NASA/TM–2015–218796

An Automated DAKOTA and

VULCAN-CFD Framework with

Application to Supersonic Facility

Nozzle Flowpath Optimization

Erik L. Axdahl

NASA Langley Research Center, Hampton, Virginia

August 2015
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NASA/TM–2015–218796

An Automated DAKOTA and

VULCAN-CFD Framework with

Application to Supersonic Facility

Nozzle Flowpath Optimization

Erik L. Axdahl

NASA Langley Research Center, Hampton, Virginia

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-2199

August 2015
Acknowledgments

The author wishes to thank Dr. Richard Ga↵ney for his help with running the Irrotational

Method of Characteristics for Nozzle Design code used in the design of the nozzle profiles as well

as prior discussions on the philosophy of nozzle design. Additional acknowledgement is made to

Dr. Robert Baurle and Mr. Michael Bynum for project planning activities leading up to the

execution of this work.

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not

constitute an ocal endorsement, either expressed or implied, of such products or manufacturers by the

National Aeronautics and Space Administration.

This report is available in electronic form at

http://ntrs.nasa.gov
Abstract

Removing human interaction from design processes by using automation may lead to

gains in both productivity and design precision. This memorandum describes e↵orts

to incorporate high fidelity numerical analysis tools into an automated framework

and applying that framework to applications of practical interest. The purpose of

this e↵ort was to integrate VULCAN-CFD into an automated, DAKOTA-enabled

framework with a proof-of-concept application being the optimization of supersonic

test facility nozzles. It was shown that the optimization framework could be deployed on a high performance computing cluster with the flow of information handled e↵ectively to guide the optimization process. Furthermore, the application of

the framework to supersonic test facility nozzle flowpath design and optimization

was demonstrated using multiple optimization algorithms.

Contents

1 Nomeclature 2

2 Introduction 4

3 Methodology 4

3.1 Computational Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2 High Performance Computing Job Structure . . . . . . . . . . . . . . 7

3.3 Reference Conditions and Requirements . . . . . . . . . . . . . . . . 8

3.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Results and Discussion 10

5 Summary and Conclusions 11

6 Directions of Future Work 13

A DAKOTA Driver Script 16

B DAKOTA Input Files 18

C Optimization Problem Definition 21

D Representative VULCAN-CFD Solution 23

E DAKOTA Iteration Data 28

E.1 DIRECT Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

E.2 COBYLA Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

E.3 Quasi-Newton Method . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1
1 Nomeclature

