SEM approximations to s(x, 1.5).

Figure 5.6. Approximation of

with N = 3.

finite element spatial derivative is a piecewise constant function. This function is used to approximate the spatial derivative at the boundary point £ = 1. Figure 5.6 shows a piecewise constant approximation used to obtain an approximate boundary condition . Hence, the error in (l) results in sensitivity errors that can be attributed to the poor approximation of this boundary condition. There are techniques that can be used to obtain better approximations to the spatial derivative along the boundary. Higher-order

5.1. Linear Model

47

Figure 5.7. Finite element approximations to p( , 1.5). elements can be used in the transformed state calculation, but this can be costly for twodimensional and three-dimensional problems. As an alternative, projection techniques have been developed to enhance the accuracy of the spatial derivative for nominal expense. This technique is explored in detail in section 5.3 and the references therein.

5.1.3

A-SAM

We turn our attention to numerical results obtained by using the A-SAM algorithm for sensitivity calculations. First, recall that (x, q) is constructed from and using the relationship.

Note that the subscript A is used to denote the sensitivity approximation obtained using the A-SAM approach. The finite element approximations to p( , 1.5) are shown in Figure 5.7 for various values of N. Since the sensitivity equation (4.33) is linear, the approximations pN( , 1.5) converge as expected. When constructing (x, 1.5) from (5.1), the piecewise constant approximation of produces discontinuities in , as shown in Figure 5.8. These discontinuities occur at points of the physical domain that correspond to mesh nodes of the computational domain lying in the interval . Note that the expressions for the mesh derivatives in (5.1) are hardwired, continuous functions, and the finite element approximations to p( , q) are continuous. It follows that as . However, one does not get convergence in the energy norm since does not belong to . Even if one computes the Hl error of the sensitivity approximation over each individual element of the mesh and computes the total error by

48

Chapter 5. Numerical Results

Figure 5.8. A-SAM approximations to s(x, 1.5).

Figure 5.9. Hl errors for sensitivity calculations. summing these local errors, we see that this error tends to a nonzero constant with mesh refinement; see Figure 5.9. The numerical results presented in the previous sections indicate that each SEM implemented here suffers from computational difficulties. Both algorithms require accurate gradient information from the state approximation to accurately approximate the sensitiv-

5.2. Nonlinear Model

49

ity. The H-SEM method requires accurate gradient information only along the boundary, whereas the A-SAM approach relies on accurate gradient approximations over the entire spatial domain. We also see that the use of mapping techniques in conjunction with SEMs must be done with care, and the mathematical analysis of the sensitivity equations can be useful in determining approximation errors related to the choice of numerical scheme. Moreover, the choice of computational scheme may also depend on the needs of the designer. The hybrid method provides sensitivity approximations that converge in the energy norm, and this issue may be important to a designer who is using the sensitivity approximations to analyze the influence of the design parameter on the underlying mathematical model. However, if the designer is approximating sensitivities for use in an optimization algorithm, then the L2 accuracy of the sensitivity approximations may be sufficient.

5.2

Nonlinear Model

This section presents numerical approximations for the nonlinear model problem using the H-SEM. We also evaluate the effect of accurate sensitivity approximations within the context of an optimization algorithm constructed to solve the inverse design problem outlined in section 3.3.1. 5.2.1

Convergence of the State and the Sensitivity

In this section, we compare the finite element approximations of the solutions to the state and sensitivity equations with their exact, or true, solutions. All the figures shown in this section depict the approximation to the state w and the sensitivity s obtained using the relations in (4.25) and (4.26) along with the approximations ZN and UN'M that satisfy (4.53) and (4.54), respectively. First, we note that the finite element scheme converges to the exact solution of the nonlinear state equation (for each q > 1). Figure 5.10 shows the finite element approximations to the solution of the boundary value problem at two parameter values: q = 2 and q = 1.2. Notice that at q = 2, the N = 4 finite element model provides an excellent match to the exact solution. However, when q = 1.2 one sees that a finer mesh (N = 8) is required to obtain the same order of accuracy. This convergence is expected because the gradient of the solution becomes singular as q —> 1+ and hence the problem becomes stiff in this parameter region. This is also the case for the sensitivity equation. Consider the corresponding finite element approximations of the sensitivity equation. Recall that N and M define the meshes for the transformed state and transformed sensitivity equations, respectively, and (4.54) is coupled to (4.53) through the appearance of the spatial derivative in (4.54). Our first approach to dealing with this term is to simply use the piecewise constant spatial derivative of the finite element approximation for ZN . The use of the piecewise constant (PWC) derivative is noted explicitly in the following figures in order to distinguish them from other numerical results presented in section 5.3. Figure 5.11 shows the finite element approximations for the sensitivity s(x, 2) with N = M ranging from 2 to 16. Observe in Figures 5.10,5.11, and 5.12 that although the finite element scheme produces excellent solutions to the state equation when N = 4, the error in the corresponding sensitivity does not diminish to a comparable level until N = M = 16. It should be noted for this example that any error in the state approximation propagates into errors in

50

Chapter 5. Numerical Results

Figure 5.10. Numerical approximations to the solution of the nonlinear model at q = 2 and q = 1.2.

Figure 5.11. Numerical approximations to the solution of the sensitivity equation at q = 2 using PWC derivatives.

5.2. Nonlinear Model

51

Figure 5.12. L2 error of the state and sensitivity approximations at q = 2 with N = M. the corresponding sensitivity approximations. Hence, the sensitivity approximations for a given grid are always less accurate than the state approximations computed on the same grid. This is a numerical issue that is inherent to the SEM approach, and Figure 5.12 provides a visualization of this phenomenon for the nonlinear example. At this stage, the reader may wonder if there are ways to improve the accuracy of the sensitivity approximations for a given amount of error in the state approximation. One possible fix is to use mesh refinement in M. Figure 5.13 shows the results for this technique when N = 2 and M ranges from 2 to 16. Note that improvements in the accuracy of the sensitivity approximation are limited by the accuracy of the state approximation and, more precisely, the accuracy of the piecewise constant derivative approximation for When N = 4 is used, the errors in the sensitivity approximations decreased significantly, especially for the coarse grid sizes M = 4 and M — 8 (see Figure 5.14). The stiffness of the problem near q = 1 increases the difficulty of computing accurate sensitivity approximations. Figures 5.15 and 5.16 show that the sensitivity approximations become unreliable as q —> 1+ although the state approximations are quite reasonable. Figure 5.17 displays both the L2 error in the state (flow) approximation and the L2 error in the sensitivity approximation for a range of mesh sizes. As N = M increases, we obtain convergence of the scheme, but for small values of N, the sensitivity approximations contain large errors and the convergence of the finite element approximation to the analytical solution is not at all monotone; see Figure 5.16. Figure 5.18 is a graph of the L2 error of the sensitivity approximations for various values of q, and we can see that the behavior of the error calculations discussed above is observed for various parameter values near q = 1. Again, one observes that a considerable amount of grid refinement is required in order to obtain accurate sensitivity approximations for parameter values near q = 1. We observe similar behavior when the H-SEM is applied to two-dimensional flow problems in Chapter 7.

52

Chapter 5. Numerical Results

Figure 5.13. Numerical approximations to the solution of the sensitivity equation at q = 2 using PWC derivatives with N = 2 and mesh refinement in M.

5.2.2

Sensitivities in Optimal Design

We now return to the inverse design problem presented in section 3.3.1 and evaluate the convergence properties of an optimization scheme using the H-SEM. After discretization, the infinite-dimensional inverse problem (3.18) becomes as follows. Find q* > 1 such that

where WN(X, q) is obtained using (4.25). Notice that the gradient has the form

It is important for the reader to observe that the H-SEM applied to the optimization problem replaces , the sensitivity of the discrete solution, with an approximation to , in particular , from (4.26).

5.2. Nonlinear Model

53

Figure 5.14. Numerical approximations to the solution of the sensitivity equation at q = 2 using PWC derivatives with N = 4 and mesh refinement in M.

Figure 5.15. Numerical approximations to the solution of the sensitivity equation atq = 1.4 using PWC derivatives.

54

Chapter 5. Numerical Results

Figure 5.16. Numerical approximations to the solution of the sensitivity equation at q = 1.2 using PWC derivatives.

Figure 5.17. L2 error of the solution and sensitivity approximations atq = 1.2.

5.2. Nonlinear Model

55

Figure 5.18. L2 error of sensitivity approximations using PWC derivatives.

Figure 5.19. Gauss-Newton algorithm.

56

Chapter 5. Numerical Results

Figure 5.20. Data generated at q = 2.

Figure 5.21. Data generated at q = 1.4.

