design sensitivity analysis

Design SensitivityAnalysis

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F R O N T I E R S INA P P L I E D M A T H E M A T I C S

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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

B O O K S P U B L I S H E D IN F R O N T I E R SIN A P P L I E D M A T H E M A T I C S

Stanley, Lisa G. and Stewart, Dawn L, Design Sensitivity Analysis: Computational Issues ofSensitivity Equation Methods

Vogel, Curtis R., Computational Methods for Inverse Problems

Lewis, F. L; Campos,].; and Selmic, R., Neuro-Fuzzy Control of Industrial Systems with ActuatorNonlinearities

Bao, Gang; Cowsar, Lawrence; and Masters, Wen, editors, Mathemot/co/ Modeling in OpticalScience

Banks, H.T.; Buksas, M.W.; and Lin,T., Electromagnetic Material Interrogation Using ConductiveInterfaces and Acoustic Wavefronts

Oostveen, Job, Strongly Stabilizable Distributed Parameter Systems

Griewank, Andreas, Evaluating Derivatives: Principles andTechniques of AlgorithmicDifferentiation

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: ComputationalAspects 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 SensitivityAnalysisComputational Issues ofSensitivity Equation Methods

Lisa G. StanleyMontana State UniversityBozeman, Montana

Dawn L. StewartAir Force Institute of TechnologyWright Patterson AFB, Ohio

siamSociety for Industrial and Applied Mathematics

Philadelphia

Copyright © 2002 by the Society for Industrial and Applied Mathematics.

1 0 9 8 7 6 5 4 3 2 I

All rights reserved. Printed in the United States of America. No part of this book may bereproduced, stored, or transmitted in any manner without the written permissionof 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 DataStanley, Lisa G.

Design sensitivity analysis : computational issues of sensitivity equation methods / LisaG. 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 52.1 Preliminaries 5

2.1.1 Notation 62.1.2 Function Space Tools 82.1.3 Bilinear Forms and Variational Tools 9

2.2 Regularity of Elliptic Boundary Value Problems 112.2.1 Strong Solutions 122.2.2 Weak Solutions 12

2.3 Mappings and Implications to Sensitivity Analysis 142.3.1 Implicit Function Theorem 142.3.2 Operator Formulations of Sensitivity Equations 15

3 Model Problems 173.1 Heat in a Thin Rod with a Parameter-Dependent Forcing Term . . . . 173.2 Heat in a Thin Rod with a Parameter-Dependent Boundary 19

3.2.1 State Equation 203.2.2 Sensitivity Equation 20

3.3 Nonlinear Model 223.3.1 State Equation 223.3.2 Sensitivity Equation 23

4 Computational Algorithms 254.1 The Method of Mappings 25

4.1.1 Transformation Techniques 254.1.2 An Algebraic Transformation 27

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viii Contents

4.2 SEMs 284.2.1 Hybrid SEM 284.2.2 Abstract Version of the Semianalytic Method 314.2.3 Applying the H-SEM to the Nonlinear Model 33

4.3 Approximation Framework 354.3.1 Variational Formulations 354.3.2 Piecewise Linear Finite Elements 37

5 Numerical Results 435.1 Linear Model 43

5.1.1 State Approximations 435.1.2 H-SEM 445.1.3 A-SAM 47

5.2 Nonlinear Model 495.2.1 Convergence of the State and the Sensitivity 495.2.2 Sensitivities in Optimal Design 52

5.3 State Gradient Approximations 605.3.1 A Global Projection Scheme 615.3.2 A Local Projection Scheme 625.3.3 Numerical Results 62

6 Mathematical Framework for Navier-Stokes Equations 716.1 The Homogeneous Dirichlet Problem 71

6.1.1 Function Spaces and Notation 716.1.2 Existence and Uniqueness of Solutions to the Variational

Form 726.2 The Nonhomogeneous Dirichlet Problem 736.3 An Abstract Framework for Navier-Stokes 75

6.3.1 The Framework 756.3.2 Using the Framework 766.3.3 Continuity of Solutions with Respect to Data 77

6.4 Analysis of the Sensitivity Equations 786.4.1 A General Formulation of the Sensitivity Equations ... 786.4.2 Existence and Uniqueness of Solutions to the Sensitivity

Equations 796.5 Differentiability of Solutions with Respect to q 80

7 Two-Dimensional Flow Problems 817.1 Flow around a Cylinder 817.2 Flow over a Bump 827.3 A Finite Element Formulation 84

7.3.1 Adaptive Methodology 847.4 Some Numerical Results 87

7.4.1 Flow around a Cylinder 877.4.2 Flow over a Bump 88

Contents ix

8 Adaptive Mesh Refinement Strategies 958.1 A Local Projection for Higher Dimensions 958.2 Numerical Results for Two-Dimensional Problems 98

8.2.1 Flow around a Cylinder 988.2.2 Flow over a Bump 119

Bibliography 135

Index 139

List of Figures

5.1 Finite element approximations to w( , 1.5) 445.2 Approximations to w(x, 1.5) 445.3 Hl error of z N ( x , q) for q ranging from 1.1 to 1.9 455.4 Finite element approximations tos(x, 1.5) 455.5 H-SEM approximations to s(x, 1.5) 465.6 Approximation of 1.5) with N = 3 465.7 Finite element approximations to p( , 1.5) 475.8 A-SAM approximations to s(x, 1.5) 485.9 Hl errors for sensitivity calculations 485.10 Numerical approximations to the solution of the nonlinear model at q = 2

and q = 1.2 505.11 Numerical approximations to the solution of the sensitivity equation at

q — 2 using PWC derivatives 505.12 L2 error of the state and sensitivity approximations at q = 2 with N = M. 515.13 Numerical approximations to the solution of the sensitivity equation at

q = 2 using PWC derivatives with N = 2 and mesh refinement in M. . . 525.14 Numerical approximations to the solution of the sensitivity equation at

q = 2 using PWC derivatives with N = 4 and mesh refinement in M. . . 535.15 Numerical approximations to the solution of the sensitivity equation at

q = 1.4 using PWC derivatives 535.16 Numerical approximations to the solution of the sensitivity equation at

q = 1.2 using PWC derivatives 545.17 L2 error of the solution and sensitivity approximations at q = 1.2 545.18 L2 error of sensitivity approximations using PWC derivatives 555.19 Gauss-Newton algorithm 555.20 Data generated at q = 2 565.21 Data generated at q = 1.4 565.22 The cost function and its approximations for p = 16 and q* ~ 2 585.23 The cost function and its approximations for p = 16 and q* ~ 1.4. . . . 585.24 Finite element derivatives with projections at N = 4 and q = 2 635.25 Finite element derivatives with projections at N = 8 and q = 1.2. . . . . 645.26 L2 error on each element for N = 4 and q = 2 645.27 L2 error on each element for N = 8 and q = 1.2 65

xi

xii List of Figures

5.28 Sensitivity approximations at q = 2 665.29 Sensitivity approximations at q = 1.4 665.30 Sensitivity approximations at q = 1.2 675.31 Model problem—L2 error of sensitivity approximations (PWC and local). 675.32 Model problem—L2 error of sensitivity approximations (global and local). 68

6.1 Sample domain with boundaries 74

7.1 Geometry for two-dimensional flow around a cylinder. 827.2 Geometry for flow over a bump 837.3 Crouzier-Raviart element 857.4 Initial and adapted meshes for a cylinder problem 887.5 w-velocity contours for flow around a cylinder. 897.6 u-velocity contours for flow around a cylinder. 907.7 w-velocity sensitivity contours for flow around a cylinder. 907.8 v-velocity sensitivity contours for flow around a cylinder. 917.9 Initial and adapted meshes for a bump problem 917.10 u, v-velocity contours for flow over a bump 927.11 u, v-velocity sensitivity contours for flow over a bump 93

8.1 Typical subdomain of an element vertex 968.2 Element with three quadratic expressions for g* 978.3 Meshes for cylinder problem at Re = 100 998.4 u-velocity sensitivities on initial mesh for Re = 100 1008.5 v-velocity sensitivities on initial mesh for Re = 100 1018.6 Error of sensitivity approximations on initial mesh for Re = 100 1028.7 w-velocity sensitivities on first adapted mesh for Re = 100 1038.8 y-velocity sensitivities on first adapted mesh for Re = 100 1048.9 Error of w-velocity sensitivity approximations on first adapted mesh for

Re = 100 1068.10 Error of v-velocity sensitivity approximations on first adapted mesh for

Re = 100 1078.11 Initial meshes and u, v-velocity contours for L = 6, 15 and Re = 350. . 1088.12 w-velocity sensitivity contours for L = 6, 15 and Re = 350 1098.13 v-velocity sensitivity contours for L = 6, 15 and Re = 350 1108.14 Meshes for cylinder problem at Re = 350 1ll8.15 u-velocity sensitivities on initial mesh for Re = 350 1128.16 v-velocity sensitivities on initial mesh for Re = 350 1138.17 Error of sensitivity approximations on initial mesh for Re = 350 1148.18 u-velocity sensitivities on first adapted mesh for Re = 350 1158.19 y-velocity sensitivities on first adapted mesh for Re = 350 1168.20 Error of w-velocity sensitivity approximations on first adapted mesh for

Re = 350 1178.21 Error of u-velocity sensitivity approximations on first adapted mesh for

Re = 350 118

List of Figures xiii

8.22 Flow about a cylinder—L2 error of u -sensitivity approximations on initialmesh 120

8.23 Row about a cylinder—L2 error of v-sensitivity approximations on initialmesh 120

8.24 Sensitivity vectors, L = 10, flow over a bump 1218.25 u-velocity contours and sensitivity vectors, Re = 500, flow over a bump. 1228.26 Flow over a bump, initial and adapted meshes for Re = 1000, L = 20. . 1238.27 u-velocity contours for flow over a bump 1248.28 v-velocity contours for flow over a bump 1258.29 u-velocity sensitivity contours for flow over a bump 1268.30 v-velocity sensitivity contours for flow over a bump 1278.31 Sensitivity vector plots on initial and adapted meshes 1288.32 Values of Jh/(q(S)) along a line 1328.33 Values of along a line 1328.34 Values of Jh(q(S)) along a line 1338.35 Values of along a line 133

List of Tables

5.1 Matching data for the optimization 575.2 Optimization results for p = 16, qopt ~ 2, qinit = 1.2, PWC derivatives. 595.3 Optimization results for p = 16, qopt ~ 1.4, qinit — 2.0, PWC derivatives. 595.4 L2 errors for the derivative approximations 655.5 Optimization results for Case 1, global projection scheme 685.6 Optimization results for Case 1, local projection scheme 695.7 Optimization results for Case 2, global projection scheme 695.8 Optimizations results for Case 2, local projection scheme 70

8.1 L2 errors for flow and sensitivities at Re = 100 1088.2 L2 errors for flow and sensitivities at Re = 350 1158.3 Values of Jh along a line, Re = 1, and A = 0.5 131

xv

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 ofthe system. Continuous sensitivity methods construct a sensitivity equation whose solutionprovides the sensitivity of the state variable with respect to a parameter. To utilize suchinformation, computational methods are typically applied. This book discusses certainpitfalls that can arise when sensitivities are computed for systems whose sensitivities existin a function space. In addition, it carefully presents the theory and motivating examplesfor avoiding these pitfalls.

As computing power has progressed, computational methods have opened the way tomaking sensitivity analysis a tractable design tool. It is this aspect of sensitivity analysis thatthis book addresses. A first (and perhaps obvious) step to computing sensitivities might be toapproximate 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 showthat errors can arise that are sometimes catastrophic with respect to computational accuracy.

The book presents illuminating examples of what can go wrong in computing sensi-tivities when the domain is a function space. Although the examples are for spatial variablesin one and two dimensions, they give insight that should lead to further research in morecomplicated domains. This work is application unspecific; the authors provide concreteexample problems to point out some of the problems that can arise and what improve-ments can be obtained. There are some simple one-dimensional partial differential equationproblems and some more sophisticated two-dimensional steady state flow problems. Theauthors consider how to obtain accurate state gradient calculations, and why accuracy maybe compromised if one computes sensitivities of computational models without consider-ing the underlying distributed parameter system. The authors present examples in whichcomputing the sensitivity by approximating the continuous sensitivity equation and solvingcan 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 greatincrease in computational speed.

The intended audience for this work is engineers, scientists, and applied mathemati-cians who want to use continuous sensitivity equations in a computational setting. Anundergraduate knowledge of analysis and numerical analysis is assumed along with a fa-miliarity with finite element methods. Necessary concepts and results for Hilbert spaces areprovided 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 provemost helpful to the reader who wants to use sensitivity analysis in a computational context.

xvii

Preface

This book provides an introduction to the computational aspects of continuous sen-sitivity equation methods (CSEMs) for partial differential equations. It is intended foradvanced undergraduate and graduate students in the areas of numerical analysis and ap-plied and computational mathematics as well as other scientists and engineers who have aparticular interest in modeling, design, control, and optimization of physical systems.

With the significant advances in computer technology during the twentieth century, in-dustries that design and manufacture high-performance products are increasingly interestedin exploiting the advantages of computer-aided design, numerical analysis, and optimaldesign methods. For example, a detailed analysis of aerodynamic systems, automated man-ufacturing processes, and casting or molding processes can be performed using a softwarepackage prior to time-consuming, expensive, and labor-intensive physical experiments.During the last decade, considerable effort has been devoted to the development of com-putational tools that allow designers to mathematically analyze such physical processes.Sensitivity analysis plays an important role in this effort. Many simulation tools can beenhanced by using local information provided by state sensitivities to obtain nearby solu-tions. Hence, one is able to develop a qualitative picture of how small changes in designparameters might affect the performance of a physical system.

