DESIGN OPTIMIZATION OF PERIODIC FLOWS USING A TIME

DESIGN OPTIMIZATION OF PERIODIC FLOWS USING A

TIME-SPECTRAL DISCRETE ADJOINT METHOD

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF AERONAUTICS &

ASTRONAUTICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Ki Hwan Lee

June, 2010

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/zf671jy0354

© 2010 by Ki Hwan Lee. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/zf671jy0354

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Juan Alonso, Primary Adviser


Antony Jameson


Robert MacCormack

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

Standard methods for unsteady optimization carry heavy computational costs and

large storage requirements, mostly due to the lengthy time integration involved in

the unsteady flow simulations. Such difficulties limit its practical application to cases

where the time integration is performed over only a smaller segment of the entire

period. The result is a loss of accuracy in the representation of the physical model.

For certain unsteady flows with periodicity, a dramatic reduction in both compu-

tational cost and required storage is realized through implementing the Time Spectral

method. Furthermore, by introducing an adjoint-based method as an alternative way

of obtaining gradient information, computational cost is further reduced. This combi-

nation of Time-Spectral and adjoint-based methodology therefore allows for unsteady

optimization within a reasonable time frame while maintaining accuracy.

In this dissertation, the Discrete Adjoint method is implemented and applied to

unsteady flows with periodicity, in the context of the Time Spectral Method. The

acquired adjoint gradient information is fed into an optimizer and truly unsteady

optimization work is carried out for the first time on a realistic test case. The devel-

opment and implementation of necessary boundary conditions prove crucial for the

successful implementation of the Discrete Adjoint method.

As a simple test case, the NACA 0012 airfoil is selected for simulation in steady

inviscid, unsteady inviscid, steady viscous, and unsteady viscous flows. In each case,

the resulting gradient information obtained from both the adjoint and finite difference

method is compared. Upon completion of the airfoil test case, the adjoint-based

method is applied to a helicopter blades, UH60, for both steady and unstaedy inviscid

flows. The gradient information obtained by the adjoint-based method shows good

iv

agreement with the conventional, Finite Difference gradient information.

The design methodology was developed for a single processor, however, multi-

processor capability is also implemented. In order to accommodate realistic meshes,

multi-block capability is added as well. With all of the necessary components im-

plemented, optimization is carried out on the UH60 helicopter blade. The objective

function is time-averaged torque over all time instances and the optimized result

shows an improvement of 5 % over the current configuration. Stanford University

Multi-block (SUmb), while implementing the unsteady Reynolds-Averaged Navier

Stokes equations with multi-block and multi-processor algorithms, is the chosen flow

solver. PETSc is employed as the adjoint solver.

Successful implementation of the Discrete Adjoint method to unsteady fluids with

periodicity provides the gradient information more easilty than the traditional finite

difference method which is hindered by its heavy computational cost and large stor-

age requirements. This research establishes a new optimization methodology which

utilizes Discrete Adjoint gradient information derived from flow solutions, obtained

using the Time Spectral method.

v

Acknowledgements

This research has been made possible by the ASC project funded by the Department

of Defense and their dedication to set up an infrastructure for massively parallel

computation.

I would like to express my deep gratitude to Professor Alonso, my advisor, for

providing me the opportunity to work in this interesting field of research and for his

continual support and guidance, encouragement, and numerous intangibles through-

out the research. I also would like to thank Professor Jameson, who pioneered the

use of adjoint-based methods in Aerospace engineering, for his valuable insight in the

the subject and his inputs.

All of my friends at Stanford supported me by enriching my life. Their valuable

insight in the research also helped me to achieve my goal. My family members has

been essential in carrying me up to this point of my life. Without them, nothing

would have been possible.

vi

Contents

Abstract iv

Acknowledgements vi

1 Introduction 1

1.1 Introduction to Research . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Gradient Based Optimization . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Adjoint Gradient Calculation . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Time Spectral Method . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Goals of Research: Time Spectral Adjoint

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Applications of the Time Spectral Adjoint

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Governing Equations 10

2.1 Navier-Stokes Equations and RANS . . . . . . . . . . . . . . . . . . . 10

2.2 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Development of A Time-Spectral Discrete Adjoint Solver 16

3.1 Benefits of Time-Spectral Discrete Adjoint

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Flow Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Time Spectral Method . . . . . . . . . . . . . . . . . . . . . . 18

3.2.2 URANS Flow Solver . . . . . . . . . . . . . . . . . . . . . . . 20

vii

3.3 Discrete Adjoint Solver . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.1 Discrete vs. Continuous Adjoint . . . . . . . . . . . . . . . . . 23

3.3.2 General Derivation of the Adjoint Equation . . . . . . . . . . 25

3.3.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.4 Derivatives of the Time-Spectral Terms . . . . . . . . . . . . . 30

3.3.5 Inviscid Formulation . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.6 Viscous Formulation . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.7 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 50

3.3.8 Unsteady Boundary Conditions . . . . . . . . . . . . . . . . . 70

3.3.9 Multiblock/Multiprocessor Implementation Issues . . . . . . . 72

3.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.4.1 Hick-Henne Bump Functions . . . . . . . . . . . . . . . . . . . 76

3.4.2 Implementation Using a Gradient Based Optimizer

with Adjoint Gradients . . . . . . . . . . . . . . . . . . . . . . 78

4 Results 81

4.1 Description of the Test Cases . . . . . . . . . . . . . . . . . . . . . . 81

4.1.1 NACA 0012 Airfoil Mesh . . . . . . . . . . . . . . . . . . . . . 82

4.1.2 UH60 Helicopter Rotor Mesh . . . . . . . . . . . . . . . . . . 85

4.1.3 Flat Channel Mesh . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2 Verification of Inviscid Gradients . . . . . . . . . . . . . . . . . . . . 88

4.2.1 Steady Gradient Comparison . . . . . . . . . . . . . . . . . . 89

4.2.2 Unsteady . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2.3 Multiblock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.3 Verification of Viscous Gradients . . . . . . . . . . . . . . . . . . . . 96

4.4 Adjoint Solver Performance Analysis . . . . . . . . . . . . . . . . . . 100

4.5 Optimization of the UH60 Rotor . . . . . . . . . . . . . . . . . . . . . 103

4.5.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.5.2 CFD/CSD Analysis . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5.3 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . 105

viii

5 Conclusions & Future Work 117

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

A Optimization 121

A.1 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

A.2 Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . . 124

Bibliography 127

ix

List of Tables

4.1 Summary Table of Test Cases . . . . . . . . . . . . . . . . . . . . . . 82

4.2 Summary Table of Applications . . . . . . . . . . . . . . . . . . . . . 82

4.3 Convergence comparison between the original and modified meshes . 94

4.4 Accuracy of Time Spectral Adjoint Method on NACA 0012

- Viscous Flow/Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.5 Design parameters and two set of nonlinear constraints . . . . . . . . 107

4.6 UH60 blade optimization with no trim angle constraints . . . . . . . 108

4.7 Design parameters and four sets of constraints . . . . . . . . . . . . . 110

4.8 UH60 blade optimization with four sets of constraints . . . . . . . . . 111

4.9 Computational time comparison for the inviscid UH60 test case . . . 115

4.10 Computational time comparison for viscous calculations on a NACA

0012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

x

List of Figures

1.1 Optimization Process Schematics . . . . . . . . . . . . . . . . . . . . 8

2.1 Boundary Layer Thickness . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Flux Discretization in 2 D . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Cell Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Flux Discretization in 2 D . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Inviscid Flux Discretization in Three-dimensions . . . . . . . . . . . . 36

3.5 Artificial Dissipation Discretization in Two-dimensions . . . . . . . . 37

3.6 Artificial Dissipation Discretization in Matrix . . . . . . . . . . . . . 41

3.7 Stencil depicting the required cells for inviscid flux and artifical dissi-

pation calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.8 Viscous Flux in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.9 Imaginary Box for Nodal Value . . . . . . . . . . . . . . . . . . . . . 47

3.10 Average Nodal Value . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.11 Surface Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.12 Stencil for the viscous flux, inviscid flux and artificial dissipation cal-

culations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.13 Supersonic Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.14 Supersonic Outlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.15 Euler Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.16 Euler Boundary – Pressure Extrapolation . . . . . . . . . . . . . . . 54

3.17 Viscous Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.18 Symmetric Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

xi

3.19 Farfield Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.20 Multiblock Mesh Configuration . . . . . . . . . . . . . . . . . . . . . 73

3.21 Mutiblock With Halo Cell . . . . . . . . . . . . . . . . . . . . . . . . 73

3.22 Multiblock With Corresponding Jacobian . . . . . . . . . . . . . . . . 74

3.23 Multiblock Horizontal Halo Exchange . . . . . . . . . . . . . . . . . . 75

3.24 Multiblock Vertical Halo Exchange . . . . . . . . . . . . . . . . . . . 75

3.25 Hicks-Henne perturbation function . . . . . . . . . . . . . . . . . . . 77

3.26 UH60 airfoil section with applied Hicks-Henne perturbation function . 78

4.1 NACA 0012 Airfoil Mesh . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2 NACA 0012 Single Block Mesh . . . . . . . . . . . . . . . . . . . . . 84

4.3 Comparison of Single and Multiblock Topologies for the NACA 0012

Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4 UH60 Helicopter Blade Mesh . . . . . . . . . . . . . . . . . . . . . . 85

4.5 UH60 Modified Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.6 Viscous Flat Channel Mesh . . . . . . . . . . . . . . . . . . . . . . . 87

4.7 Inviscid, Steady Gradient Comparison for the NACA 0012 . . . . . . 89

4.8 Sensitivity comparison for angular frequency = 5 . . . . . . . . . . . 90

4.9 Sensitivity comparison for angular frequency = 35 . . . . . . . . . . . 91

4.10 Sensitivity comparison for angular frequency = 140 . . . . . . . . . . 92

4.11 Inviscid, Unsteadiness Sensitivity Captured with Time Spectral Dis-

crete Adjoint Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.12 Time Spectral Adjoint Gradient Comparison for the UH60 . . . . . . 94

4.13 Arbitrary Multi-Block Halo Cells . . . . . . . . . . . . . . . . . . . . 95

4.14 Multi-Block Adjoint Solver Study . . . . . . . . . . . . . . . . . . . . 96

4.15 Viscous Flat Channel Sensitivity Study . . . . . . . . . . . . . . . . . 97

4.16 NACA Airfoil 121 × 33 × 13 mesh . . . . . . . . . . . . . . . . . . . . 97

4.17 Viscous NACA Airfoil Mesh Refinement Study . . . . . . . . . . . . . 98

4.18 NACA Airfoil Time Spectral Adjoint Sensitivity Comparison . . . . . 99

4.19 Time Spectral Discrete Adjoint run on UH60 Mesh, part 1 . . . . . . 100

4.20 Time Spectral Discrete Adjoint run on UH60 Mesh, part 2 . . . . . . 101

xii

4.21 Modified CFD/CSD Analysis Diagram . . . . . . . . . . . . . . . . . 105

4.22 UH60 base and optimized configurations . . . . . . . . . . . . . . . . 106

4.23 Cp plots at different radial locations . . . . . . . . . . . . . . . . . . . 108

4.24 Geometric changes along the radial span . . . . . . . . . . . . . . . . 109

4.25 Optimized geometry with full trim constraints . . . . . . . . . . . . . 111

4.26 Geometric changes along the radial span with full trim constrants . . 112

4.27 Cp plot comparison at different radial locations

with full trim constraints . . . . . . . . . . . . . . . . . . . . . . . . . 113

xiii

Chapter 1

Introduction

1.1 Introduction to Research

Shape optimization is an important part of engineering design and manufacturing. In

aerospace engineering, shape optimization is particularly important because geometry

holds a great influence over the overall performance and stability of the aircraft.

Traditionally, optimization is performed based on the experience and intuition of the

engineer.

Typical objective functions in aerospace engineering have complicated dependence

relatioinships due to the nonlinear interaction of many input variables. Hence, rea-

sonable ways to approach optimization problems are either heavily dependent on the

judgement of experienced eyes or based on reduced models for which a set of analytical

solutions exist.

Recent developments[42, 55, 35, 53, 49, 12, 57, 19, 37, 56, 45, 46] in Numerical

Science facilitate the use of Computational Fluid Dynamics (CFD) in aerodynamic

shape optimization, and accordingly, significant performance improvements have been

achieved. However, most of this work has been confined to steady cases or unsteady

cases using vastly simplified models. The objective of the present work is to develop

a high-order shape optimization method for unsteady flow.

Although the computer industry has made remarkable technological leaps in recent

years, accurate resolution of unsteady fluid phenomena still requires a prohibitively

1

CHAPTER 1. INTRODUCTION 2

large amount of computing power. In addition, the large number of design parameters

used in the optimization can become an additional hindrance. In order to estimate

the gradient with respect to each parameters by the finite difference method, the

process of solving the set of governing equations must be repeated a number of times

equal to the number of independent parameters as will be discussed in the following

sections. An adjoint-based method for determining gradients[24, 50] can obviate this

process, requiring additional computational cost equivalent to only a flow solution.

As a result, the time savings with the adjoint-based gradients is proportional to the

number of independent parameters used in the optimization.

Even with the incorporation of adjoint-based gradients, the unsteady solution still

often requires a large number of time steps for a reasonably accurate representation

of the unsteady flow phenomena. It is not unusual to require 1,000 to 100,000 time

steps of unsteady integration, depending on the nature of the flow being considered.

In addition to the CPU time, the large amount of time spent storing the entire flow

solution history make this approach practically infeasible to implement. In order to

alleviate these costs, the Time-Spectral Method for unsteady flow solutions has been

adopted in this research.

Employing the Time Spectral Method[10] together with the Discrete Adjoint for-

mulation for gradient information yields the following advantages:

• Reduction in the computational costs for flow solutions at each design iteration

• Reduction in the computational costs for determining gradients at each design

iteration

• Ability to optimize with a large number of design parameters

• Provision of error estimation[5] information for unsteady flows (as a by-product

of solving the adjoint equations).

Recent works by Nadarajah and Tatossian[45, 46] describe parallel efforts to reduce

the cost of unsteady shape optimization using the nonlinear frequency domain method

in adjoint approach.


In this work, after implementing and verifying all of these necessary components,

optimization has been performed on a helicopter rotor to verify the new adjoint

gradient-based, unsteady optimization methodology. In the following sections, the

background and objectives of the research are discussed more extensively.

1.2 Gradient Based Optimization

Before computational methods were used in the design process, the primary tool for

the development of aerodynamic configurations was the wind tunnel. With the tech-

nological advances in both computer and numerical science, computational methods

are now widely accepted in the aircraft industry. Much of the early effort was made

by Murman and Cole[42], Jameson[17, 19] and Caughey[27], to name a few, although

advancements have been pursued throughout the years. Even though the introduc-

tion of computational methods for shape optimization was widely recognized as an

idea with great potential, the computational cost for design work posed a formidable

challenge.

Gradient-based optimization[20], by nature, requires evaluation of the objective

and constraint functions and their gradients at each design iteration. Suppose the

cost function depends on N parameters.

I = I(α1, α2, · · · , αN)

A direct approach to evaluating the gradients is to perturb each parameter in turn,

recalculate the cost, and use finite differences to estimate the gradient.

∂I

∂αi

=I(α1, · · · , αi + δαi, · · · , αN) − I(α1, · · · , αi, · · · , αN)

δαi

Consequently, the cost of finding a descent direction at each design iteration, when

using finite differences to compute the necessary gradients, is equivalent to

(N + 1) ∗ Tflow,


where N is the number of design parameters and Tflow is the time required to solve

the flow equations and evaluate the objective and constraint functions. Considering

that a viscous solution on a medium sized mesh for a three dimensional wing costs

approximately 5 hours of CPU time, the total time to find a descent direction at each

design iteration would take more than 100 CPU hours for 20 design parameters.

1.3 Adjoint Gradient Calculation

While the introduction of new computational tools presented a new method for op-

timization, its practical application is limited by the manageability of the problem

size. As discussed in the previous section, obtaining gradient information for each

parameter at every design iteration is very time consuming. Assuming a quadratic

behavior of the objective function and constraints, a Sequential Quadratic Program-

ming algorithm[9, 48] will find its optimum after a number of iterations equal to the

number of design parameters. Assuming N parameters, it will require at least N2

evaluations of the flow solution if the gradients are calculated by finite differences.

This proves to be a serious bottleneck in the optimization process, particularly for

design problems with a large N .

In 1988, Jameson successfully introduced a concept of control theory into the field

of aerodynamic shape optimization[20]. The crux of this idea is to use control theory

in order to obtain gradients for every parameter at each design iteration at a cost

equivalent to capturing a single flow solution. The significance of this method is that,

at each design iteration, the computational time required to determine a descent

direction is of the same order as the time required to evaluate one flow solution.

Instead of requiring N2 evaluations, a time equivalent to N evaluations of the flow

solution would suffice to attain an optimal point in the ideal case. The method is

termed the Adjoint Method.

Following earlier applicatioins of this approach to airfoil and wing design [24,

26, 50, 51, 28], Kim[33] applied a continuous adjoint formulation for aerodynamic


shape optimization to multi-element airfoils, using the compressible Reynolds Aver-

aged Navier-Stokes equations. Drag minimization and lift maximization were per-

formed on multi-element airfoils using a multi-block, multi-processor RANS solver.

As design variables, Kim chose the airfoil shape, element configuration and angle

of attack. In his work, the application of the Adjoint Method to the compressible

RANS equation was demonstrated successfully in the aerodynamic shape optimiza-

tion of high lift systems.

Nadarajah[47] compared a continuous and a discrete Adjoint formulation. The

difference between these two formulations is in the way of applying control theory

directly to the actual discretized form of the governing equation. He also applied the

discrete Adjoint Method to the inverse problem in which both drag minimization and

reduced near field pressure minimization were achieved.

1.4 Time Spectral Method

In order to reduce the inherently large computational costs, more efficient ways of

solving unsteady flows have garnered much attention. One approach is the Harmonic

Balance Method by Hall[58]. Another is the Non-Linear Frequency Domain (NLFD)

method developed by McMullen[40]. This takes a pseudo-spectral approach to recast

the non-linear, unsteady equations in the time domain into an equivalent set of equa-

tions in the frequency domain. Once in the frequency domain, only the number of

modes used to approximate the solution matters. Selecting a finite number of modes

represents a loss of information. However, McMullen shows by using NLFD methods

that the first few modes are often sufficient to approximate an unsteady solution with

reasonable accuracy.

Although the advantages of the NLFD method are significant, it incurs an addi-

tional cost of computing Fourier transforms to the frequency domain and the inverse

Fourier transform to the time domain. Not only does the additional source of com-

putational cost slightly offset its advantages, it requires an additional set of complex

variables which causes extra complexity in the code and demands more storage space.