5.2. Nonlinear Model

57

Table 5.1. Matching data for the optimization. p =4 q=2

0.1250 0.2500 0.5000 0.7500

0.4495 0.8284 1.4641 2.0000

0.4516 0.7722 1.4080 1.8338

Perturbation 0.0021 0.8248 1.3541 -0.0563 2.1394 -0.0561 -0.1662 2.7553 p = 16

q =2

0.0312 0.0625 0.0938 0.1250 0.1562 0.1875 0.2188 0.2500 0.3125 0.3750 0.4375 0.5000 0.5625 0.6250 0.6875 0.7500

0.1213 0.2361 0.3452 0.4495 0.5495 0.6458 0.7386 0.8284 1.0000 1.1623 1.3166 1.4641 1.6056 1.7417 1.8730 2.0000

0.1126 0.2207 0.3670 0.4372 0.5916 0.6611 0.7651 0.8687 0.9091 1.2674 1.2486 1.6086 1.6483 1.9067 1.9668 2.0554

q = 1.4 Perturbation -0.0817 0.7431 1.3875 0.0335 1.9848 -0.1546 2.5758 -0.1795 q = 1.4

Perturbation -0.0088 -0.0154 0.0218 -0.0123 0.0421 0.0154 0.0265 0.0403 -0.0909 0.1051 -0.0680 0.1445 0.0427 0.1651 0.0938 0.0554

0.2677 0.4806 0.6629 0.8248 0.9720 1.1079 1.2347 1.3541 1.5749 1.7768 1.9641 2.1394 2.3048 2.4619 2.6117 2.7553

0.2690 0.4480 0.6375 0.7563 0.9772 1.0335 1.2820 1.4106 1.4821 1.6265 1.9460 2.0504 2.0950 2.3973 2.8040 2.9296

Perturbation 0.0013 -0.0326 -0.0254 -0.0685 0.0052 -0.0744 0.0473 0.0565 -0.0928 -0.1503 -0.0181 -0.0890 -0.2098 -0.0646 0.1922 0.1743

58

Chapter 5. Numerical Results

Figure 5.22. The cost function and its approximations for p = 16 and q* ~ 2.

Figure 5.23. The cost function and its approximations for p = 16 and q* ~ 1.4.

5.2. Nonlinear Model

59

Table 5.2. Optimization results for p = 16, qopt ~ 2, qinit = 1.2, PWC derivatives.

N 2 2 4 4 8 8 16 16 32 32 64 64 128

M 2 5 4 9 8 17 16 33 32 65 64 129 128

CONV/DNC

Iterations

DNC

20 17 20 15 20 11 20 12 20 20 20 12 10

CONV

DNC CONV

DNC CONV

DNC CONV

DNC CONV

DNC CONV CONV

Time 31.4

3.1 68.3

6.6

143.1

7.4

334.7 16.0 916.6 42.1 4323.1 191.9 252.2

2.4088 0.32610 2.6252 0.26469 3.1942 0.24332 3.0308 0.24235 3.1796 0.24278 0.24276 0.242783 0.24285

q

1.21993 1.94862 1.15545 1.94795 1.08641 1.94328 1.16909 1.93902 1.18975 1.93771 1.93477 1.93747 1.93584

Table 5.3. Optimization results for p = 16, qopt ~ 1.4, qinit = 2.0, PWC derivatives.

N 2 2 4 4 8 8 16 16 32 32 64 64 128

M 2 5 4 9 8 17 16 33 32 65 64 129 128

CONV/DNC

Iterations

Time

DNC DNC DNC DNC DNC DNC DNC

20 20 20 20 20 20 20 18 11 16 13 17 12

0.6

CONV CONV CONV CONV CONV CONV

32.5 63.3 68.4 134.9 61.3 293.2 18.8

6.8 13.2 33.7 58.0 192.7

0.86393 0.72436 0.54447 0.52754 0.60045 0.50601 0.78813 0.41994 0.41001 0.41046 0.40820 0.40858 0.40774

q

1.83807 1.33907 1.48191 1.4419 1.38030 1.47419 1.33031 1.43638 1.41986 1.42342 1.41681 1.42048 1.41605

The standard Gauss-Newton algorithm is used to approximate q*; see [18] for details. The algorithm solves a least-squares problem at each iteration and proceeds as described in Figure 5.19. The data to be matched, denoted by in (5.2), are indicated by pluses in Figures 5.20 and 5.21. This data set was generated by randomly perturbing the value of w( ) in (3.17) using q = 2 and q — 1.4 with p = 4,16 data points. Table 5.1 shows the numerical values of the data for comparison purposes. The exact cost functional, J(q), and several approximations to it, JN(q), are plotted

60

Chapter 5. Numerical Results

in Figures 5.22 and 5.23 for the case where p = 16 and q* ~ 2 and q* ~ 1.4, respectively. Although several simulations were constructed with various data sets, we present the results of two of these: • Case 1. The true state was calculated for q = 2.0 and the noise vector in Table 5.1 was added to obtain data for optimization. Here, the optimal value of the design parameter is approximately q* ~ 2. The optimization algorithm was started at qinit = 1.2. • Case 2. In this case, the true state was calculated for q = 1.4 and the noise vector in Table 5.1 was again added to obtain data for optimization. Here, the optimal value of the design parameter is approximately q* ~ 1.4. Here, the optimization algorithm was started at qinit = 2.0. The scheme was considered converged when the norm of the gradient of the cost functional was less than 10~7. Notice that the simulations were performed using sensitivities calculated using the natural piecewise constant finite element gradient approximations. Tables 5.2 and 5.3 show the results of these simulations as N, M ranged from 2 to 128. The runs were measured in seconds and were performed on a Silicon Graphics Onyx2. Notice the effect of the inaccurate sensitivity approximations on the convergence of the optimization scheme for Case 1 when N = M. As expected, larger values of N, M were required for convergence of the optimization algorithm for Case 2. In the following section, we return to this problem and use smoothing projections to enhance sensitivity computations and convergence.

5.3

State Gradient Approximations

The goal of this section is to describe two projection techniques that can be used to improve the accuracy of sensitivity approximations by obtaining better state gradient approximations. In the context of the nonlinear model, we seek to improve the numerical approximations of • In the previous section, the gradient approximations computed using the finite element derivative were discontinuous across element faces. Here, we analyze global and local projection techniques for calculating continuous gradient approximations and evaluate their impact on sensitivity approximations in the context of the nonlinear model problem. The local projection technique is one that is used for obtaining a posteriori error estimates in adaptive finite element codes. The local projection technique is also discussed in Chapter 7. Recall that the variational form of the approximate transformed sensitivity equation (4.54) for the nonlinear model problem has the form

where 7 = 1, 2 , . . . , M and

5.3. State Gradient Approximations

61

Here, gN ( ) is a piecewise step function providing an approximation of the spatial derivative needed in (5.4). The goal is to replace gN( ) with a continuous function, denoted by gN( ), which is a more accurate approximation to true state gradient. The following sections describe global and local projection schemes for constructing this function. Remark. Even in higher dimensions, the finite element derivatives are often discontinuous across element faces. It is possible to construct higher-order basis functions that provide continuous derivatives across element faces. However, from the standpoint of CFD, this construction also must satisfy the inf-sup conditions in order to guarantee convergence, and the computational cost of using higher-order basis functions often outweighs the benefits of improved accuracy. The projection techniques surveyed here provide the scientist with the convenience of low-order basis functions along with improved accuracy for a relatively little computational work.

5.3.1

A Global Projection Scheme

In its simplest form, this approach replaces the discontinuous piecewise constant function gN( ) by its projection onto the space of piecewise linear splines on the mesh defined by the nodes . Thus, we consider the space

where gi,

and (•) are the "hat functions" defined in section 4.3. Observe that and contains functions with nonzero trace on the boundary of [0, 1]. Define gN( ) to be the orthogonal projection of gN( ) onto SN. In particular,

where and mation of zN ( ). Since

is the solution of a linear system of the form contains the coefficients defined by the finite element approxi-

we have MG is the (N + 2) x (N + 2) global mass matrix and F is the (N + 2) x N matrix

for i = 1, 2 , . . . , N and j, k = 0, 1, 2,... , N + 1.

62

5.3.2

Chapter 5. Numerical Results

A Local Projection Scheme

In addition to the global projection scheme, we consider a local projection scheme, which involves performing a series of local projections on subdomains of = [0,1]. At each element vertex, , we define the subdomain , to be the union of all elements for which is a vertex. In , define to be the least-squares projection of gN( ) | onto the space of linear polynomials spanned by monomial basis functions. For the nonlinear example, the basis functions are P1 ( ) = 1 and • On the subdomain , we express the projection as

where the vector a = (a 1 ,a 2 ) T contains the coefficients of the basis functions. These coefficients are determined by solving the normal equation system

The matrix ML is of the form

for i, j = 1,2. The vector on the right side of the equation is

for i = l,2. Then, on

we define the continuous local projection to be

With higher-order finite elements and in higher dimensions, one must resolve the value of at a nonvertex node. An averaging technique is generally used. The local projection technique is described in more detail in Chapter 8.

5.3.3

Numerical Results

The following sections address some of the numerical issues associated with the projection techniques. Examples of state gradient approximations are given using the piecewise continuous finite element derivatives, the local projections as well as the global projections. We also explore how each of the state gradient approximations affects the accuracy of the corresponding sensitivity approximations. A brief section exploring the effects of using these projection techniques within an optimization algorithm is included.

5.3. State Gradient Approximations

63

Figure 5.24. Finite element derivatives with projections at N = 4 and q = 2. Derivative Approximations

Figure 5.24 shows the exact spatial derivative along with the finite element derivative and its local and global projections for the case when q = 2.0 and N = 4. Figure 5.25 shows a similar set of functions and approximations for the case when q = 1.2 and N = 8. Clearly, the global and local projections give different results. Since we have an exact solution for the nonlinear example, and thus its spatial derivative, we can calculate element by element L2 errors as well as overall L2 errors over the entire domain for each derivative approximation. Figures 5.26 and 5.27 show the L2 element errors for the two cases q = 2.0, N = 4 and q = 1.2, N = 8, respectively. The L2 errors over the spatial domain for each of these cases are summarized in Table 5.4. Figures 5.26 and 5.27 demonstrate that the errors for the local projection technique are highest near the boundary, but away from the boundary the local projection technique actually has less error than the global projection technique for these two cases. We now analyze these techniques in calculating sensitivities for varying discretizations (N, M) and parameter values (q) to gain a better understanding of how the improved derivative approximations affect our numerical sensitivities. Sensitivity Approximations

Figures 5.28,5.29, and 5.30 display the sensitivity approximations using the three different state gradient approximations against the exact sensitivities for q — 2.0, 1.4, and 1.2, respectively (with varying mesh sizes). It is clear that the use of a projection technique greatly improves the accuracy of our sensitivity approximations, especially as q —> 1. Recall that at q = 1.4 and q = 1.2 our sensitivity approximations obtained using PWC derivatives were extremely inaccurate so that the approximations do not even show up on

64

Chapter 5. Numerical Results

Figure 5.25. Finite element derivatives with projections at N = 8 and q = 1.2.