This book is intended to give the reader an overview of applications and computationalissues regarding sensitivity calculations performed using CSEMs. The focus is on theconstruction and analysis of algorithms for computing sensitivities. Consequently, adjointmethods and other methods used primarily in the context of optimization applications areomitted from this text. Myriad algorithms have been developed for the computation ofsensitivities. In Chapter 1, a number of these methods are briefly discussed. The methodsare 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 appliedto a variety of problems. The term sensitivity equation methods refers to a broad class ofcomputational methods that compute a sensitivity by deriving and solving an equationknown as the sensitivity equation. This text focuses on the use of CSEMs, which refers tothe formal mathematical differentiation of governing equations (usually partial differentialequations) to derive the sensitivity equations (which also take the form of partial differentialequations). Chapter 1 gives a general overview of this approach and Chapter 2 providesthe appropriate definitions to answer the question of what exactly is meant by the termsensitivity. Essentially, the sensitivity is interpreted as a partial derivative with respect to adesign parameter. Chapters 2 and 6 present the mathematical framework developed by theauthors for the purpose of rigorous mathematical justification of the formal differentiation

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xx Preface

techniques that form the base of the CSEM concept. These chapters are very mathematicaland are intended for readers with a background in real analysis and function analysis. Ageneral framework is given for a class of linear elliptic boundary value problems in Chapter2, 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 techniquesin 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 whichthey are based. In the course of outlining these problems, many of the fundamental char-acteristics of CSEMs are discussed. For example, an inherent by-product of CSEMs is thecoupling of the sensitivity equation (partial differential equation) to the original state equa-tion (partial differential equation). This coupling can take the form of sensitivity boundaryconditions 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 ofthe state variable.

Chapters 4 and 5 focus on the computational algorithms used for sensitivity calcu-lations throughout the book. Computational schemes are developed for both linear andnonlinear elliptic problems. The method of mappings is incorporated into the CSEM as anapproach to computing shape sensitivities for a class of heat transfer applications. Numeri-cal 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 numbersituations and for applications involving boundary layers, considerable effort is concentratedon the analysis of a nonlinear model problem. In the case of nonlinear state equations, termsinvolving state gradients often appear as coefficients in the sensitivity equations. This phe-nomenon produces a special set of obstacles when the goal is to quickly and cheaply obtainaccurate sensitivity approximations. Indeed, one of the model problems illustrates that ob-taining accurate state gradient approximations is of fundamental importance for sensitivitycomputations in the presence of a boundary layer. Hence, an entire section of Chapter 5 isdevoted to the topic of extracting and improving state gradient approximations. Projectiontechniques first developed for a posteriori error estimators in finite element codes are usedto improve the accuracy of these state gradient approximations. In addition, Chapter 5contains a brief section that explores the usefulness of sensitivity approximations within theframework of an optimal design problem.

Finally, Chapters 7 and 8 investigate the validity of the CSEM approach for someexamples of two-dimensional fluid flow problems. In Chapter 7, the continuous sensitivityequation method is shown to be numerically efficient for computing sensitivities for flowaround a cylinder and for flow over a bump. However, Chapter 7 points out that, in certaininstances, the accuracy of the sensitivity calculations can be greatly improved by takingadvantage of adaptive refinement strategies that may already be available within the flowsolver producing the state approximations. Chapter 8 presents an example that combinesadaptive refinement with projection techniques in order to provide exceptionally accuratesensitivity approximations for nominal extra computational expense.

The authors would like to take this opportunity to thank the many people and agencieswho have made this book possible. Lisa Stanley would like to express her appreciationto 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 tothe Air Force for granting its permission for this book to be written. The InterdisciplinaryCenter for Applied Mathematics provided both authors with the computer facilities, theoffice facilities, and, most important, the mentoring necessary to carry out the researchcomponent of their dissertations. These mentors include the thesis advisor Dr. John A. Burnsand other friends and colleagues, such as Dr. Jeff T. Borggaard, Dr. Gene M. Cliff, andDr. Belinda B. King. A special note of thanks to Dr. Dominique Pelletier for allowing theauthors the opportunity to use and modify the code used to generate steady state solutionsand sensitivities for Navier-Stokes equations. The authors are also extremely grateful toeveryone at SLAM, especially Dr. Tom Banks, Editor-in-Chief of the Frontiers series, andMarianne Will, Acquisitions Editor, for providing the opportunity to publish work in thewell-respected Frontiers series. Lisa Stanley would also like to acknowledge the BurnsTelecommunication Center, in particular Ritchie Boyd, on the campus of Montana StateUniversity for the computer facilities needed for some graphics editing.

Finally, this section would not be complete without the recognition of our families fortheir constant encouragement, support, patience, and never-ending love and understandingthroughout the years.

Lisa G. StanleyDawn L. Stewart

1 Introduction

As modern computing capabilities increase, considerable effort is devoted to the develop-ment of computational tools for the analysis, design, control, and optimization of complexphysical systems [4], [19], [29]. A critical, and sometimes expensive, first step in the pro-cess of computer-aided analysis and design is the formulation of a detailed and accuratemodel of the system. The physical performance of the design is modeled by a system ofmathematical equations, usually ordinary or partial differential equations. For many engi-neering applications, analysis of the design requires efficient construction and manipulationof complex geometries along with fast and accurate numerical methods for solving partialdifferential equations. Generally, there are a number of physical parameters (or designvariables) that designers can adjust (either manually or within a computer simulation) toimprove the design. In applications such as aerodynamic design, growth and control of thinfilms, or design of smart materials, some design variables may influence the shape (geom-etry) of the design. Consequently, designers then become interested in how sensitive thestate variables are to small changes in the design variables. For example, when analyzinga composite material, one may be interested in the sensitivity of the heat flow through thematerial to small changes in the thickness of the film. Sensitivity analysis is a mathematicaltool that provides a methodology for investigating such questions.

This book has two main goals. The first is to describe and illustrate the applicationof continuous sensitivity equation methods (CSEMs) to several examples. The exampleswere chosen to demonstrate that there are many computational schemes available withinthe context of CSEMs. Furthermore, the choice of a specific algorithm should be guidedby the overall objectives of the problem. The second goal is to provide the foundation of amathematical framework that can be used to derive sensitivity equations. We provide math-ematical 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 rigorouslyanalyzing sensitivity equations. Once a sufficient amount of background is constructed, wepoint out the mathematical as well as the numerical issues that are inherent to CSEMs.

To make effective use of sensitivities, accurate and efficient computational algorithmsare 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. Forexample, if sensitivity calculations are used to enhance a simulation tool, accuracy maybe more important than speed. However, when used for gradient computations within anoptimization algorithm, speed may be more crucial than high accuracy. Hence, the scientificmotivation behind the sensitivity calculation can often influence the choice of computationalalgorithm.

Many numerical methods have been developed for the purpose of sensitivity compu-tation. The first, and maybe the best known, is the finite difference technique. The method isvery simple to implement and has a background that is fundamentally established in mathe-matics. However, for complex fluid flow problems where each state approximation requiresconsiderable computational resources, this method requires two flow solves, each at differ-ent parameter values, to form a difference quotient. In large aerospace flow problems, thisis sometimes an extremely severe requirement, since mesh generation itself can take weeksto months. In addition, a step-size sufficiently small to ensure accurate approximations isnot generally known a priori. Such obstacles can make finite difference techniques imprac-tical in many situations and difficult to incorporate into optimization schemes. A procedurefor estimating an optimal step-size is described in [36]. The optimal step-size attemptsto balance truncation error and round-off error. Unfortunately, since the discretization isgenerally nonuniform, the truncation error varies from point to point so that the results mustbe 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 toapproximate the solution to a partial differential equation and then implicitly differentiatingthis result to obtain a sensitivity approximation scheme. Recent advances in this areainclude the development of automatic differentiation packages (see [25], [47]), such asADIFOR. Given source code for state calculations, this package generates source code forsensitivity calculations. Once a computer code for generating a finite-dimensional stateapproximation has been constructed, the code is viewed as a sequence of compositionsof elementary functions, and differentiation is achieved by clever and repeated use of thechain rule. Automatic differentiation has been applied to shape optimization problems andoptimal design problems (see [16] and [34]), and the ideas have been considered and refinedfor a variety of other applications; see [26], [27], [33]. For more references and a generaldiscussion comparing this method with other shape optimization algorithms, see [22]. Adisadvantage of the discretize-then-differentiate technique is that in cases in which the meshis parameter dependent, as is the case in shape-optimization problems, then differentiationof the discrete PDF leads to mesh sensitivities on the right-hand side. Although much workhas been done in recent years to get a handle on these quantities (see [53]), calculatingmesh derivatives is still not well understood, particularly in cases in which the meshes areprescribed adaptively.

Several new techniques have been developed in an attempt to alleviate some of the dis-advantages of the two approaches mentioned above. The idea is to employ a "differentiate-then-discretize" scheme, or a so-called CSEM, which consists of implicitly differentiatinga partial differential equation to obtain a sensitivity equation and choosing an appropriatediscretization scheme based on that equation's structure. Then both the partial differentialequation and the sensitivity equation are discretized to obtain finite-dimensional equationsfor 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. Earlyuse of the sensitivity equation for optimal design purposes is seen in [38], and theoreticalframework describing sensitivity functions can be found, e.g., [29] and [42]. Variants ofsensitivity equation methods have been applied to a variety of problems. In [23], Godfreyinvestigates several uses of sensitivity equation methods (SEMS) for the analysis of chemi-cally reacting flows in aerodynamic applications. SEMs are used to compute aerodynamicstability derivatives in [24] and [45]. Optimal control of fluid flows and shape optimizationalgorithms 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 asa starting point for the interested reader.

As is the case with most new research, the CSEM approach provides its own set ofadvantages and disadvantages. Although the CSEM has a very mathematical foundation,derivation of the sensitivity equation can be difficult at times. Parameters can enter amodel in a variety of ways, and much of the analysis still must be done on an individualproblem basis for many cases. Identifying the correct boundary conditions for the sensitivityequation can be tricky. The mathematical constructions in Chapters 2 and 6 certainly providea platform for further analysis. However, developing a comprehensive mathematical theoryfor CSEMs will require much more sophisticated tools than those presented here. Even for arelatively small class of elliptic partial differential equations, there is no blanket approach tosharply 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 thesmoothness of the state to that of the sensitivity. Once a firm mathematical foundation isdeveloped, the advantage of the CSEM approach is that it will provide insight into situationswhere the sensitivity equation must be interpreted in a weak sense. This may be the casewith discontinuous coefficients or forcing functions when the discontinuities are somehowdependent on the parameter of interest. Another area where CSEMs may be useful is inoptimal design. CSEMs can provide cheap and accurate sensitivity approximations forgradient 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 makea contribution. This topic has long been linked to variational problems through study inthe area of the calculus of variations; see [39], [40], and [41] for applications focusingon structural design optimization. Optimal shape design encompasses a mathematicallychallenging set of problems as the domains of differential operators become parameterdependent. In this setting, the concept of differentiation of an operator becomes quitedifficult to characterize in certain instances.

Clearly, the advantages, disadvantages, and potential applications of CSEMs are stillrevealing themselves as the research continues. This book is a compilation of the authors'dissertation work while attending Virginia Polytechnic Institute and State University. Someof the numerical results in this book were generated using MATLAB on various UNIX plat-forms, including SUN workstations and Silicon Graphics Indigo and Indigo 2 workstations.The two-dimensional fluid flow results were generated using an adaptive finite element codedeveloped by Dominique Pelletier and other researchers at Ecole Polytechnique de Mon-treal. The code was modified to calculate sensitivities at Virginia Polytechnic Institute andState University by Dr. Jeff Borggaard and Dr. Dawn Stewart.

2 Mathematical Framework forLinear Elliptic Problems

This chapter surveys the mathematical background needed to analyze the regularity of aparticular class of linear elliptic partial differential equations, which includes some of themodel problems introduced in Chapter 3. In particular, the state equation takes the form ofa 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 thenpresent the necessary notation and framework to provide the existence and uniqueness ofboth strong and weak solutions for the general form. Finally, we discuss the implicationsthis framework has for the derivation of sensitivity equations. Readers who are interestedin an overview of the general approach of SEMs may choose to skip this chapter on a firstreading; it contains a number of technical mathematical details and is not essential readingfor those who seek a general exposure to SEMs. However, readers who are already familiarwith the topic may find that the details of this chapter help to solidify their understandingof 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, assumethat C Rn, n = 1,2, or 3, is a domain satisfying the cone condition with theboundary (q) such that locally, (q) lies on one side of (q). Consider the ellipticboundary value problem

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

2.1 PreliminariesThis section introduces some preliminary concepts and results that are used throughoutthe 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, andtheir related properties are reviewed, and the results from the theory of partial differentialequations relevant to this work are surveyed. The authors concede that the material inthis 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 basicconcepts from functional analysis, such as linear operators, inner-product spaces, and dualitypairings. 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]. Webegin with some general geometric assumptions. Sobolev spaces as well as their innerproducts and related norms are described. Let Q represent a bounded, connected, opensubset 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 ofUnless 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, ands to indicate that this variable may represent a vector of scalar variables. We assume a finitenumber 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 spatialvariable 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 frameworkconstructed in later sections: Let (X, \\ • \ \X ) , (Y, || • ||y), and (Q, \\ • \\Q) be normed linearspaces, 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 acontinuous linear operator DxG(x0) 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 —> YbyG1(x) = G(x,qo). If G 1 ( x ) has a Frechet derivative at x = XQ , then G (x, q0) has a partialFrechet derivative with respect to x at (XQ, qo), and this derivative is denoted

2.1. Preliminaries 7

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

is a Hilbert space. Let Hm denote the subspace of functions in L2 whosegeneralized partial derivatives up to order m also belong to L2 In this chapter, we willmake 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 byIt is well known that the set C Hm is dense in Hm and the completion ofC with respect to the norm \\ • \\m is defined to be the set Hm

0 . The dual space ofHm

0 , that is, the set of all bounded linear functionals defined on H0 m is denoted asH-m . For a general Hilbert space, V, we denote its dual space as V*. Recall that thenorm 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 productsand duality pairings. The subscripts are used only when this notation could be confusedwith that of an L2 inner product, and we remark on the specific meaning of the notationwhenever the usage is ambiguous.