More specifically, it is equivalent to decreasing the available memory by half. Even


with these drawbacks, the NLFD method by McMullen opened up the possibility of

incorporating spectral solution approaches for periodic unsteady flows.

As an alternative, Gopinath[11], Jameson, and Van der Weide proposed the Time

Spectral Method. The Time Spectral Method utilizes a clever formulation of the time

derivative terms in the time domain which captures the effect of the spectral solution.

Typical approaches for evaluating time derivatives use two or more time instances of

the solution that had been previously computed. The Backward Difference Formula

(BDF) scheme is one such approach that uses three time instances of the solutions and

is second order accurate in time. However, the Time Spectral method makes use of all

time instances, and it is this set of previously known solutions which contributes to

capturing the spectral effect. By cleverly exploiting the concept of a geometric series,

the final form of the derived equation hides all details of Fourier and inverse Fourier

transforms such that the use of complex variables is not necessary. Thus, it solves the

two difficulties in using the NLFD method. First, no additional computational cost

is required. Second, no additional memory is required. Hence, the Time Spectral

method achieves a better performance with lower CPU costs and smaller memory

requirements.

1.5 Goals of Research: Time Spectral Adjoint

Method

The application of the Adjoint Method to unsteady flow has been attempted previ-

ously by Nadarajah[45, 46, 47]. In Nadarajah’s work, in order to obtain time-accurate

unsteady gradients, unsteady adjoint equations were constructed at every time step

by using flow solutions around the current time step. Because obtaining the time-

accurate solution for unsteady flows can easily require thousands of time steps, the

unsteady adjoint equations must also be constructed and solved for the same large

number of time steps at each design iteration. Knowing that a Sequential Quadratic

Programming (SQP) algorithm[48] requires a number of iterations equivalent to the

number of design parameters to find an optimum, Nadarajah’s approach leaves no


choice but to either employ reduced order models or sacrifice the integrity of the un-

steady flow solution. Another serious bottleneck is due to storing unsteady solutions

at every time step. For example, assuming it takes about 10,000 time steps in order

to obtain unsteady solutions within the desired accuracy and the size of each solution

file is 20 MBytes, storing all of the solution would require 10,000 times 20 MBytes,

which is 200 GBytes. In a more realistic case, not only would storage space much

larger than 200 GBytes to be required, but also the input and output of that much

data renders the whole procedure unrealistic. It is well known both in theory and

in practice that I/O time, not CPU time, is the biggest bottleneck of computational

works.

The Time Spectral Adjoint Method, which combines the Time Spectral Method

and the Adjoint Method, can provide a viable solution to the unsteady design prob-

lem. The Time Spectral Adjoint Method can be applied to any unsteady flow with

periodicity to obtain unsteady gradients at the desired design iterations. As previ-

ously discussed, a successful implementation of this method would not only reduce

the convergence time of the unsteady flow solution, it would also deliver gradient

information for each design iteration at a computational cost equivalent to a single

additional time-spectral flow solution.

In the implementation of the Time Spectral Discrete Adjoint Method, all neces-

sary adjoint derivations are developed for both the Euler and Navier-Stokes equations,

along with different boundary conditions. In the multi-block, multi-processor imple-

mentation for the helicopter blade, a way of updating halo cells has been devised and

presented in the papers by the present author[34], [2], [1].

1.6 Applications of the Time Spectral Adjoint

Method

For initial inviscid test cases, a NACA 0012 airfoil configuration is chosen for simu-

lations in both steady and unsteady flow for a more practical application, a model of

the UH-60 helicopter rotor is chosen. Unsteady flow calculations are performed and


Figure 1.1: Optimization Process Schematics


once the gradient information obtained by the Adjoint Method is verified in compar-

ison with finite differences, it can be applied in the optimization process outlined in

figure 1.1

For the calculation of the flow solution, the Euler and Reynolds-Averaged Navier-

Stokes (RANS) solver is used. The solver, named SUmb, is multi-block and multi-

processor capable and utilizes implicit residual averaging, dual time stepping and

enthalpy damping convergence acceleration techniques. It also has a full implemen-

tation of the time-spectral formulation of Gopinath that is used in this research. For

the solution of the time-spectral adjoint system, the Portable, Extensible Toolkit for

Scientific Computation (PETSc), developed by Argonne National Laboratory is em-

ployed. A discrete adjoint formulation is applied, which offers a distinct advantage

in that there is no limitation in the choice of cost function. With the continuous

adjoint approach, the only available choice for the cost function are those functions

that explicitly depends on pressure or shear stress alone. For multi-processor imple-

mentation, MPI and MPI2 standard is used.

The drag coefficient is the chosen objective function for the steady case, and a

total drag coefficient over all the time instances of the time-spectral solution is chosen

for the unsteady case. For the helicopter blade, a torque is chosen with nonlinear

constraints on thrust and drag forces. Refere to 4.5 for more details.

Chapter 2

Governing Equations

2.1 Navier-Stokes Equations and RANS

The Navier-Stokes equations encompass the necessary physics for modeling the com-

pressible and viscous nature of a fluid. The three-dimensional Navier-Stokes equations

[14, 15, 36] can be written in indicial notation as,

∂w

∂t+∂fi

∂xi

=∂fvi

∂xi

, (2.1)

where the state vector w, inviscid flux vector f , and viscous flux fv are represented

by

w =

ρ

ρu1

ρu2

ρu3

ρE

, fi =

ρui

ρuiu1 + pδi1

ρuiu2 + pδi2

ρuiu3 + pδi3

ρuiH

, fvi =

0

σijδj1

σijδj2

σijδj3

ujσij + k ∂T∂xi

.

The viscous stress tensor, σ, can be written as

σij = µ

(

∂ui

∂xj

+ λδij∂uk

∂xk

)

,

10

CHAPTER 2. GOVERNING EQUATIONS 11

with µ, λ, and δ as the first and second coefficients of viscosity and the Kronecker delta

function, respectively. The coefficient of thermal conductivity, k, can be calculated

from the following relationship,

k =cpµ

Pr,

where Pr is the Prandtl number and cp is the specific heat at a pressure given by

p = (γ − 1)ρ

E −1

2(uiuj)

Temperature can be determined using the perfect gas law as,

T =P

Rρ.

A relationship between energy and enthalpy is

ρH = ρE + p,

where γ is the ratio of specific heats. E and H represent the total energy, the

stagnation enthalpy, respectively.

When applying the Navier-Stokes equations to physical engineering problems, a

transformation between the physical and computational coordinate systems (xi and

ξi) is desired. This transformation can be described using

Kij =

[

∂xi

∂ξj

]

, J = det(K), K−1ij =

[

∂ξi

∂xj

]

and

S = JK−1.

The elements of the matrix S are nothing but the face area of the given cell projected

into the x1, x2 and x3 directions in the finite volume formulation. Using the above

relationships and the physical observation that the sum of the projected face area

in each direction becomes zero in each cell, the Navier-Stokes equations[43] in the


computational coordinate system become

∂Jw

∂t+∂Fi − Fvi

∂ξi= 0, (2.2)

where the inviscid and viscous flux contributions are now defined with respect to the

computational cell faces given by Fi = Sijfj and Fvi = Sijfvj .

To ensure numerical stability, artificial dissipation is added. The artificial dissipa-

tion scheme used in this research is a blended first and third order flux which was first

introduced by Jameson, Schmidt, and Turkel [22, 23, 29]. The artificial dissipation is

defined as

Di+ 1

2,j = ǫ2

i+ 1

2,j(wi+1,j − wi,j) − ǫ4

i+ 1

2,j(wi+2,j − 3wi+1,j + 3wi,j − wi−1,j). (2.3)

The first term in the above equation is a first order scalar diffusion term, where ǫ2i+ 1

2,j

is scaled by the normalized pressure difference and serves to damp oscillations near

shock waves. ǫ4i+ 1

2,j

is the coefficient for the third derivative of the artificial dissipation

flux. The coefficient is scaled so that it is zero in regions with large gradient, such as

shock waves.

2.2 Euler Equations

The three-dimensional Euler equations[14, 15, 36] can be written as,

∂w

∂t+∂fi

∂xi

= 0, (2.4)

where the state vector w and inviscid flux vector are represented by

w =

ρ

ρu1

ρu2

ρu3

ρE

, fi =

ρui

ρuiu1 + pδi1

ρuiu2 + pδi2

ρuiu3 + pδi3

ρuiH

.


A relationship for calculating pressure, p, is given by

p = (γ − 1)ρ

E −1

2(uiuj)

,

and a relationship between total energy and enthalpy is

ρH = ρE + p,

where γ is the ratio of specific heats. E,H, and δ represent the total energy, the

stagnation enthalpy and the Kronecker delta function, respectively.

The Euler Equations are equivalent to an inviscid form of the Navier-Stokes equa-

tions. While the inviscid assumption might seem to limit their usefulness, in general,

the Euler Equations are widely applicable. This is because in viscous flow, nearly

all of the viscous behavior is concentrated in thin regions near surfaces (boundary

layers) where large velocity gradients exist. In flows where the boundary layer thick-

ness is negligible compared to the length scale of the body, inviscid flow can be safely

assumed around the effective body that is formed by the presence of boundary layers

on the surface of the original geometry, as illustrated in the figure 2.1.

For two-dimensional, steady flow while neglecting the normal stress in the y-

direction, τyy, the following equation is obtained,

ρu∂u

∂x+ ρv

∂v

∂y= −

∂p

∂y+

∂

∂x[µ(

∂u

∂x+∂u

∂y)].

In terms of non-dimensional variables,

ρ′u′∂u′

∂x′+ ρ′v′

∂v′

∂y′= −

1

γM2∞

∂p′

∂y′+

1

Re∞

∂

∂x′[µ′(

∂u′

∂x′+∂u′

∂y′)].


Figure 2.1: Boundary Layer Thickness


The order of magnitude of each non-dimensionalized variable can be summarized as,

ρ = O(1) (2.5)

x′ = O(1)

y′ = O(δ)

u′ = O(1)

v′ = O(δ).

Performing an order of magnitude analysis on the above non-dimensionalized equation

results in the equation

O(δ) +O(δ) = −1

γM2∞

∂p′

∂y′+O(δ2)[O(δ) +O(

1

δ)]. (2.6)

From the equation (2.6), it is clear that ∂p′

∂y′= O(δ). From this, one can approximate

that∂p

∂y= 0. (2.7)

Equation (2.7) has a significant implication; at a given position in the x-direction,

the pressure is constant throughout the boundary layer in a direction normal to the

surface. In other words, the pressure at the outer edge of the boundary layer can

be applied directly to the surface beneath the boundary layer. Since most of the

aerodynamic quantities of interest are direct functions of pressure, the Euler equa-

tions produce a reasonably accurate pressure distribution over the original body with

an error comparable to the difference in size between the effective and the original

bodies.

Chapter 3

Development of A Time-Spectral

Discrete Adjoint Solver

3.1 Benefits of Time-Spectral Discrete Adjoint

Method

The process of aerodynamic optimization, as discussed in the chapter 1, requires at

least two major components: an optimizer and a flow solver. The optimizer receives

objective function and constraint values from the flow solver at the given design iter-

ation and these values are fed into the optimizer in order to determine feasibility and

a descent direction for the set of parameters. Once a descent direction is found, a

proper step length is determined as well. In traditional gradient-based optimization,

these two components are sufficient for reaching a local optimum under a given opti-

mization problem statement. In this research, the concept of adjoints is employed as

the final component required to complete an adjoint gradient-based optimization. As

discussed and compared in sections 1.2 and 1.3, deriving gradient information from

adjoints reduces the number of flow solution evaluations by N − 1, where N is the

number of independent parameters in the objective function. The decrease reduces

the total number of runs from N + 1 to 2, regardless of the size of N .

Applying the same logic, the effect of utilizing adjoint on the total number of flow

16

CHAPTER 3. DEVELOPMENT OF A TSM ADJOINT SOLVER 17

solution evaluation can be summarized as,

N + 1 ⇒ 2 (3.1)

for steady equations, where N is the number of parameters, and

(N + 1) ∗ Tflow ⇒ 2 ∗ Tflow (3.2)

for unsteady equations, where N is the number of parameters and Tflow is the to-

tal number of time steps required to resolve the given unsteady flow phenomenon.

Equations 3.1 and 3.2 both illustrate the benefits of using adjoints in the opti-

mization process, and it has already been implemented and documented in previous

studies[30, 31, 32, 43, 59, 44, 26].

The Time Spectral Method, pioneered by Arathi, Van der Weide and Jameson[3,

10], reduces the total number of time steps required for periodic flows. Therefore,

Tflow above can be replaced by Ttime−instances, where Ttime−instances represents the total

number of time instances required to resolve a periodic unsteady flow phenomenon.

The savings in computational time are significant. One specific study for Stage 35

found that the method required 6 times fewer multigrid cycles, reducing the 400,000

required for the traditional Backward Difference Formula, BDF, to 70,000 cycles[3].

By combining the Time Spectral Method with the Adjoint Method, CPU times for

obtaining gradients at each design iteration can be greatly reduced. From equation

3.2, the total number of flow solver evaluations can be further summarized as

• for optimization using time accurate unsteady simulation and finite difference

gradients,

(N + 1) ∗ Tflow (3.3)

• for optimization using time accurate unsteady simulation and adjoint gradients,

2 ∗ Tflow (3.4)


• optimization using the Time-Spectral Method for unsteady simulation and ad-

joint gradients,

2 ∗ TTS (3.5)

where TTS is defined as

TTS = RTS ∗ Tflow

and RTS designates the effectiveness of the Time Spectral Method for different appli-

cations. Finally, the effect of Time Spectral Adjoint Method is written as

(N + 1) ∗ Tflow ⇒ 2 ∗ Tflow ∗RTS = 2 ∗ TTS (3.6)

and the gain in CPU time is represented as

(N + 1) ∗ Tflow − 2 ∗ Tflow ∗RTS = Tflow ∗ (N + 1 − 2 ∗RTS). (3.7)

Development of the Time Spectral Adjoint Method is described in the following

three sections in more detail, including the governing equation for the Time-Spectral

Method, the implementation of the Discrete Adjoint Mehthod, and a description of

the optimization problem.

3.2 Flow Solver

3.2.1 Time Spectral Method

The Navier-Stokes equations in semi-discrete form can be written as

V∂w

∂t+R(w) = 0. (3.8)

When the flow follows a periodic pattern with known period T, the flow solution, w,

can be decomposed into a Fourier series, and as a result, V ∂w∂t

can be represented as


a function of w as follows,

wn =

N2−1∑

k=−N2

wkeiktn

Dtwn =

2π

T

N2−1∑

k=−N2

ikwkeiktn ,

where tn = n∆t. At this stage, Dtwn is a function of wk, so it must be converted into

a function of wk using an inverse transform,

wk =N−1∑

n=0

wne−ikn∆t.

Now, for the even number of N , Dtwn can be represented in terms of wn,

Dtwn =

2π

T

N2−1∑

m=−N2

+1

dmwn+m, (3.9)

where the coefficients[41] are given by

dm =

12(−1)m+1cot(πm

N) : m 6= 0

0 : m = 0.(3.10)

For the odd number of N ,

Dtwn =

2π

T

N−1

2∑

m= 1−N2

dmwn+m, (3.11)

and

dm =

12(−1)m+1cosec(πm

N) : m 6= 0

0 : m = 0.(3.12)

With both Dtwn and R(wn) as functions of wn, a new residual, R∗(wn), can be


defined:

R∗(wn) = V Dtwn +R(wn), (3.13)

Then, R∗(wn) is iteratively solved in pseudo-time until its convergence. The Time-

Spectral form of the governing equation is thus

V∂wn

∂τ+R∗(wn) = 0 (3.14)

where τ represents a pseudo time. The implementation of the time spectral adjoint

method requires an efficient solver for this equation.

3.2.2 URANS Flow Solver

A flow solver communicates with an optimizer in two ways: by returning objective

values – values of the cost function – as well as by returning constraint information,

as will be explained in section 3.4. Choosing a flow solver is no less important than

choosing an optimizer itself; most of the computational time involved in optimization

is spent evaluating the objective function and obtaining constraint information. Both

of the tasks are completed using flow solutions obtained by the solver[6, 8, 7, 9, 48].

Accuracy, CPU time, and memory storage space are important factors to consider

when choosing a flow solver. The solver selected for this work, SUmb, provides 2nd

order accuracy in space and 2nd order accuracy in time. It is an explicit solver, and

its required storage is approximately 100 bytes per node. CPU time depends mostly

on the computer architecture. However, due to its explicit, iterative nature, several

convergence techniques are implemented within SUmb[36, 41, 14, 15, 19, 18, 17, 21].

Faster convergence, low memory requirement, and reasonable accuracy makes it an

excellent choice for flows accompanying a shock.

SUmb

Due to its explicit nature, SUmb is better suited than an implicit solver for a large-

scale computation. This stems from a requirement that an implicit solver must store

information about the entire domain in a matrix which is subsequently inverted. An


explicit solver, on the other hand, needs only the bare minimum information required

to set up the problem and solve a solution.

One disadvantage of explicit solvers, however, is the prolonged time to conver-

gence. While the implicit solver was uses information from every cell in the domain,

the explicit solver knows of only those cells in its stencil at the time of each iteration.

Information must then travel through each stencil to reflect changes in the far right

cell from the far left cell and vice versa. This is one of the biggest penalties of the

explicit solver.

SUmb employs several techniques to overcome the slow convergence, namely:

• Runge-Kutta type Hybrid ODE solver

• Multiblock

• Parallelization.

Each of these will be further detailed in the following sections.

Hybrid ODE Solver Runge-Kutta ODE solvers are widely used because of their

simplicity of implementation and ability to handle large time-steps. The formulation

of the hybrid scheme is as follows:

w(0) = wn

w(1) = w(0) − αm∆tR(w0)

...

w(m+1) = w(0) − αm∆tR(w(m−1))

wn+1 = w(p).

(3.15)


The power of the hybrid scheme lies in the separate treatment of convective and

dissipative terms.

R(w(m)) = C(w(m)) +D(w(m))

D(w(m)) = βmD(w(m)) + (1 − βm)D(w(m−1)).

(3.16)

where C and D represent the convective and dissipative parts of the residual, respec-

tively. Separation of the convective and the dissipative part allows for the application

of different or optimized damping coefficients for different sets of governing equatios.