Figure 5.26. L2 error on each element for N = 4 and q = 2.

5.3. State Gradient Approximations

65

Figure 5.27. L2 error on each element for N = 8 and q = 1.2. Table 5.4. L2 errors for the derivative approximations.

PWC Global Local

L2 errors N = 4 and q = 2 W = 8 and q = 1.2 1.1432 0.4330 0.2165 0.8760 0.2954 1.0501

the graphs in Figures 5.29 and 5.30 for some values of N. When compared with the original discontinuous state gradient approximations, the local projections clearly decrease the L2 error of the sensitivities for a given N and q, as shown in Figure 5.31. Moreover, the most promising result is that the projections stabilize the calculations over the parameter range. The local projections also give slightly more accurate sensitivity approximations for some values of q when compared to the global projection technique. Figure 5.32 compares the L2 error of sensitivities calculated using local projections with the error calculated using global projections. Note that as N increases, the global projections do a better job than local projections over a wider range of q. This is not surprising since the global projection is the "best" least-squares linear approximation (using the finite element basis) of the piecewise continuous finite element derivative in the limit. Next, we briefly examine the effect that the improved sensitivity approximations have on the optimization problem considered in section 5.2.2.

66

Chapter 5. Numerical Results

Figure 5.28. Sensitivity approximations at q = 2.

Figure 5.29. Sensitivity approximations at q = 1.4.

5.3. State Gradient Approximations

67

Figure 5.30. Sensitivity approximations at q = l .2.

Figure 5.31. Model problem—L2 error of sensitivity approximations (PWC and local).

68

Chapter 5. Numerical Results

Figure 5.32. Model problem—L2 error of sensitivity approximations (global and local).

Table 5.5. Optimization results for Case \, global projection scheme.

N 2 2 4 4 8 8 16 16 32 32 64 64 128

M 2 5 4 9 8 17 16 33 32 65 64 129 128

CONV/DNC CONV CONV CONV CONV CONV CONV CONV CONV CONV CONV CONV CONV CONV

Iter 20 16 15 15 12 13 13 13 12 12 12 11 11

Time 9.43 9.06 17.80 12.91 27.01 25.84 27.95 31.62 77.32 78.90 288.06 302.73 355.10

.32586 .32548 .26463 .26468 .24327 .24326 .24235 .24235 .24272 .24273 .24274 .24274 .24285

q

1.9604 1.9514 1.9509 1.9481 1.9423 1.9421 1.9375 1.9376 1.9361 1.9361 1.9359 1.9359 1.9358

5.3. State Gradient Approximations

69

Table 5.6. Optimization results for Case I, local projection scheme.

N 2 2 4 4 8 8 16 16 32 32 64 64 128

M 2 5 4 9 8 17 16 33 32 65 64 129 128

CONV/DNC CONV CONV CONV CONV CONV CONV CONV CONV CONV CONV CONV CONV CONV

Iter 20 16 15 15 12 13 13 13 12 12 12 11 11

Time 9.43 9.06 17.80 12.91 27.01 25.84 27.95 31.62 77.32 78.90 288.06 302.73 355.10

.32586 .32548 .26463 .26468 .24327 .24326 .24235 .24235 .24272 .24273 .24274 .24274 .24285

q

1.9604 1.9514 1.9509 1.9481 1.9423 1.9421 1.9375 1.9376 1.9361 1.9361 1.9359 1.9359 1.9358

Table 5.7. Optimization results for Case 2, global projection scheme.

N 2 2 4 4 8 8 16 16 32 32 64 64 128

M 2 5 4 9 8 17 16 33 32 65 64 129 128

CONV/DNC DNC DNC DNC DNC DNC DNC CONV CONV CONV CONV CONV CONV CONV

Iterations 20 20 20 20 20 20 14 15 15 11 13 14 13

Time 2.06 64.37 133.51 132.27 127.18 143.56 28.69 33.07 13.22 12.75 45.43 48.74 258.93

.93423 .72262 .53248 .52703 .48242 .50490 .41967 .41973 .41001 .41001 .40819 .40819 .40774

q

2.3054 1.3530 1.4557 1.4405 1.5029 1.4791 1.4329 1.4322 1.4197 1.4196 1.4168 1.4167 1.4160

70

Chapter 5. Numerical Results

Table 5.8. Optimizations results for Case 2, local projection scheme.

N 2 2 4 4 8 8 16 16 32 32 64 64 128

M 2 5 4 9 8 17 16 33 32 65 64 129 128

CONV/DNC DNC DNC DNC DNC DNC DNC CONV CONV CONV CONV CONV CONV CONV

Iterations 20 20 20 20 20 20 14 16 11 15 13 12 12

Time 60.82 64.60 130.65 131.56 133.27 133.92 30.00 32.03 11.21 14.34 44.45 45.70 253.86

.73574 .72273 .53216 .52991 .50451 .50470 .41968 .41974 .41000 .41001 .40820 .40819 .40774

q

1.4154 1.3634 1.4558 1.4513 1.4830 1.4807 1.4328 1.4322 1.4197 1.4196 1.4168 1.4168 1.4160

Optimization Results

We evaluated the same two cases considered in section 5.2.2: qinit = 1.2 with q* ~ 2 (Case 1) and qinit = 2.0 with q* ~ 1.4 (Case 2). This time the simulations were performed using the piecewise linear derivative approximations. Tables 5.5-5.8 show the results of these simulations as N, M range from 2 to 128. Note that the use of the piecewise linear derivative approximations clearly improve the results of the optimization algorithm for Case 1 and that the global and local projection schemes provided very similar results. The improvement was not as marked for Case 2; however, the scheme did converge for the case N — M = 16 with the improved gradient approximations and did not with the piecewise constant derivatives. In addition, the local scheme required slightly less time to converge than the global projection scheme (see Tables 5.7 and 5.8).

6 Mathematical Framework for Navier-Stokes Equations

This chapter presents the mathematical framework used to assert the existence of the sensitivity for Navier-Stokes equations. The necessary preliminaries, such as function spaces and notation, are given, and the Dirichlet problem is treated in detail. We use, throughout, the following nondimensional form of the Navier-Stokes equations for the two-dimensional steady state flow of an incompressible, viscous fluid in a bounded domain with a Lipschitz-continuous boundary :

Recall that v = where Re is the Reynolds number, for the nondimensional form of the equations. See [44] and [51] for a more detailed derivation of the Navier-Stokes equations.

6.1

The Homogeneous Dirichlet Problem

We begin by considering the case of the homogeneous Dirichlet boundary condition

To discuss existence and uniqueness and to introduce the variational form of the problem, we introduce some standard function spaces as well as some necessary bilinear and trilinear forms.

6.1.1

Function Spaces and Notation

In addition to the function spaces and tools given in Chapter 2, we use the following notation:

Recall that the trace theorem proves the existence of a trace operator, and, as noted in [54], the range of the trace operator is . The norm for functions g belonging to 71

72

Chapter 6. Mathematical Framework for Navier-Stokes Equations

can be defined by

The vector-valued counterparts of these spaces in

are denoted by boldface symbols,

i.e.,

The norm for H1 ( ) is defined by

The divergence free subspace of

, Z0, is given by

We define the following bilinear form:

where

Also, let

and

6.1.2

Existence and Uniqueness of Solutions to the Variational Form

We present some basic results regarding the existence and uniqueness of solutions to the variational form of the homogeneous Navier-Stokes problem as presented in [21]. To begin, we note that the homogeneous partial differential equation (6.1)-(6.2) can be written in variational form as follows. Variational Problem 6.1. Given

find a pair

such that

6.2. The Nonhomogeneous Dirichlet Problem Recall that the trilinear form, following lemma. Lemma 6.1. Let the trilinear form

73

has some nice properties described in the

and let is continuous on

with and and satisfies

Then,

Now, the existence result from [21] follows. Theorem 6.2. For satisfies (6.3).

there exists at least one pair

that

In order to discuss the uniqueness of the solutions (u, p) to (6.3), we introduce the norm of the trilinear form a1 ( • ; - , - ) , denoted , and defined by

We also set

Theorem 6.3. If

and

then Variational Problem 6.1 has a unique solution

6.2

The Nonhomogeneous Dirichlet Problem

Now consider the more general case of a nonhomogeneous Dirichlet boundary condition

the connected components of the boundary Denote by Figure 6.1. We henceforth assume that

as depicted in

The variational form of the nonhomogeneous partial differential equation (6.1) with (6.9) is obtained using standard weak formulations. In particular, we consider the following problem.

74

Chapter 6. Mathematical Framework for Navier-Stokes Equations

Figure 6.1. Sample domain with boundaries.