2.1.2 Function Space Tools

Let represent a bounded, connected, open subset of Rn, n = 1,2, or 3. We beginby defining the geometric properties needed for the assumptions on the domain . Thefollowing definitions along with a more general presentation of these concepts can be foundin Wloka's book [54]. A function belongs to the class C if and all its partialderivatives up to order k are continuous, differentiable, and bounded on and the kthpartial 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 -diffeo-morphism, or T if the following conditions hold:

1. The coordinate functions of the map T denoted byeach belong to the class

2. The coordinate functions of the inverse map T-l denoted by xi = Mi ( y 1 , . . . , yn),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 one-to-one correspondence with the central plane xn = 0 and so that Ux lies on one sideof 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 diffeomorphismas a mapping that maintains an orientation with respect to the boundary . Now that thegroundwork has been laid, we assume that the domain belongs to the class C . Theappropriate values of k and are assigned as we consider each class of solutions in thefollowing 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 > , withm a nonnegative integer and l + 1 < k + . Then there exists a continuous linear traceoperator

with the property that

Here denotes differentiation in the direction of the outward normal.

2.1. Preliminaries 9

For the case that we are only interested in the mapping T0 ,we can relax the restrictionl + 1 < k + to l < k + X. One can also characterize the kernel of the trace operatorintroduced above. We condense this theorem to concentrate only on the particular resultneeded for the analysis in the following sections, and we also need the inverse theoremfound in [54].

Theorem 2.3. Suppose that is (0, 1)-smooth. IfT0: Hl(ti) -> H1/2( is the traceoperator given above, then

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

with the property that

More detailed results concerning Sobolev spaces, trace operators, and related subjectscan 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 formula-tions rely on the concept of a Gelfand triple. We also need some elementary definitions andresults concerning bilinear forms and their properties.

Definition 2.5. Let V and H be Hilbert spaces. Suppose that the mapping i: V —> H is acontinuous, 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 followingsections, 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 innerproduct 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


This provides a continuous extension of the inner product on H to a continuous linearfunctional on V* x V, and it gives a representation of the linear functionals in V* by meansof the inner product in H.

Definition 2.6. Let V be a Hilbert space; then a mapping a: V x V —> R is called abilinear 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 c1,c2 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 uniquelydetermines a continuous linear mapping L: V —> V by

and || a || = ||L||. The proof of this statement is a consequence of the Riesz representationtheorem, and a more general version can be found in Theorem 17.8 of [54]. Moreover, let(h1 , i (v))H denote the representation (2.4) of an arbitrary linear functional l V*. Then bythe Riesz representation theorem, we have the relation

where R is the Riesz isomorphism R: V* V. Using this relation along with (2.9), weobtain

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

And A is said to be V-elliptic if the bilinear form a(•, •) is. The construction given aboveleads us to the Lax-Milgram theorem, which provides the existence and uniqueness of weaksolutions 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 thecorresponding operator A: V —> V* given in (2.11) is a linear topological isomorphismbetween 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 dependscontinuously on f.

Many forms of this theorem are given in the literature; see [5] and [21]. This particularversion is found in [54]. Once these fundamental results have been introduced, we are readyto 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 ProblemsWe note that for a fixed q, the problem (2.1)-(2.2) defines an elliptic boundary value prob-lem on a fixed domain, = (q). Also, as we show in Chapter 4, the method of mappingsis often used to transform the problem (2. l)-(2.2) on (q) to a problem on a fixed compu-tational domain In either case, without loss of generality, we may study the regularityof solutions by fixing the domain .

This section contains a survey of theoretical results characterizing the regularity ofelliptic boundary value problems of the general form

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

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


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 ellipticequation (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 functionsuch that and

Ifz is a strong solution to (2.16)-(2.17) and if both andthen 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 thatA0 is a strongly elliptic, self-adjoint differential operator; see [54]. The following theoremcharacterizes 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 strongsolution.

Theorem 2.10. Assume that , and let If the data andthen there exists a unique solution z D(A0) 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 unique-ness of weak solutions to a state equation of the form given in (2.16)-(2.17). The variationalform of the state equation is expressed, and Theorem 2.8 can be used to assert the existenceof the weak solution to the state equation. Note that

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

2.2. Regularity of Elliptic Boundary Value Problems 13

R by the following:

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

and

Here, m2 is the upper bound placed on the coefficient k(x, q) for all xq Q. The constant c involves the lower bound on the coefficient k(k, q) as well as theconstant prescribed by the Poincare inequality. See [1], [13], and [54] for details regardingthese results as well as many other theoretical results related to variational formulations ofpartial 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

and for all

such that

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

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 directconsequence of the construction given above and the Lax-Milgram theorem.

Theorem 2.12. Let the domain ) and the boundary datathen there exists a unique solution to the operator equation

(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 varioustypes of boundary conditions treated using variational techniques. The reader may also


choose to consult several other classical texts, such as [2], [20], and [37], for generalexpositions on the theory of partial differential equations.

In the previous sections, we constructed a mathematical framework for analyzingthe regularity of elliptic boundary value problems on a fixed domain. However, we want toapply these concepts to equations defined on parameter-dependent domains in order to derivesensitivity equations for shape parameters. With this goal in mind, we now transition to themethod of mappings that provides a mechanism for transforming equations on parameter-dependent domains to equations defined on fixed domains, thereby allowing us to use theresults of this chapter to analyze sensitivity equations.

2.3 Mappings and Implications to Sensitivity AnalysisIn Chapter 4, we discuss a technique for transforming an elliptic boundary value problemdefined on a parameter-dependent domain to a "transformed" boundary value problem thatis also elliptic but is now defined on a domain independent of the design parameter. Theprocess of transforming can introduce an explicit dependence of the transformed equationon 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 thesensitivity equation for the transformed state can be derived in a mathematically rigorousfashion. In this section, we introduce the necessary mathematical framework and derivethe (operator) sensitivity equation. The relationship between the operator equation and thedifferential 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, andlet 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(z0 , q0), and

3. [ zG(z0 , 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 Q such thatfor any q W, the equation

G(z, q) = 0

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 qG(z0 , q0) exists, thenu(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 15

2.3.2 Operator Formulations of Sensitivity Equations

In this section, we outline an approach for deriving sensitivity equations that relies on theimplicit function theorem. For the analysis of elliptic boundary value problems in section2.2, each of the constructions ultimately takes the form of an operator equation such thatfor each q Q

where C: Q (Z, Y) is a continuous linear operator, Z and Y are Banach spaces, andF: Q —> Y. One can further generalize this equation by defining the operator G: Zx QY 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 operatorequation (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 operatorDqz(q0): 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 Theorem2.13 to derive an operator sensitivity equation for Dqz(q0). In the following paragraphs, weprovide the construction and conditions necessary to apply the theorem. We then move tosome 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 abounded linear operator [Dq£(qo)]: Q B(Z, Y). If we define the operator [L(qo)]: QY by

then it is straightforward to show that L (q0) is a bounded linear operator from Q to Y. Furthersuppose that D qF(q 0): Q Y exists in the Frechet sense. Then the partial derivative ofG 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.


Theorem 2.14. Let q0 int( Q) be fixed. Suppose that there exists a unique Z0(q0) int(Z)so that G(z0(g0), q0) = 0. Further suppose that [L(q0)]-1 exists in B(Y, Z). If the Frechetderivatives Dq£(q0) and DqF(q0) 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) ofTheorem 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) belongsto the function space Z, and it is the solution to the operator equation (2.33). At thebeginning of the following chapter, we present an example to demonstrate the derivationof the sensitivity equation. It is important to note that the choice of function spaces Z andY is nontrivial. Moreover, one can construct relatively simple state equations for whichthe 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 explorethe concept of a sensitivity and to illustrate several of the mathematical issues involved inits computation. For clarity of presentation, we have chosen state equations whose spatialdomains are one-dimensional. In addition to their simplicity, these examples are convenientfor comparing numerical results since both the state and the sensitivity equations haveanalytical solutions. They also provide the reader insight into several aspects of sensitivitycomputation that are inherent to our approach. The first example is used to illustrate theuse of the mathematical framework outlined in the previous chapter. A state equation isintroduced, 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 valueproblem—provide us with one-dimensional examples of state equations with parameter-dependent domains. The state and sensitivity equations are outlined along with their respec-tive closed-form solutions. In subsequent chapters, several aspects pertaining to sensitivityanalysis and the implementation of CSEMs are explored for each of these problems. Com-putational methods for numerically approximating the solutions to the sensitivity equationsare described, and numerical results are given.

3.1 Heat in a Thin Rod with a Parameter-DependentForcing Term

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

with boundary conditions

17

18 Chapter 3. Model Problems

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 thecorresponding sensitivity equation for p(., q). Here, we illustrate the importance of choosingthe function spaces Z and Y correctly, and we note that a formal derivation can often providethe 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, £ = £(q0) = A0

is constant with respect to the parameter q. Furthermore, the operator F: Q L2(0, 1) isdefined by

Using this framework, the operator G: Z x Q L2(0, 1), and the operator equation givenby

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 sec-

tion 2.2.1. The structure of L implies that its Frechet derivative is the zero operator inB(Q, B(Z, Y)). However, the derivative DqF(q) does not exist in the L2-sense, and Theo-rem 2.14 is not applicable for this construction.

However, if one chooses the spaces Z = H10(0, 1)and Y = H - l ( ), then the operator

£(q): Q B(H10(0,1), H-l is defined by

The A operator is defined in (2.23). It follows from the framework presented in section 2.2.2that A-l = [£(q) - 1 exists in B (H - 1 H1

0(0, 1)), and the Frechet derivative DqL(q0)is the zero operator in B(H1

0(0, 1), H - l Furthermore, one now defines the operatorF: 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 Frechetderivative 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

for all u(•) H10(0, 1) and for each h Q. Here, is the Dirac delta function

with mass at x = The hypotheses of Theorem 2.14 are now satisfied. Therefore, the

sensitivity p exists (in H10(0, 1)) and satisfies the operator equation (in

H-l (0,1)) given by

That is, for any qo Q, the sensitivity equation is well defined and is given by the variationalequation

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 operatordescribing the state equation, A in the above paragraphs, is also the differential operatorassociated with the sensitivity equation. Hence, the operator sensitivity equation is derivedwith the appropriate mathematical rigor, and a corresponding differential equation, oursensitivity equation, is constructed using the operator expression. In particular, p( , q)satisfies the following differential equation

where belongs to H- l (0,1) for each (1,2). More-over,

where is the Dirac delta function with mass at The boundary conditions for

are clear since p(•, q) H10(0,1).

In the following sections, we consider other examples of state equations and theirassociated sensitivity equations. Once the examples are introduced, we examine specificmathematical and computational issues related to sensitivity analysis for these types ofproblems. Although they are somewhat simple in nature, these model problems illustratemany of the fundamental issues addressed in this book.

3.2 Heat in a Thin Rod with a Parameter-DependentBoundary

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


3.2.1 State Equation

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

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 determinethe sensitivity of the state to small changes in the parameter. The forcing function, f(•), isdiscontinuous on (0, 2) and belongs to the space L2(0, 2). For a fixed q (1, 2), the statew(•, q) is a function belonging to H2(0, q) D H1

0(0, q). In particular, one can verify thatthe 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 thesensitivity

From (3.10), we see that for each q Q = (1, 2), the sensitivity is a linear function of xand has the following form:

Note that for a fixed q Q, s(x, q) is a C function on (0, q). We now derive a differentialequation for which the function given in (3.12) is a solution.