The convective term is dominant in the Euler equations, whereas in the Navier-Stokes

equations, the dissipative term is dominant. The separate adjustment of each term

produces an efficient amplification factor suitable for each set of equations. A five

stage scheme is widely used, and its coefficients are:

α1 =1

4, α2 =

1

6, α3 =

3

8, α4 =

1

2, α5 = 1,

β1 = 1, β2 = 0, β3 =14

25, β4 = 0, β5 =

11

25.

(3.17)

Multiblock Certain physical geometries are best represented by multiblock config-

urations. The inner boundary interface of a block becomes a set of halo cells which

need to be updated at every iteration to reflect changes in the flow variables. In an

explicit solver, this update process follows that of a single block configuration exactly,

except slight numerical errors due to the different paths for obtaining the same value.

In an implicit solver, however, the difference in the numbering system causes differ-

ent convergence behavior. This different numbering system is equivalent to having

different locations for off-diagonal terms.

Parallelization Parallelization is achieved by employing more than a single pro-

cessor to solve the discretized governing equation. Present-day computer technology

allows for machines containing large number of processors and to make the best use


of the available resources, multiple processors can execute code in parallel. Explicit

schemes are more suitable for parallelization than implicit schemes because with ev-

ery iteration each processor needs information from only those cells included in the

stencil. Parallelization should be built into the flow solver itself, because the logic of

the parallelized code is different from that of the sequential code.

For multi-CPU parallelization, the Message Passing Interface (MPI), is used. MPI

has long been the standard for parallel computing and has gained wide acceptance in

the CFD community. The MPICH programming library is very robust and available

for many different operating systems and compilers, including all major versions of

unix and linux, MAC OSX, and Microsoft Windows. In this study, MPICH for SGI

Origin 2000, SGI compiler, and Intel compiler has been used.

3.3 Discrete Adjoint Solver

Many realistic design problems involve a large number of parameters such that ob-

taining gradients by finite difference becomes prohibitively expensive. The adjoint

method, alternatively, can provide cheap gradient information reducing computa-

tional time roughly equivalent to evaluating a single flow solution. Therefore the total

computational cost for obtaining gradient information does not vary significantly for

a reasonably large number of design parameters.

3.3.1 Discrete vs. Continuous Adjoint

Adjoint solutions can be obtained through two different approaches: continuous ad-

joint and discrete adjoint formulations[43]. In the continuous adjoint approach, the

adjoint equation is first derived as a partial differential equation and then discretized.

In the discrete adjoint approach the discretized adjoint equation is derived directly

from the discretized cost function and flow equations. Hence, a discrete adjoint im-

plementation is inherently tied to the way in which the residual of the flow solver

is constructed. The continuous adjoint formulation has more freedom in the choice

of the complexity of discretization, while the discrete adjoint formulation has more


freedom in the choice of the cost function. In the limit of infinite grid resolution,

with sufficiently smooth geometries and smooth functions, both converge to the same

analytical solution as explored in detail in Nadarajah’s previous work[47],[43],[33].

In the continuous formulation, the adjoint solutions satisfy a set of governing

equations in the integral form, obtained after applying integration by parts. Thus, no

restriction is given to the discretization of the adjoint equations in the integral form

as long as the integral equations are satisfied. Because integration by parts involves

a conversion between a volume integral and a surface integral in the domain, only

cost functions that are a direct function of the flow variables on the surface can be

selected.

In the discrete formulation[4],[52], a discrete counterpart of integration by parts

is employed on the discretized governing equations. This has an implication in that

the order of discretization of the adjoint equations is directly tied to the order of

discretization of the flow solver. However, no restriction on the choice of the cost

function is required, as long as the adjoint variables satisfy the given set of equations.

The freedom to choose any cost function is the main reason for using the discrete

adjoint instead of the continuous adjoint in this research.

Implementation of the discrete adjoint requires obtaining exact jacobians for ev-

ery flow quantity in the discretized equations, which is not only very tedious but also

prone to human errors due to the complexity of certain quantities, such as the vis-

cosity coefficient, heat transfer coefficient and turbulent variables. Hand-derivations

of the flux jacobians proved a daunting task for some of the jacobians of the bound-

ary fluxes. In principle, automatic differentiation[54] can solve this problem since it

produces exact jacobians of any quantity by performing line-by-line differentiation

directly on the equations itself as long as its derivative is defined. However, even

though automatic differentiation reduces the possibilities of human errors consider-

ably, proper identification of inputs and outputs for each jacobian still needs careful

attention.


3.3.2 General Derivation of the Adjoint Equation

Let I represent a cost function of interest, such as drag or lift. In this case, I would

depend on the flow-field variables, w, and the geometry of the domain, F .

I = I(w,F).

The change of I[52] can be represented as

δI =∂IT

∂wδw +

∂IT

∂FδF, (3.18)

which depends on both δw and δF . The component of δI involving δw is much more

costly to calculate than δF of component. With this in mind, the concept of Lagrange

Multiplier, ψ, can be applied to the equation (3.18) so that the final form does not

contain any terms involving δw.

The residual can be defined as

R(w, F ) = 0,

which is also a function of w and F and its variable δR can be split into two parts

using the chain rule,

δR =

[

∂R

∂w

]

δw +

[

∂R

∂F

]

δF = 0. (3.19)

Noticing that δR is equivalent to zero, it can be freely added or subtracted to equation

3.18, which is represented as,

δI =∂IT

∂wδw +

∂IT

∂FδF − ψT

([

∂R

∂w

]

δw +

[

∂R

∂F

]

δF

)

=

∂IT

∂w− ψT

[

∂R

∂w

]

δw +

∂IT

∂F− ψT

[

∂R

∂F

]

δF.


As long as ψ is chosen to satisfy the equation

[

∂R

∂w

]T

ψ =∂I

∂w, (3.20)

the quantity inside the first set of brackets on the right hand side of the equa-

tion(number) is zero. The result is

δI =

∂IT

∂F− ψT

[

∂R

∂F

]

δF, (3.21)

which is commonly represented as,

δI = GδF,

where

G =

∂IT

∂F− ψT

[

∂R

∂F

]

.

In equation 3.21, δI is expressed in the form in which the explicit dependence on δw

is removed, after ψ is chosen from equation 3.20.

3.3.3 Implementation

The general platform for the adjoint equation is shown in figure 3.1, and specific

formulations depend on the way the residual is constructed as well as the type and

number of neighboring cells from which its components are coming. In some cases,

the derivative does not even exist according to the mathematical definition, and an

approximation should be made.

Specific formulations can be divided into Inner-Cell and Outer-Cell parts as seen

in figure 3.2. Inner cells have residuals composed of contributions from the cells that

lie entirely within the boundary. Outer cells are those with residuals coming from

either the boundary cells or cells that belong to a different computational domain

(defined as 1-to-1 internal connection cell).

The residual formulation for all of the inner cells is exactly the same, regardless of


Figure 3.1: Flux Discretization in 2 D


Figure 3.2: Cell Types


location, so the same derivative formulation can also be applied to all of the inner cells.

A description with further detail can be found in subsection 3.3.5 for inviscid flow and

3.3.6 for viscous flow. For the outer cells, residual formulations vary, depending on

the types of physical boundaries. The 1-to-1 internal connection cells follow the same

residual formulation as the inner cells, but have different indices as they are essentially

the same inner cells just separated by a different domain definition. Physical boundary

cells require much more attention and call for their own residual formulations.

While many physical boundary conditions exist, the following six conditions were

needed in this study, and their derivatives are developed in subsection 3.3.7:

• Farfield Boundary

• Euler Wall Boundary

• Viscous Wall Boundary

• Supersonic Inlet Boundary

• Supersonic Outlet Boundary

• Symmetric Boundary.

The biggest obstacle in developing the derivatives in equations 3.19 - 3.21 is de-

termining a reliable method of verification for the derived quantity. Three different

methods can be considered for evaluating the derivatives: Finite Difference, Complex

Derivative[38], and Direct Differentiation.

Finite differences are the simplest to implement, but a known drawback of the

method is cancellation error which can arise whenever a Jacobian matrix has near-

zero elements. Complex derivatives[39], on the other hand, produce accurate deriva-

tives, but their implementation is significantly more involved. The direct approach

was adopted in this work. The differentiation can be performed by hand or automat-

ically with dedicated software packages such as Tapenade. Automatic differentiation

involves treating a source code as a list of sequential functions (lines) which can be

differentiated in order following the chain rule. Both approaches were applied, be-

cause the results from Tapenade require the correct specification of a set of input


and output variables. When the result from the jacobian formulation matches that

of Tapenade as well as the result of line-by-line differentiation, the jacobian has been

verified.

Once the jacobian matrices and the objective function derivatives have been for-

mulated, the system is solved using PETSc, a matrix solver from Argonne National

Laboratory. While there may be a rather steep learning curve with PETSc, it has

proven to be a very effective software package. More information can be found in the

following website addresses:

• http://www.mcs.anl.gov/petsc/petsc-as/ for PETSc

• http://tapenade.inria.fr:8080/tapenade/index.jsp for Tapenade (automatic dif-

ferentiation program).

3.3.4 Derivatives of the Time-Spectral Terms

The derivatives of the time spectral term in equation 3.11,

Dtwn =

2π

T

N−1

2∑

m= 1−N2

dmwn+m,

play a crucial role in the successful implementation of the Time-Spectral Discrete

Adjoint Method. The time spectral term gathers contributions of the flow variables

over all the time instances, each of which is multiplied by the corresponding coefficient

in equation 3.12,

dm =

12(−1)m+1cosec(πm

N) : m 6= 0

0 : m = 0.

The sum is then added to the spatial residual as shown in equation 3.13,

R∗(wn) = V Dtwn +R(wn).


Following the definition of residual for the adjoint equations in equation 3.20,

[

∂R

∂w

]T

ψ =∂I

∂w,

the time spectral term needs to be differentiated at each time instance with respect

to each conservative flow variables at all the time instances. Thus, when the time

spectral term is differentiaed with respect to its own flow variables at different time

instances, the resulting jacobian becomes unity and only the coefficients survive.

When the derivatives are applied to the time spectral term assuming 5 time instances,

the following vectors are formed, :

−−→Dtw

1

wi = [ 0 d1w2 d2w

3 −d2w4 −d1w

5 ]

−−→Dtw

2

wi = [ −d1w1 0 d1w

3 d2w4 −d2w

5 ]

−−→Dtw

3

wi = [ −d2w1 −d1w

2 0 d1w4 d2w

5 ]

−−→Dtw

4

wi = [ d2w1 −d2w

2 −d1w3 0 d1w

5 ]

−−→Dtw

5

wi = [ d1w1 d2w

2 −d2w3 −d1w

4 0 ]

(3.22)

where the power of w denotes the specified time instance of the flow variable, w and

i spans from 1 to 5. For the coefficient, dm, refer to equation 3.12.

In the implementation of the vector in equation 3.22, each element of the vector is

located in the index location of its corresonding time instance. Assuming 100 spatial

nodes for the computational mesh and five conservative flow variables for each node,

the matrix for a single time instance is a 500x500 sparse matrix. If 5 time instances

are used, the global matrix becomes a 2500x2500 sparse matrix. The second element

of the vector in equation 3.22 is 500 indices away from the first element of the vector

in the global matrix. As a result, each row of the 2500x2500 global matrix has 5

elements in the row, spaced 500 indices apart.


3.3.5 Inviscid Formulation

The Euler equations in differential form can be written as

∂w

∂t+∂f

∂x+∂g

∂y= 0. (3.23)

The equations put into integral form are[25]

∂

∂t

∫

S

wdS +

∫

∂S

[f(w)dy − g(w)dx] = 0 (3.24)

in two dimensions. In a full three-dimensional representation, surface vectors, Sx, Sy,

and Sz, replace dx and dy as follows,

ddt

∫

ΩWdV +

∫

∂Ω(FxdSx + FydSy + FzdSz) = 0

ddt

∫

ΩWdV +

∫

∂ΩΣi

−→Fi ·

−→dSi = 0,

(3.25)

wherep = (γ − 1)ρ(E − q2

2),

H = E + p

ρ= c2

γ−1+ q2

2.

(3.26)

The vector q is the flow velocity,

q2 = u2 + v2 + w2, c2 =γp

ρ, (3.27)

and c is the speed of sound. From the definitions of above, equation 3.25 is discretized,

and the fluxes are represented as

W =

ρ

ρu

ρv

ρw

ρE

, Fx =

ρu

ρuu+ p

ρuv

ρuw

ρuH

, Fy =

ρv

ρvu

ρvv + p

ρvw

ρvH

, Fz =

ρw

ρwu

ρwv

ρww + p

ρwH

.

(3.28)


Figure 3.3: Flux Discretization in 2 D

Euler Flux

As shown in figure 3.3, the flux through each face of a cell in the positive i directioin

can be represented as

−→F 1

2

·−→S 1

2

=

ρ(uSx + vSy + wSz)

ρu(uSx + vSy + wSz) + pSx

ρv(uSx + vSy + wSz) + pSy

ρw(uSx + vSy + wSz) + pSz

ρE(uSx + vSy + wSz) + p(uSx + vSy + wSz)

, (3.29)

where F 1

2

signifies the average flux value over the i and i+ 1 cell,

Fi+ 1

2

=1

2Fi + Fi+1, (3.30)


and S 1

2

signifies the surface connecting i and i + 1 cell, as in figure 3.3. In the i

direction, the residual at cell i is

R(w) = Σ−→F ·

−→S

= 12

Fi+1S+ 1

2

+ FiS+ 1

2

− FiS− 1

2

+ Fi−1S− 1

2

= 12

S+ 1

2

Fi+1 + (S+ 1

2

− S− 1

2

)Fi + S− 1

2

Fi−1

.

(3.31)

By summing up the fluxes in three directions (i, j, and k with fluxes F,G, and H,

respectively),

R(w) = Σ−→F n

1

2

·−→Sn

1

2

= 12

Si+ 1

2

Fi+1 + (Si+ 1

2

− Si− 1

2

)Fi + Si− 1

2

Fi−1

+ 12

Sj

+ 1

2

Gj+1 + (Sj

+ 1

2

− Sj

− 1

2

)Gj + Sj

− 1

2

Gj−1

+ 12

Sk+ 1

2

Hk+1 + (Sk+ 1

2

− Sk− 1

2

)Hk + Sk− 1

2

Hk−1

,

(3.32)

where Si, Sj, and Sk represent cell faces and F,G, and H represent fluxes in the i, j,

and k directions of the cell, respectively.

In terms of the flow variables, each flux is divided into two parts, each of which

has a set of equations as in equation 3.29. In the three-dimensional representation,

there are seven cells involved, and the residuals on each cell are

cell(i, j, k) = 12

(Si+ 1

2

− Si− 1

2

)Fi + (Sj

+ 1

2

− Sj

− 1

2

)Gj + (Sk+ 1

2

− Sk− 1

2

)Hk

cell(i+ 1, j, k) = 12

Si+ 1

2

Fi+1

cell(i− 1, j, k) = 12

Si− 1

2

Fi−1

cell(i, j + 1, k) = 12

Sj

+ 1

2

Gj+1

cell(i, j − 1, k) = 12

Sj

− 1

2

Gj−1

cell(i, j, k + 1) = 12

Sk+ 1

2

Hk+1

cell(i, j, k − 1) = 12

Sk− 1

2

Hk−1

,

(3.33)

where the definitions of F,G, and H resemble equation 3.29, with the correct faces

and flow variables chosen for the corresponding fluxes. Figure 3.4 shows the row


arrangement for the corresponding ∂R∂w

in matrix form.

Since the entire equation 3.32 constitutes a residual for one cell, in the derivative

of each cell with respect to the flow variables at every other cells, ∂R(w)∂w

, only those

cells involved in the residual calculation contributes to each row of the matrix. Hence,

each row of the matrix looks like those in figure 3.4, where

A = coefficient of the cell(i, j, k − 1)

B = coefficient of the cell(i, j − 1, k)

C = coefficient of the cell(i− 1, j, k)

D = coefficient of the cell(i, j, k)

E = coefficient of the cell(i+ 1, j, k)

F = coefficient of the cell(i, j + 1, k)

G = coefficient of the cell(i, j, k + 1)

as in equation 3.33.

For all of the inner cells whose contributing cells are within the computational

boundaries, the residual equation 3.32 is applied. However, the flux definition changes

for the cells beyond the boundaries. As a consequence, equation 3.29 needs modifi-

cations which make it suitable for the unique physical nature of each boundary. The

index of the contributing cell, as in equation 3.33 changes as well. Thus, the deriva-

tion presented in this section, equation 3.32, and equation 3.33 apply only to the

formulation of the inner cell flux. For special treatment of physical boundary cells,

refer to the 3.3.7

Artificial Dissipation

For inviscid equations, an implementation of artificial dissipation is mandatory, be-

cause it contributes heavily to the speed of convergence[25, 16] and is required for

the stability of the explicit, iterative methods used in SUmb. SUmb could be roughly

categorized as a central difference, explicit solver.

The flux balance in the residual can also be represented as

∂wj

∂t+ hj+ 1

2

+ hj− 1

2

= 0, (3.34)


Figure 3.4: Inviscid Flux Discretization in Three-dimensions


Figure 3.5: Artificial Dissipation Discretization in Two-dimensions


wherehj+ 1

2

= 12(fj+1 + fj) − dj 1

2

hj− 1

2

= 12(fj + fj−1) − dj− 1

2

dj+ 1

2

= ǫ2i+ 1

2

αi+ 1

2

(wi+1 − wi)

−ǫ4i+ 1

2

αi+ 1

2

(wi+2 − 3wi+1 + 3wi − wi−1)

dj+ 1

2

= ǫ2i− 1

2

αi− 1

2

(wi − wi−1)

−ǫ4i− 1

2

αi− 1

2

(wi+1 − 3wi + 3wi−1 − wi−2).

(3.35)

The expressionǫ2i+ 1

2

and ǫ4i+ 1

2

are adaptive coefficients, and αi+ 1

2

gives the proper scale

to the dissipative term. The expression ǫ is defined as follows,

νi,j =∣

∣

∣

pi+1,j−2pi,j+pi−1,j

pi+1,j+2pi,j+pi−1,j

∣

∣

∣

νi+ 1

2,j

= max(νi+2,j , νi+1,j , νi,j, νi−1,j)

ǫ2i+ 1

2,j

= min(12, k2ν

i+ 1

2,j)

ǫ4i+ 1

2,j

= min(0, k4 − ανi+ 1

2,j) ,

(3.36)

wherek2 = 1

k4 = 132

α = 2.

It should be noticed that, in equation 3.35, the artificial dissipation term spans across

four adjacent cells for each side of flux. With two artificial dissipation terms involved

in two-dimensional calculation, a five-cell stencil is involved in the flux balance equa-

tion for the given cell in one direction.