Variational Problem 6.2. Given that

find a pair

such

To prove existence for the nonhomogeneous problem, we need the following technical result, due to Hopf (see Lemma 2.3 in [21]). Lemma 6.4. Let such that

satisfy (6.10). For any

there exists a function

The following existence theorem may be found in [21]. Theorem 6.5. Let pair

and satisfying (6.10). There exists at least one which is a solution of (6.11).

Before stating the uniqueness result again, we make a few definitions. For any function

6.3. An Abstract Framework for Navier-Stokes

75

, define

and

where

Define

as in

by

The basic uniqueness result for (6.11) is found in [21]. We state it below for convenience. Theorem 6.6. Assume the hypothesis of Theorem 6.5. If Problem 6.2 has a unique solution

then Variational

Remark. Although the results above provide existence and uniqueness for the basic NavierStokes problems, they do not address the continuity and differentiability of these solutions with respect to the parameter q. For example, in the problems considered below we have , and/or g = g(q) so that v0 = = v0( ), where q is some parameter defining the flow. We need to establish the smoothness of these mappings to address the existence and uniqueness of solutions to the sensitivity equations. This is the subject of the following sections.

6.3

An Abstract Framework for Navier-Stokes

In this section we present an abstract framework for analyzing the dependence on q of solutions to the nonhomogeneous Dirichlet problem for the Navier-Stokes equations. We show the continuity of solutions with respect to parameters for two specific cases, and we conclude with results about the differentiability of those solutions. We extend the framework in [21] to certain parameter-dependent flows. 6.3.1

The Framework

Let X and X be two Banach spaces and Given a Cp mapping (P 1)

, where

is open and A is compact.

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Chapter 6. Mathematical Framework for Navier-Stokes Equations

we are interested in solutions to the state equation

Let {(q, u(q)}; q

] be a. branch of solutions of (6.19). This means that

Moreover, we suppose that these solutions are nonsingular in the sense that

As an immediate consequence of (6.22), it follows from the implicit function theorem (see, e.g., [56]) that q u(q) is a Cp function from A into X. 6.3.2

Using the Framework

We now show that the parameter-dependent Dirichlet problem for the Navier-Stokes equations in the velocity-pressure formulation (6.1) with (6.9) fits into this abstract framework. We assume that any or all of the following hold:

where

is a design parameter for the flow. Define

and the intermediate space

where as follows: Given we denote by Dirichlet problem for the Stokes equations:

In addition, let P : Q

Y be defined by

and the nonlinear operator

be given by

Next we define a linear operator T the solution of the

6.3. An Abstract Framework for Navier-Stokes as before. where v is the constant Now finally, with the data Y defined by

77

we associate a C1 mapping G from Q x X into

and we set

The following result follows directly from Lemma 3.1 in [21]. Lemma 6.7. The pair is a solution of (6.1)-(6.9) if and only if(q, u(q)) is a solution of (6.19), where w(q) = (u(q), p(q)/v) and where the spaces X and X are defined by (6.25) and the compound mapping, F, is defined by (6.31).

6.3.3

Continuity of Solutions with Respect to Data

We now address the continuity of solutions (u(q), p(q)) to (6.1) with (6.9) with respect to changes in the parameter q. We assume the map is continuously Frechet differentiable. Note, that this is certainly true for the cylinder problem presented in section 7.1. To begin, we need Lemma 1.3.2 from [21]. Lemma 6.8. There exists a continuous linear function D : V g V, we have that w = D(g) satisfies

H1 ( ) such that for each

The following corollary is a direct consequence of Lemma 6.8. Corollary 6.9. The map from

\ is Frechet differentiable.

To analyze the parameter-dependent solution to the weak form of the nonhomogeneous Navier-Stokes problem, we need a result analogous to Lemma 6.4 for parameter-dependent boundary functions. The following result may be established by a straightforward extension of Lemma 2.3 in [21], Lemma 6.10. There exists a continuous linear function for each

where U 0 (q) is defined by Lemma 6.4 and satisfies

such that

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Chapter 6. Mathematical Framework for Navier-Stokes Equations

and

for each We now consider continuity with respect to the right-hand-side function f. We again consider an abstract framework for the Navier-Stokes equations. We define a map : X x Z. Here, X is the set of forcing functions, f H-1( ), and Y is defined as follows:

The range of

and

is contained in

is defined by

Note that given by

and the Frechet derivative at a point

is

It is also clear that (u, p) satisfies the homogeneous Navier-Stokes equations, (6.1) and (6.2), with right-hand-side f if and only The implicit function theorem implies that (u, p) is a continuously differentiable function of f if the linear map is an isomorphism. But this is equivalent to the condition that the homogeneous sensitivity equation has a unique solution in y for each f H-1( ). We show in the following that the sensitivity equation does indeed have a unique solution in y so that we have (u, p) is a C1 function of f. We then return to the smoothness of solutions to the parameter-dependent Navier—Stokes equations.

6.4

Analysis of the Sensitivity Equations

We begin by stating a general form of the sensitivity equations for the parameter-dependent Navier-Stokes equations. We show that if we have a unique solution (u, p) of the NavierStokes problem, then there exists a unique solution (s, r) of our sensitivity equation. Lastly, we use an abstract formulation of the Navier-Stokes problem and the implicit function theorem to show that the solution (u, p) is in fact a nonsingular solution of the NavierStokes problem.

6.4.1

A General Formulation of the Sensitivity Equations

reoresent some fixed sensitivity oarameter Let sensitivity forcing function, by and the boundary function,

Now denote the Lastly, by

6.4. Analysis of the Sensitivity Equations

79

recall that in some cases . As in the case of the one-dimensional model problem, the sensitivity equations are obtained by implicitly differentiating the flow equations and their associated boundary conditions. The sensitivity equations fit into the following general form. Given and satisfying (6.10) and (u, p) a solution of (6.1) with (6.9), find a pair (s, r) H1 x such that (s, r) satisfies

We can write the variational form of (6.40) as follows. Variational Problem 6.3. Given (u, p) a solution (6.1)-(6.9), find a pair (s, r)

and x

satisfying (6.10) and such that

for all z 6.4.2

Existence and Uniqueness of Solutions to the Sensitivity Equations

We state and prove the following existence and uniqueness result following [21]. Theorem 6.11. Assume the hypotheses of Theorem 6.6. If(u, p) is the unique solution of Variational Problem 6.2, then there exists a unique solution, (s, r), to Variational Problem 6.3. Proof. To check that the equations (6.41) have a unique solution, it is sufficient to prove that the bilinear form

is V-elliptic. But it follows from (6.5) that

Now, assume that v > V0, where V0 is defined by (6.17). Then, there exists a function such that

Setting u = U0 + w, we have by (6.6) and (6.14)

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Chapter 6. Mathematical Framework for Navier-Stokes Equations

Since

we obtain

so that the ellipticity property holds.

6.5

Differentiability of Solutions with Respect to q

We return now to the smoothness of solutions to the parameter-dependent Navier-Stokes equations. Note that the parameter V0 = v0(q) is a continuous function of q in this case since the map from q U0 is continuous and the map from U0 p is continuous. With this, the continuity of the map (f, g) (u, p), and the fact that q (f (q), g(q)) is C1, we can now establish the following result. Theorem 6.12. Assume that q is such that v > vo . There exists a neighborhood of such that for all q the solution of the variational, parameterdependent Navier-Stokes equations (6.11), with f = f(q) and g = g(q), exists and is unique. Moreover, the solution is a Cl function of q. Proof. We consider the map F defined by (6.31). We have shown that there exists Rn so that we have a branch of solutions {(q, u(q)); q }, i.e., that both (6.20) and (6.21) hold for q , for both cases presented in section 6.3.3. We now turn our attention to the Frechet derivative of F, Du F. We have

Again, the fact that DuF is an isomorphism can be shown to be equivalent to the fact that the homogeneous sensitivity equation has a unique solution for all q . We showed in section 6.4.2 that the sensitivity equation does indeed have a unique solution. Then, by the implicit function theorem, we have that q (u(q), p(q)) is a C1 function from into X.

7Two-DimensionalFlow Problems

We now return to analyzing design sensitivities, and we focus on two specific parameterdependent flow problems in CFD. We present flow equations for these problems and use formal implicit differentiation to derive the continuous sensitivity equations. The reader should observe that the coupling of the sensitivity equations to the state equations through the appearance of state and state gradient terms can be seen in both examples. Hence, many of the numerical issues addressed for the nonlinear model of section 3.3 are also addessedfor the examples introduced in this chapter. An adaptive finite element technique is described and used to solve the state and sensitivity equations. The numerical results are presented to investigate the convergence of the adaptive grids.

7.1

Flow around a Cylinder

We begin by considering the standard problem of two-dimensional flow around a cylinder. This problem, a nonhomogeneous Dirichlet problem on a bounded domain, is modeled on the problem in which the boundary is an infinite strip. We assume parabolic inflow into a channel containing a cylindrical obstruction whose geometry is shown in Figure 7.1. Here, = [—2, L] x [—1, 1], where L R+ is a fixed number. We assume that L is large enough for the outflow to have returned to the same parabolic velocity profile present at the inflow. The governing equations are the two-dimensional incompressible Navier-Stokes equations presented earlier (see (6.1)) with f = 0. The boundary conditions at the inflow and outflow are given by

for — 1 y 1, where q R+ is a parameter describing the strength of the inflow. Nopenetration and no-slip conditions are applied on the top, bottom, and cylinder sidewalls (i.e., u = 0). We are interested in calculating sensitivities with respect to the inflow parameter q. As before with the case of the one-dimensional model problems, define the sensitivity as 81

82

Chapter 7. Two-Dimensional Flow Problems

Figure 7.1. Geometry for two-dimensional flow around a cylinder. follows:

The sensitivity equations are obtained by implicitly differentiating the system (6.1) and its associated boundary conditions with respect to the parameter q. Assuming the order of differentiation can be interchanged, we obtain the following:

The sensitivity boundary conditions at the inflow and outflow are obtained by differentiating (7.1), producing

After differentiation, the no-penetration, no-slip conditions for u imply no-penetration, noslip conditions for s. In section 6.4 we present the weak form of these sensitivity equations and prove an existence result.