Beginning with the state equation defined on the interval (q), one can implicitlydifferentiate (3.7)-(3.8) to obtain a sensitivity equation. At this stage, it is important tonote 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 thiscase 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


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

Remark. Observe that the condition at x = q in (3.14) requires boundary informationfrom the spatial derivative of the state. This coupling arises as a result of the domain of thestate equation, [0, q] in this case, depending explicitly on the parameter q. Even for verysimple linear elliptic state equations of this type, the corresponding sensitivity equationsare coupled to the state through the boundary conditions. This feature is characteristic ofsensitivity 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 math-ematical as well as computational issues. In the case of an elliptic sensitivity equation withDirichlet boundary conditions, the function determining the parameter-dependent boundarycondition w(q) will be required to belong to H ( (q)) (for each value of q) to ensurethat the sensitivity equation possesses a unique, classical solution. To guarantee that a weaksolution exists, the function w(q) should belong to H ( (q)). Further details regardingthe regularity of solutions to elliptic partial differential equations are presented in Chapter2 and references therein. Also noteworthy is the fact that the data on the right side of thesensitivity equation are very smooth. That is, the sensitivity equation is homogeneous sincethe source term f(x) in (3.7) does not explicitly depend on the parameter q. As shownin the previous example, when the source term, or forcing function, depends explicitly onthe parameter, derivation of the sensitivity equation needs to be done with care but can berigorously achieved using variational techniques.

Recall that the state w(x, q) in (3.10) has a piecewise linear, discontinuous secondderivative. For this particular problem, the sensitivity is very smooth as compared to thestate. This fact is somewhat counterintuitive, since one might expect to lose smoothnesswhen examining derivatives of the state. Over the course of our research, we have found thatsmoothness may be gained, lost, or even preserved when moving from the state equationto the sensitivity equation. Indeed, a general rule for this phenomena is yet unclear. Thisis partly because there are so many different ways in which a parameter can appear in amathematical model of a physical system. Furthermore, the study of linear elliptic systemsis only a first step since many of the most complicated physical systems are modeled usinghyperbolic or parabolic partial differential equations. For our work, we continue to answerthe question of smoothness case by case. In the next section, we examine a nonlinear examplethat allows us to gain significant insight into computation of sensitivities for Navier-Stokesequations presented in later chapters.


3.3 Nonlinear ModelIn this section, we consider a simple one-dimensional nonlinear boundary value problemwith spatial domain in one dimension. As in the case of the linear model, the designparameter is denoted by q, and the design parameter determines the length of the intervalover which the differential equation is defined. The corresponding sensitivity equation isgiven, and we note that the sensitivity equation for this model problem is linear althoughthe 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, nonlineardifferential equation


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, inorder to comment on the use of continuous sensitivity equation methods in conjunction withoptimization algorithms. Let 0 < x1 < x2 < • • • < xp < 1 be fixed locations and assumethat 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 ofdata 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 tocompute the state, and the sensitivity, and to form the computation(3.19). This involves first solving (3.15)-(3.16) for w(x, q) and then computing s(x, q).

3.3. Nonlinear Model 23

3.3.2 Sensitivity Equation

One benefit of using the model problem is that we can calculate the sensitivityby 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) satisfiesthe linear differential equation


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

4 Computational Algorithms

This chapter describes two computational methods for solving sensitivity equations. Notethat 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 contextof numerical sensitivity approximations are given. In particular, mapping techniques arediscussed, and we show how these techniques can be blended with SEMs to numerically ap-proximate sensitivities. We also show that there are some pitfalls associated with combiningthese techniques.

4.1 The Method of MappingsThe examples introduced in Chapter 3 belong to a class of elliptic boundary value problemsdefined on domains that depend on a parameter. For many engineering applications, a typicalapproach to such problems is to begin by transforming the problem to a fixed computationaldomain that is not parameter dependent. This computational domain is often more regularin shape, which simplifies grid generation and can improve the accuracy of numericalcalculations. This mapping technique is very common for problems involving complexgeometries that occur in fluid dynamics. The book [53] is an excellent source of informationabout 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 isused extensively in the theory of partial differential equations. We briefly summarize thetheoretical results that provide the mathematical foundation for the use of this technique. Thetransformation theorem on page 80 of [54] gives the conditions under which Sobolev spacesdefined on the physical domain, are equivalent to those defined on the computationalspace Let T +: be a Ck diffeomorphism between the domains. Here, we areimplicitly assuming that the domain is smooth enough to construct the transformation

25

26 Chapter 4. Computational Algorithms

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 theoryof elliptic differential operators from [54] also gives the important result that under theappropriately smooth diffeomorphism, (strongly) elliptic operators defined on transformto (strongly) elliptic operators defined on . This is also true for hyperbolic and parabolicoperators as noted in [32]. A general form for the relationship between the original operatorand the transformed operator is given on page 143 of [54]. We now address some of thepractical aspects of using transformations.

From a computational standpoint, transforming can be a complex process for partialdifferential equations with spatial domains in R2 or R3. In addition to the issue of regular-ity, one is concerned with loss of accuracy. The mappings are implemented by defining anew coordinate system that maps to a fixed computational space, . Transformationsthat produce coordinate systems that are orthogonal at boundaries are preferred. Other-wise, accuracy decreases as orthogonality declines (see [53, pp. 3-5]). A two-dimensionalcoordinate system on takes the form

where (x, y) represents the coordinate system in The transformation isgiven by

Aside from the issue of orthogonality, T must also satisfy other mathematical properties. Inparticular, T is required to be one-to-one, and the mapping should maintain some smoothnessin the distribution of grid points. Ideally, T should also provide some mechanism forclustering grid points once a discretization has been constructed.

For problems involving complex geometries, determining the mappings can be acomplicated process. The transformations can be constructed using conformal mappingmethods, partial differential equation methods, or algebraic methods. Conformal mappingmethods are based on the theory of complex variables and are used only when working witha physical domain given in two spatial dimensions. Partial differential equation methodsare also widely used. With this approach, the computational domain is rectangular with auniform grid, and the location of corresponding points in the physical domain is determinedthrough the solution of a system of linear, or nonlinear, partial differential equations. Ellipticgrid generators are the most commonly used. However, there are methods that use hyperbolicor parabolic equations to determine the mappings. To determine a mapping, these equationsare usually solved numerically. Consequently, this method may introduce numerical errorsinto the grid-generation aspect of the overall computational scheme. As we will see later, thespatial derivatives of the transformations are needed to construct the transformed equations.Hence, questions of invertibility and differentiability become more tedious to resolve whenthe mappings are computed numerically. For this reason, computing the transformations

4.1. The Method of Mappings 27

numerically can be a drawback. However, for very complicated geometries, this approachmay be the only one suitable for numerical computation.

Finally, algebraic methods determine an algebraic expression which relates pointsin with those in . Interpolation schemes are used to represent the interior points ofthe domain in terms of the points along the boundaries. Implementation of these methodsis often simple and fast. Another advantage of algebraic methods is that the derivativesof the transformations can be computed analytically, thereby reducing computational timeand avoiding the introduction of additional numerical inaccuracies into the computationalmethod. However, these methods may be difficult to implement for very complex ge-ometries. An algebraic approach is used to construct the transformations presented in thischapter.

Once the transformation is determined, the process of transforming the differentialequation begins. We remark here that the crucial step is to derive the relationship betweenthe differential operator defined on and the corresponding operator defined on £2. Thisrequires the computation of the derivatives of the mappings with respect to the spatial vari-ables. An example of this process is given in the following section, and the details ofthis relationship will be presented in the context of the example. For equations posed onparameter-dependent domains, the mappings are also parameter dependent. In order to con-struct the transformed equations on the computational domain, derivatives of the mappingwith respect to the spatial variables are computed, and these derivatives will also involvethe parameter. Hence, the design parameter often appears explicitly in the transformeddifferential 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 computationalmethod can result in the need to also compute the partial derivative of the mapping withrespect to the parameter. Usually referred to as a "mesh sensitivity," this derivative isoften difficult or even impossible to compute in two-dimensional and three-dimensionalproblems. In particular, this derivative is difficult to obtain when the transformations arecomputed numerically or when adaptive algorithms are used. The mesh sensitivity appearsin the example presented in the following section. We comment further on this issue as weexamine the following example.

4.1.2 An Algebraic Transformation

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

by

Note that the function M defined by


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

4.2 SEMsThis section presents two examples of SEMs used for the numerical approximation ofsensitivities. We give a detailed description of the schemes used to obtain the numericalcalculations presented in subsequent chapters. The transformation techniques described insection 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 prob-lem given in (3.7)-(3.8) as a platform for investigating the implementation of the mappingtechniques, in combination with SEMs, for the numerical approximation of sensitivities. Insection 4.2.3, we demonstrate how one of the SEMs can be applied to the nonlinear modelgivenin(3.15)-(3.16).

As illustrated in this chapter, there are several ways to implement SEMs, and thesevariations can yield algorithms with different convergence properties. We consider twospecific SEMs used in conjunction with the transformation techniques described in section4.1. The first is based on transforming both the state and the sensitivity equations to thecomputational domain, solving the transformed equations, and mapping these solutions backto the physical domain. The second approach transforms the state equation and then derivesits sensitivity equation. Once the state and sensitivity systems are solved, the solutions aremapped 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 questionsneed to be answered before a complete theory can be developed, and we address some ofthese issues with the computational schemes and numerical results presented here. We nowdescribe 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-discretizemethod. Given the state equation defined on the physical domain, the corresponding sensi-tivity 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 re-quired the use of differential operators and subsequent differentiation of those operators.However, in many instances, the formal implicit differentiation of the boundary value prob-lem and boundary conditions is acceptable. Once the state and sensitivity equations areestablished, the transformation techniques are used to derive the corresponding transformedstate and sensitivity equations posed on the computational domain. The discretization isthen applied to these transformed systems, and numerical calculations are performed.

The preceding discussion outlines the general structure of the H-SEM, and in thefollowing paragraphs we construct a particular example by applying the H-SEM approachto 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 insection 3.2.2. Once the state and sensitivity equations are described, the method of mappingsis applied. For the sake of clarity, the equations are given here. The state equation is givenby

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 transformedfunctions. The transformed state is denoted by (-, •); likewise, the notation (-, •) refers tothe transformed sensitivity. For and q (1,2), define

and

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

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

and


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


Likewise, the transformed sensitivity satisfies the differential equation


Note that equations (4.7)-(4.8) and (4.9)-(4.10) remain weakly coupled through the bound-ary 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 discretiza-tion for this particular example is discussed in section 4.3. The H-SEM is summarized belowto 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 givenin (4.2), the closed-form solutions to the transformed state and transformed sensitivityequations 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 fixedvalue of (1, 2), it follows from Theorem 2.10 that the transformed state equation given

Theorem 2.2 implies that thein (4.7)-(4.8) has a unique strong solutiontrace of the function at the boundary can only be interpreted as a functionbelonging to the function space Hence, the boundary data (4.10) of the transformedsensitivity equation only satisfy the requirements of Theorem 2.12, and the transformedsensitivity equation is only guaranteed to possess a weak solution for afixed value of the parameter. In this one-dimensional example, the transformed sensitivityis actually a strong solution to the differential equation. In particular,However, in higher dimensions, the appearance of the trace of the gradient of the statewithin the boundary conditions of the sensitivity equation is an issue that presents not onlytheoretical but also computational dilemmas. The computational issues are addressed inthis as well as in later chapters. First, we describe an alternative approach for obtaining asensitivity equation.

4.2.2 Abstract Version of the Semianalytic Method

The preceding section developed the transformed sensitivity and its relationship to theoriginal sensitivity under the transformation M. One can also investigate the sensitivity ofthe transformed state. An advantage of this approach is that the derivation of a sensitivityequation can be mathematically justified, and the details of this justification are presentedin 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 frameworkwhich mathematically justifies the formal argument given below.

This section describes an abstract version of the semianalytical method. The processoutlined here differs from the hybrid method in the order in which the sensitivity equation isderived and the transformations are performed. This approach to the computation of the sen-sitivity is similar in spirit to the semianalytical method (SAM), a technique often used in theengineering community [6]. Roughly speaking, the SAM begins by first transforming thestate equation to the computational domain. The second step is to discretize the state equa-tion, thereby producing an algebraic system. This discrete equation is then differentiated toobtain a discrete sensitivity equation, which is solved using special techniques.

An abstract version of this method (A-SAM) may be constructed by deriving a sen-sitivity equation after transforming but before discretizing the state equation. We focus onthis 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 firststep 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 toobtain an equation for the sensitivity of the transformed state. The framework for the rigor-ous mathematical derivation of the sensitivity equation is given in Chapter 2, and the detailsfor this particular example can be found in section 3.1. In the following, the mechanics ofthe approach are described and an overview of the method is given.

On the computational domain, define the sensitivity of the transformed state byThe goal is to derive a differential equation for which is a


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 differ-ential equation

where is the Dirac delta function with mass at The reader is referred to section3.1 for the details.