Another important fact is that the definitions of ν and ǫ involve absolute value

and min/max functions. By the definition of differentiation, a derivative of neither

of them exists. In the present development, this value is held constant.


The artificial dissipation balance term alone can be expressed as,

dj+ 1

2

− dj− 1

2

= −B+wj+2

+ (A+ + 3B+ −B−)wj+1

+ (−A+ − 3B+ − A− − 3B−)wj

+ (A− + 3B− −B+)wj+1

+ −B−wj−2,

(3.37)

whereA+ = ǫ2

j+ 1

2

αj+ 1

2

B+ = ǫ4j+ 1

2

αj+ 1

2

A− = ǫ2j− 1

2

αj− 1

2

B− = ǫ4j− 1

2

αj− 1

2

.

Figure 3.5 shows the five cell stencils involved in the calculation. In the full three-

dimensional representation, the complete artificial dissipation balance would appear

as,


di+ 1

2

− di− 1

2

+ dj+ 1

2

− dj− 1

2

+ dk+ 1

2

− dk− 1

2

= −Bi+wi+2,j,k

+ (Ai+ + 3Bi

+ −Bi−)wi+1,j,k

+ (−Ai+ − 3Bi

+ − Ai− − 3Bi

−)wi,j,k

+ (Ai− + 3Bi

− −Bi+)wi−1,j,k

+ −Bi−wi−2,j,k

+ −Bj+wi,j+2,k

+ (Aj+ + 3Bj

+ −Bj−)wi,j+1,k

+ (−Aj+ − 3Bj

+ − Aj− − 3Bj

−)wi,j,k

+ (Aj− + 3Bj

− −Bj+)wi,j−1,k

+ −Bj−wi,j−2,k

+ −Bk+wi,j,k+2

+ (Ak+ + 3Bk

+ −Bk−)wi,j,k+1

+ (−Ak+ − 3Bk

+ − Ak− − 3Bk

−)wi,j,k

+ (Ak− + 3Bk

− −Bk+)wi,j,k−1

+ −Bk−wi,j,k−2

+ −Bk−wi,j,k−2 ,

(3.38)

where the superscript i indicates i-direction and the subscript i indicates the cell

index.

Figure 3.6 shows the matrix representation of the equation 3.38. Within the matrix,

each coefficient corresponds to a specific cell. The general relationships are


Figure 3.6: Artificial Dissipation Discretization in Matrix


A = coefficient of the cell(i, j, k − 2)

B = coefficient of the cell(i, j, k − 1)

C = coefficient of the cell(i, j − 2, k)

D = coefficient of the cell(i, j − 1, k)

E = coefficient of the cell(i− 2, j, k)

F = coefficient of the cell(i− 1, j, k)

G = coefficient of the cell(i, j, k)

H = coefficient of the cell(i+ 1, j, k)

I = coefficient of the cell(i+ 2, j, k)

J = coefficient of the cell(i, j + 1, k)

K = coefficient of the cell(i, j + 2, k)

L = coefficient of the cell(i, j, k + 1)

M = coefficient of the cell(i, j, k + 2).

Notice that the cell with indices i, j, k collects information for the artificial dissipation

term from all three directions:

Coefficient of wi,j,k

=

(−Ai+ − 3Bi

+ − Ai− − 3Bi

− −Aj+ − 3Bj

+ − Aj−

−3Bj− − Ak

+ − 3Bk+ −Ak

− − 3Bk−)

By putting together equations 3.38 and 3.32, the stencil in three-dimensions can

be represented by the schematic in figure 3.7. The stencil consists of five cells required

for the inviscid flux terms, while an additional six cells are needed for the artificial

dissipation terms.


Figure 3.7: Stencil depicting the required cells for inviscid flux and artifical dissipationcalculations


3.3.6 Viscous Formulation

Viscous Jacobian Calculation

The calculation of the viscous flux is based on the Divergence Theorem,

∫ ∫ ∫

V ol(−→ ·

−→F )dV =

∫ ∫

Surf

−→F · d−→a ,

where F represents flux vectors. In a discretized form, the right hand side of the

equation becomes∫ ∫

Surf

−→F · d−→a = ΣN

i=1(ΣMj=1FjSj)i,

where S,N, and M represent surfaces, the number of surfaces, and the number of

physical dimensions, respectively.

Terms for the viscous stress tensor from the Navier-Stokes equations(in one direc-

tion, u) can be written as

∫ ∫ ∫

V ol

∂(λ−→ ·

−→V + 2∂u

∂x)

∂x+µ∂(∂v

∂x+ ∂u

∂y)

∂y+µ∂(∂u

∂z+ ∂w

∂x)

∂zdV,

which can be written equivalently as

∫ ∫ ∫

V ol

−→ · (λ

−→ ·

−→V + 2

∂u

∂x, µ∂v

∂x+∂u

∂y, µ∂u

∂z+∂w

∂x)dV.

In this form, the Divergence Theorem can be directly applied to produce the following

result∫ ∫

Surf

(λ−→ ·

−→V + 2

∂u

∂x, µ∂v

∂x+∂u

∂y, µ∂u

∂z+∂w

∂x) · d−→n dS.

Each of the remaining viscous terms can be formulated in the same manner and are

listed here for completion;

∫ ∫

Surf

(λ−→ ·

−→V + 2

∂u

∂x, µ∂v

∂x+∂u

∂y, µ∂u

∂z+∂w

∂x) · d−→n dS

∫ ∫

Surf

(µ∂v

∂x+∂u

∂y, λ

−→ ·

−→V + 2

∂v

∂y, µ∂w

∂y+∂v

∂z) · d−→n dS


∫ ∫

Surf

(µ∂u

∂z+∂w

∂x, µ∂w

∂y+∂v

∂z, λ

−→ ·

−→V + 2

∂w

∂z) · d−→n dS

∫ ∫

Surf( uτxx + vτxy + wτxz + k ∂T

∂x,

uτyx + vτyy + wτyz + k ∂T∂y,

uτzx + wτzy + wτzz + k ∂T∂z

) · d−→n dS,

(3.39)

whereτxx = λ

−→ ·

−→V + 2∂u

∂x

τyy = λ−→ ·

−→V + 2∂v

∂y

τzz = λ−→ ·

−→V + 2∂w

∂z

τxy = µ ∂v∂x

+ ∂u∂y

τxz = µ∂u∂z

+ ∂w∂x

τyz = µ∂w∂y

+ ∂v∂z

τyx = τxy

τzx = τxz

τzy = τyz .

The formulation of the viscous term of the Navier-Stokes equations in equation

3.39 can be visually interpreted in figure 3.8. Red dots represent the nodal points

of the mesh, while black dots are the mid-surface points for each of the six surfaces

of the cube. Directional derivatives of each velocity component are obtained at the

nodal points, and their average values are obtained at each surface. In mathematical

terms, on each surface of the volume,

∂−→V i

∂−→x i

at the black dots = Σ1

4

∂−→V i

∂−→x i

over the four red dots, (3.40)

where the left hand side is the value computed at the black dot and the right hand

side consists of an average value over each of the red dots at the nodes of the surface.

Obtaining an Average Value at nodal points

In obtaining the nodal values, the same divergence theorem is applied to the velocity

vector itself rather than to the derivative of the velocity vector as in equation 3.39.


Figure 3.8: Viscous Flux in 3D

Thus, the volume-averaged divergence at a node can be represented as the sum of the

quantity itself multiplied by the surface area. Figure 3.9 shows an imaginary volume

over which the divergence theorem is applied, and figure 3.10 shows the location of

the quantity at each surface.

The process involves the following steps:

• Identify four cells which enclose the surface of interest and identify the rectan-

gular cube, as in figure 3.11

• Calculate the surface area of the imaginary volume by taking an average of the

eight sub-surfaces surrounding itself

• Calculate the average quantity of the velocity component by taking an average

of the four cells surrounding itself

• Using the average surface area and the average value of the velocity components,

apply the divergence theorem.

The procedure discussed above adds considerable difficulty to the derivation of

the jacobian, because each nodal velocity component involves eight surrounding cells

and four of those values contribute to one of the surfaces in the flux computation.

The same procedure must be applied three times, once for each velocity component.


Figure 3.9: Imaginary Box for Nodal Value

In the end, the flux on each surface depends on eighteen surrounding cells and the

full flux in each direction depends on twenty-seven surrounding cells. The detailed

procedure is derived below. Let

−→F = (u(x, y, z), 0, 0)

be the function of interest, which can be represented in the alternative form

∫ ∫ ∫

∂u

∂xdV =

∫ ∫ ∫

−→ · (u(x, y, z), 0, 0)dV.

After applying the divergence theorem,

∫ ∫ ∫

−→ · (u(x, y, z), 0, 0)dV =

∫ ∫

(u(x, y, z), 0, 0) · −→n dS,

a surface integral replaces the volume integral. In discretized form,

∫ ∫

(u(x, y, z), 0, 0) · −→n dS = ΣiuiSi

x.

This result can be substituted back into the original volume integeral form with

the result∂u(x, y, z)

∂xV = Σiu

iSix.


Figure 3.10: Average Nodal Value

Finally, an average value of the derivative in x-direction becomes,

∂u(x, y, z)

∂x=

ΣiuiSi

x

V.

Similarly, the average values of the derivatives in all directions become,

∂u(x, y, z)

∂x=

ΣiuiSi

x

V

∂u(x, y, z)

∂y=

ΣiuiSi

y

V

∂u(x, y, z)

∂z=

ΣiuiSi

z

V

∂v(x, y, z)

∂x=

ΣiviSi

x

V


Figure 3.11: Surface Calculation


∂v(x, y, z)

∂y=

ΣiviSi

y

V

∂v(x, y, z)

∂z=

ΣiviSi

z

V

∂w(x, y, z)

∂x=

ΣiwiSi

x

V

∂w(x, y, z)

∂y=

ΣiwiSi

y

V

∂w(x, y, z)

∂z=

ΣiwiSi

z

V.

Once these values are obtained at each node, they are averaged in order to obtain a

value at the mid surface. The values obtained in this manner can be used to construct

normal and shear stresses at the surface. Finally, the viscous flux, which requires the

values of stress at the surface, can be calculated.

A full implementation of the inviscid flux, viscous flux and artificial dissipation

calculations requires twenty seven cells for forming the inviscid and viscous flux terms

and additional six cells for the artificial flux, as shown in figure 3.12. Thus, the full vis-

cous flux in each cell depends on a total of thirty three surrounding cells. Accordingly,

developing viscous flux jacobians is a more involved process than that of developing

inviscid flux jacobians. This more involved process is also much more susceptible to

the introduction of human error. With the help of an automatic differentiation tech-

nique, assessing the accuracy of the flux jacobians is made feasible. Finite different

methods fail to provide reliable comparisons whenever cancellation errors are present.

To assure the accuracy of a viscous flux jacobian, it needs to be tested in a situation

where any contribution from the inviscid flux jacobian is minimal.

3.3.7 Boundary Conditions

Description

Obtaining flux jacobians of ghost cells with respect to the inner cells is not a trivial

process to implement. Sometimes, updating boundary conditions can be as easy as


Figure 3.12: Stencil for the viscous flux, inviscid flux and artificial dissipation calcu-lations


forming a jacobian matrix whose elements are zeros. However, in more complicated

cases, boundary updates can only be achieved by freezing certain flow variables,

since the dependence between the boundary cells and inner cells can be prohibitively

complex or it is possible that derivatives are not clearly defined.

The ultimate goal of boundary updates is to calculate flux jacobians at the bound-

ary surfaces of the corresponding ghost cell with respect to the inner cells. The

complexity of this process becomes apparent when one considers the difficulty of

obtaining the exact dependence of the ghost cells on the corresponding inner cells.

Certain boundary conditions require a complicated process of updating ghost cells

with each step which can be challenging. The whole tracking process should be ap-

plied at each step of the calculation for each term of the five-by-five jacobian matrix

in the three-dimensional case.

The whole process can be conceptually divided into two parts, to make the task

more manageable. First, flow variable derivatives of the ghost cell with respect to the

flow variables of the corresponding inner cell are calculated and saved. This results in

a five-by-five matrix if each of the conservative variables is composed of a five-by-one

column vector. Once a matrix is obtained, the process of calculating jacobian fluxes

can be modified so that it utilizes not only the flow variables in the ghost cell but also

their derivatives with respect to the flow variables of the inner cell. This approach

simplifies the verification of the final jacobian matrix, because each smaller step of

the process is simpler to verify individually than the final jacobian itself.

Jacobian matrices are obtained using the chain rule of differentiation and verified

with values obtained by taking a direct derivative of the fluxes with respect to the

inner cell variable:∂Fouter

∂winner

=Fouter

wouter

wouter

winner

. (3.41)

Some of the jacobians have a very complex dependence on the inner cell variable

such that a direct hand-calculation is almost futile. The application of equation 3.41

proved very useful in such cases and especially so in the farfield boundary case. Each

boundary condition used in this research is listed below and further discussed.


Figure 3.13: Supersonic Inlet

Figure 3.14: Supersonic Outlet

Supersonic Inflow

By the theory of characteristics at an inflow location, it is clear that the flow variables

in the ghost cell do not depend on those of the adjacent inner cell, as shown in figure

3.13. In fact, it is the inner cell variables that depend on the variables of the ghost

cells. This means that the derivative of an upstream ghost cell with respect to the

inner cells is zero, meaning that the jacobian is also zero. Hence, the flux jacobian

on the incoming surface is a five-by-five matrix of zeroes.

Supersonic Outflow

At supersonic outflow locations, the values of the flow variables are exact copies of

those of the inner cells, as shown in figure 3.14. The derivative of the flux jacobian

on the ghost cell with respect to the inner cell becomes the same as the flux jacobian


Figure 3.15: Euler Boundary

Figure 3.16: Euler Boundary – Pressure Extrapolation

on the inner cell itself.

Euler Wall

The outer cell updates for the Euler wall boundary condition are based on the physical

observation that the velocity vector on the face should be parallel to the face, as shown

in figure 3.15. This is true if the tangential component of the velocity in the ghost

cell is the same as that of the inner cell while the normal components are equal in

magnitude and opposite in direction. This relationship is illustrated by the following


equations,

u‖ = v‖ with respect to the surface

u⊥ = −v⊥with respect to the surface.

Halo cell pressures must also be represented in terms of inner cell flow variables. As

shown in figure 3.16,

p1 = p2 + (p2 − p3)

= 2p2 − p3,

and

p2 = (γ − 1)(ρE)2 −12ρ2(

u22+v2

2+w22

2)

p3 = (γ − 1)(ρE)3 −12ρ3(

u23+v2

3+w2

3

2),

where γ is the ratio of specific heats and u, v, and w are the i, j, and k components

of the velocity vector at inner cells 2 and 3. Thus, the wall boundary condition for

inviscid flow can be represented as,

0

p1Sx

p1Sy

p1Sz

0

= 2

0

p2Sx

p2Sy

p2Sz

0

−

0

p3Sx

p3Sy

p3Sz

0

, (3.42)

and it is possible to take a derivative of equation 3.42 with respect to the flow variables

in the two adjacent inner cells.

Viscous Wall

In the cell-centered approach of discretization schemes, the viscous boundary con-

dition imposes zero velocity at a wall surface, which requires the velocity vector of

the outer cell to be exactly opposite to that of the corresponding inner cell in each

direction. For a viscous boundary, pressures of the outer cell are often equated to

the pressures of the inner cell. Temperatures are linearly extrapolated so that the


Figure 3.17: Viscous Boundary

Figure 3.18: Symmetric Boundary

average of T1 and T2 is equal to Twall, as in equation 3.43,

p

ρu

ρv

ρw

2Twall − T2

1

=

p

−ρu

−ρv

−ρw

T

2

. (3.43)


Figure 3.19: Farfield Boundary

Symmetric Boundary

A symmetric boundary condition is depicted in figure 3.18. One may notice its

resemblance to the inviscid wall boundary condition in figure 3.15. In fact, the velocity

vector is updated in a manner similarly to that of the inviscid boundary condition:

u‖ = v‖ with respect to the reflection surface

u⊥ = −v⊥with respect to the reflection surface.

The difference between the two updates is that the normal component, u‖, follows

the reflection surface rather than the physical surface. Other than u⊥ = −v⊥, all of

the other flow variables share exactly the same values.

Farfield

The main difficulty in obtaining outer cell updates for a farfield boundary condition

comes from the complexity involved in using Riemann invariants. In order to obtain

flux jacobians, representations of the outer cell flow variables with respect to those of

the inner cell must first be established. However, the process of updaing boundaries

using Riemann invariants is a function of the speed of sound and normal velocity

of the inner and outer cells at the mid-surface. Consequently, the calculation of


jacobians from Riemann invariants inherently requires performing derivatives on a set

of complex functions involving square root and power functions. The derivation and

implementation of the flux jacobian for a farfield boundary condition is an important

component of this research and is explained in detail throughout the rest of the

section.

Based on the theory of characteristics, extrapolated quantities can be categorized

as incoming and outgoing in a simple one dimensional analysis. Accordingly, Riemann

invariants are represented as,

R∞ = v∞ · −→n −2c∞γ − 1

Re = ve ·−→n +

2ceγ − 1

,

where e represents information coming from inner cells and ∞ represents extrapolated

information from the far field condition (outer cells). These two equations can be

manipulated to obtain

va · n⊥ =1

2(R∞ +Re)

ca =γ − 1

4(−R∞ +Re).

Further information on the tangential velocity is required in order to get the full

velocity vector. This information depends on the direction of the flow. For inflow, a

tangential velocity from the inner cell is used. For outflow, the tangential component

of the farfield velocity is used as follows,

va = ve · n‖ + va · n⊥ for outflow

va = v∞ · n‖ + va · n⊥ for inflow,(3.44)

where v∞ is an undisturbed free stream velocity and ve is that of the inner cell.

Now that we have a complete description of −→v and c, the speed of sound, only one

additional piece of information is required to close the system of equations. Entropy


can also be obtained by looking at the direction of the flow and as in equation 3.44,

sa = se for outflow

sa = s∞ for inflow.

In summary, one must obtain the following:

• ce and vn using the equations of Riemann invariants

• vt using the proper velocity vectors(based on the direction of the flow)

• S, entropy, based on the direction of the flow.

To apply this boundary condition to the Discrete Adjoint equation, one needs to

take derivatives of the farfield flux with respect to the flow variables of the inner cells.

A methodical application of the process can be summarized as follows:

• obtain the necessary flow variables using Riemann invariant extrapolation

• using the updated flow variables, obtain the flux jacobian at the farfield interface

• differentiate the flux with respect to the flow variables of the inner cell.