7.2

Flow over a Bump

We also consider a problem examined by Burkardt in [14]. In particular, the problem is two-dimensional incompressible flow over a bump in a channel. The geometry of the channel is indicated in Figure 7.2 with = [0, L] x [0, 3], where L 0 is a fixed number representing the length of the channel. As before, the governing equations for this problem are the two-dimensional incompressible Navier-Stokes equations presented earlier (see (6.1)) with f = 0. The boundary conditions for the flow are as follows:

for 0 y 3, where = 0.5 was a constant parameter describing the strength of the inflow. Again, no-penetration and no-slip conditions are applied on the top and bottom channel sidewalls as well as on the bump.

7.2. Flow over a Bump

83

Figure 7.2. Geometry for flow over a bump. For this application, we examine a shape sensitivity. Here, the shape of the bump is a cubic spline determined by a parameter, q. We examine two cases: q = q1 R1 and q = (q1, q2, q3)T R3. We seek to find the sensitivity of the flow in the channel to changes in q, which we denote

We must derive and solve a separate set of linear sensitivity equations for each sensitivity. The sensitivity equations are

In this case, the sensitivity boundary conditions are given by

where h(x, q) denotes the y-coordinate of the boundary of for 1 x 3. We generate h(x, q) using a cubic spline with free boundary conditions. If q R1, q1 specifies the height of the spline at x = 2.0. In this case, h(x, q) satisfies the following:

For q

R3, q1, q2, and q3 specify the height of the spline at x = 1.5, 2.0, and 2.5,

84

Chapter 7. Two-Dimensional Flow Problems

respectively. Here, h(x, q) satisfies

7.3

A Finite Element Formulation

Since exact solutions to the Navier-Stokes equations are unavailable except for the simplest of problems, we now turn our attention to finding good methods for computing approximations to the solutions of the flow equations. We use a finite element approach to solve the variational problems and, later, the weak formulations of the sensitivity equations as well. The weak form of the Navier-Stokes equations (6.2) is discretized using the CrouzierRaviart triangular element (see Figure 7.3), which is a type of "bubble" element described in [15]. This element uses a so-called enriched quadratic velocity interpolant and a discontinuous linear pressure. The discretized variational equations are solved in the primitive variables using a penalty method to solve for the pressure degrees of freedom. Once the state approximations are obtained, the weak form of the sensitivity equations (6.41) can be similarly solved with one additional iteration of the flow solver. For a more detailed explanation of the finite element implementation, see [51]. Additionally, an adaptive methodology is used to strategically refine the mesh, thus improving the accuracy of flow approximations. Because the local projection technique used in the error-estimation step of the adaptive methodology is also employed to improve the sensitivity approximations in Chapter 8, we now take a closer look at the adaptive technique.

7.3.1

Adaptive Methodology

The basic idea of adaptive gridding is to use an error-estimation technique to evaluate the quality of the finite element approximation and to strategically modify the grid based on that evaluation. The grid modification scheme allows the user control over element size and grading. This process has been shown to be successful in resolving shear, stagnation points, jets, and wakes (see [30], [31], [35], [48], and [49]). The two main elements of the adaptive process are error estimation and grid generation. Error Estimation

The error estimation is performed using an approach introduced by Zhu and Zienkiewicz [3], [57], [58] and involves the postprocessing of stresses and strains. Recall that the energy

7.3. A Finite Element Formulation

85

Figure 7.3. Crouzier-Raviart element. norm of u is

or, given in Cartesian coordinates,

Note that the energy norm has a form very similar to the H1 seminorm. In fact, it can be shown that they are equivalent norms. Using the energy norm for the velocity, define the so-called Stokes norm of the solution as

As pointed out in [50], the use of the energy norm over the H1 seminorm offers some advantages, especially to the engineering community. Note that both the velocity and the pressure norms are expressed in terms of surface forces, which are the quantities of prime interest in engineering fluid mechanics. Second, errors computed in these norms can be interpreted as errors in the stresses, which can then easily be related to errors in global quantities such as lift and drag. The Zhu-Zienkiewicz approach uses the Stokes norm to measure the error, e(u, p) = (uex — uh, pex — ph), where (uex, pex) is the exact solution of the flow problem and (uh, ph)

86

Chapter 7. Two-Dimensional Flow Problems

is the finite element approximation. Then the norm of the error is

We concentrate on forming an approximation to uex — uh E • Since the exact solution, and more particularly the gradients of the exact solution, are not known, the approach is to use the finite element approximation to construct approximations to these gradients. Note that the finite element approximations to the gradients are discontinuous across element faces while the exact gradients are, in most cases, continuous across the domain. Thus, the first goal in error estimation is to obtain continuous approximations to the discontinuous finite element gradients. Two methods have been evaluated for this process: global projections and local least-squares projections. Global projections are performed over the entire domain, , and involve finding the best approximation to the discontinuous finite element gradients in the original continuous finite element space. For example, if a piecewise linear approximation of the flow solution is calculated, then the finite element gradient is piecewise constant and discontinuous across elements. The global projection would replace the piecewise constant gradient approximation with a piecewise linear approximation calculated by projecting the piecewise constant function onto the original finite element basis functions. The local leastsquares projections, however, are done node by node and consider only gradient information from subdomains of that contain the current node. Thus, a series of smaller projections is done in combination with some averaging techniques to obtain a continuous gradient projection for the entire domain, . The details of each of these projection techniques are described in the context of the one-dimensional nonlinear model problem in Chapter 5.3. The details of the local projection technique employed for two-dimensional flow problems are given in Chapter 8. The term pex — ph is similarly approximated except that we construct a continuous, quadratic approximation of pex using local projections of the discontinuous linear finite element approximations. Once this is done the L2 norm can be calculated as usual. We now return to the issue of adaptive gridding to briefly describe the remeshing strategy. Remeshing Strategy

Once error estimates are obtained for each element, say, ei, a new mesh density (or element size), d, is calculated which requires equidistribution of the element errors across all the elements. For example, if we wish to reduce the error in each element by a factor of y, then the target error, eT, for an element in the new mesh can be given by

where e is the error over the entire domain and N is the number of elements. If one assumes that the finite element method is of order k, then it is reasonable to write

where h is the current element size and d is the ideal element size we seek. Clearly, we have assumed that we are in the asymptotic range of the finite element method and that the

7.4. Some Numerical Results

87

convergence constant, c, is the same for both meshes. This may or may not be the case. Nevertheless, the system (7.18)-(7.19) can be solved for the new mesh density, d, obtaining

The new element size computation is done for each of the dependent variables in the problem, e.g., velocity, pressure, and an ideal mesh density is obtained for each. Finally, the minimum of these is used for the generation of the new mesh. The details of the actual redefinition of the mesh are omitted here; we refer the interested reader to [30] and [31]. Despite the rather major assumptions made above, this adaptive remeshing strategy works remarkably well. The strategy has been verified in a series of numerical experiments for a variety of flow types (see [30], [31], [35], [48], [49]). We use this adaptive strategy in the problems investigated below.

7.4

Some Numerical Results

In this section, we apply the computational techniques outlined above to the two-dimensional flow problems presented at the beginning of this chapter, and we investigate convergence of the approximate solution as the mesh is refined.

7.4.1

Flow around a Cylinder

Consider first the cylinder problem presented in section 7.1 with the state equations given in (6.1). We again assume parabolic inflow into the channel as follows:

The outflow boundary condition is modified, however, since applying the Dirichlet boundary condition at the outflow causes numerical instability (see [17]). A free boundary condition is used that requires

where is the outward normal at the end of the channel. No-flow and no-slip conditions are applied on the top, bottom, and cylinder sidewalls. As noted earlier, we wish to approximate the sensitivity of the solution with respect to the inflow parameter q. The sensitivity equations are given by (7.3), and the sensitivity boundary conditions at the inflow are obtained as before, with

The computational outflow boundary conditions for the sensitivity equations become

88

Chapter 7. Two-Dimensional Flow Problems

Figure 7.4. Initial and adapted meshes for a cylinder problem. Numerical results for this flow problem were generated using the approximation techniques outlined in section 7.3. The Reynolds number for the calculations is Re = 100, the length of the channel is 8 (L = 6), and the sensitivity parameter value is q = 0. Contour plots of the u, v-velocity fields as well as the u, u-velocity sensitivities are given in Figures 7.5 through 7.8. In Figure 7.4, the initial and adapted grids are shown. It is clear that the mesh is refined around the cylinder and in areas of large velocity gradients. This gives improved approximations of the velocity field, as can be seen in Figures 7.5 and 7.6. Since the mesh refines on the velocity field, it is convenient that, for this problem, the sensitivity flow field is similar to the velocity field, so that as the mesh refines we obtain improved sensitivity approximations as well (see Figures 7.7 and 7.8). It is important to note that this is not always the case, as is shown in [11]. The code was modified by Jeff Borggaard to adapt on the sensitivity field as well, and some results for this are shown later. 7.4.2