For this example, can be calculated directly from (4.11). In particular,

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 statecan be derived using (4.5)-(4.6) and the chain rule. Beginning with the definition of thesensitivity of the transformed state, it follows that

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

Observe the appearance of the mesh sensitivitv. and the spatial derivative ofin 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 differ-ential equation method discussed earlier. Obtaining derivatives of these maps can be verydifficult, 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 ofthe 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 generallyneeds to compute not only the sensitivity of the transformed state but also the spatial deriva-tive of the transformed state and the mesh derivative As mentioned, the meshderivative 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 theoriginal sensitivity s(-, •) by means of the sensitivity of the transformed state p(-, •)• Thatis, we may analyze the existence and regularity of s(-, •) by first examining these issuesfor p(-, •) and then applying those results to s(-, •) through the equation (4.18). In somesituations, such as the current example, this can be advantageous. Recall that the existenceof and the rigorous derivation of the corresponding sensitivity equation are discussedin section 3.1. Before moving to the numerical solution of the state and sensitivity equationsfor each of the methods previously detailed, we first illustrate how the H-SEM approachcan 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 morerealistic nonlinear problems, such as the Navier-Stokes equations discussed in Chapter7. 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 constructFor this problem, we also include an intermediate step

for homogenizing the boundary conditions of the transformed state equation. Hence, wedefine 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 thecomputational domain and given by the Dirichlet problem



In a similar manner, we transform the sensitivity equation (3.21)-(3.22) to the com-putational domain. In particular, we let and defineby

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 informationfrom 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 theterms involving In practice, one must use some numerical scheme to solve the boundaryvalue problem (3.15)-(3.16), and the computation of s (x , q) must be accomplished by usingthis 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 wediscuss several schemes, we concentrate on a projection method approach in later chapters.The basic idea can be extended to complex aerodynamic flow problems. However, manytheoretical 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)of the transformation . In particular, one needs or else a numencalapproximation 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 SAMfrom section 4.2.2, then the mesh sensitivity, or an approximation, is required to recoverthe original sensitivity; see (4.18). The numerical approximation of a mesh sensitivity isthe source of considerable computational complexity for many realistic CFD applications.Consequently, one advantage of mapping both the state and the sensitivity equation is thatthe computation of this gradient can be eliminated.

Finally, we note that once and are computed on the stateand sensitivity can be recovered on through the expressions

4.3. Approximation Framework 35

and

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

4.3 Approximation FrameworkIn this section, we outline the finite element methods used for the numerical implementationof the algorithms discussed in the previous sections. Variational formulations for each ofthe equations are given. Brief descriptions of the finite element spaces, discretization, andgrid construction are also included.

4.3.1 Variational Formulations

For the following discussion, note that the underlying function space, V, is defined to bethe 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 anarbitrary 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) andthe L2 inner product defined in section 2.1.1, produces the weak form of the transformedstate 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 thevariational 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


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

If is

then and satisfies the boundary conditions s*(0) = 0 andIt follows that belongs to V and solves the

differential equation


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 recoveredusing the relationship Details regarding this technique fornonhomogeneous boundary conditions can be found in [5]. In this particular case of theone-dimensional problem, the solution to (4.32) is the trivial solution, and the precedingconstruction of s* and v is somewhat unnecessary. However, the definition of allowsus to clearly point out a numerical issue inherent to continuous SEMs. With the reader'sindulgence, we discuss the issue in some detail in the following sections.

For the implementation of the A-SAM method, one must numerically approximatethe sensitivity of the transformed state, p(., q). The derivation of the sensitivity equationis 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 thesensitivity 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 linearfunctional is given by

Nonlinear Model

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


and the transformed sensitivity equation

with homogeneous boundary conditions

and

respectively. Observe that we have homogenized boundary conditions prior to any varia-tional 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 outlinedin the sections above. Note that for each and exist and belong to

Then for each V the transformed state z satisfies the variationalequation

Likewise, the transformed sensitivity equation satisfies

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

4.3.2 Piecewise Linear Finite Elements

In the following sections, we briefly describe the discretization and numerical implementa-tions used to compute state and sensitivity approximations.

Linear Model

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


where is the standard continuous piecewise linear basis function.For the computations presented for the linear model, we use the partition of de-

veloped above for both state and sensitivity approximations. Hence, the meshfor 7 = 0, 1, . . . , N is used to calculate approximations to ( , q), v( ), and p( , q). Thefunction 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 mannerand The matrix K is given by

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

and


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

where is the approximation to given by

Note that the subscript notation used on refers to the use of the H-SEM approach forthe sensitivity calculation. The approximation to the original sensitivity is obtained usingthe mapping T and the relation Likewise, we use the notation

to denote the sensitivity approximation obtained using the abstract version of theSAM method. For this approach, the approximation and the relationship

are used to calculate the sensitivity approximation. Note that the spatial derivative and thesensitivity of the mapping M are hardwired into the relation. Therefore, no error is incurredfrom the mapping terms when we recover using the expression above.

Comment. The use of for introduces an error into that is independentof 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 toreconstruct through the expression above. These issues play an important role inthe numerical results presented in section 5.1.3.

Nonlinear Model

Although there are several possible choices for finite element spaces, we limit our discussionto the simplest (convergent) scheme. Recall that the nonlinear model motivates the algorithmwe apply to two-dimensional flow problems in later chapters; hence, we note that in morecomplex problems one must choose these spaces with care to ensure the algorithm satisfiesthe appropriate convergence criteria (inf-sup conditions, etc.).

For this example, we construct a finite element scheme that allows for more flexibilitythan in the previous linear model. In particular, the weak coupling of the transformed stateand transformed sensitivity equations allows one to choose different mesh sizes for eachof these numerical calculations. We remark that such an option is certainly viable for thelinear model; however, it is not explored in this work. In this section, N denotes the numberof basis functions used for the finite element solution of the transformed state equation, andM 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


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

and

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

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 ofequations as

Also note that the coefficients of the unknowns involve the computation of several formsof inner products dictated by (4.53). With a certain amount of algebra and patience, thesecoefficients can be determined analytically. This was our approach; however, these coeffi-cients can also be generated numerically if one prefers. The nonlinear system given abovecan be solved using a black box nonlinear solver, or one can generate one's own code as anexercise in numerical methods. The approximations given later in the book were generatedusing Newton's method with a line search. Although it might seem a bit impervious, theequation in (4.54) yields an N x N linear system of equations. Once ZN is known, then theterms involving can be used to determine the matrix and right-hand-side vectors forthis equation.

Note that UM (•) depends on ZN (•) and its spatial derivative (•) on To empha-size the dependence, we let uN,M(.) denote the solution of (4.54), given that zN(.) obtainedfrom (4.53) is used in (4.54). At this point, two important observations play a key role inthe 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


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 becombined to construct numerical solutions of ZN ( , q) and uN,M ( , q) so that the errorin uN,M( , q) is sufficiently small to ensure convergence of optimal design algorithmsbased 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 (discon-tinuous) function. However, the actual transformed sensitivity u( ) is smooth, andone might expect to lose at least one order of accuracy in uN,M( ). In fact, things canbe much worse unless special care is exercised.

There are two obvious fixes with which to address these issues. One could use higher-order 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 alsoused for the state equation. The other obvious fix is to use mesh refinement in M (assumingaccuracy in N). A third approach makes use of smoothing projections. The idea is similarto 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 themodel problem and to a two-dimensional fluid flow problem.

5 Numerical Results

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

5.1 Linear ModelHere we present numerical approximations to w(x, q) and s(x, q) obtained by using thecomputational methods described in sections 4.2.1 and 4.2.2. All computations presentedfor the linear model use the same grids for both state and sensitivity approximations. Sincethe transformed state must be used in the application of both H-SEM and A-S AM, we beginwith a brief section reporting state approximations including error calculations. Sensitivityapproximations 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 ap-

proximations to for various values of N; these approximations converge rapidlywith grid refinement. Similar behavior is observed over a range of parameter values. Thecorresponding approximations to , obtained by transforming the finite elementapproximation, ( , q), back to the physical domain, are shown in Figure 5.2. ComparingFigures 5.1 and 5.2, one can see that convergence of the approximations is preserved underthe domain transformation.

Figure 5.3 shows the Hl error in WN(X, q) for values of q between 1.1 to 1.9. Thevalues of N range from 3 to 33 and are indicated in the legend. Note that the rate ofconvergence is better for q 1 as the quadratic term (see (4.11)) in the transformed statebecomes 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 algo-rithm. 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 conver-gence of (x, 1.5) to (x, 1.5). Although numerical results are given only for q = 1.5, theerror in the approximate boundary condition and the qualitative behavior of the convergenceare similar over the entire parameter range. We take a moment to clarify this issue in thefollowing paragraph.

Recall that ( ) is obtained using a piecewise linear approximation. Thus, the


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 usedto approximate the spatial derivative at the boundary point £ = 1. Figure 5.6 showsa piecewise constant approximation used to obtain an approximate boundary condition

. Hence, the error in (l) results in sensitivity errors that can be attributed to thepoor approximation of this boundary condition. There are techniques that can be used toobtain 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 two-dimensional and three-dimensional problems. As an alternative, projection techniques havebeen developed to enhance the accuracy of the spatial derivative for nominal expense. Thistechnique 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 forsensitivity calculations. First, recall that (x, q) is constructed from andusing the relationship.

Note that the subscript A is used to denote the sensitivity approximation obtained using theA-SAM approach. The finite element approximations to p( , 1.5) are shown in Figure 5.7for various values of N. Since the sensitivity equation (4.33) is linear, the approximationspN( , 1.5) converge as expected. When constructing (x, 1.5) from (5.1), the piecewiseconstant approximation of produces discontinuities in , as shown in Figure5.8. These discontinuities occur at points of the physical domain that correspond to meshnodes of the computational domain lying in the interval . Note that the expressionsfor the mesh derivatives in (5.1) are hardwired, continuous functions, and the finite elementapproximations to p( , q) are continuous. It follows thatas . However, one does not get convergence in the energy norm since

does not belong to . Even if one computes the Hl error of the sensitivityapproximation over each individual element of the mesh and computes the total error by


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 meshrefinement; see Figure 5.9.

The numerical results presented in the previous sections indicate that each SEM im-plemented here suffers from computational difficulties. Both algorithms require accurategradient information from the state approximation to accurately approximate the sensitiv-


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 entirespatial domain. We also see that the use of mapping techniques in conjunction with SEMsmust be done with care, and the mathematical analysis of the sensitivity equations canbe 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 approximationsto 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 ModelThis section presents numerical approximations for the nonlinear model problem using theH-SEM. We also evaluate the effect of accurate sensitivity approximations within the contextof an optimization algorithm constructed to solve the inverse design problem outlined insection 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 stateand sensitivity equations with their exact, or true, solutions. All the figures shown in thissection depict the approximation to the state w and the sensitivity s obtained using therelations 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 exactsolution of the nonlinear state equation (for each q > 1). Figure 5.10 shows the finiteelement approximations to the solution of the boundary value problem at two parametervalues: q = 2 and q = 1.2. Notice that at q = 2, the N = 4 finite element model providesan 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 expectedbecause the gradient of the solution becomes singular as q —> 1+ and hence the problembecomes 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 sensitivityequations, respectively, and (4.54) is coupled to (4.53) through the appearance of the spatialderivative in (4.54). Our first approach to dealing with this term is to simply use thepiecewise constant spatial derivative of the finite element approximation for ZN . The useof the piecewise constant (PWC) derivative is noted explicitly in the following figures inorder to distinguish them from other numerical results presented in section 5.3. Figure 5.11shows the finite element approximations for the sensitivity s(x, 2) with N = M rangingfrom 2 to 16. Observe in Figures 5.10,5.11, and 5.12 that although the finite element schemeproduces excellent solutions to the state equation when N = 4, the error in the correspondingsensitivity does not diminish to a comparable level until N = M = 16. It should benoted for this example that any error in the state approximation propagates into errors in


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

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


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 agiven 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 avisualization of this phenomenon for the nonlinear example.

At this stage, the reader may wonder if there are ways to improve the accuracy ofthe sensitivity approximations for a given amount of error in the state approximation. Onepossible fix is to use mesh refinement in M. Figure 5.13 shows the results for this techniquewhen N = 2 and M ranges from 2 to 16. Note that improvements in the accuracy ofthe sensitivity approximation are limited by the accuracy of the state approximation and,more precisely, the accuracy of the piecewise constant derivative approximation forWhen 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 accuratesensitivity approximations. Figures 5.15 and 5.16 show that the sensitivity approximationsbecome 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 errorin the sensitivity approximation for a range of mesh sizes. As N = M increases, weobtain convergence of the scheme, but for small values of N, the sensitivity approximationscontain large errors and the convergence of the finite element approximation to the analyticalsolution is not at all monotone; see Figure 5.16. Figure 5.18 is a graph of the L2 error ofthe sensitivity approximations for various values of q, and we can see that the behavior ofthe 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 toobtain accurate sensitivity approximations for parameter values near q = 1. We observesimilar behavior when the H-SEM is applied to two-dimensional flow problems in Chapter 7.


Figure 5.13. Numerical approximations to the solution of the sensitivity equationat 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 theconvergence 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 problemreplaces , the sensitivity of the discrete solution, with an approximation to

, in particular , from (4.26).


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

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


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

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


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

Figure 5.19. Gauss-Newton algorithm.


Figure 5.20. Data generated at q = 2.

Figure 5.21. Data generated at q = 1.4.


Table 5.1. Matching data for the optimization.

p = 4

0.12500.25000.50000.7500

q=2

0.44950.82841.46412.0000

0.45160.77221.40801.8338

Perturbation0.0021-0.0563-0.0561-0.1662

q = 1.4

0.82481.35412.13942.7553

0.74311.38751.98482.5758

Perturbation-0.08170.0335-0.1546-0.1795

p = 16

0.03120.06250.09380.12500.15620.18750.21880.25000.31250.37500.43750.50000.56250.62500.68750.7500

q = 2

0.12130.23610.34520.44950.54950.64580.73860.82841.00001.16231.31661.46411.60561.74171.87302.0000

0.11260.22070.36700.43720.59160.66110.76510.86870.90911.26741.24861.60861.64831.90671.96682.0554

Perturbation-0.0088-0.01540.0218-0.01230.04210.01540.02650.0403-0.09090.1051-0.06800.14450.04270.16510.09380.0554

q = 1.4

0.26770.48060.66290.82480.97201.10791.23471.35411.57491.77681.96412.13942.30482.46192.61172.7553

0.26900.44800.63750.75630.97721.03351.28201.41061.48211.62651.94602.05042.09502.39732.80402.9296

Perturbation0.0013-0.0326-0.0254-0.06850.0052-0.07440.04730.0565-0.0928-0.1503-0.0181-0.0890-0.2098-0.06460.19220.1743


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.