In order to minimize the probability of human error in complicated derivations, a

good practice for managing the complexity is the utilization of the chain rule of

differentiation. First, represent the flux derivative with respect to the generic flow

variables as a vector of partial derivative,

∂F

∂W= (

∂F

∂w1,∂F

∂w2,∂F

∂w3,∂F

∂w4,∂F

∂w5).

If the flux vector F is expanded into its five components, the derivatives of the flux

vector with respect to conservative variables can be represented in a more complete

form as∂F1

∂W= (

∂F1

∂w1,∂F1

∂w2,∂F1

∂w3,∂F1

∂w4,∂F1

∂w5)

∂F2

∂W= (

∂F2

∂w1,∂F2

∂w2,∂F2

∂w3,∂F2

∂w4,∂F2

∂w5)


∂F3

∂W= (

∂F3

∂w1

,∂F3

∂w2

,∂F3

∂w3

,∂F3

∂w4

,∂F3

∂w5

)

∂F4

∂W= (

∂F4

∂w1,∂F4

∂w2,∂F4

∂w3,∂F4

∂w4,∂F4

∂w5)

∂F5

∂W= (

∂F5

∂w1,∂F5

∂w2,∂F5

∂w3,∂F5

∂w4,∂F5

∂w5).

The above equation can then be represented in a more generic way by utilizing the

chain rule,

∂F1

∂W= (

∂F1

∂g1

∂g1

∂w1,∂F1

∂g1

∂g1

∂w2,∂F1

∂g1

∂g1

∂w3,∂F1

∂g1

∂g1

∂w4,∂F1

∂g1

∂g1

∂w5)

∂F2

∂W= (

∂F2

∂g2

∂g2

∂w1,∂F2

∂g2

∂g2

∂w2,∂F2

∂g2

∂g2

∂w3,∂F2

∂g2

∂g2

∂w4,∂F2

∂g2

∂g2

∂w5)

∂F3

∂W= (

∂F3

∂g3

∂g3

∂w1,∂F3

∂g3

∂g3

∂w2,∂F3

∂g3

∂g3

∂w3,∂F3

∂g3

∂g3

∂w4,∂F3

∂g3

∂g3

∂w5)

∂F4

∂W= (

∂F4

∂g4

∂g4

∂w1,∂F4

∂g4

∂g4

∂w2,∂F4

∂g4

∂g4

∂w3,∂F4

∂g4

∂g4

∂w4,∂F4

∂g4

∂g4

∂w5)

∂F5

∂W= (

∂F5

∂g5

∂g5

∂w1

,∂F5

∂g5

∂g5

∂w2

,∂F5

∂g5

∂g5

∂w3

,∂F5

∂g5

∂g5

∂w4

,∂F5

∂g5

∂g5

∂w5

)

where g is a set of intermediate functions that describe the complicated relation-

ship between F and W . This general relationship can be further broken down into

intermediate steps depending on the actual functions represented by g’s,

∂F

∂W=∂F

∂g1

∂g1

∂g2

∂g2

∂g3

∂g3

∂g4· · ·

∂gn

∂W

Derivation of Farfield Flux Jacobian

Using the chain rule, a full derivation of the farfield jacobian is here carried out with

the following definitions:

• the subscript B denotes values from a ghost cell

• the subscript JL denotes values from an internel cell


• the subscript S denotes values at the flux surface between the ghost and internal

cells.

With these definitions, the conservative variables at the ghost cell can be written as

functions of the variables in the internal cells:

ρB = f1( ~WJL)

PB = f2( ~WJL)

uB = f3( ~WJL)

vB = f4( ~WJL)

wB = f5( ~WJL),

where ~W is the vector of conservative variables. To determine the values at the flux

surface, a linear interpolation is used:

ρS =1

2(ρB + ρJL)

PS =1

2(PB + PJL)

uS =1

2(uB + uJL)

vS =1

2(vB + vJL)

wS =1

2(wB + wJL) .

From the values at the surface, a flux can be obtained using the following formulation,

G =

ρSVS

ρSVSuS + PSSx

ρSVSvS + PSSy

ρSVSwS + PSSz

ρSVSES + PSVS

,


whereVS = usSx + vsSy + wsSz

= 12VB + 1

2VJL.

The jacobian matrix for G, in terms of its own variables, is written as,

∂G1

∂w=

1

4

∂ (ρBVB + ρBVJL + ρJLVB + ρJLV JL)

∂w∂G2

∂w=

∂G1

∂wuS + ρSVS

(

1

2

)(

∂uB

∂w+∂uJL

∂w

)

+ Sx

∂P

∂w

∂G3

∂w=

∂G1

∂wvS + ρSVS

(

1

2

)(

∂vB

∂w+∂vJL

∂w

)

+ Sy

∂P

∂w

∂G4

∂w=

∂G1

∂wwS + ρSVS

(

1

2

)(

∂wB

∂w+∂wJL

∂w

)

+ Sz

∂P

∂w

∂G5

∂w=

1

2

(

VB

∂w+∂VJL

∂w

)(

γ

γ − 1PS +

ρS

2

(

u2S + v2

S + w2S

)

)

+

1

2(VB + VJL)

γ

γ−1

(

12

) (

∂PB

∂w+ ∂PJL

∂w

)

+14(u2

S + v2S + w2

S)+

ρS

uS12

(

∂uB

∂w+ ∂uJL

∂w

)

+

vS12

(

∂vB

∂w+ ∂vJL

∂w

)

+

wS12

(

∂wB

∂w+ ∂wJL

∂w

)

.

The above jacobian contains various terms of ~WB, ~WJL, ~WS and their partial deriva-

tives with respect to ~WJL, where ~WS is defined above as an average of ~WB and ~WJL.

At this point, more information about ~WB and ~WJL is needed. Using the relationships

between the primitive and conservative variables,


ρJL = WJL1

uJL =WJL2

WJL1

vJL =WJL3

WJL1

wJL =WJL4

WJL1

PJL = (γ − 1)WJL5−

1

2

W 2JL2

+W 2JL3

+W 2JL4

WJL1

.

The partial derivatives of the primitive variables with respect to ~WJL can be easily

obtained as

∂ρJL

∂w=

[

1 0 0 0 0]

∂uJL

∂w=

[

−WJL2

W 2JL1

1WJL1

0 0 0]

∂vJL

∂w=

[

−WJL3

W 2JL1

0 1WJL1

0 0]

∂wJL

∂w=

[

−WJL4

W 2JL1

0 0 1WJL1

0]

∂PJL

∂w1= (γ − 1)

(

1

2

)

W 2JL2

+W 2JL3

+W 2JL4

WJL1

∂PJL

∂w2= (γ − 1) (−1)

WJL2

WJL1

∂PJL

∂w3

= (γ − 1) (−1)WJL3

WJL1

∂PJL

∂w4= (γ − 1) (−1)

WJL4

WJL1

∂PJL

∂w5= (γ − 1) ,

∂VJL

∂w= Sx

∂uJL

∂w+ Sy

∂vJL

∂w+ Sz

∂wJL

∂w.


In order to get the jacobian for ~WB with respect to ~WJL, one must represent

~WB in terms of ~WJL. It is here where the concept of Riemann invariants enter the

analysis. The application of Riemann invariants results in values for CB and VB⊥,

which by themselves are insufficient to fully represent the five conservative variables.

To completely describe the conservative variables, SB and VB‖are extrapolated from

the internal or external cells, depending on the direction of the velocity vector at the

surface of the boundary.

To proceed further, we need to express ~WB in terms of CB, SB, VB⊥, and VB‖

, as

well as CB, SB, VB⊥, and VB‖

in terms of ~WJL. Using the well known expressions

SB = PB

ρBγ and CB =

√

γPB

ρB,

ρB =

(

1

γSB

) 1

γ−1

CB

2

γ−1

PB = SB

(

1

γSB

)γ

γ−1

CB

2γ

γ−1 ,

where CB is determined from Riemann invariants. Also, it is clear that ~VB is equal

to the vector sum ~VB⊥+ ~VB‖

. While ~VB‖receives the value of either ~VJL‖

or ~V∞‖,

~VB⊥is determined from Riemann invariants, similar to the manner in which CB was

obtained. For incoming flow,

SB = S∞

~VB‖= ~V∞ − ~V∞⊥

,

where

~V∞⊥=∣

∣

(

V∞xNx + V∞y

Ny + V∞zNz

)∣

∣

(

~Nx + ~Ny + ~Nz

)

while for the outgoing flow,

SB = SJL

~VB‖= ~VJL − ~VJL⊥

,


where

~VJL⊥=∣

∣

(

VJLxNx + VJLy

Ny + VJLzNz

)∣

∣

(

~Nx + ~Ny + ~Nz

)

.

It is clear that SB, and ~VB‖are easily obtained without any additional calculations.

The only remaining unknowns are CB, ~VB⊥, which can be obtained via Riemann

invariants. J+ and J− are defined as

J+ = VJL⊥+

2CJL

γ − 1

J− = V∞⊥−

2C∞

γ − 1.

CB, and VB⊥can be computed from the Riemann invariants,

VB⊥=

J+ + J−

2

CB =

(

γ − 1

4

)(

J+ − J−

2

)

.

V∞⊥and C∞ are constants, but VJL⊥

and CJL are functions of ~WJL,

VJL⊥= ~VJL · ~N

= VJLxNx + VJLy

Ny + VJLzNz

=WJL2

Nx +WJL3Ny +WJL4

Nz

WJL1

,

and

CJL =

√

γPJL

ρJL

=

√

(γ (γ − 1))

(

WJL5

WJL1

−1

2

W 2JL2

+W 2JL3

+W 2JL4

W 2JL1

)

,


which are both scalar values. Finally, after substituting the appropriate expressions,

~WB can be represented as a function of ~WJL,

ρB =

(

1

γSB

)1

γ−1(

γ − 1

4

)2

γ−1

WJL2Nx+WJL3

Ny+WJL4Nz

WJL1

− V∞⊥

+

2γ−1

(

C∞ +

√

(γ (γ − 1))(

WJL5

WJL1

− 12

W 2JL2

+W 2JL3

+W 2JL4

W 2JL1

)

)

2

γ−1

(3.45)

PB = SB

(

1

γSB

)γ

γ−1(

γ − 1

4

)2γ

γ−1

WJL2Nx+WJL3

Ny+WJL4Nz

WJL1

− V∞⊥

+

2γ−1

(

C∞ +

√

(γ (γ − 1))(

WJL5

WJL1

− 12

W 2JL2

+W 2JL3

+W 2JL4

W 2JL1

)

)

2γ

γ−1

(3.46)

uB = V∞x− |V∞⊥

|Nx +

Nx

2

WJL2Nx+WJL3

Ny+WJL4Nz

WJL1

+ V∞⊥

+

2γ−1

(

−C∞ +

√

(γ (γ − 1))(

WJL5

WJL1

− 12

W 2JL2

+W 2JL3

+W 2JL4

W 2JL1

)

)

(3.47)

vB = V∞y− |V∞⊥

|Ny +

Ny

2

WJL2Nx+WJL3

Ny+WJL4Nz

WJL1

+ V∞⊥

+

2γ−1

(

−C∞ +

√

(γ (γ − 1))(

WJL5

WJL1

− 12

W 2JL2

+W 2JL3

+W 2JL4

W 2JL1

)

)

(3.48)

wB = V∞z− |V∞⊥

|Nz +

Nz

2

WJL2Nx+WJL3

Ny+WJL4Nz

WJL1

+ V∞⊥

+

2γ−1

(

−C∞ +

√

(γ (γ − 1))(

WJL5

WJL1

− 12

W 2JL2

+W 2JL3

+W 2JL4

W 2JL1

)

)

.(3.49)


In order to express the above relationships in a most compact form, the following

shorthand notations is applied:

JJP =

WJL2Nx+WJL3

Ny+WJL4Nz

WJL1

− V∞⊥+

2γ−1

(

C∞ +

√

(γ (γ − 1))(

WJL5

WJL1

− 12

W 2JL2

+W 2JL3

+W 2JL4

W 2JL1

)

)

JJM =

WJL2Nx+WJL3

Ny+WJL4Nz

WJL1

+ V∞⊥+

2γ−1

(

−C∞ +

√

(γ (γ − 1))(

WJL5

WJL1

− 12

W 2JL2

+W 2JL3

+W 2JL4

W 2JL1

)

)

.

With this simplification, the jacobian for each variable can be more cleanly repre-

sented as

∂ρB

∂w=

(

1

γSB

)1

γ−1(

γ − 1

4

)2

γ−1(

2

γ − 1

)

JJP3−γγ−1

∂JJP

∂w

∂PB

∂w= SB

(

1

γSB

)γ

γ−1(

γ − 1

4

)2γ

γ−1(

2

γ − 1

)

JJPγ+1

γ−1∂JJP

∂w

∂uB

∂w=

Nx

2

∂JJM

∂w∂vB

∂w=

Ny

2

∂JJM

∂w∂wB

∂w=

Nz

2

∂JJM

∂w.

Thus, the task of obtaining the jacobians of ~WB comes down to obtaining jacobians

for ~JJM, and ~JJP in terms of ~WJL. After differentiating the expressions above, the

jacobians of JJP and JJM are


∂JJP

∂w1

= −W2Nx +W3Ny +W4Nz

W 21

+

2

γ − 1

γ (γ − 1)

2√

W5

W1−

W 22+W 2

3+W 2

4

W 21

(

−W5

W 21

+W 2

2 +W 23 +W 2

4

W 31

)

∂JJP

∂w2

=Nx

W1

+2

γ − 1

γ (γ − 1)

2√

W5

W1−

W 22+W 2

3+W 2

4

W 21

(

−W2

W 21

)

∂JJP

∂w3=

Ny

W1+

2

γ − 1

γ (γ − 1)

2√

W5

W1−

W 22+W 2

3+W 2

4

W 21

(

−W3

W 21

)

∂JJP

∂w4=

Nz

W1+

2

γ − 1

γ (γ − 1)

2√

W5

W1−

W 22+W 2

3+W 2

4

W 21

(

−W4

W 21

)

∂JJP

∂w2=

2

γ − 1

γ (γ − 1)

2√

W5

W1−

W 22+W 2

3+W 2

4

W 21

(

1

W 21

)

,

and∂JJM

∂w1=

∂JJP

∂w1

∂JJM

∂w2=

∂JJP

∂w2

∂JJM

∂w3=

∂JJP

∂w3

∂JJM

∂w4=

∂JJP

∂w4

∂JJM

∂w5=

∂JJP

∂w5.

The simple relationship between the jacobians of JJP and JJM exists because only

a single sign differs between the two equations, and this sign difference disappears

with differentiation.

For outgoing flow, SB and VB‖are now functions of ~WJL which requires that ∂SB

∂w

and∂VB‖

∂ware included in the calculation of the jacobian. This is in contrast to the

incoming flow, where ∂SB

∂wand

∂VB‖

∂ware entirely zero since SB and VB‖

are constant.

Thus, the jacobian of the outgoing flow can be described as that of the incoming flow


plus the contributions coming from SB and ~WB‖, or

∂ρB

∂w=

(

∂ρB

∂w

)

incoming

+ γ−1

γ−1

(

γ − 1

4

)2

γ−1

JJP2

γ−1

(

−1

γ − 1

)

S−γ

γ−1

B

∂SB

∂w

∂PB

∂w=

(

∂PB

∂w

)

incoming

+ γ−γ

γ−1

(

γ − 1

4

)2γ

γ−1

JJP2γ

γ−1

(

−1

γ − 1

)

S−γ

γ−1

B

∂SB

∂w

∂uB

∂w=

(

∂uB

∂w

)

incoming

+∂uB‖

∂w

∂vB

∂w=

(

∂vB

∂w

)

incoming

+∂vB‖

∂w

∂wB

∂w=

(

∂wB

∂w

)

incoming

+∂wB‖

∂w.

Again, calculating the jacobian requires obtaining two more sets of partial derivatives:

those for SB and ~VB‖.

SB =P

ργ

=(γ − 1)

(

ρE −(

12

)

ρ (u2 + v2 + w2))

ργ

= (γ − 1)

[

WJL5

WγJL1

−

(

1

2

)

W 2JL2

+W 2JL3

+W 2JL4

W1+γJL1

]

~VB‖=

WJL2

WJL1

−|NxWJL2

+NyWJL3+NzWJL4|Nx

WJL1

WJL3

WJL1

−|NyWJL2

+NyWJL3+NzWJL4|Ny

WJL1

WJL4

WJL1

−|NzWJL2

+NyWJL3+NzWJL4|Nz

WJL1

,


and their derivatives are given by,

∂uB‖

∂w=

[

−WJL2

W 2JL1

+(NxWJL2

+NyWJL3+NzWJL4)Nx

W 2JL1

1WJL1

− NxNx

WJL1

−NyNx

WJL1

−NzNx

WJL1

0

]

∂vB‖

∂w=

[

−WJL3

W 2JL1

+(NxWJL2

+NyWJL3+NzWJL4)Nx

W 2JL1

−NxNy

WJL1

1WJL1

− NyNy

WJL1

−NzNy

WJL1

0

]

∂wB‖

∂w=

[

−WJL4

W 2JL1

+(NxWJL2

+NyWJL3+NzWJL4)Nx

W 2JL1

−NxNz

WJL1

−NyNz

WJL1

1WJL1

− NzNz

WJL1

0

]

∂SB

∂w=

(γ − 1)

(

−γWJL5

Wγ+1

JL1

+ γ+12

W 2JL2

+W 2JL3

+W 2JL4

W2+γJL1

)

(γ − 1)

(

−WJL2

W1+γJL1

)

(γ − 1)

(

−WJL3

W1+γJL1

)

(γ − 1)

(

−WJL4

W1+γJL1

)

(γ − 1)(

1W

γJL1

)

T

.

3.3.8 Unsteady Boundary Conditions

The integral form of the Navier-Stokes equations can be formulated as

ddt

∫

VρdV +

∫

Sρ(−→u · −→n )dS = 0

ddt

∫

Vρ−→u dV +

∫

Sρ−→u (−→u · −→n )dS = −

∫

Spn⊥dS +

∫

S−→σ · −→n dS

ddt

∫

VρEdV +

∫

SρE(−→u · −→n )dS = −

∫

Sp−→u · −→n dS +

∫

S

−→u · −→σ · −→n dS

−∫

S−→q · −→n dS,

(3.50)


where

ρ = the density

p = the pressure−→u = the velocity vector−→σ = viscous stress tensor−→q = heat flux vector

E = total energy−→n = surface unit normal

V and S = volume and surface of the control volume, respectively

p = (γ − 1)ρ

E − 12(−→u · −→u )

.