Flow over a Bump

We next consider the problem presented in section 7.2 characterized by flow over a bump. The outflow boundary condition for the computations is again taken to be a free boundary condition as follows:

The Reynolds number for the calculations is Re = 100 and the length of the channel L = 8. The initial and adapted grids are shown in Figure 7.9. Contour plots for u, v velocities and sensitivities are displayed in Figures 7.10 and 7.11. Note that the mesh refines in the area of the bump and the elements are allowed to become larger downstream in the channel where the flow is again quadratic. The bump problem is similar to the cylinder problem in that the sensitivities are largest in the same areas where the mesh refines. Thus we obtain improved

7.4. Some Numerical Results

89

Figure 7.5. u-velocity contours for flow around a cylinder. approximations for sensitivities as we refine the mesh for the flow. So far we have developed sensitivity equations for both one-dimensional and twodimensional parameter-dependent differential equations. We have obtained accurate numerical approximations for the continuous sensitivity equations using an adaptive finite element technique. We have observed that refining on the flow provided improved sensitivity approximations, as well, when the sensitivity field was similar to the flow field. In Chapter 8, we explore the idea of adapting the mesh based on error estimation for both the flow and the sensitivity. In Chapter 5, we found that accurate sensitivity approximations for the one-dimensional problems previously described were more difficult to obtain in certain parameter ranges. A projection technique was used to obtain more accurate state gradient approximations, thus decreasing errors in sensitivity calculations. A similar phenomenon occurs as the Reynolds number is increased for the two-dimensional model problems described in this chapter. Chapter 8 describes a local projection technique for two-dimensional problems and evaluates the effectiveness of this technique in improving sensitivity approximations.

90

Chapter 7. Two-Dimensional Flow Problems

Figure 7.6. v -velocity contours for flow around a cylinder.

Figure 7.7. u-velocity sensitivity contours for flow around a cylinder.

7.4. Some Numerical Results

Figure 7.8. v-velocity sensitivity contours for flow around a cylinder.

Figure 7.9. Initial and adapted meshes for a bump problem.

91

92

Chapter 7. Two-Dimensional Flow Problems

Figure 7.10. u, v-velocity contours for flow over a bump.

7.4. Some Numerical Results

Figure 7.11. u, v-velocity sensitivity contours for flow over a bump.

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8 Adaptive Mesh Refinement Strategies

This chapter describes a local projection technique (analogous to that given in Chapter 5.3) for a two-dimensional case. We demonstrate that this projection technique, which provides improved state gradient approximations, can be combined with adaptive grid refinement, on both the state and sensitivities, to improve the accuracy of sensitivity approximations. A case study is presented to evaluate the use of state and sensitivity approximations for the evaluation of cost functionals and their gradients in optimization.

8.1

A Local Projection for Higher Dimensions

In this section, we describe the local projection scheme employed in the error estimation module of the finite element code described in Chapter 7. This local projection scheme is virtually the same as the local scheme for the one-dimensional model problem discussed in section 5.3.2, yet it is presented for a two-dimensional problem below since the use of higher-order finite elements in higher dimensions adds some complexities not previously discussed. Again, the local projection scheme involves performing a series of projections on subdomains of . The projected gradients are given as polynomial expansions around a given vertex, , of a finite element mesh. The subdomain, , over which the projection is defined consists of all elements having = (xi, yi) as a vertex. Figure 8.1 illustrates a typical subdomain, a finite element mesh of quadratic triangles. In principle, the choice of the degree of the polynomial expansion for the improved gradient approximation is independent of the selection of the finite element basis being used. However, in practice, the degree of the polynomial expansion is chosen to match the degree of the finite element basis employed. This leads to an order of accuracy improvement in the gradient approximations. For all the numerical results presented in section 8.2 below, quadratic triangular finite elements were used and so the locally projected gradients are written as polynomials of degree two on . At element vertices, and we define g* to be the local least-squares projection of the finite element derivative, uh, onto the space of quadratic polynomials. For ease of notation, we denote uh by gh. Letting P = [1, x, y, x2, xy, y2] denote the basis functions of this space, we can express each component of the gradient projection, and , 95

96

Chapter 8. Adaptive Mesh Refinement Strategies

Figure 8.1. Typical subdomain of an element vertex as

where the vectors ax, ay R6 contain the coefficients of the basis functions. These coefficients are obtained by solving the following least-squares problems:

Thus, for each component of g*, we solve the following 6x6 fora x and ay:

system of linear equations

The finite element fluxes, and , are obtained in the usual manner by differentiating the finite element basis functions. Note that the left-hand matrix is independent of the quantity being projected and thus can be viewed as the projection matrix for node, , and can be used for obtaining locally projected derivative approximations for all the dependent variables (e.g., u and v) as long as the projection basis, P, is not changed. Once the linear systems are solved, we have a quadratic expression for the locally projected derivative, g*, at each

8.1. A Local Projection for Higher Dimensions

97

Figure 8.2. Element with three quadratic expressions for g*. vertex. Let denote the expression for g* obtained by solving the systems (8.3)-(8.4), where is the subdomain associated with element vertex, . Define and similarly for the remaining element vertices, and . We need a unique definition of g* for any point, inside the element (see Figure 8.2). For quadratic elements, there are several ways to do this. We describe the technique employed in the current version of the code. Unique nodal values of g* at the element vertices, which we denote and , are simply defined as follows:

Nodal values for the midside nodes are obtained by averaging the values of the polynomial expressions for g* at the endpoints of element side. For example,

Then, at any point in the element, the value of the locally projected derivative g* at is

where the Nj are the quadratic basis functions for the finite element space and the are the nodal values of the locally projected derivative obtained as described above. In the following section, we apply the described local projection technique to obtain improved sensitivity approximations for two flow problems.

98

8.2

Chapter 8. Adaptive Mesh Refinement Strategies

Numerical Results for Two-Dimensional Problems

We now return to the two specific flow problems described in Chapter 7 to show that the projection techniques described in section 8.1 can be used to obtain improved sensitivity approximations. We begin by considering the cylinder problem discussed in section 7.1.

8.2.1

Flow around a Cylinder

Numerical approximations to the state and sensitivities for this problem are calculated over a range of Reynolds numbers. The sensitivities are calculated using both the natural discontinuous (unprojected) state gradient approximations and the locally projected continuous state gradient approximations. In each case, the two sensitivity approximations (using unprojected and projected gradients) obtained for the initial and first adapted meshes are compared to an approximation generated by adapting to a very fine, final mesh and then interpolating the "true" solution from the final mesh onto the initial or first adapted meshes. On this final mesh, both schemes converged to solutions differing by less than 10-3. For comparison purposes, we chose the unprojected solution on the final mesh as the true solution. Note that the length of the channel was 8 (i.e., L = 6) for all the results in this section. This length was not sufficient for the flow at the outflow to return to the parabolic inflow, especially over the range of Reynolds numbers being considered; however, it was sufficient to meet the computational free outflow boundary condition. We use the Re = 350 case to show that the results for the higher Reynolds cases were not affected by the length of the channel. In this chapter, we also use an adaptive technique that refines on sensitivity errors as well as flow errors. This technique is completely analogous to the technique used to refine on the flow errors (see [12]). In the results that follow, we are careful to identify meshes that were adapted on approximations of flow errors and meshes that were adapted on approximations of both flow and sensitivity errors. We present a detailed error analysis of the velocity sensitivities s for two Reynolds numbers, Re = 100 and Re = 350, beginning with the results at Re = 100. Figure 8.3 shows the initial mesh, the first adapted meshes, and final mesh for this case. Notice that we present three first adapted meshes: one adapted on the flow only, one adapted on both the flow and the sensitivities calculated with unprojected derivatives, and one adapted on both the flow and the sensitivities calculated with projected derivatives. At this Reynolds number, the difference between these three meshes is less dramatic than it is at Re = 350, yet we present all of them for consistency and comparison purposes. As expected, the scheme employing the locally projected derivatives gives better sensitivity results, as seen in Figures 8.4 and 8.5. In these figures, recall that the true solution is the unprojected solution on the final mesh interpolated onto the initial mesh. Once this interpolation is made, node-by-node errors can be calculated. These errors are shown in Figure 8.6. From these plots, we can see that the local projection scheme reduces the maximum error by about a factor of 2. We also look at the sensitivity solutions using schemes on the first adapted mesh. The first adapted mesh (for the flow) is shown in Figure 8.3(b). The sensitivity results for the unprojected and locally projected derivative schemes for this mesh are shown in Figures 8.7 and 8.8. In these figures, the true solution is now the unprojected solution on the final mesh

8.2. Numerical Results for Two-Dimensional Problems

Figure 8.3. Meshes for cylinder problem at Re = 100.

99

100

Chapter 8. Adaptive Mesh Refinement Strategies

Figure 8.4. u-velocity sensitivities on initial mesh for Re = 100.

8.2. Numerical Results for Two-Dimensional Problems

Figure 8.5. v-velocity sensitivities on initial mesh for Re = 100.

101

102

Chapter 8. Adaptive Mesh Refinement Strategies

Figure 8.6. Error of sensitivity approximations on initial mesh for Re = 100.

8.2. Numerical Results for Two-Dimensional Problems

Figure 8.7. u-velocity sensitivities on first adapted mesh for Re = 100.