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

N224488161632326464128

M25498171633326564129128

CONV/DNCDNC

CONVDNC

CONVDNC

CONVDNC

CONVDNC

CONVDNC

CONVCONV

Iterations20172015201120122020201210

Time31.43.1

68.36.6

143.17.4

334.716.0

916.642.1

4323.1191.9252.2

2.40880.326102.6252

0.264693.1942

0.243323.0308

0.242353.1796

0.242780.24276

0.2427830.24285

q1.219931.948621.155451.947951.086411.943281.169091.939021.189751.937711.934771.937471.93584

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

N224488161632326464128

M25498171633326564129128

CONV/DNCDNCDNCDNCDNCDNCDNCDNC

CONVCONVCONVCONVCONVCONV

Iterations20202020202020181116131712

Time0.6

32.563.368.4

134.961.3

293.218.86.8

13.233.758.0

192.7

0.863930.724360.544470.527540.600450.506010.788130.419940.410010.410460.408200.408580.40774

q1.838071.339071.481911.44191.380301.474191.330311.436381.419861.423421.416811.420481.41605

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 describedin Figure 5.19. The data to be matched, denoted by in (5.2), are indicated by pluses inFigures 5.20 and 5.21. This data set was generated by randomly perturbing the value ofw( ) in (3.17) using q = 2 and q — 1.4 with p = 4,16 data points. Table 5.1 shows thenumerical values of the data for comparison purposes.

The exact cost functional, J(q), and several approximations to it, JN(q), are plotted


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 wasadded to obtain data for optimization. Here, the optimal value of the design parameteris 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 vectorin Table 5.1 was again added to obtain data for optimization. Here, the optimal valueof the design parameter is approximately q* ~ 1.4. Here, the optimization algorithmwas started at qinit = 2.0.

The scheme was considered converged when the norm of the gradient of the cost functionalwas less than 10~7. Notice that the simulations were performed using sensitivities calculatedusing the natural piecewise constant finite element gradient approximations. Tables 5.2 and5.3 show the results of these simulations as N, M ranged from 2 to 128. The runs weremeasured in seconds and were performed on a Silicon Graphics Onyx2. Notice the effect ofthe inaccurate sensitivity approximations on the convergence of the optimization scheme forCase 1 when N = M. As expected, larger values of N, M were required for convergence ofthe optimization algorithm for Case 2. In the following section, we return to this problemand use smoothing projections to enhance sensitivity computations and convergence.

5.3 State Gradient ApproximationsThe goal of this section is to describe two projection techniques that can be used to improvethe accuracy of sensitivity approximations by obtaining better state gradient approximations.In the context of the nonlinear model, we seek to improve the numerical approximationsof • In the previous section, the gradient approximations computed using the finiteelement derivative were discontinuous across element faces. Here, we analyze global andlocal projection techniques for calculating continuous gradient approximations and evaluatetheir 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 estimatesin adaptive finite element codes. The local projection technique is also discussed in Chapter7.

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 derivativeneeded 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 de-scribe global and local projection schemes for constructing this function.

Remark. Even in higher dimensions, the finite element derivatives are often discontinuousacross element faces. It is possible to construct higher-order basis functions that providecontinuous derivatives across element faces. However, from the standpoint of CFD, thisconstruction also must satisfy the inf-sup conditions in order to guarantee convergence, andthe computational cost of using higher-order basis functions often outweighs the benefitsof improved accuracy. The projection techniques surveyed here provide the scientist withthe convenience of low-order basis functions along with improved accuracy for a relativelylittle computational work.

5.3.1 A Global Projection Scheme

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

where gi, and (•) are the "hat functions" defined in section 4.3. Observe thatand 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 is the solution of a linear system of the form

and contains the coefficients defined by the finite element approxi-mation of zN ( ). Since

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.


5.3.2 A Local Projection Scheme

In addition to the global projection scheme, we consider a local projection scheme, whichinvolves performing a series of local projections on subdomains of = [0,1]. At eachelement vertex, , we define the subdomain , to be the union of all elements for whichis a vertex. In , define to be the least-squares projection of gN( ) | onto the spaceof linear polynomials spanned by monomial basis functions. For the nonlinear example, thebasis 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. Thesecoefficients 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 thevalue of at a nonvertex node. An averaging technique is generally used. The localprojection 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 projectiontechniques. Examples of state gradient approximations are given using the piecewise con-tinuous 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 thecorresponding sensitivity approximations. A brief section exploring the effects of usingthese projection techniques within an optimization algorithm is included.


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 andits local and global projections for the case when q = 2.0 and N = 4. Figure 5.25 shows asimilar 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 thenonlinear 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 andq = 1.2, N = 8, respectively. The L2 errors over the spatial domain for each of thesecases are summarized in Table 5.4. Figures 5.26 and 5.27 demonstrate that the errors for thelocal projection technique are highest near the boundary, but away from the boundary thelocal projection technique actually has less error than the global projection technique forthese two cases. We now analyze these techniques in calculating sensitivities for varyingdiscretizations (N, M) and parameter values (q) to gain a better understanding of how theimproved derivative approximations affect our numerical sensitivities.

Sensitivity Approximations

Figures 5.28,5.29, and 5.30 display the sensitivity approximations using the three differentstate 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 techniquegreatly 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 PWCderivatives were extremely inaccurate so that the approximations do not even show up on


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.


Figure 5.27. L2 error on each element for N = 8 and q = 1.2.

Table 5.4. L2 errors for the derivative approximations.

PWCGlobalLocal

L2 errorsN = 4 and q = 2

0.43300.21650.2954

W = 8 and q =1.21.14320.87601.0501

the graphs in Figures 5.29 and 5.30 for some values of N. When compared with the originaldiscontinuous 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 mostpromising result is that the projections stabilize the calculations over the parameter range.The local projections also give slightly more accurate sensitivity approximations for somevalues of q when compared to the global projection technique. Figure 5.32 compares theL2 error of sensitivities calculated using local projections with the error calculated usingglobal projections. Note that as N increases, the global projections do a better job than localprojections 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 piecewisecontinuous finite element derivative in the limit. Next, we briefly examine the effect thatthe improved sensitivity approximations have on the optimization problem considered insection 5.2.2.


Figure 5.28. Sensitivity approximations at q = 2.

Figure 5.29. Sensitivity approximations at q = 1.4.


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

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


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

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

N224488161632326464128

M25498171633326564129128

CONV/DNCCONVCONVCONVCONVCONVCONVCONVCONVCONVCONVCONVCONVCONV

Iter20161515121313131212121111

Time9.439.06

17.8012.9127.0125.8427.9531.6277.3278.90

288.06302.73355.10

.32586

.32548

.26463

.26468

.24327

.24326

.24235

.24235

.24272

.24273

.24274

.24274

.24285

q1.96041.95141.95091.94811.94231.94211.93751.93761.93611.93611.93591.93591.9358


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

N224488161632326464128

M25498171633326564129128

CONV/DNCCONVCONVCONVCONVCONVCONVCONVCONVCONVCONVCONVCONVCONV

Iter20161515121313131212121111

Time9.439.06

17.8012.9127.0125.8427.9531.6277.3278.90

288.06302.73355.10

.32586

.32548

.26463

.26468

.24327

.24326

.24235

.24235

.24272

.24273

.24274

.24274

.24285

q1.96041.95141.95091.94811.94231.94211.93751.93761.93611.93611.93591.93591.9358

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

N224488161632326464128

M25498171633326564129128

CONV/DNCDNCDNCDNCDNCDNCDNC

CONVCONVCONVCONVCONVCONVCONV

Iterations20202020202014151511131413

Time2.06

64.37133.51132.27127.18143.5628.6933.0713.2212.7545.4348.74

258.93

.93423

.72262

.53248

.52703

.48242

.50490

.41967

.41973

.41001

.41001

.40819

.40819

.40774

q2.30541.35301.45571.44051.50291.47911.43291.43221.41971.41961.41681.41671.4160


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

N224488161632326464128

M25498171633326564129128

CONV/DNCDNCDNCDNCDNCDNCDNC

CONVCONVCONVCONVCONVCONVCONV

Iterations20202020202014161115131212

Time60.8264.60

130.65131.56133.27133.9230.0032.0311.2114.3444.4545.70

253.86

.73574

.72273

.53216

.52991

.50451

.50470

.41968

.41974

.41000

.41001

.40820

.40819

.40774

q1.41541.36341.45581.45131.48301.48071.43281.43221.41971.41961.41681.41681.4160

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 performedusing the piecewise linear derivative approximations. Tables 5.5-5.8 show the results ofthese simulations as N, M range from 2 to 128. Note that the use of the piecewise linearderivative approximations clearly improve the results of the optimization algorithm forCase 1 and that the global and local projection schemes provided very similar results. Theimprovement was not as marked for Case 2; however, the scheme did converge for the caseN — M = 16 with the improved gradient approximations and did not with the piecewiseconstant derivatives. In addition, the local scheme required slightly less time to convergethan the global projection scheme (see Tables 5.7 and 5.8).

6 Mathematical Framework forNavier-Stokes Equations

This chapter presents the mathematical framework used to assert the existence of the sen-sitivity for Navier-Stokes equations. The necessary preliminaries, such as function spacesand 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-dimensionalsteady state flow of an incompressible, viscous fluid in a bounded domain with aLipschitz-continuous boundary :

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

6.1 The Homogeneous Dirichlet ProblemWe 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 trilinearforms.

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 thevariational form of the homogeneous Navier-Stokes problem as presented in [21]. Tobegin, we note that the homogeneous partial differential equation (6.1)-(6.2) can be writtenin variational form as follows.

Variational Problem 6.1. Given find a pair such that

6.2. The Nonhomogeneous Dirichlet Problem 73

Recall that the trilinear form, has some nice properties described in thefollowing lemma.

Lemma 6.1. Let and let with and Then,the trilinear form is continuous on and satisfies

Now, the existence result from [21] follows.

Theorem 6.2. For there exists at least one pair thatsatisfies (6.3).

In order to discuss the uniqueness of the solutions (u, p) to (6.3), we introduce thenorm 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 ProblemNow consider the more general case of a nonhomogeneous Dirichlet boundary condition

Denote by the connected components of the boundary as depicted inFigure 6.1. We henceforth assume that

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 followingproblem.


Figure 6.1. Sample domain with boundaries.

Variational Problem 6.2. Given find a pair suchthat

To prove existence for the nonhomogeneous problem, we need the following technicalresult, due to Hopf (see Lemma 2.3 in [21]).

Lemma 6.4. Let satisfy (6.10). For any there exists a functionsuch that

The following existence theorem may be found in [21].

Theorem 6.5. Let and satisfying (6.10). There exists at least onepair 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 conve-nience.

Theorem 6.6. Assume the hypothesis of Theorem 6.5. If then VariationalProblem 6.2 has a unique solution

Remark. Although the results above provide existence and uniqueness for the basic Navier-Stokes problems, they do not address the continuity and differentiability of these solutionswith respect to the parameter q. For example, in the problems considered below we have

, and/or g = g(q) so that v0 = = v0( ), whereq is some parameter defining the flow. We need to establish the smoothness of thesemappings 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-StokesIn this section we present an abstract framework for analyzing the dependence on q ofsolutions to the nonhomogeneous Dirichlet problem for the Navier-Stokes equations. Weshow the continuity of solutions with respect to parameters for two specific cases, and weconclude with results about the differentiability of those solutions. We extend the frameworkin [21] to certain parameter-dependent flows.

6.3.1 The Framework

Let X and X be two Banach spaces and , where is open and A is compact.Given a Cp mapping (P 1)


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 equa-tions 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 Next we define a linear operator Tas follows: Given we denote by the solution of theDirichlet problem for the Stokes equations:

In addition, let P : Q Y be defined by

and the nonlinear operator be given by

6.3. An Abstract Framework for Navier-Stokes 77

where v is the constant as before.Now finally, with the data we associate a C1 mapping G from Q x X into

Y defined by

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 andonly if(q, u(q)) is a solution of (6.19), where w(q) = (u(q), p(q)/v) and where the spacesX 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 tochanges in the parameter q. We assume the map iscontinuously Frechet differentiable. Note, that this is certainly true for the cylinder problempresented 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 H1 ( ) such that for eachg V, we have that w = D(g) satisfies

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 nonhomogeneousNavier-Stokes problem, we need a result analogous to Lemma 6.4 for parameter-dependentboundary functions. The following result may be established by a straightforward extensionof Lemma 2.3 in [21],

Lemma 6.10. There exists a continuous linear function such thatfor each

where U0(q) is defined by Lemma 6.4 and satisfies


and

for each

We now consider continuity with respect to the right-hand-side function f. We againconsider 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 asfollows:

The range of is contained in

and is defined by

Note that and the Frechet derivative at a point isgiven by

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 differentiablefunction of f if the linear map is an isomorphism. But this is equivalent tothe condition that the homogeneous sensitivity equation has a unique solution in y for eachf H-1( ). We show in the following that the sensitivity equation does indeed have aunique solution in y so that we have (u, p) is a C1 function of f. We then return to thesmoothness of solutions to the parameter-dependent Navier—Stokes equations.