For an unsteady flow, equation 3.50 requires the following specifications,

ρ(−→u · −→n ) => ρ(−→u −−→u surface ·−→n )

ρ−→u (−→u · −→n ) => ρ−→u (−→u −−→u surface ·−→n )

ρE(−→u · −→n ) => ρE(−→u −−→u surface ·−→n )

−∫

Sp−→u · −→n dS => −

∫

Sp−→u −−→u surface ·

−→n dS,

(3.51)

where −→u surface refers to the velocity of any moving surfaces.

In most cases with unsteady flow, a moving surface has a fixed velocity, such that

its derivative with respect to the flow variables becomes zero. However, the effect

of a moving surface is propagated into the flow field through multiplication with the

conservative flow variables, ρ, ρ−→u , and ρE. At a solid wall, (−→u · −→n ) becomes zero,

so contributions from the flux must also be. As explained in section 3.3.7, the only

meaningful contribution at a wall comes from p, as in the equation 3.42. The presence

of a wall velocity adds an additional term to equation 3.42,

∫

S

p(−→u surface)dS,


such that the equation becomes

Gwall =

0

pSx

pSy

pSy

p(−→u sruface ·−→S )

, (3.52)

where−→S is equal to (Sx, Sy, Sz), a surface vector in the x,y, and z directions.

In summary, the presence of a moving surface modifies the flux through that

surface. An effective velocity, −→u −−→u wall, must be applied to the flux contribution to

the inner cells to compensate. At the wall boundary, it also adds an additional term,

p(−→u surface ·−→S ), to the energy balance, as seen in equation 3.52.

3.3.9 Multiblock/Multiprocessor Implementation Issues

The quality of a computational mesh is a very important in determining the overall

quality of a CFD solution. However, different geometrics demand different meshing

schemes for generating a scientific mesh. In most cases, variations in the geometry

of interest make it extremely difficult to obtain high quality meshes using only one

meshing strategy for the entire geometry. In a multiblock approach, suitable meshing

schemes are chosen and applied to subsets of the total geometry. These individual

blocks are then grouped together, so the mesh quality of the entire geometry is as

good as that of each part. Thus, the implementation of a multiblock capability is an

important developement when dealing with realistic geometries.

The actual implementation of a multiblock, however, requires extra effort in book-

keeping boundary information as well as additional storage space. Within the flow

solver, the exchange of information between the blocks occurs through the updates

in the halo cells. The adjoint solver, however, requires an additional exchange of

index information for the halo cells, as shown in figure 3.20, 3.21, and figure 3.22. As

seen in figure 3.21, the contributions from donor cells must be properly accounted for

in the construction of each row of the matrix. This task is not necessarily straight


Figure 3.20: Multiblock Mesh Configuration

Figure 3.21: Mutiblock With Halo Cell


Figure 3.22: Multiblock With Corresponding Jacobian

forward, because the numbering system of multiblock cells is different from that of

single-block cells, even though the exact same overall configuration is represented.

The two types of halo cells seen in two-dimensional multiblock meshes are edge and

corner halo cells. Disserning between these two types becomes important due to the

different stencils that each requires for flux calculations. As discussed in section 3.3.5

and shown in figure 3.3, the calculation of an inviscid flux jacobian can involve one

edge cell halo at the most. An artificial viscosity jacobian can require two edge halo

cells as in figure 3.5. A viscous flux jacobian calculation must require contributions

from a handful of the edge halo cells as well as additional contribution from the corner

halo cells. It is relatively eash to keep track of edge halo cell indices since adjacent

block numbers are directly available. This information for corner halo cells is not as

easy to extract, however. For a multiblock mesh with arbitrary topology, one of the

methods for determining the indices of corner halo cells is to systematically check

through every edge cell in all of the connected blocks.

Another clever way to approach this problem is with a directional exchange of halo

cell information. In the first step of this exchange, shown in figure 3.23, information


Figure 3.23: Multiblock Horizontal Halo Exchange

Figure 3.24: Multiblock Vertical Halo Exchange


is passed horizontally, although the corner cell information is not yet available. The

key idea of the directional update process is that the corner cell of any given block

is also an edge cell for a different block. Therefore, for a two-dimensional mesh,

once all of the edge cells have been updated in one direction, all of the required

corner cell information is available for exchange in the other direction, as shown in

figure 3.24. In three-dimensions, the corner cell information is available only after

completing exchanges in two of the three directions. The order of the updates can

be arbitrary, as long as all of the edges are updated in a given direction and all of

the directions are covered. Another thing to notice is the increase in the number of

off-diagonal elements due to the index shift in the multiblock representation. This

increased number of off-diagonal elements causes an increase in the convergence time.

The implementation of a multiprocessor capability was the final component de-

veloped for this research. The MPICH library, which follows the MPI standard,

is utilized. Adding multiprocessor capability also requires additional bookkeeping,

similar to these required for multiblock. The main difficulty in implementing this

capability is in interfacing with PETSc. PETSc, an external package, provides a

robust and scalable library, but learning its capabilities and limitations can require

large amounts of time and effort.

3.4 Optimization

3.4.1 Hick-Henne Bump Functions

In aerodynamic shape optimization using adjoint gradients, each grid point can be

used as a design variable. Perturbing each grid point, however, introduces disconti-

nuities. This difficulty can be overcome by applying a set of smooth functions that

perturb the initial geometry. In choosing such a function, the existence of continuous

and smooth derivatives is important. Abrupt changes in the derivative of the function,

although the function itself is continuous, could have adverse effects on any quantities

in the flow solver that require contributions from the gradient of the quantity. Once

a smooth curve is generated, it can be stretched or contracted locally to simulate a


Figure 3.25: Hicks-Henne perturbation function

smooth bump around the desired grid point. For this purpose, the Hicks-Henne bump

function, has been selected[13],[31].

The Hicks-Henne bump function is a composite function of the sine function that

offers finer control in the localization of bump without losing continuity of its deriva-

tives. It is represented as

b(x) = a[

sin(πxlog5

logt1 )]t2

, 0 ≤ x ≤ 1. (3.53)

In equation 3.53, a represents the maximum bump height, t1 shows the location of the

maximum bump (at t1), and t2 determines the bump width. By adjusting t1 and t2,

a set of smooth bump functions is generated, each of which has the maximum bump

at the desired grid point, as shown in figure 3.25. Upon the generation of the smooth

curve, the bump is added on top of the geometry of the airfoil in the direction that

is normal to the surface as shown in figure 3.26.


Figure 3.26: UH60 airfoil section with applied Hicks-Henne perturbation function

3.4.2 Implementation Using a Gradient Based Optimizer

with Adjoint Gradients

While the gradient information is supplied by the Time Spectral Adjoint Method, a

Sequential Quadratic Programming (SQP) method[48, 9] is employed as an optimiza-

tion algorithm. The SQP method implements a set of nonlinear constraints during

the optimization process, which allows for more realistic test cases to be studied.

The Hicks-Henne function, as explained in section 3.4.1, is applied on nine airfoil

sections along each span. For each airfoil section, ten locations on the surface are

chosen as locations of maximum perturbation. Thus, a total of ninety design param-

eters are used for the current optimization problem. Further details are presented in

section 4.5. When applying the Hicks-Henne bump function to the airfoil surface, the

height of the bump function at different airfoil sections is scaled proportionally to the

thickness of the corresponding section.


NPSOL

NPSOL[7] is a suite of software for solving constrained optimization problems and it

executes a dense SQP algorithm. The SQP method is especially effective for nonlin-

ear problems with objective and gradients which are expensive to evaluate. As its

name suggests, the SQP algorithm utilizes a quadratic representation of the objective

function and solves it sequentially.

Given a nonlinear optimization problem described by

minimize f(x)

subject to g(x) = 0,(3.54)

a new function, called a Lagrangian, can be found from the function and constraints

as

L(x, λ) = f(x) − λTg(x), (3.55)

where λ is a vector of Lagrange multipliers. This formulation converts a constrained

optimization problem into an unconstrained problem where the optimality conditions

(Karush-Kuhn-Tucker, KKT, Necessary conditions)[48] are given by

L(x, λ) = 0.

Value of x and λ are updated as

xk+1 = xk + pk

λk+1 = λk + νk

, (3.56)

where pk and νk are obtained by solving the linear equation,

2L(xk, λk)

pk

νk

= − L(xk, λk).

As a result, the original nonlinear objective function with nonlinear constraints,


equation 3.54, is recast as a new, quadratic objective function with linear constraints,

minimize over p 12pT [2

xxL(xk, λk)]p+ pT [xL(xk, λk)]

subject to [g(xk)]Tp+ g(xk) = 0,

(3.57)

where νk is the vector of Lagrange multipliers. Thus, the problem is transformed into

a minimization of a quadratic equation. In the Sequential Quadratic Programming

method, the above quadratic equation is solved at each iteration to find a new set of

points, (pk, νk), which are used to update (xk, λk), as shown in equation 3.56. Refer

to Appendix A for further details.

Chapter 4

Results

4.1 Description of the Test Cases

Upon completing the Time Spectral Adjoint Method implementation, it has been

validated using gradient information obtained from both an Automatic Differentiation

Method as well as a Finite Difference Method. Test cases were studied using three

different geometries: a NACA 0012 airfoil, a flat channel, and a UH60 helicopter

blade, as summarized in table 4.1. After verifying the gradient information obtained

from the Time Spectral Adjoint Method, a shape optimization was attempted on the

UH60 blade with a realistic set of constraints.

A NACA 0012 airfoil was used as the testbed geometry throughout the whole

development process due to its simple O-mesh topology. Once the adjoint gradient

information for the airfoil test case was validated against the jacobians generated

by automatic differentiation, the adjoint method was applied to the UH60 which

represents a realistic test case. Then, a shape optimization was attempted in order

to validate the new optimization methodology which relies on the adjoint gradients.

In practice, the UH60 test case was much more difficult to execute than that of the

NACA 0012 airfoil. A C-H mesh topology of the blade presented sub-facing problems

as well as bad quality meshes near both tips of the blade. To account for these issues,

multi-block and multi-processor capabilities were implemented in the early phase of

the research. Upon adding these capabilities, the complexity grew larger again, as

81

CHAPTER 4. RESULTS 82

Geometry SensitivityVerifications

Inviscid Viscous Steady Unsteady

NACA 0012 Yes Yes Yes Yes

UH60 (modified) Yes No Yes Yes

UH60 (original) Yes No Yes Yes

Flat Channel No Yes Yes No

Table 4.1: Summary Table of Test Cases

Geometry Multi- Multi- Sequential Optimizationblock processor Halo Update

NACA 0012 Yes Yes Yes No

UH60 (modified) Yes Yes Yes No

UH60 (original) No Yes No Yes

Flat Channel No No No No

Table 4.2: Summary Table of Applications

the communication between halo cells had to be addressed. The directional halo cell

update scheme in section 3.3.9 was devised for this specific requirement. By tapping

into internal communication structures of the flow solver, SUmb, a shape optimization

on the UH60 was carried out utilizing multi-processor capability on single block mesh.

Application of the developments is summarized in table 4.2.

4.1.1 NACA 0012 Airfoil Mesh

A NACA 0012 airfoil is shown in figure 4.1. Initially, a two-dimensional airfoil was

used as a preliminary check on the gradient information. Figure 4.2 shows a tapered


X

Y

Z

(a) Far View

X

Y

Z

(b) Close Up View

X

Y

Z

(c) Surface View

Figure 4.1: NACA 0012 Airfoil Mesh


Z

X

Y

(a) Far View

Z

X

Y

(b) Close View

Figure 4.2: NACA 0012 Single Block Mesh

X

Y

Z

(a) Multiblock

X

Y

Z

(b) Single block

Figure 4.3: Comparison of Single and Multiblock Topologies for the NACA 0012 Mesh


Y

X

Z

(a) Single Block

X

Y

Z

(b) Multiblock

Figure 4.4: UH60 Helicopter Blade Mesh

wing which incorporates fully three-dimensional features. A taper ratio of fifty percent

was chosen. In order to test the multiblock capability of SUmb, the wing mesh is split

into 27 blocks which are shown in figure 4.3. The mesh is divided into three blocks in

each of the i, j and k directions, which allowed for the testing of all possible multiblock

configurations. The mesh has an O-shape topology and is generated from conformal

mapping procedures. The O-mesh topology is known to provide good quality results.

The farfield boundary stretches to approximately fifty times the chord length.

This is considered far enough from the airfoil surface, as it ensures the absence of

a reflecting wave from the farfield boundary. Figure 4.1 displays the internal 1-to-1

boundary condition for the spanwise boundaries.

4.1.2 UH60 Helicopter Rotor Mesh

Figure 4.4 shows UH60 blade meshes in both single block and multi block topologies.

They use a C-H mesh topology to capture the wake that is generated by the blade

rotation. The nature of the C-H topology causes the wing tip region to have several

quad cells with near-flat corners. Its non-periodic shape along with these flat corners

contributes to a prolonged convergence time.

In an effort to circumvent these issues, the surface definition of the original UH60


(a) Top View (b) Skewed View

X

Y

Z

(c) Surface View

Figure 4.5: UH60 Modified Mesh


Figure 4.6: Viscous Flat Channel Mesh

mesh in figure 4.4 was extracted and a new multiblock mesh was created. The farfield

boundaries for the modified mesh are far enough away to ensure better residual con-

vergence. The distance is between 10− 20 times blade span, with a tradeoff existing

between the farfield distance and the number of grid points required. Originally, the

mesh encompassed the entire circular path, but this required too many grid points.

Furthermore, the blade hub has an elliptical shape which prevented a symmetric rep-

resentation in a half-circle. Hence, three-quater circle was chosen as a compromise

between the number of grid points and better representation of the flow physics.

4.1.3 Flat Channel Mesh

A flat channel with four viscous walls, as in figure 4.6, was chosen for the viscous test

case. The absence of any wall curvature allows for an accurate test of the adjoint

gradient method, whereby any gradients are generated solely due to viscous effects.

Upon a verification of the viscous adjoint gradients, it is applied to NACA 0012 airfoil

for both of steady and unsteady cases.


4.2 Verification of Inviscid Gradients

Implementing the inviscid adjoint method was the first major development. The

treatment of boundary conditions offered major difficulties. The adjoint formula for

the inner cells, once developed, can be uniformly applied throughout the domain.

Physical boundaries, however, need their own set of equations, which require pro-

cedures ranging from the simple to the extremely lengthy. Refer to section 3.3 for

detailed information.

To verify the inviscid adjoint gradients, the NACA 0012 pitching airfoil was chosen

for steady and unsteady cases. It requires the following boundary conditions:

• inviscid wall

• symmetric

• 1-to-1 inner cell

• farfield

The UH60 was chosen as the more realistic case for steady, unsteady, and opti-

mization studies. It requires the following boundary conditions :

• inviscid wall

• 1to1 inner cell

• farfield

• multiblock

The UH60 application utilized a multiblock capability which was discussed in subsec-

tion 3.3.9.

In order to obtain gradients for all input parameters, the Hicks-Henne bump

function is applied to a set of nodal points along the airfoil chord on the upper and

lower surfaces. Rather than applying two sets of Hicks-Henne functions, one set is

applied starting from the lower trailing edge and ending at the upper trailing edge so

that a smooth surface is retained.


Figure 4.7: Inviscid, Steady Gradient Comparison for the NACA 0012

4.2.1 Steady Gradient Comparison

Figure 4.7 shows a comparison between the adjoint and finite difference gradients of

a drag coefficient for NACA 0012 airfoil. A transonic case with a mach number of

0.8 was simulated and produced a very good match. The Hicks-Henne function was

applied to the lower and upper surface.

4.2.2 Unsteady

Figure 4.8-4.10 shows an application of the Time Spectral Adjoint method to the

NACA 0012 for the same transonic conditions. The gradient shown is that of a

summation of cost function, a drag coefficient, over all time instances. Again, a very

good match is obtained between the adjoint and finite difference gradients. In figure

4.8 - 4.10, gradient comparisons are shown for varying angular frequencies, from 5 to

140. In figure 4.11, the three subfigures, 4.8 - 4.10, are plotted together to visualize


Figure 4.8: Sensitivity comparison for angular frequency = 5






Figure 4.11: Inviscid, Unsteadiness Sensitivity Captured with Time Spectral DiscreteAdjoint Method


Order of Convergence

Original 1.0e-3Modified 1.0e-8

Table 4.3: Convergence comparison between the original and modified meshes

0 2 4 6 8 10 12 14 16

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2LowerSurf_FiniteDifference_E5

LowerSurf_FiniteDifference_E7

LowerSurf_FiniteDifference_E8

LowerSuf_Adjoint

(a) Gradient Comparison on the BladeLower Surface

0 2 4 6 8 10 12 14 16

Location of Maximum Bump along Chord

-2

-1

0

1

2

Gradient of Cm/Cl

UpperSurf_FDGradient_E5

UpperSurf_FDGradient_E7

UppserSurf_FDGradient_E8

UpperSurf_AdjointGradient

(b) Gradient Comparison on the BladeUpper Surface

Figure 4.12: Time Spectral Adjoint Gradient Comparison for the UH60

the unsteady effect.

4.2.3 Multiblock

In the application of the Time Spectral Adjoint Method to the UH60 helicopter blade,

many different approaches were considered, partly due to mesh quality and partly due

to physical memory limitations. Section 4.1.2 details the process by which the UH60

mesh was modified from an initial configuration. As a result, much better convergence

in both of the flow solver and the adjoint solver was achieved compared to the original

mesh. The convergence rates of the flow solver are compared in table 4.3.

Sequential halo updates were applied to account for the unknown configuration of

the multiblocks. Identifying adjacent blocks is a relatively simple task, but identifying

indices of the corner halo cells needs careful consideration. A general schematic of this

situation appears in figure 4.13. The method of section 3.3.9 solves this issue correctly


(a) Arbitrary Multiblock Topology (b) Halo Cells

Figure 4.13: Arbitrary Multi-Block Halo Cells

for a structured mesh block without subfacing. It appears that even the presence of

subfacing might be handled correctly by the sequential update scheme, but it is left

for future research. Figure 4.12 shows the results of applying the Time Spectral

Adjoint Method with sequential halo cell updating to the modified multiblock UH60

mesh. The lower and upper surfaces are separately treated with independent sets of

Hicks-Henne functions. The adjoint gradients fall well within the envelope of the finite

difference gradients which are obtained using different degrees of perturbation. From

these results, it is clear that the adjoint gradients fall within the accuracy bounds set

by the cancellation error of the finite difference method.