103

104

Chapter 8. Adaptive Mesh Refinement Strategies

Figure 8.8. v-velocity sensitivities on first adapted mesh for Re = 100.

8.2. Numerical Results for Two-Dimensional Problems

105

interpolated onto the first adapted mesh. Note that the difference between the approximations using the projected gradients and those using the natural finite element derivatives is no longer as apparent. Again, a node-by-node comparison was made. These errors are shown in Figures 8.9(a) and (b) and 8.10 (a) and (b) and are plotted on the same scales used for the initial mesh for easy comparison. We also present nodal errors for the two schemes in the cases where we adapt on the flow and the sensitivities (see Figures 8.9(c) and (d) and 8.10(c) and (d)). At this Reynolds number, there seems not to be a significant advantage gained by adapting on both the flow and sensitivity errors. To get an overall evaluation of the errors, we used the node-by-node errors to calculate an L2 error over the entire domain for the flow and the sensitivities (see Table 8.1). This more clearly shows the 50% reduction of the error for the local projection scheme on the initial mesh. Note that the errors for u, v on the first adapted meshes are the same for the unprojected and projected schemes on the mesh that was refined only on the flow, but they are slightly different (since the meshes are slightly different) when the meshes were also refined on the sensitivity. Note that adapting on the flow and sensitivity errors not only improved the sensitivity approximations over those obtained by adapting on the flow alone but also improved the numerical approximations for the flow. We now turn to the results for Re = 350. We begin by comparing the state and sensitivity approximations for L = 6 and L = 15 to ascertain whether the length of the channel affects our results. Figure 8.11 shows the initial meshes for the two channel lengths as well as the u, v velocity contours. It is clear that the flow more nearly returns to the inflow for the longer channel. However, the results for the shorter channel are very similar to those for the longer channel in the areas where they overlap. In fact, Figures 8.12 and 8.13 show that the dramatic difference between the sensitivities obtained with unprojected derivatives and those obtained using locally projected derivatives occurs in both the long and the shorter channels. This gives us reasonable confidence that the length of the channel is not affecting our results. We now return to complete the same error analysis for Re = 350 that we did for Re = 100. The initial and first adapted meshes are shown in Figure 8.14. Note that the difference between the three first adapted meshes is definitely greater at this Reynolds number. This is due, at least in part, to the greater discrepancy between the flow error, the unprojected sensitivity error, and the projected sensitivity error. For Re = 350, the differences between the sensitivities calculated with the two different derivative schemes is now dramatic, as can be seen in Figures 8.15 and 8.16. The node-by-node error analysis (see Figure 8.17) shows that locally projected derivatives are definitely better, although at this Reynolds number, both approximations contain fairly large errors. Using the locally projected derivatives to calculate sensitivities reduces the overall L2 error by about 600% (see Table 8.2, initial mesh section). The same analysis is done on the first adapted meshes (See Figures 8.18 through 8.21). Here, adapting on the flow as well as the sensitivity makes a greater difference in the error reduction from the initial mesh to the first adapted mesh. This is easily seen in the overall L2 errors for the approximate flow and sensitivities displayed in Table 8.2. It is interesting to note that the errors for all the quantities are less for Projected (Flow & Sensitivity) than for Unprojected (Flow & Sensitivity), although the Unprojected (Flow & Sensitivity) mesh is quite a bit finer.

106

Chapter8. Adaptive Mesh Refinement Strategies

Figure 8.9. Error ofu-velocity sensitivity approximations on first adapted mesh for Re = 100.

8.2. Numerical Results for Two-Dimensional Problems

107

Figure 8.10. Error of v-velocity sensitivity approximations on first adapted mesh for Re = 100.

108

Chapter8. Adaptive Mesh Refinement Strategies

Table 8.1. L2 errors for flow and sensitivities at Re = 100.

u V

su sv u V

su sv

Initial Mesh Unproj Proj 1.7397E-01 1.7397E-01 9.0160E-02 9.0160E-02 5.0463E-01 2.8227E-01 2.2453E-01 1.3050E-01 First Adapted Meshes Unproj (Flow) Proj (Flow) Unproj (Flow & Sens) 1.0313E-01 1.0313E-01 7.2637E-02 4.5997E-02 4.5997E-02 3.3000E-02 3.4173E-01 3.5800E-01 2.5988E-01 1.4487E-01 1.1039E-01 9.1075E-02

Proj (Flow & Sens) 5.3305E-02 2.8115E-02 2.7273E-01 7.6292E-02

Figure 8.11. Initial meshes and u, v-velocity contours for L = 6, 15 and Re = 350.

8.2. Numerical Results for Two-Dimensional Problems

Figure 8.12. u-velocity sensitivity contours for L = 6, 15 and Re = 350.

109

110

Chapter 8. Adaptive Mesh Refinement Strategies

Figure 8.13. v-velocity sensitivity contours for L = 6, 15 and Re = 350.

8.2. Numerical Results for Two-Dimensional Problems

Figure 8.14. Meshes for cylinder problem at Re = 350.

111

112

Chapter 8. Adaptive Mesh Refinement Strategies

Figure 8.15. u-velocity sensitivities on initial mesh for Re = 350.

8.2. Numerical Results for Two-Dimensional Problems

Figure 8.16. v-velocity sensitivities on initial mesh for Re = 350.

113

114

Chapter 8. Adaptive Mesh Refinement Strategies

Figure 8.17. Error of sensitivity approximations on initial mesh for Re = 350.

8.2. Numerical Results for Two-Dimensional Problems

115

Table 8.2. L2 errors for flow and sensitivities at Re = 350.

u V

su Sv

u V

su Sv

Initial Mesh Unproj Proj 1.4808E-00 1.4808E-00 4.1024E-01 4.1024E-01 1.9438E+01 3.6791E-00 3.0582E-00 5.8348E-01 First Adapted Meshes Unproj (Flow) Proj (Flow) Unproj (Flow & Sens) 9.6580E-01 9.6580E-01 3.5985E-01 2.6044E-01 2.6044E-01 1.0257E-01 3.1614E-00 3.8080E-00 1.1145E-00 3.4686E-01 1.5339E-00 8.7266E-01

Proj (Flow & Sens) 2.8674E-01 7.7747E-02 1.0423E-00 2.9430E-01

Figure 8.18. u-velocity sensitivities on first adapted mesh for Re = 350.

116

Chapter 8. Adaptive Mesh Refinement Strategies

Figure 8.19. v-velocity sensitivities on first adapted mesh for Re = 350.

8.2. Numerical Results for Two-Dimensional Problems

117

Figure 8.20. Error ofu-velocity sensitivity approximations on first adapted mesh for Re = 350.

118

Chapter 8. Adaptive Mesh Refinement Strategies

Figure 8.21. Error of v-velocity sensitivity approximations on first adapted mesh for Re = 350.

8.2. Numerical Results for Two-Dimensional Problems

119

To conclude, we observe that using the locally projected derivatives clearly stabilizes the calculations over a larger range of Reynolds numbers, as it did in section 5.3.3 for the one-dimensional model problem. Plots of the overall L2 sensitivity errors on the initial mesh are shown in Figures 8.22 and 8.23. In addition, the mesh refinement very effectively reduces the errors for the flow and the sensitivities. At most Reynolds numbers, after the mesh is refined once, there is little difference between the sensitivities calculated with the unprojected and the locally projected techniques. At higher Reynolds numbers, however, using locally projected derivatives to calculate sensitivities remains advantageous. We now turn our attention to obtaining numerical approximations to the design sensitivities for the flow over a bump problem.

8.2.2

Flow over a Bump

We begin with a qualitative comparison of the sensitivity values we obtain with those presented in [14]. We point out some of the difficulties of calculating shape sensitivities and show how the process of error estimation and grid refinement is extremely important in obtaining accurate numerical approximations of the sensitivities for these problems. Figure 8.24 displays vector sensitivity plots for a = = 0.5, = 0.5, and L = 10. These plots are qualitatively very comparable to those presented by Burkardt on page 172 in [14]. As Burkardt notes, the shape sensitivities for smaller Reynolds numbers appear as "whirlpools" and are predominately localized to the region above the bump. As the Reynolds number increases, however, the effect of the bump is carried downstream. This is especially apparent in Figure 8.24(c). Another effect of increasing the Reynolds number on flow calculations is that the task of meeting the outflow boundary conditions becomes more challenging numerically. We clearly see this effect in the Re = 500 case shown in Figure 8.25. With a channel length of L = 10, the flow does not reach the parabolic flow profile of the inflow. For this case, we lengthened the channel to L = 20. Notice also that we are getting a large secondary whirlpool further downstream from the bump. This phenomenon does not appear in the sensitivities presented in [14]. We examine this further for the case Re = 1000. The case of Re = 1000 is used to examine a number of issues relating to our sensitivity approximations. First, we investigate the secondary whirlpool that appears in sensitivity calculations. We also evaluate the accuracy of the sensitivity approximations near the bump using the error estimation and grid refinement process presented in section 7.3.1. The mesh is adapted only on flow errors. Figure 8.26 shows the initial mesh, which is similar in density to the one used in [14], and the adapted meshes. Note that the mesh refines where one would expect, in the regions of large velocity gradients around and downstream from the bump. It is also refining at the outflow in an effort to accurately meet the outflow boundary condition. Figures 8.27 and 8.28 display u and v contours for the flow on the initial and adapted meshes. It is important to note that as the mesh refines, the contours smooth and the accuracy of the gradients in u and v are greatly improved, especially in the vicinity of the bump. This lack of accuracy of the velocity gradients on the initial mesh greatly hampers our ability to obtain good sensitivity approximations. This is seen by noting that the sensitivity boundary conditions on the bump, (7.9)-(7.10), require approximations to the velocity gradients on the boundary. If there are large errors in the velocity gradients, there will be large errors in the sensitivity approximations. This effect can be seen in Figures 8.29 through 8.31. As the

120

Chapters. Adaptive Mesh Refinement Strategies

Figure 8.22. Flow about a cylinder—L2 error of u-sensitivity approximations on initial mesh.