Acronyms/

Initialisms

Description

AHSTF Arc-Heated Scramjet Test Facility

BFGS Broyden, Fletcher, Goldfarb, Shanno

CFD computational fluid dynamics

CFL Courant, Freidrichs, Lewy

COBYLA constrained optimization by linear approximation

DAKOTA design analysis kit for optimization and terascale applications

DIRECT dividing rectangles

DAF diagonal approximate factorization

HPC high performance computing

IMOCND irrotational method of characteristics for nozzle design

LDFSS low dissipation flux vector split scheme

MOC method of characteristics

MUSCL monotonic upstream-centered scheme for conservation laws

NASA National Aeronautics and Space Administration

PBS portable batch system

VULCAN viscous upwind algorithm for complex flow analysis

Latin

Symbols

Description Units

Acore nozzle exit core area m2

A ̃core normalized nozzle exit core area 1

Aexit Nozzle exit area m2

ai weighting coecient 1

e error 1

F objective function 1

F ̃ pseudo-objective function 1

feval function evaluations 1

Hexit nozzle exit height m

i number of algorithm iterations 1

Lnozzle nozzle length m

L ̃nozzle nondimensional nozzle length 1

Lref reference nozzle length m

M Mach number 1

Mcenterline nozzle exit centerline Mach number 1

M ̃centerline scaled nozzle exit centerline Mach number 1

Mdesign targeted nozzle exit design Mach number 1

MMOC nozzle exit Mach number from Method of Characteristics 1

M ̃MOC scaled nozzle exit Mach number from Method of Characteristics 1

P penalty function 1

p ̃ scaled pressure 1

p0 stagnation pressure Pa

2
p ̃0 scaled stagnation pressure 1

Q number of processors required per function evaluation 1

R penalty parameter 1

Rc nozzle throat radius of curvature m

R ̃c nondimensional nozzle throat radius of curvature 1

Rt nozzle throat half-length from centerline m

S number of processors available 1

T static temperature K

T ̃ scaled static temperature 1

T0 stagnation temperature K

X, Y, Z cartesian coordinates m

y+ nondimensional distance to the wall 1

Greek

Symbols

Description

 variable placeholder

Superscripts Description

* optimized condition

3
2 Introduction

E↵orts have been made in the Hypersonic Airbreathing Propulsion Branch at NASA

Langley Research Center to improve the automation of design and analysis tools and

methods used at multiple levels of fidelity ranging from engine cycle analysis to high

fidelity CFD. By gradually removing the human from the design loop, productivity can be increased by exploring larger design spaces and achieving designs that

are closer to the global optimum or that possess improved robustness. Furthermore, automating the tools used in the design process will ease studies that provide

quantification of uncertainty and sensitivity. This will allow for improved design

knowledge for decision making by project leaders, designers, and analysts alike.

The design of facility nozzles is a good application area for tool automation

because the design process can be generalized to an automated framework. The

method of characteristics (MOC) is a method by which one may design a nozzle

wall contour for a given exit Mach number, but because the assumptions underlying

the method do not account for viscosity, boundary layer buildup in the nozzle flow

will produce both vorticity due to boundary layer viscous e↵ects as well as an exit

Mach number that is o↵ design because the boundary layer inhibits expansion of

the flow by reducing the e↵ective cross-sectional area. Additionally, the presence of

the boundary layer has an adverse impact on exit flow uniformity. Therefore, the

designer would typically be required to adjust the nozzle design using MOC manually

until the desired exit Mach number is achieved in a viscous solution within some

specified tolerance.

The purpose of the e↵ort described in this memorandum was to demonstrate

a capability for integrating the high-fidelity VULCAN-CFD code in an automated

framework driven by the DAKOTA toolkit. To that end, a proof-of-concept supersonic facility nozzle optimization study was initiated to demonstrate the practical

methodology of executing the framework on a high performance computing cluster.

Three optimization algorithms were chosen in order to demonstrate the automated

framework varying levels of algorithmic fidelity and eciency. At the conclusion

of the e↵ort, a capability was gained in automated supersonic facility nozzle optimization, which may provide a starting point for problems in other research areas

requiring a similar automated framework.

3 Methodology

3.1 Computational Tools

Computer codes used in an automated framework require the capability to have

variables passed to them without direct human interaction. Therefore, each code

must have the ability to gather its inputs from a text-based input file, via options

on the command line, an application programming interface, or a combination of all

three. In order to adjust design variables and simulation parameters appearing in

input files, template input files for each code were defined with placeholder strings

occupying locations in each template where the framework driver may substitute a

variable value. At the conclusion of the sequence of codes to be run, numerical values

4
Figure 1. Flow of information for the described simulation framework.

representing the final results were extracted and placed in a results file that was read

by the framework driver. For the proof-of-concept supersonic facility nozzle design

problem, codes were run sequentially in order to define the nozzle profile, generate

the three-dimensional numerical grid, run the CFD analysis, and extract the results

that were fed back to the framework driver in order to record results and assign

values to the design variables for subsequent iterations. The diagram in fig. 1 maps

to the following discussion in order to visualize the flow of information from one

code to the next.