Figure 4.14(a) contains a comparison of the resulting adjoint gradients obtained

from using different numbers of blocks on the NACA 0012 mesh, and gives a sense

of accuracy while utilizing multiblocks. Figure 4.14(b) shows the effect of multiblock

on the convergence. When forming jacobian matrices, the index of the cell deter-

mines the matrix bandwidth. In a single block, adjacent cells occupy predetermined

locations in the matrix, organized by the i, j and k directions. However, the multi-

block configuration requires different indexing schemes that follow a block-by-block

sequence. The block-by-block, global indexing, as opposed to the local simple index-

ing for a single block, brings additional complexity. With global indexing, an adjacent


0 2 4 60

5

10

15

20

25

30Single Block Single CPU − (1,1)

FD Gradient

ADJ Gradient

0 2 4 60

0.1

0.2

0.3

0.4Percentage Difference

0 2 4 60

0.1

0.2

0.3

0.4Comparison of Normalized Gradient

0 2 4 60

5

10

15

20

25

30Single Block Multi CPU − (1,2)

0 2 4 60

0.1

0.2

0.3

0.4

0 2 4 60

5

10

15

20

25

30Multi Block Multi CPU − (2,2)

0 2 4 60

0.1

0.2

0.3

0.4

(a) Verification of Multiblock Multipro-cessor

0 200 400 600 800 1000 1200 1400−10

−5

0

5Convergence History of CL/CM

Iteration

LOG

10 (

Adj

oint

Sol

ver

Res

idua

l )

UH60 MultiblockMach Number of 0.585Mach Number of 0.885Mach Number of 0.542Mach Number of 0.458

Residual History

(b) Convergence History of MultiblockUH60 Blade

Figure 4.14: Multi-Block Adjoint Solver Study

cell could be located at an index much larger or smaller than current location. As a

result, the stretched bandwidth in the resulting matrix requires additional iterations

for convergence, and its effect is clearly reflected in figure 4.14(b).

The sequential halo update is mainly devised for the construction of the viscous

jacobian of the multiblock topology. Although its effect is throughly tested and

verified, it is not applied to the calculation of the viscous jacobian. The viscous

results in the following section, 4.3, are obtained from a single block topology for

both of the flat channel, in figure 4.6, and the NACA 0012 airfoil, in figure 4.2-4.3.

Shape optimization on the UH60 blade, in section 4.5, is also carried out in the single

block topology of the UH60 mesh without utilizing the sequential halo update.

4.3 Verification of Viscous Gradients

A flat channel with four viscous walls in figure 4.6 is used to validate viscous jaco-

bians. Since the flat surface minimizes gradient contributions generated by non-zero

curvatures of the wall, this approach proves effective in isolating and validating the

viscous jacobians. A major difficulty in dealing with the viscous flux jacobians, beside

the complexities in the derivation, is in handling the corner halo cells. The physical


0 10 20 30 40 50 60

0

1

2

3

4

5

6

7

8

9

10

x 10−4

Perturbation point along the surface

Gra

dien

t

Comparison of Gradient on Cd

Viscous Channel

Mach = 1.66Re = 1.95e+4Mesh Size = 65x33x17Perturbation Size = 1e−8Free Stream T = 288 Kelvin

FD

ADJ

Figure 4.15: Viscous Flat Channel Sensitivity Study

X

Y

Z

Figure 4.16: NACA Airfoil 121 × 33 × 13 mesh

representation of an inner cell at an outer-most corner consists of a combination of

flow variables for the surrounding halo cells, and the final product should be differ-

entiated. This ambiguity adds another source of variability to the viscous adjoint

solution. In figure 4.15, the adjoint solution shifted up and down slightly depending

on the choice of flow direction at the halo cells of the outer-most corner of the interior

cells.

The viscous adjoint solution was also tested on a finer, inviscid grid, as shown in

figure 4.16, and a grid sensitivity comparison for the steady case is presented in figure

4.17. Three cases are presented to illustrate the effect of grid refinement along the

chord and span directions. The three mesh sizes are:


0 10 20 30 40 50 60 70

0

2

4

6

8

10

12


Gra

dien

t


NACA AirfoilMach = 1.73Re = 5e+5Mesh Size = 81x33x7Perturbation Size = 1e−8Free Stream T = 278 Kelvin

FD

ADJ

(a) 81 × 33 × 7

0 10 20 30 40 50 60 70

0

2

4

6

8

10

12


Gra

dien

t


FD

ADJ

NACA Airfoil

Mach = 1.73Re = 5e+5T

FS = 288.15 K

Perturbation = 1e−8

(b) 81 × 33 × 13

0 20 40 60 80 100

0

5

10

15

20


Gra

dien

t


NACA AirfoilMach = 1.73Re = 5e+5Mesh Size = 121x33x13Perturbation Size = 1e−8Free Stream T = 288 Kelvin

FD

ADJ

(c) 121 × 33 × 13

Figure 4.17: Viscous NACA Airfoil Mesh Refinement Study

• 81x33x7 mesh

• 81x33x13 mesh

• 121x33x13 mesh

Agreement of the gradient improves as more refined meshes are used. The results

of the Time Spectral Adjoint Method as shown in figure 4.18. There is a reasonable

agreement between the viscous adjoint gradients and finite difference gradients. The

difference is less than a few percent in all cases and within engineering approximations,

the method provides accurate gradient information.

With an availability of automatic differentiation packages, it would be feasible to

utilize computational resources for generating the exact jacobians for different flow

solvers. However, users are still held responsible for providing insight and identifying


0 10 20 30 40 50 60−10

0

10

20

30

40

50

Pertubation location along the airfoil surface

Gra

dien

t on

Dra

g C

oeffi

cien

t

Viscous Time Spectral Adjoint Validation against Finite Difference Gradient

data1data2Finite DifferenceAdjoint Calculation

Step Size = 1.8x10−4

Figure 4.18: NACA Airfoil Time Spectral Adjoint Sensitivity Comparison

Difference Between Finite Difference and Adjoint Gradient

Difference (%) 1.5 0.5 2.4 3.9 4.9 1.5Chord point 2 11 22 31 45 52

Table 4.4: Accuracy of Time Spectral Adjoint Method on NACA 0012- Viscous Flow/Adjoint


0 10 20 30 40 50 60 70

400

600

800

1000KSP Iteration vs. Processors

Number of Processor

KS

P It

erat

ion

0 1 2 3 4 5 6 7

400

600

800

1000

KSP Iteration vs. Log2(Processors)

Log2 (Number of Processor)

KS

P It

erat

ion Adjoint Convergence Comparison on Linux

Residual = 1e−8

(a) Convergence Comparison for MultiProcessor for 1e-8

0 2 4 6 8 10 12 14 16300

400

500

600

KSP Iteration vs. Processor

Number of Processor

KS

P It

erat

ion

0 2 4 6 8 10 12 14 16

1000

2000

3000

4000

5000

6000

Time (sec) vs. Processor

Number of Processor

Tim

e (s

ec)

Adjoint Convergence Comparison on SGI Origin 2000

Residual = 1e−8

(b) Convergence Comparison for MultiProcessor for 1e-8

0 5 10 15 20 25 30 35400

500

600

700

800KSP Iteration vs. CPU

CPU

KS

P It

erat

ion

Flow Convergence = 1e−7, Adjoint Convergence = 1e−8

Ablation

0 5 10 15 20 25 30 35100

200

300

400Time(sec) vs. CPU

CPU

Tim

e(se

c)

(c) KSP Total Iteration vs. Convergenceof Flow Solution

Figure 4.19: Time Spectral Discrete Adjoint run on UH60 Mesh, part 1

the correct inputs and outputs of the program. Manual effort is still required, but the

development would be much less rigorous. Accounting for multiblock compatibility in

the construction of the viscous flux jacobians requires additional book-keeping. Cur-

rent implementation of the viscous adjoint method could correctly handle multiblock

meshes, but this capability has not been verified as stated in table 4.2).

4.4 Adjoint Solver Performance Analysis

A performance analysis was completed on the Time Spectral Discrete Adjoint Solver.

Again, the flow solutions were obtained with SUmb, a Reynolds-Averaged Navier-

Stokes, multiblock and multiprocessor capable solver. Three different platforms were


0 1 2 3 4 5 6 7650

660

670

680

690

700

710

720

730

740KSP Iteration for different flow convergence order

Flow solution convergence order

KS

P It

erat

ion

Flow Convergence = various, Adjoint Convergence = 1e−8

Using 32 CPU

(a) KSP Total Iteration vs. Convergenceof Flow Solution

0 100 200 300 400 500 600 7000

2000

4000

6000KSP for different restart iteration numbers using 32 CPU

Restart Iteration

KS

P It

erat

ion

requ

ired

Flow Convergence = 1e−7, Adjoint Convergence = 1e−8

10,20 for 32 CPU and 10, 20, 40 for 4 CUP stalled

0 100 200 300 400 500 600 7000

2000

4000

6000KSP for different restart iteration numbers using 4 CPU

Restart Iteration

KS

P It

erat

ion

requ

ired

(b) KSP Total Iteration vs. Convergenceof Flow Solution

Figure 4.20: Time Spectral Discrete Adjoint run on UH60 Mesh, part 2

used in the analysis:

• Linux Operating System, Fedora Release 7 (Moonshine)

– Intel Xeon CPU 3.20 GHz, 8 processors

– 8 GB of memory

– intel fortran version 11.0

– mpich2-1.05p4

– cgns-2.5-2

– petsc-2.3.2-p10

• SGI Origin 2000, IRIX64 6.5 IP35

– 600 MHZ IP35, 32 CPU’s , MIPS R14000 Processor Chip Revision: 2.4

– 16 GB of memory

– MIPSpro Compilers: Version 7.3.1.3m

– SGI inhouse, implemeted MPICH 1.2 standard

– cgns-2.5-3

– petsc-2.3.2-p10


• Ablation, Centos

– Itainium, 96 processors

– 2 GB per processor

– Intel compiler

– mpich2

– cgns-2.5-3

– petsc-2.3.2-p10

The analysis was performed on the UH60 rotor blade running in parallel on multiple

processors. A single block topology was chosen rather than multiblock, and the com-

munication of the processors was achieved through the SUmb native communication

channel. Figure 4.19 shows the variation in the number of iterations and CPU time

required for a given level of convergence as the number of processors is changed. KSP

Iteration in the figure refers to the total number of iterations required for convergence

with PETSc. Figure 4.20 also shows variations in the iteration number, using various

restart numbers and various orders of convergence for the flow solution.

First, a study was performed to determine the number of iterations PETSc requires

to achieve a minimum residual of 1e−8. On the linux machine, the study was carried

out for 2, 4, 8, 16, 32 and 64 processors. The linux box only has eight CPU’s. However,

for the purpose of measuring the number of iteration, not the actual run time, the

given configuration produces meaningful results. Figure 4.19(a), on the linux box,

shows two plots of the results obtained using up to 64 processors. The second plot

simply took the logarithm on both axes, so that more features can be revealed, if

present. It does show slightly exponential or parabolic patterns. Figure 4.19(b), on

the SGI, used 1, 2, 4, 6, 8, 10, 12 and 14 processors. This figure also compares the

convergence time for different numbers of processors. It shows a dramatic reduction

in CPU time from one processor to two processors and a slightly inverse relationship

between the CPU time and the processor count can be spotted throughout the entire

plot. The total run time has plateaued around 1000 iterations, which signifies a

tradeoff between the number of processors and the overhead cost of communication.


Figure 4.19(c) can be compared directly to the figure 4.19(b) in the sense that the

exact same process is repeated on two different platforms: SGI Origin 2000 and

Ablation. One thing to notice here is that Ablation is roughly 4 − 5 times faster

than SGI Origin 2000. When running on both 4 and 8 processors, the two platforms

require almost the same number of iterations.

Figure 4.20(a) shows the required iterations to achieve a residual convergence of

1e − 8 starting from residuals of a flow solution with various orders of convergence.

This value fluctuates around 700 ± 30 iterations. Thus, the iterations required to

achieve a 1e − 8 order of residual convergence for the adjoint solution is fairly inde-

pendent of the convergence level of the flow solution. Figure 4.20(b) compares the

iterations required using different restart numbers. PETSc utilizes a restart capabil-

ity to ease memory requirements. A larger restart number implies that PETSc has

retained more sets of vectors such that a better direction is chosen with improved

performance. The result clearly shows that using a restart number less than 100

causes serious performance degradation. When the restart capability is not used,

approximately 700 iterations are required. For a restart value near 40, it requires

approximately 4500 iterations, which is about 6 − 7 times of the number required

without restart. However, when the test case is large, especially with the Time Spec-

tral Discrete Adjoint Method where the mesh size is multiplied by the number of the

total instances, memory space can become a serious bottleneck.

4.5 Optimization of the UH60 Rotor

4.5.1 Description

After verifying the accuracy of the Time Spectral Discrete Adjoint Method, it was

applied to the design optimization of a UH60 rotor blade (single block), in figure 4.4,

in the forward flight condition. Viscous effects and wake phenomena are not dominant

under the high advance ratio of the UH60 during flight. Under the given conditions,

the invisicd solver without wake coupling was used for this design application on a

UH60 mesh with 65x33x33 grid points. The optimization was carried out in two


separate stages:

1. Minimize torque with limited trim constraints

2. Minimize torque with full trim constraints

In the first case, the design variables are chosen so that the outboard planform

shape and twist distribution could be adjusted for best performance. Additionally,

ten airfoil sections were chosen and Hicks-Henne bump functions, shown in figure

3.25, were applied to the airfoil surface. An exaggerated form of the bump function

appears in figure 3.26. In figure 3.26, the bump function spans only the upper surface,

but in the optimization run, the bump function spans the entire surface of the airfoil.

For the second case, a set of four control angles were added to the set of design

parameters and a full set of trim conditions were applied. Yaw control is neglected in

the analysis because its effects are not as dominant as other forces and moments in

the trim condition. Further studies including the side force would be desirable, but

were not attempted in tihs project.

NPSOL oversaw the process as the main optimization software package. NPSOL

implements a Sequential Quadratic Programming technique and its robustness and

accuracy have been demonstrated repeatedly in previous literature. More details on

NPSOL can be found in the section 3.4.

4.5.2 CFD/CSD Analysis

The overall optimization process is carried out with an inviscid flow solver and invis-

cid adjoint solver, as viscous effects and wake coupling are not dominant. However,

the results of the optimization are checked under a CFD/CSD analysis with the help

of Choi [1, 2] who had an access to Aeroflightdynamics Directorate in NASA Ames

Research Center, where the CFD/CSD analysis tool is available. A CFD/CSD analy-

sis, as shown in figure 4.21(a), ensures the transfer of airloads from the computational

fluid dynamics (CFD) solver to the computational structural dynamics (CSD) solver.

Upon receiving a more accurate set of airloads after evaluating the flow solution, the

CSD solver can produce more accurate results for structural deformation. This newly


(a) Comprehensive CFD/CA Analysis (b) Modified CFD/CA Analysis

Figure 4.21: Modified CFD/CSD Analysis Diagram

obtained geometry is used to find a correct set of trim angles, which will set appropri-

ate orientations for the geometry. At this point, the CFD analysis is performed again,

and the previous steps are repeated until the entire process converges. This process,

however, is extremely time consuming and not suitable for applying the method of

adjoints. Figure 4.21(b) presents a modified CFD/CA analysis that is more suitable

for the Time Spectral Discrete Adjoint Method.

For this test case, obtaining the adjoint gradients due to the structural deformation

is not trivial and would require a comprehensive set of mathematical derivations on

the effects of shape changes on the flow variables. This could be considered an entirely

new research topic in its own right, and these changes in geometry have been left out

of the analysis. However, based on the given geometry, a set of boundary conditions

are carefully chosen such that the effect of trim angles is reflected in the balance of

forces and moments. Albeit a simplified version, the modified CFD/CSD analysis

presents a unique opportunity to test the feasibility of unsteady blade optimization

utilizing the Time Spectral Discrete Adjoint Method.

4.5.3 Optimization Results

The minimization of torque was pursued subject to various nonlinear constraints on

the design variables. A total of 106 design variables for the limited trim condition and


(a) Base UH60 (b) Optimized UH60

Figure 4.22: UH60 base and optimized configurations

110 design variables for the full trim condition were used, where outboard planform

shape, sectional airfoil shape, and sectional twist adjustment were allowed as the

design variables changed. The main difference between the limited and full trim

conditions is the addition of four control angles:

• One collective angle

• Two cyclic angles

• One shaft angle

These additions are necessary to retain the balance of forces and moments about all

three axes while blade shape changes occur.

For both trim conditions, the resulting planform shape changes are assumed to

effect the leading edge and chord length at roughly the 85%, 95% and 100 % radial

stations. The outboard section is where most of the dynamic and various other

flow phenomena occur, such as blade-wake vortex interactions, transonic flow, and

dynamic stall. Twist variations are allowed throughout the entire span, with an

emphasis on the inboard and outboard sections. The Hicks-Henne function is also

applied at various sections along the span, where the bump locations are chosen for

each section.

UH60 Optimization without Control Angles

Table 4.5 lists the aforementioned design variables along with the appropriate nonlin-

ear constraints. Optimization terminated after 38th iteration and the final results are

shown in figure 4.22(a), with a comparison to the base configuration in figure 4.22(b).


Parameters Description Lower bound Upper bound

x1 leading edge at 85% R x1 − 0.15 x1 + 0.15x2 leading edge at 95% R x1 − 0.2 x1 + 0.2x3 leading edge at 100% R x1 − 0.5 x1 + 0.5x4 chord length at 85% R 0.7x4 1.3x4

x5 chord length at 95% R 0.7x5 1.3x5

x6 chord length at 100% R 0.7x6 1.3x6

x7 twist () at 13% R x7 − 0.2 x7 + 0.2x8 twist () at 17% R x8 − 0.2 x8 + 0.2x9 twist () at 19% R x9 − 0.2 x9 + 0.2x10 twist() at 26% R x10 − 3.4 x10 + 3.4x11 twist () at 42% R x11 − 7.84 x11 + 7.84x12 twist () at 65% R x12 − 7.7 x12 + 7.7x13 twist () at 85% R x13 − 7.8 x13 + 7.8x14 twist () at 95% R x14 − 4.9 x14 + 4.9x15 twist () at 98% R x15 − 1.8 x15 + 1.8x16 twist () at 99% R x16 − 1.0 x16 + 1.0

x17 ∼ x106 bump amplitude -0.8% chord length 0.8% chord length

Table 4.5: Design parameters and two set of nonlinear constraints


(a) Cp at 81.7% R, ψ = 80 (b) Cp at 91.8% R, ψ = 80

(c) Cp at 96.7% R, ψ = 80 (d) Cp at 99.0% R, ψ = 80

Figure 4.23: Cp plots at different radial locations

CMzCL CMx

CMy

Optimization 7.4% reduction Almost Constant 30% increase 15 % increaseCFD/CSD 5% reduction Almost Constant 3% increase 2 % increase

Table 4.6: UH60 blade optimization with no trim angle constraints

At radial locations of 81.7%, 91.8%, 96.7%, and 99.0%, section Cp plots are shown.