Figure 8.23. Flow about a cylinder—L2 error of v-sensitivity approximations on initial mesh.

8.2. Numerical Results for Two-Dimensional Problems

Figure 8.24. Sensitivity vectors, L = 10, flow over a bump.

121

122

Chapter8. Adaptive Mesh Refinement Strategies

Figure 8.25. u-velocity contours and sensitivity vectors, Re = 500, flow over a bump.

8.2. Numerical Results for Two-Dimensional Problems

123

Figure 8.26. Flow over a bump, initial and adapted meshes for Re = 1000, L = 20.

124

Chapter8. Adaptive Mesh Refinement Strategies

Figure 8.27. u -velocity contours for flow over a bump.

8.2. Numerical Results for Two-Dimensional Problems

Figure 8.28. v -velocity contours for flow over a bump.

125

126

Chapter 8. Adaptive Mesh Refinement Strategies

Figure 8.29. u-velocity sensitivity contours for flow over a bump.

8.2. Numerical Results for Two-Dimensional Problems

Figure 8.30. v-velocity sensitivity contours for flow over a bump.

127

128

Chapter 8. Adaptive Mesh Refinement Strategies

Figure 8.31. Sensitivity vector plots on initial and adapted meshes.

8.2. Numerical Results for Two-Dimensional Problems

129

gradient approximations improve in the second and third adapted meshes, the values of the sensitivities become much more accurate, not only in the local area of the bump but downstream as well. Note also that the secondary whirlpool, which appears in the Re = 500 case, is seen on the initial mesh in this case also. As the mesh is refined, however, the size of the whirlpool is diminished until it is almost gone, as can be seen in Figure 8.31(c). Clearly, having accurate state gradient approximations on the boundary is necessary for accurate sensitivity approximations, especially shape sensitivities. This is an area requiring future research. Analysis of Cost Functionals and Gradients—An Optimization Issue

Finally, we turn to the issue of using numerical approximations of the state and sensitivities to approximate cost functionals and their gradients. A comparison of our results with a problem considered by Burkardt in Chapter 11 of [14] is given. Burkardt presents what he refers to as a discretized sensitivity failure. We evaluate his results and show that the process of adaptive mesh refinement is key to obtaining good cost function and gradient evaluations in order to prevent inaccurate results from an optimization code. We examine the numerical experiment carried out in section 11.2 of [14]. We evaluate the following cost functional:

Here, P is the number of matching points and for the results we present herein P was fixed at 15. The matching points, (3, yi), i = 1, 2 , . . . , P, were evenly distributed along the line x = 3, y [0, 3]. Also, the target profile, uTarg(3, y), was obtained by computing a finite element solution with the shape parameter, qTarg = (0.375, 0.5,0.375)T, and = 0.5, and then interpolating to obtain the values for uT(3, yi), i = 1, 2,... , P. The gradient of the cost function with respect to q is expressed as

The SEM uses the discrete approximation of the continuous sensitivity equation as before to approximate the value of the cost gradient. We denote this approximation of the cost gradient by qJh; thus we have

In order to nearly duplicate the numerical experiment carried out by Burkardt, we have L = 10, Re = 1 and we generate the matching profile from the initial mesh for qTarg. We calculate the values of the cost functional Jh and its gradient along a line parameterized by S, connecting q = (-0.117, 0.419, -0.149)T at S = 0 and q = (0.375,0.5,0.375)T at S = 25. This is the same line along which Burkardt explored. His results are presented in Table 11.2 on page 153 and Figures 11.3 and 11.4 on page 145 of [14].

130

Chapter 8. Adaptive Mesh Refinement Strategies

Our results are displayed in Table 8.3 and Figures 8.32 and 8.33. Burkardt reported a local maximum at S = 5. As seen in Figure 8.32, we do not get the same results on the initial mesh. To determine if the difference was due to not having sufficient accuracy on the initial mesh, we also present the results for meshes which were refined on the flow field. It is possible that the difference between our results and those of Burkardt could be due to our having fixed the value of = 0.5. It is also possible that the difference is a result of numerical inaccuracies due to discretization differences. We perform another numerical experiment similar to the one described above with one exception: we now use Re = 100. Figures 8.34 and 8.35 show the cost function and gradient along the same line explored above. Note that adapting the mesh is even critical to obtaining accurate, smooth cost and gradient approximations at this Reynolds number.

8.2. Numerical Results for Two-Dimensional Problems

131

Table 8.3. Values of Jh along a line, Re = 1, and

s 0

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

q1

q2

q3

-0.117 -0.097 -0.077 -0.057 -0.038 -0.018 0.001 0.020 0.040 0.060 0.079 0.099 0.119 0.138 0.158 0.178 0.197 0.217 0.237 0.256 0.276 0.296 0.316 0.335 0.355 0.375 0.394 0.414 0.434 0.453 0.473

0.419 0.422 0.425 0.428 0.432 0.435 0.438 0.441 0.444 0.448 0.451 0.454 0.457 0.461 0.464 0.467 0.470 0.474 0.477 0.480 0.483 0.487 0.490 0.493 0.496 0.500 0.503 0.506 0.509 0.513 0.516

-0.149 -0.128 -0.107 -0.086 -0.065 -0.044 -0.023 -0.002 0.018 0.039 0.060 0.081 0.102 0.123 0.144 0.165 0.186 0.207 0.228 0.249 0.270 0.291 0.312 0.333 0.354 0.375 0.396 0.417 0.438 0.459 0.480

Jh.103 16.9 16.3 15.5 14.6 13.7 12.9 12.2 11.4 10.6 9.72 9.02 8.35 7.46 6.21 5.62 5.00 4.38 3.70 3.08 2.47 1.89 1.33 0.72 0.32 0.11 0.00 0.06 0.31 0.81 1.63 2.72

(

qJ

h.

S).103

Initial Mesh -0.99 -1.03 -1.00 -0.94 -0.91 -0.92 -1.04 -1.00 -0.96 -0.99 -1.15 -1.07 -1.08 -0.77 -0.78 -0.79 -0.78 -0.76 -0.74 -0.70 -0.64 -0.56 -0.50 -0.35 -0.21 0.00 0.16 0.38 0.63 0.92 1.22

Jh.103 12.1 11.7 11.3 10.8 10.3 9.95 9.49 9.02 8.56 8.00 7.48 6.95 6.39 5.76 5.17 4.56 3.94 3.27 2.66 2.07 1.51 0.99 0.57 0.25 0.05 0.01 0.16 0.53 1.16 2.08 3.30

= 0.5. (qJh

-

03 Mesh -0.56 -0.57 -0.58 -0.60 -0.61 -0.62 -0.63 -0.65 -0.67 -0.68 -0.70 -0.72 -0.74 -0.75 -0.76 -0.77 -0.76 -0.75 -0.72 -0.68 -0.62 -0.53 -0.42 -0.29 -0.13 0.05 0.28 0.53 0.81 1.13 1.48

).103

132

Chapter 8. Adaptive Mesh Refinement Strategies

Figure 8.32. Values ofJh(q(S))

along a line.

Figure 8.33. Values of

alone a line.

8.2. Numerical Results for Two-Dimensional Problems

Figure 8.34. Values of

along a line.

Figure 8.35. Values of

along a line.

133

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Index adaptive remeshing strategy, 87

model problems, 17-23, 81-84

Banach spaces, 75 bilinear forms, 10,13, 72, 79 boundary value problems examples of linear elliptic, 17-21 examples of nonlinear, 22-23, 8189 linear elliptic, 1, 5-16 mathematical theory for nonlinear, 71-80 regularity of linear elliptic, 11-14

Navier-Stokes equations, 1, 21, 33 examples of, 81-89 mathematical framework, 71-80 variational formulation of, 71-75 optimal design, 22, 52-60, 129-130 Preface, xix

Hilbert spaces, 14

sensitivity analysis, 1 sensitivity equations computational algorithms for solving, 28-35 differential forms, 19-21,23,29,30, 32, 34, 37, 79, 82, 83 numerical calculations, 43-70 operator forms, 15-16, 18, 19 variational forms, 36, 37, 79 Sobolev spaces, 6-7 state gradient approximations finite element derivatives for, 44-51 projection techniques for, 60-70,95130 their affect on sensitivity approximation, 63-65 stiffness matrix, 38 Stokes norm, 85

implicit function theorem, 14, 76, 78 Introduction, 1

trace theorems, 8-9 trilinear forms, 72-73

Lax-Milgram theorem, 11, 13

Zhu-Zienkiewicz error estimator, 85

computational algorithms, 25 continuous sensitivity equation methods, 1, 2, 22, 82, 83, 129 Crouzier-Raviart element, 84-85 diffeomorphism, 8, 25 finite element formulations adaptive mesh refinement, 84-87 Crouzier-Raviart element, 84-85 piecewise linear, 37-41 Frechet differentiability, 6,15,16,19,77 Gelfand triple, 9,12

mass matrix, 61 mathematical frameworks, 5-16, 71-80 method of mappings, 1, 25-28 139