Non-dimensionalized, two-dimensional nozzle profiles were generated using a

code written by Richard Ga↵ney of the Hypersonic Airbreathing Propulsion Branch

at NASA Langley Research Center. The IMOCND code1 uses an MOC procedure

to design an inviscid nozzle profile given input parameters such as target nozzle exit

Mach number and non-dimensional throat radius of curvature. Because the method

produces non-dimensionalized nozzle profiles, the resulting nozzle wall contour was

scaled such that the exit height matched the design requirement.

Final nozzle wall profile definition and grid generation were completed via a

custom script that took as input the nozzle wall profile from the IMOCND code

and produced a three dimensional computational grid along the entire nozzle length

from the subsonic inflow to the supersonic exit. The profile was scaled given a

user-defined exit size and the subsonic nozzle contour was curve fit to the subsonic

entry point in order to achieve a fixed nozzle entrance wall angle. Grid spacing was

defined by a user-specified number of longitudinal and transverse points as well as

clustering parameters near the walls and near the throat.2 Clustering was enforced

to achieve sucient resolution in the wall boundary layer as well as in areas where

rapid area changes occur.

The nozzle flow properties were simulated using the VULCAN-CFD3,4 code, developed and maintained at NASA Langley Research Center. VULCAN-CFD uses a

5
Figure 2. A notional schematic is shown of the definition of the core flow area in

the nozzle exit. The core area is defined here as the largest rectagular area that fits

completely within the region of flow outside of the boundary layer.

finite-volume, cell-centered scheme for solving flows on structured grids. The nozzle

flow for this study was assumed to be viscous, thermally perfect, and chemically

frozen with non-vititated air. The flow fluxes were computed using LDFSS5 and

were recombined with a 3rd order MUSCL interpolation strategy. The flow was

integrated temporally using the ILU scheme6 with local time stepping and an increasing CFL number schedule. Turbulence was modeled using the two-equation

Menter k-! model7 with a turbulent Prandtl number of 0.9. Wall matching functions8 were used for wall-bounded areas of the solution domain to relax boundary

layer grid resolution requirements. Finally, a coarse, medium, and fine grid sequencing technique was used to speed up the time required for solution convergence.

Solution quantities of interest were extracted from the VULCAN-CFD flow field

using Tecplot⇤. At the conclusion of each simulation, the exit plane data from

the nozzle was extracted and exported to a data file that was read and processed

by a custom script. Regions in the plane where the exit flow Mach number were

within 1% of the design Mach number were identified so that the core size could

be determined as an engineering estimate. A notional schematic of the definition of

the nozzle core is shown in fig. 2. The ratio of the core size to the nozzle exit size

was used to constrain the optimization process with an ideal value of unity. The

centerline Mach number was also extracted to drive the optimization process.

The codes used in the nozzle profile definition, grid generation, flow simulation,

and parameter extraction were integrated using the Sandia National Laboratorydeveloped and -maintained code DAKOTA.9 DAKOTA is a toolkit for integrating

codes in order to conduct higher order analyses such as optimization, uncertainty

⇤Tecplot is a trademark of Tecplot, Inc.

6
Figure 3. A schematic of the high performance computing architecture is shown. A

master DAKOTA instance is run from its own, long-running PBS job while spawning

independent nozzle simulations on their own PBS jobs with unique design variables.

quantification, parameter estimation, and design space exploration. The flow of

information from one code to the next was controlled in a black-box manner using

shell scripts, in this case written in tcsh, and analysis outputs were fed back into

DAKOTA in order to generate new sets of design variable values for subsequent

simulation runs. Appendix A shows an example of the shell script used to drive

the analysis in this work. A powerful aspect of DAKOTA is that di↵erent types

of analyses may be run simply by altering the master input file fed to DAKOTA

without making any further changes to the driver scripts controlling the execution

of codes, flow of information, and parameter extraction in the analysis framework.

Appendix B shows an example of a DAKOTA input file used in this work.