The green line represents the section shape. The solid lines represent the baseline, and

the dashed lines represents the optimized configuration. Figure 4.24 shows variations

in the leading edge location, chord length, and the difference of twist angles between

the base and optimized configurations. Table 4.6 shows the results of the optimiza-

tion as well as the result of a CFD/CSD verification analysis. This verification consists

of a complete viscous analysis with turbulent effect performed on the optimized blade

geometry. Notice that, even though an inviscid analysis with no wake coupling was

performed, the comprehensive analysis with viscous and turbulent effects produced

not only a reduction in the cost function, but also almost no deviations in the forces

and moments balance. This result justifies the previous assumption that viscous and

turbulent phenomena have a minimal effect in the forward flight condition.


(a) Chord length (b) Leading Edge location

(c) Difference in twist along the radialspan

Figure 4.24: Geometric changes along the radial span


Parameters Description Lower bound Upper bound

x1 = −0.2486 leading edge at 85% R x1 − 0.15 x1 + 0.15x2 = −0.2200 leading edge at 95% R x1 − 0.2 x1 + 0.2x3 = 0.0879 leading edge at 100% R x1 − 0.5 x1 + 0.5x4 = 0.9884 chord length at 85% R 0.7x4 1.3x4

x5 = 1.0481 chord length at 95% R 0.7x5 1.3x5

x6 = 1.0497 chord length at 100% R 0.7x6 1.3x6

x7 = 9.6700 twist () at 13% R x7 − 0.2 x7 + 0.2x8 = 9.6700 twist () at 17% R x8 − 0.2 x8 + 0.2x9 = 9.6280 twist () at 19% R x9 − 0.2 x9 + 0.2x10 = 8.6000 twist() at 26% R x10 − 3.4 x10 + 3.4x11 = 5.8000 twist () at 42% R x11 − 7.84 x11 + 7.84x12 = 1.7000 twist () at 65% R x12 − 7.7 x12 + 7.7x13 = −1.0790 twist () at 85% R x13 − 7.8 x13 + 7.8x14 = −3.5640 twist () at 95% R x14 − 4.9 x14 + 4.9x15 = −1.9540 twist () at 98% R x15 − 1.8 x15 + 1.8x16 = −1.3030 twist () at 99% R x16 − 1.0 x16 + 1.0x17 = 0.0 collective angle() x17 − 8.6 x13 + 1.4x18 = 0.0 lateral cyclic angle() x18 − 2.4 x14 + 2.6x19 = 0.0 longitudinal cyclic() x19 − 6.4 x15 + 3.6x20 = −7.31 shaft angle() x20 − 10.0 x20 − 4.0

x21 ∼x110 bump amplitude -0.8% chord length 0.8% chord length

Table 4.7: Design parameters and four sets of constraints

UH60 Optimization with Full Trim Constraints

Table 4.7 lists the full set of design variables with their upper and lower bounds.

The addition of the four control angles, x17 − x20, enables the implementation of full

trim conditions which provides a realistic representation of the UH60 rotor in the

forward flight condition. The optimization terminated after 10th iteration with the

improvements shown in table 4.8. Once the optimized geometry is supplied to the

CFD/CSD analysis, a reduction of 14.62 % in the torque is achieved at the expense

of a small percentage of the lift and drag coefficients. However, the current results

show a noticeable deviation in the pitching moment. Further research involving the

pitching moment balance would a topic of interest for future work.


CMzCL CD CMx

CMy

Original -28.86 21.24 -2.52 23 12.97Optimized -24.64 20.72 -2.34 12 11.34

Improvement 14.62% reduction reduction reduction reduction

Table 4.8: UH60 blade optimization with four sets of constraints

(a) Base UH60 (b) Optimized UH60

Figure 4.25: Optimized geometry with full trim constraints

The four control angles are necessary to control the hub moments of the heli-

copter in the forward flight condition. The successful implementation of full trim

conditions validates the viability of the unsteady design methodology that makes

use of adjoint-based gradient information and flow solutions obtained from the Time

Spectral Method. A comparison of the base and optimized blades is made in figure

4.25(a) and 4.25(b). Figure 4.26 compares the geometry changes in detail. Figure

4.27 compares Cp plots for the original and optimized blades.

Computational Time Comparison

Table 4.9 summarizes the amount of computational time saved by applying the Time

Spectral Discrete Adjoint Method rather than employing the traditional Finite Differ-

ence Method for obtaining the gradient of each design parameter. Only the case with

the limited trim conditions is considered in illustrating the advantages of employing


(a) Chord length (b) Leading edge location

(c) Difference in twist along radial span

Figure 4.26: Geometric changes along the radial span with full trim constrants


(a) Cp plot comparison at 85.13 % radiallocation, ψ = 80

(b) Cp plot comparison at 91.0 % radiallocation, ψ = 80

(c) Cp plot comparison at 97 % radiallocation, ψ = 80

(d) Cp plot comparison at 99.5 % radiallocation, ψ = 80

Figure 4.27: Cp plot comparison at different radial locationswith full trim constraints


the Time Spectral Adjoint methodology for unsteady optimization.

For each design parameter, the finite difference gradient must be calculated at

each design point, whereas the adjoint gradients can be determined for all param-

eters simultaneously at each design point. The cost associated with the adjoint

method is therefore equivalent to having a single parameter. As an example, with

finite difference gradients and a residual convergence of 1e − 6, it would require

5185.4 ∗ 100 sec = 144 hours per iteration for solving a flow solution 100 times. For

38 iterations, it would take 144 hours ∗ 38 iterations = 228 days , and this is just to

complete the unsteady design. In summary, by using adjoint gradients, 4.5 days of

computational time would be enough for 100 design parameters and 38 iterations for

what would have taken 228 days of CPU time with finite difference gradients. Not

only does it save tremendous amount of computational time, but the new method-

ology is solely responsible for making computationally infeasible problem completely

tractable.

Table 4.10 contains a run time comparison for the NACA0012 airfoil. Optimiza-

tion was not performed, but listed is the computational time that could be saved if

the optimization is to be carried out. Again, not only does it save almost 98% of the

CPU time, it enables computation that was previously not possible.


Design Variables 100Design Iterations 38

Finite Difference

Order of Convergence 1e-3 1e-4 1e-5 1e-6

Inviscid Run Time (sec) 62.1 126.7 3464.9 5185.4per iteration per variable

Adjoint


Inviscid Run Time (sec) 1000 1000 1000 1000per iteration for all variables

Saved Run Time per Iteration 5120 11670 345490 517540Saved Run Time per Iteration (in %) 84 92 99 99

Table 4.9: Computational time comparison for the inviscid UH60 test case


Design Variables 100Design Iterations 38

Finite Difference


Viscous Run Time (sec) 441 871 1442 2016per iteration per variable

Adjoint


Viscous Run Time (sec) 3680 3680 3680 3680per iteration for all variables

Saved Run Time per Iteration 40420 83240 140520 197920Saved Run Time per Iteration (in %) 92 96 97 98

Table 4.10: Computational time comparison for viscous calculations on a NACA 0012

Chapter 5

Conclusions & Future Work

5.1 Conclusions

Unsteady flows in nature appear in numerous forms: flapping wings, turbulence,

turbomachinery flows, rotating stall, and surge, just to name a few. In engineering

sysystem, unsteady phenomena, not only shorten the life-cycle of the components,

but also may lead to operational failures which cost human lives. Consequenty, there

has been an enormous need for the ability to perform optimization in unsteady flows,

but due to the difficulties in implementation, only a minimal amount work has been

carried out so far. In this thesis, an unsteady optimization methodology utilizing

adjoint-based gradients along with the Time Spectral Method for periodic flows has

been implemented and validated.

Traditionally, many developed algorithms have contributed to solving unsteady

flows. Starting from explicit methods, which include Runge-Kutta and various other

multi-step methods, all the way to implicit methods, which include ADI, DD-ADI and

BDF, many algorithms have been applied to unsteady optimization. Most explicit

schemes face the problem of traversing through a large number of time steps. The

problem is aggravated by the presence of transient solutions, which tend to grow

longer for flows with high frequency components, such as high-RPM compressors.

Implicit algorithms often face storage problems, as well. High-order models such as

the Nonlinear Frequency Domain Method and the Harmonic Balance Method have

117

CHAPTER 5. CONCLUSIONS & FUTURE WORK 118

been applied to bypass transient solutions, but the computational time spent on

converting variables between the time and frequency domains has proved to be quite

significant. The extra storage space required for complex variables also presents an

additional problem. In a typical unsteady optimization process, with each design

iteration, the optimizer requires gradient information which is obtained by solving

the unsteady flow equations as many times as the number of design parameters plus

one. But with adjoint gradients based on the time spectral flow solutions, the number

of required unsteady flow evaluations is reduced to two: one for solving the unsteady

flow around the current geometry and the other for solving the adjoint equation with

perturbations applied to a set of design parameters. As a result, the Time Spectral

Adjoint Method provides gradient information with such a low computational cost

that it enables optimization within certain categories of periodic unsteady flows that

have not been previously attempted.

The unsteady adjoint gradient method proposes a way to obtain gradient infor-

mation at low computational cost at a given design iteration. However, since the

adjoint requires backward integration in time, the entire flow solution at every time

step needs to be stored. A typical number of time steps in an engineering application

might range between a few hundred to tens of thousands, which renders the storage

of the entire flow solution very problematic. Using adjoint gradients with the Time

Spectral Method for unsteady flows with periodicity solves not only the storage prob-

lem, but also bypasses the time required to step through transient solutions. Entire

periodic flow solutions are solved at once.

Finally, this research proposed, implemented, and validated a design methodology

for unsteady flows with periodicity. It has been tested on a NACA 0012 airfoil ex-

tensively, and upon a validation of the adjoint implementation, it was applied to the

design optimization of a UH60 blade in forward flight with trim conditions. Along

the development timeline, a partial set of trim conditions were applied first to the

optimization process, which resulted in 5 % improvement in torque. A full set of trim

condition were also applied, resulting in a 14.6 % improvement, but at the expense of

violating the moment balance. Both cases demonstrated improvements after applying

a CFD/CA analysis (cowork with Choi).


The successful implementation and verification of the proposed design methodol-

ogy opens up many areas of engineering applications. Flutter often shortens the life

expectancy of an aircraft wing or other components. Rotating stall in a compressor

not only damages the compressor blades, but also exposes humans to serious danger

during engine failure events. Turbulent flows also exhibit periodicity. From flapping

bird wings to the engine cycle of a car, there are numerous examples of periodic,

unsteady flows in nature and the Time Spectral Adjoint approach would be a good

starting point for tackling these problems.

As a final list of technical developments in this thesis, the following concepts were

derived, implemented and validated:

• Euler flux jacobians for discrete adjoints

• Artificial viscosity jacobians

• Viscous flux jacobian

• Various boundary conditions

– Farfield

– Supersonic Inlet/Outlet

– Subsonic Inlet/Outlet

– Euler wall

– Viscous wall

– Symmetric wall

– Internal 1-to-1

• Multi-processor capability using MPI and PETSc

• Multi-block capability through a Sequential Halo Update procedure.


5.2 Future Work

Adjoint solutions, mostly used to obtain gradient information, can also provide an

error indicator without any additional computational cost. Mesh quality becomes

one of the primary factors that helps to achieve a faster flow convergence. However,

a better quality grid requires longer computational times. In time spectral solutions,

the total number of grid points is multiplied by the number of total time instances, so

that the final run time is prolonged by the same multiplicative factor. Error indicators

could be used to redistribute or refine the mesh in the areas where a reduction of error

is maximized with a minimum amount of mesh modification.

The application of unsteady optimization to a turbomachinery flow would be a

good example, as well. An advantage of using discrete adjoints is that it allows the

selection of any flow quantity for the cost function, so optimization could be applied

to quantities such as the total pressure loss across the compressor or the prevention

of surge under given conditions.

The addition of different trim conditions to the UH60 test case would be a useful

exercise. With the successful implementation of the current optimization methodol-

ogy, comparing optimization results over various trim conditions could present phys-

ical insights that would otherwise hard to obtain.

Flutter could be another useful application. Much research has already been done

on the subject and a great deal of information is available. Utilizing this knowledge

in the context of the Time Spectral Adjoint optimization methodology could provide

wing shapes that are beyond the scope of current physical insight.

Appendix A

Optimization

Optimization problems can be divided into three categories[48, 9]: constrained linear,

unconstrained , and constrained nonlinear optimization. Problems with a linear ob-

jective function fall under linear programming and those with a nonlinear objective

function fall under nonlinear programming. Most of the objective functions, CL, CM ,

and CP and their variants fall under the nonlinear programming category.

A.1 Linear Programming

Constrained linear programming has a set of known coefficients, which defines a set

of known spaces in which an optimal solution is sought. The set of coefficients can be

interpreted in an algebraic sense or a geometric manner. The algebraic point of view

comes from the study of linear algebra, where the optimization problem is cast in a

particular way, called the ”standard form”. Optimization problems in the standard

121

APPENDIX A. OPTIMIZATION 122

form are written as,

minimize z = cTx,

subject to Ax = b,

x ≥ 0,

b ≥ 0,

x and c = vector of length n,

b = vector of length n,

A and c = m by m constraint matrix.

(A.1)

Since the coefficients are known quantities, the solution space is fixed. This im-

plies that known tools of linear algebra can be readily applied using the concept of

the null and the solution space. In the geometric point of view, an extreme point

is defined geometrically using convexity. The concepts of basic solution and basic

feasible solution are introduced depending on the types of constraints it satisfies.

The simplex method is widely used for linear programming and its general solu-

tion, x, can be written as,

x =

(

xB

xN

)

where xB is a vector of basic variables and xN is a vector of nonbasic variables. The

objective function, z, can be written as,

z = cTBxB + cTNxN

where cB represents coefficients for the basic variables and cN for the nonbasic vari-

ables. Consequently, the solution, xB is represented as a function of xN as in

BxB +NxN = b

which can be re-written as

xB = B−1b− B−1NxN .


Unconstrained Optimization

Unconstrained optimization problems can be written as

minimize f(x) with respect to x

where no constraint is placed on the variables, x = (x1, ..., xn)T . A local minimum

can be assured if the following conditions are met at x∗

f(x∗) = 0

2f(x∗) is positive definite(A.2)

which is called second-order sufficient conditions. The above claim can be justified in

the following arguments. Assuming x∗ is a local minimum, the Taylor series around

x∗ can be written as

f(x∗ + ζ) = f(x∗) + f(x∗)Tp+

1

2pT 2 f(ζ)p, (A.3)

where p is a nonzero vector and lies between x∗ and x∗ + p. If x∗ is indeed a local

minimum, then

f(x∗)Tp ≥ 0 for all feasible directions p

pT 2 f(x∗)p ≥ 0.(A.4)

Therefore, for x∗ to be a local minimum,

f(x∗) = 0

has to be satisfied, assuming the Hessian matrix, 2f(x∗), is positive definite.

Starting from the current point xk, a new search direction is found by solving

[2f(xk)]p = − f(xk),

and a new point is obtained from

xk+1 = xk + αkpk


where α is found by applying typical line search algorithms. At or within a given

tolerance of the local minimum, the conditions A.2 would be satisfied. Thus, the

magnitude of the tolerance is an important factor to determine how precisely a local

minimum can be refined; sometimes, there are limitations to the resolution of the

objective function and the user needs to be aware of the importance of the control

parameters.

The conjugate gradient method belongs to this category of optimization technique.

However, it utilizes a different approach to obtain search directions and step length,

which is further explained in section 3.4.2.

A.2 Nonlinear Programming

A general form of the nonlinear program can be written as follows,

minimize f(x)

subject to gi(x) = 0, i ∈ E

gi(x) ≥ 0, i ∈ I

(A.5)

where the equality constraints belong to E and the inequality to I. When the nonlin-

ear objective function is constrained by a set of linear equality functions, the equality

constraints can be absorbed into the objective function so that the final form becomes

that for unconstrained form. Then, all of the known techniques for unconstrained op-

timization can be applied, as long as the search direction, p, lies within the null space

of A, a matrix composed of the set of coefficients from the equality constraints. The

new objective function is called a reduced function and the new gradient a reduced

gradient.

minimize φ(v) = f(x+ Zv)

φ(v) = reduced function

φ(v∗) becomes ZT f(x∗) = 0

2φ(v∗) becomes ZT 2 f(x∗)Z = 0.


According to the property of linear algebra, this reduction implies the null space

component of the gradient should be equal to zero,

f(x∗) = Zv∗ + ATλ. (A.6)

Physical interpretations imply that the descent direction should be pointing away

from the null space, Z, so that ZTZv∗ should be equal to zero. Inequality constraints

are bound by equality constraints, with additional feasible regions mapped by the

inequality equations. Whether the movement toward feasible regions increases or

decreases the objective function is controlled by the sign of a Lagrange multiplier, as

represented in the following equation:

f(x) ≈ f(x∗) + (x− x∗)T f(x∗)

= f(x∗) + (x− x∗)TATλ∗

= f(x∗) + δTλ∗

= f(x∗) + Σδiλ∗i.

The equality constraint remains the same, but the inequality is controlled by the

sign of the Lagrange variables. As a result, the sufficient conditions for the linear

inequality constraints can be summarized as

• Ax∗ ≥ b,

• f(x∗) = ATλ∗,

• λ∗ ≥ 0,

• λT (Ax∗ − b) = 0, and

• ZT+ 2 f(x∗)Z+ is positive definite,

and each item can be interpreted as

• Satisfy the inequality constraint

• Move away from null space


• Move into the feasible region

• Zero out inactive lagrange variables

• Concave up.

For nonlinear equality and inequality constraints, the same idea applies with one

difference: instead of using the original function, f(x), define another function called

a Lagrange function,

L(x, λ) = f(x) − Σλigi(x) = f(x) − λTg(x),

where λ is a vector of Lagrange multipliers and g represents a set of constraint vectors

for gi. Thus, the sufficient conditions for the nonlinear inequality constraints can

be summarized as

• xL(x∗, λ∗) = 0,

• λ∗ ≥ 0,

• λT∗ g(x∗) = 0, and

• Z+(x∗)T 2

xx L(x∗, λ∗)Z+(x∗) is positive definite.

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DESIGN OPTIMIZATION OF PERIODIC FLOWS USING A TIME