6.4 Analysis of the Sensitivity EquationsWe begin by stating a general form of the sensitivity equations for the parameter-dependentNavier-Stokes equations. We show that if we have a unique solution (u, p) of the Navier-Stokes 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 functiontheorem to show that the solution (u, p) is in fact a nonsingular solution of the Navier-Stokes problem.

6.4.1 A General Formulation of the Sensitivity Equations

Let reoresent some fixed sensitivity oarameter Now denote thesensitivity forcing function, by and the boundary function, by Lastly,

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 andtheir associated boundary conditions. The sensitivity equations fit into the following generalform. 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 and satisfying (6.10) and(u, p) a solution (6.1)-(6.9), find a pair (s, r) x such that

for all z

6.4.2 Existence and Uniqueness of Solutions to the SensitivityEquations

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 ofVariational Problem 6.2, then there exists a unique solution, (s, r), to Variational Problem6.3.

Proof. To check that the equations (6.41) have a unique solution, it is sufficient to provethat 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 functionsuch that

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


Since

we obtain

so that the ellipticity property holds.

6.5 Differentiability of Solutions with Respect to qWe return now to the smoothness of solutions to the parameter-dependent Navier-Stokesequations. Note that the parameter V0 = v0(q) is a continuous function of q in this casesince the map from q U0 is continuous and the map from U0 p is continuous. Withthis, the continuity of the map (f, g) (u, p), and the fact that q (f (q), g(q)) is C1, wecan now establish the following result.

Theorem 6.12. Assume that q is such that v > vo . There exists a neighborhoodof such that for all q the solution of the variational, parameter-dependent Navier-Stokes equations (6.11), with f = f(q) and g = g(q), exists and isunique. 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 factthat the homogeneous sensitivity equation has a unique solution for all q . We showedin section 6.4.2 that the sensitivity equation does indeed have a unique solution. Then, bythe implicit function theorem, we have that q (u(q), p(q)) is a C1 function from intoX.

7 Two-Dimensional FlowProblems

We now return to analyzing design sensitivities, and we focus on two specific parameter-dependent flow problems in CFD. We present flow equations for these problems and useformal implicit differentiation to derive the continuous sensitivity equations. The readershould observe that the coupling of the sensitivity equations to the state equations throughthe appearance of state and state gradient terms can be seen in both examples. Hence, manyof the numerical issues addressed for the nonlinear model of section 3.3 are also addessedforthe examples introduced in this chapter. An adaptive finite element technique is describedand used to solve the state and sensitivity equations. The numerical results are presented toinvestigate the convergence of the adaptive grids.

7.1 Flow around a CylinderWe 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 onthe problem in which the boundary is an infinite strip. We assume parabolic inflow into achannel 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 enoughfor 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 equationspresented earlier (see (6.1)) with f = 0. The boundary conditions at the inflow and outfloware given by

for — 1 y 1, where q R+ is a parameter describing the strength of the inflow. No-penetration 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 itsassociated boundary conditions with respect to the parameter q. Assuming the order ofdifferentiation can be interchanged, we obtain the following:

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

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

7.2 Flow over a BumpWe also consider a problem examined by Burkardt in [14]. In particular, the problemis two-dimensional incompressible flow over a bump in a channel. The geometry of thechannel is indicated in Figure 7.2 with = [0, L] x [0, 3], where L 0 is a fixednumber representing the length of the channel. As before, the governing equations for thisproblem 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 theinflow. Again, no-penetration and no-slip conditions are applied on the top and bottomchannel 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 isa cubic spline determined by a parameter, q. We examine two cases: q = q1 R1 andq = (q1, q2, q3)

T R3. We seek to find the sensitivity of the flow in the channel to changesin q, which we denote

We must derive and solve a separate set of linear sensitivity equations for each sensi-tivity. 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 generateh(x, q) using a cubic spline with free boundary conditions. If q R1, q1 specifies theheight 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,


respectively. Here, h(x, q) satisfies

7.3 A Finite Element FormulationSince exact solutions to the Navier-Stokes equations are unavailable except for the simplestof problems, we now turn our attention to finding good methods for computing approxima-tions to the solutions of the flow equations. We use a finite element approach to solve thevariational 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 Crouzier-Raviart triangular element (see Figure 7.3), which is a type of "bubble" element describedin [15]. This element uses a so-called enriched quadratic velocity interpolant and a dis-continuous linear pressure. The discretized variational equations are solved in the primitivevariables using a penalty method to solve for the pressure degrees of freedom. Once the stateapproximations are obtained, the weak form of the sensitivity equations (6.41) can be simi-larly solved with one additional iteration of the flow solver. For a more detailed explanationof the finite element implementation, see [51]. Additionally, an adaptive methodology isused 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 adaptivemethodology is also employed to improve the sensitivity approximations in Chapter 8, wenow 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 thequality of the finite element approximation and to strategically modify the grid based onthat evaluation. The grid modification scheme allows the user control over element size andgrading. 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 adaptiveprocess 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 beshown that they are equivalent norms. Using the energy norm for the velocity, define theso-called Stokes norm of the solution as

As pointed out in [50], the use of the energy norm over the H1 seminorm offers someadvantages, especially to the engineering community. Note that both the velocity and thepressure norms are expressed in terms of surface forces, which are the quantities of primeinterest in engineering fluid mechanics. Second, errors computed in these norms can beinterpreted as errors in the stresses, which can then easily be related to errors in globalquantities 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)


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 touse the finite element approximation to construct approximations to these gradients. Notethat the finite element approximations to the gradients are discontinuous across element faceswhile the exact gradients are, in most cases, continuous across the domain. Thus, the firstgoal in error estimation is to obtain continuous approximations to the discontinuous finiteelement gradients. Two methods have been evaluated for this process: global projections andlocal least-squares projections. Global projections are performed over the entire domain, ,and involve finding the best approximation to the discontinuous finite element gradients in theoriginal continuous finite element space. For example, if a piecewise linear approximationof the flow solution is calculated, then the finite element gradient is piecewise constant anddiscontinuous across elements. The global projection would replace the piecewise constantgradient approximation with a piecewise linear approximation calculated by projecting thepiecewise constant function onto the original finite element basis functions. The local least-squares projections, however, are done node by node and consider only gradient informationfrom subdomains of that contain the current node. Thus, a series of smaller projectionsis done in combination with some averaging techniques to obtain a continuous gradientprojection for the entire domain, . The details of each of these projection techniques aredescribed 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 problemsare 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 finiteelement approximations. Once this is done the L2 norm can be calculated as usual. We nowreturn 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 elementsize), d, is calculated which requires equidistribution of the element errors across all theelements. For example, if we wish to reduce the error in each element by a factor of y, thenthe 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 assumesthat 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, wehave 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 minimumof these is used for the generation of the new mesh. The details of the actual redefinition ofthe mesh are omitted here; we refer the interested reader to [30] and [31].

Despite the rather major assumptions made above, this adaptive remeshing strategyworks remarkably well. The strategy has been verified in a series of numerical experimentsfor a variety of flow types (see [30], [31], [35], [48], [49]). We use this adaptive strategy inthe problems investigated below.

7.4 Some Numerical ResultsIn this section, we apply the computational techniques outlined above to the two-dimensionalflow problems presented at the beginning of this chapter, and we investigate convergenceof 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 givenin (6.1). We again assume parabolic inflow into the channel as follows:

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

where is the outward normal at the end of the channel. No-flow and no-slip conditionsare applied on the top, bottom, and cylinder sidewalls.

As noted earlier, we wish to approximate the sensitivity of the solution with respectto the inflow parameter q. The sensitivity equations are given by (7.3), and the sensitivityboundary conditions at the inflow are obtained as before, with

The computational outflow boundary conditions for the sensitivity equations become


Figure 7.4. Initial and adapted meshes for a cylinder problem.

Numerical results for this flow problem were generated using the approximation tech-niques outlined in section 7.3. The Reynolds number for the calculations is Re = 100, thelength of the channel is 8 (L = 6), and the sensitivity parameter value is q = 0. Contourplots of the u, v-velocity fields as well as the u, u-velocity sensitivities are given in Figures7.5 through 7.8. In Figure 7.4, the initial and adapted grids are shown. It is clear that themesh is refined around the cylinder and in areas of large velocity gradients. This givesimproved approximations of the velocity field, as can be seen in Figures 7.5 and 7.6. Sincethe mesh refines on the velocity field, it is convenient that, for this problem, the sensitivityflow field is similar to the velocity field, so that as the mesh refines we obtain improvedsensitivity approximations as well (see Figures 7.7 and 7.8). It is important to note that thisis not always the case, as is shown in [11]. The code was modified by Jeff Borggaard toadapt 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 boundarycondition 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 andsensitivities are displayed in Figures 7.10 and 7.11. Note that the mesh refines in the area ofthe bump and the elements are allowed to become larger downstream in the channel wherethe flow is again quadratic. The bump problem is similar to the cylinder problem in that thesensitivities are largest in the same areas where the mesh refines. Thus we obtain improved


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 two-

dimensional parameter-dependent differential equations. We have obtained accurate nu-merical approximations for the continuous sensitivity equations using an adaptive finiteelement technique. We have observed that refining on the flow provided improved sensi-tivity approximations, as well, when the sensitivity field was similar to the flow field. InChapter 8, we explore the idea of adapting the mesh based on error estimation for boththe flow and the sensitivity. In Chapter 5, we found that accurate sensitivity approxima-tions for the one-dimensional problems previously described were more difficult to obtainin certain parameter ranges. A projection technique was used to obtain more accuratestate gradient approximations, thus decreasing errors in sensitivity calculations. A similarphenomenon occurs as the Reynolds number is increased for the two-dimensional modelproblems described in this chapter. Chapter 8 describes a local projection technique fortwo-dimensional problems and evaluates the effectiveness of this technique in improvingsensitivity approximations.


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

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


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

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


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


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

8 Adaptive Mesh RefinementStrategies

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 providesimproved 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 theevaluation of cost functionals and their gradients in optimization.

8.1 A Local Projection for Higher DimensionsIn this section, we describe the local projection scheme employed in the error estimationmodule of the finite element code described in Chapter 7. This local projection scheme isvirtually the same as the local scheme for the one-dimensional model problem discussedin section 5.3.2, yet it is presented for a two-dimensional problem below since the use ofhigher-order finite elements in higher dimensions adds some complexities not previouslydiscussed. Again, the local projection scheme involves performing a series of projectionson subdomains of . The projected gradients are given as polynomial expansions around agiven vertex, , of a finite element mesh. The subdomain, , over which the projectionis defined consists of all elements having = (xi, yi) as a vertex. Figure 8.1 illustrates atypical subdomain, a finite element mesh of quadratic triangles. In principle, the choice of thedegree of the polynomial expansion for the improved gradient approximation is independentof the selection of the finite element basis being used. However, in practice, the degree of thepolynomial 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 thenumerical results presented in section 8.2 below, quadratic triangular finite elements wereused 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 projec-tion of the finite element derivative, uh, onto the space of quadratic polynomials. For easeof notation, we denote uh by gh. Letting P = [1, x, y, x2, xy, y2] denote the basis func-tions 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 coeffi-cients are obtained by solving the following least-squares problems:

Thus, for each component of g*, we solve the following 6x6 system of linear equationsforax and ay:

The finite element fluxes, and , are obtained in the usual manner by differentiating thefinite element basis functions. Note that the left-hand matrix is independent of the quantitybeing projected and thus can be viewed as the projection matrix for node, , and can beused 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 systemsare 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 similarlyfor the remaining element vertices, and . We need a unique definition of g* for anypoint, inside the element (see Figure 8.2). For quadratic elements, there are several waysto 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 polynomialexpressions 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 arethe nodal values of the locally projected derivative obtained as described above. In thefollowing section, we apply the described local projection technique to obtain improvedsensitivity approximations for two flow problems.


8.2 Numerical Results for Two-Dimensional ProblemsWe now return to the two specific flow problems described in Chapter 7 to show that theprojection techniques described in section 8.1 can be used to obtain improved sensitivityapproximations. 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 arange of Reynolds numbers. The sensitivities are calculated using both the natural discontin-uous (unprojected) state gradient approximations and the locally projected continuous stategradient approximations. In each case, the two sensitivity approximations (using unpro-jected and projected gradients) obtained for the initial and first adapted meshes are comparedto an approximation generated by adapting to a very fine, final mesh and then interpolatingthe "true" solution from the final mesh onto the initial or first adapted meshes. On this finalmesh, both schemes converged to solutions differing by less than 10-3. For comparisonpurposes, 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 sufficientto meet the computational free outflow boundary condition. We use the Re = 350 case toshow that the results for the higher Reynolds cases were not affected by the length of thechannel.

In this chapter, we also use an adaptive technique that refines on sensitivity errorsas well as flow errors. This technique is completely analogous to the technique used torefine on the flow errors (see [12]). In the results that follow, we are careful to identifymeshes that were adapted on approximations of flow errors and meshes that were adaptedon approximations of both flow and sensitivity errors.

We present a detailed error analysis of the velocity sensitivities s for two Reynoldsnumbers, Re = 100 and Re = 350, beginning with the results at Re = 100. Figure 8.3shows the initial mesh, the first adapted meshes, and final mesh for this case. Notice thatwe present three first adapted meshes: one adapted on the flow only, one adapted on boththe flow and the sensitivities calculated with unprojected derivatives, and one adapted onboth the flow and the sensitivities calculated with projected derivatives. At this Reynoldsnumber, 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 bettersensitivity results, as seen in Figures 8.4 and 8.5. In these figures, recall that the truesolution 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 areshown in Figure 8.6. From these plots, we can see that the local projection scheme reducesthe maximum error by about a factor of 2.

We also look at the sensitivity solutions using schemes on the first adapted mesh. Thefirst adapted mesh (for the flow) is shown in Figure 8.3(b). The sensitivity results for theunprojected and locally projected derivative schemes for this mesh are shown in Figures 8.7and 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 99

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


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


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


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


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


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


interpolated onto the first adapted mesh. Note that the difference between the approxima-tions using the projected gradients and those using the natural finite element derivativesis no longer as apparent. Again, a node-by-node comparison was made. These errors areshown in Figures 8.9(a) and (b) and 8.10 (a) and (b) and are plotted on the same scales usedfor the initial mesh for easy comparison. We also present nodal errors for the two schemesin the cases where we adapt on the flow and the sensitivities (see Figures 8.9(c) and (d) and8.10(c) and (d)). At this Reynolds number, there seems not to be a significant advantagegained 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 calculatean L2 error over the entire domain for the flow and the sensitivities (see Table 8.1). Thismore clearly shows the 50% reduction of the error for the local projection scheme on theinitial mesh. Note that the errors for u, v on the first adapted meshes are the same for theunprojected and projected schemes on the mesh that was refined only on the flow, but theyare slightly different (since the meshes are slightly different) when the meshes were alsorefined on the sensitivity. Note that adapting on the flow and sensitivity errors not onlyimproved the sensitivity approximations over those obtained by adapting on the flow alonebut also improved the numerical approximations for the flow.

We now turn to the results for Re = 350. We begin by comparing the state andsensitivity approximations for L = 6 and L = 15 to ascertain whether the length of thechannel affects our results. Figure 8.11 shows the initial meshes for the two channel lengthsas well as the u, v velocity contours. It is clear that the flow more nearly returns to theinflow for the longer channel. However, the results for the shorter channel are very similarto those for the longer channel in the areas where they overlap. In fact, Figures 8.12 and8.13 show that the dramatic difference between the sensitivities obtained with unprojectedderivatives and those obtained using locally projected derivatives occurs in both the longand the shorter channels. This gives us reasonable confidence that the length of the channelis not affecting our results.

We now return to complete the same error analysis for Re = 350 that we did forRe = 100. The initial and first adapted meshes are shown in Figure 8.14. Note thatthe difference between the three first adapted meshes is definitely greater at this Reynoldsnumber. 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, thedifferences between the sensitivities calculated with the two different derivative schemes isnow 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 atthis Reynolds number, both approximations contain fairly large errors. Using the locallyprojected 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 errorreduction from the initial mesh to the first adapted mesh. This is easily seen in the overallL2 errors for the approximate flow and sensitivities displayed in Table 8.2. It is interestingto note that the errors for all the quantities are less for Projected (Flow & Sensitivity) thanfor Unprojected (Flow & Sensitivity), although the Unprojected (Flow & Sensitivity) meshis quite a bit finer.

106 Chapter8. Adaptive Mesh Refinement Strategies

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


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


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

Initial Mesh

uV

su

sv

Unproj1.7397E-019.0160E-025.0463E-012.2453E-01

Proj1.7397E-019.0160E-022.8227E-011.3050E-01

First Adapted Meshes

uV

su

sv

Unproj (Flow)1.0313E-014.5997E-023.4173E-011.4487E-01

Proj (Flow)1.0313E-014.5997E-023.5800E-011.1039E-01

Unproj (Flow & Sens)7.2637E-023.3000E-022.5988E-019.1075E-02

Proj (Flow & Sens)5.3305E-022.8115E-022.7273E-017.6292E-02

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


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


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


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


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


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


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


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

Initial Mesh

uV

su

Sv

Unproj1.4808E-004.1024E-011.9438E+013.0582E-00

Proj1.4808E-004.1024E-013.6791E-005.8348E-01

First Adapted Meshes

uV

su

Sv

Unproj (Flow)9.6580E-012.6044E-013.8080E-001.5339E-00

Proj (Flow)9.6580E-012.6044E-013.1614E-008.7266E-01

Unproj (Flow & Sens)3.5985E-011.0257E-011.1145E-003.4686E-01

Proj (Flow & Sens)2.8674E-017.7747E-021.0423E-002.9430E-01

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


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


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


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


To conclude, we observe that using the locally projected derivatives clearly stabilizesthe calculations over a larger range of Reynolds numbers, as it did in section 5.3.3 for theone-dimensional model problem. Plots of the overall L2 sensitivity errors on the initialmesh are shown in Figures 8.22 and 8.23. In addition, the mesh refinement very effectivelyreduces the errors for the flow and the sensitivities. At most Reynolds numbers, after themesh is refined once, there is little difference between the sensitivities calculated with theunprojected and the locally projected techniques. At higher Reynolds numbers, however,using locally projected derivatives to calculate sensitivities remains advantageous. We nowturn our attention to obtaining numerical approximations to the design sensitivities for theflow 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 pre-sented in [14]. We point out some of the difficulties of calculating shape sensitivities andshow how the process of error estimation and grid refinement is extremely important inobtaining accurate numerical approximations of the sensitivities for these problems. Figure8.24 displays vector sensitivity plots for a = = 0.5, = 0.5, and L = 10. These plotsare qualitatively very comparable to those presented by Burkardt on page 172 in [14]. AsBurkardt notes, the shape sensitivities for smaller Reynolds numbers appear as "whirlpools"and are predominately localized to the region above the bump. As the Reynolds numberincreases, however, the effect of the bump is carried downstream. This is especially apparentin Figure 8.24(c). Another effect of increasing the Reynolds number on flow calculationsis that the task of meeting the outflow boundary conditions becomes more challenging nu-merically. We clearly see this effect in the Re = 500 case shown in Figure 8.25. With achannel 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 largesecondary whirlpool further downstream from the bump. This phenomenon does not ap-pear 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 sensitivityapproximations. First, we investigate the secondary whirlpool that appears in sensitivitycalculations. We also evaluate the accuracy of the sensitivity approximations near the bumpusing the error estimation and grid refinement process presented in section 7.3.1. The meshis adapted only on flow errors. Figure 8.26 shows the initial mesh, which is similar in den-sity to the one used in [14], and the adapted meshes. Note that the mesh refines where onewould expect, in the regions of large velocity gradients around and downstream from thebump. It is also refining at the outflow in an effort to accurately meet the outflow boundarycondition.

Figures 8.27 and 8.28 display u and v contours for the flow on the initial and adaptedmeshes. It is important to note that as the mesh refines, the contours smooth and the accuracyof the gradients in u and v are greatly improved, especially in the vicinity of the bump. Thislack of accuracy of the velocity gradients on the initial mesh greatly hampers our ability toobtain good sensitivity approximations. This is seen by noting that the sensitivity boundaryconditions on the bump, (7.9)-(7.10), require approximations to the velocity gradients onthe boundary. If there are large errors in the velocity gradients, there will be large errors inthe 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 oninitial mesh.

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


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


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


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


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


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


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


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


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


gradient approximations improve in the second and third adapted meshes, the values ofthe sensitivities become much more accurate, not only in the local area of the bump butdownstream as well. Note also that the secondary whirlpool, which appears in the Re = 500case, is seen on the initial mesh in this case also. As the mesh is refined, however, the size ofthe 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 accuratesensitivity approximations, especially shape sensitivities. This is an area requiring futureresearch.

Analysis of Cost Functionals and Gradients—An Optimization Issue

Finally, we turn to the issue of using numerical approximations of the state and sensitivitiesto approximate cost functionals and their gradients. A comparison of our results with aproblem considered by Burkardt in Chapter 11 of [14] is given. Burkardt presents what herefers to as a discretized sensitivity failure. We evaluate his results and show that the processof adaptive mesh refinement is key to obtaining good cost function and gradient evaluationsin order to prevent inaccurate results from an optimization code.

We examine the numerical experiment carried out in section 11.2 of [14]. We evaluatethe following cost functional:

Here, P is the number of matching points and for the results we present herein P was fixedat 15. The matching points, (3, yi), i = 1, 2 , . . . , P, were evenly distributed along the linex = 3, y [0, 3]. Also, the target profile, uTarg(3, y), was obtained by computing a finiteelement solution with the shape parameter, qTarg = (0.375, 0.5,0.375)T, and = 0.5, andthen 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 beforeto approximate the value of the cost gradient. We denote this approximation of the costgradient by qJ

h; thus we have

In order to nearly duplicate the numerical experiment carried out by Burkardt, wehave 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 parameterizedby S, connecting q = (-0.117, 0.419, -0.149)T at S = 0 and q = (0.375,0.5,0.375)T atS = 25. This is the same line along which Burkardt explored. His results are presented inTable 11.2 on page 153 and Figures 11.3 and 11.4 on page 145 of [14].


Our results are displayed in Table 8.3 and Figures 8.32 and 8.33. Burkardt reporteda local maximum at S = 5. As seen in Figure 8.32, we do not get the same results on theinitial mesh. To determine if the difference was due to not having sufficient accuracy on theinitial 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 dueto our having fixed the value of = 0.5. It is also possible that the difference is a resultof numerical inaccuracies due to discretization differences. We perform another numericalexperiment 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 exploredabove. Note that adapting the mesh is even critical to obtaining accurate, smooth cost andgradient approximations at this Reynolds number.


Table 8.3. Values of Jh along a line, Re = 1, and = 0.5.

s q1 q2 q3

0123456789101112131415161718192021222324252627282930

-0.117-0.097-0.077-0.057-0.038-0.0180.0010.0200.0400.0600.0790.0990.1190.1380.1580.1780.1970.2170.2370.2560.2760.2960.3160.3350.3550.3750.3940.4140.4340.4530.473

0.4190.4220.4250.4280.4320.4350.4380.4410.4440.4480.4510.4540.4570.4610.4640.4670.4700.4740.4770.4800.4830.4870.4900.4930.4960.5000.5030.5060.5090.5130.516

-0.149-0.128-0.107-0.086-0.065-0.044-0.023-0.0020.0180.0390.0600.0810.1020.1230.1440.1650.1860.2070.2280.2490.2700.2910.3120.3330.3540.3750.3960.4170.4380.4590.480

Jh.103 ( q J h . S ) . 1 0 3

Initial Mesh16.916.315.514.613.712.912.211.410.69.729.028.357.466.215.625.004.383.703.082.471.891.330.720.320.110.000.060.310.811.632.72

-0.99-1.03-1.00-0.94-0.91-0.92-1.04-1.00-0.96-0.99-1.15-1.07-1.08-0.77-0.78-0.79-0.78-0.76-0.74-0.70-0.64-0.56-0.50-0.35-0.210.000.160.380.630.921.22

Jh.103(qJ

h - ).103

03 Mesh12.111.711.310.810.39.959.499.028.568.007.486.956.395.765.174.563.943.272.662.071.510.990.570.250.050.010.160.531.162.083.30

-0.56-0.57-0.58-0.60-0.61-0.62-0.63-0.65-0.67-0.68-0.70-0.72-0.74-0.75-0.76-0.77-0.76-0.75-0.72-0.68-0.62-0.53-0.42-0.29-0.130.050.280.530.811.131.48


Figure 8.32. Values ofJh(q(S)) along a line.

Figure 8.33. Values of alone a line.


Figure 8.34. Values of along a line.

Figure 8.35. Values of along a line.

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Index

adaptive remeshing strategy, 87

Banach spaces, 75bilinear forms, 10,13, 72, 79boundary value problems

examples of linear elliptic, 17-21examples of nonlinear, 22-23, 81-

89linear elliptic, 1, 5-16mathematical theory for nonlinear,

71-80regularity of linear elliptic, 11-14

computational algorithms, 25continuous sensitivity equation methods,

1, 2, 22, 82, 83, 129Crouzier-Raviart element, 84-85

diffeomorphism, 8, 25

finite element formulationsadaptive mesh refinement, 84-87Crouzier-Raviart element, 84-85piecewise linear, 37-41

Frechet differentiability, 6,15,16,19,77

Gelfand triple, 9,12

Hilbert spaces, 14

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

Lax-Milgram theorem, 11, 13

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

model problems, 17-23, 81-84

Navier-Stokes equations, 1, 21, 33examples of, 81-89mathematical framework, 71-80variational formulation of, 71-75

optimal design, 22, 52-60, 129-130

Preface, xix

sensitivity analysis, 1sensitivity equations

computational algorithms for solv-ing, 28-35

differential forms, 19-21,23,29,30,32, 34, 37, 79, 82, 83

numerical calculations, 43-70operator forms, 15-16, 18, 19variational forms, 36, 37, 79

Sobolev spaces, 6-7state gradient approximations

finite element derivatives for, 44-51projection techniques for, 60-70,95-

130their affect on sensitivity approxi-

mation, 63-65stiffness matrix, 38Stokes norm, 85

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

Zhu-Zienkiewicz error estimator, 85

139

design sensitivity analysis

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