3.2 High Performance Computing Job Structure

Because design iterations were carried out on an HPC cluster with multiple simulations carried out simultaneously and independently of each other, attention was paid

toward the practical methodology of running the automated framework described

in section 3.1. Table 1 summarizes candidate methods for how to run DAKOTA

and the simulations it drives depending on whether each are run in a serial or parallel mode and the type of HPC resources and settings available to the user. As

shown schematically in fig. 3, a job architecture was chosen corresponding to Case

4 consisting of a master DAKOTA instance on a long-duration PBS job spawning

independent simulations in PBS jobs each with unique sets of design variable values.

While Case 3 may also be suitable for the present analysis, such a method requires

techniques such as processor tiling as well as possibly wasting job resources if they

are not all needed the entire time. Considering this, the ability to run many, smaller

PBS jobs asynchronously allowed individual simulations to be run as resources were

freed on the shared HPC resource. Furthermore, submitting jobs separately allowed

cluster resources to be used as needed as the number of simultaneous simulations

required do not necessarily remain constant during the optimization process.

7
Table 1. DAKOTA and application parallelism use cases (table reproduced from

information in DAKOTA user manual9 and associated presentation packaged with

source code)

Case DAKOTA Simulation Notes

1 Parallel Serial S  1 (or S if DAKOTA run on login

node) simultaneous simulation instances,

each requiring Q = 1 processors

2 Serial Parallel One simultaneous simulation instance at

a time on Q processors

3 Serial Parallel Run ⇡ (S  1)/Q or ⇡ S/Q simultaneous, Q processor simulations on a single

job using processor tiling

4 Serial Parallel Run DAKOTA on a login node or its own

job and submit simulation jobs each with

Q processors to a scheduler (e.g., qsub)

Figure 4. A notional schematic is shown of the nozzle, including the definition of

the nozzle length and throat radius of curvature.

3.3 Reference Conditions and Requirements

The nozzle design requirements were selected in order to reflect those of a previously

exercised, manual design activity for designing square nozzles for the NASA Langley

AHSTF supersonic tunnel.10,11,12 For this study a nozzle was designed with a specific

target exit Mach number, although the same procedure could be applied to a nozzle

design of any given exit Mach number. Note that while vitiated air would normally

be present in the gas flowing from the facility plenum,13,14 the air composition was

assumed to be non-vitiated in the present numerical analyses.

3.4 Optimization

For a nozzle constrained by design exit Mach number and exit size, notionally shown

in fig. 4, the principal parameters defining its design when using the IMOCND

code are the throat radius of curvature and the MOC exit Mach number. When

considering the need to produce a nozzle exit flow profile that is as uniform as

8
Nozzle Length

Core Size

Unique solution for a given 

design variable combination

Pareto Optimal 

(Non-Dominated Solutions)

Dominated 

Solutions

Ideal Solution

(Not achievable)

Optimum design

(closest design to

ideal solution)

Figure 5. A schematic is shown of notional design space solutions, including the

Pareto front and the definition of the ideal solution.

possible, increasing the throat radius of curvature is one way to accomplish this

as large radii produce a sonic line at the throat that approaches a linear shape.

Furthermore, large throat radii of curvature cause a slower expansion with reduced

flow gradients. However, increasing the radius of curvature also increases the length

of the nozzle which can have adverse e↵ects on the manufacturability of the nozzle

as well as what the facility itself can accommodate. In addition, the boundary layer

will become thicker with longer nozzles,15 which impacts exit core size and flow

uniformity. Therefore the optimization problem has two competing objectives of

nozzle length and exit flow uniformity (surrogated here as the nozzle exit core size)

with the centerline Mach number being an equality constraint. The exit centerline

Mach number was chosen as a constraint instead of part of the objective function

because there are multiple nozzle solutions that may satisfy that constraint. The

multi-objective function to be minimized was expressed as

F(R ̃c, M ̃MOC) = a1A ̃core + a2L ̃nozzle (1)

where the weighting coecients a1 and a2 may take on any values such that they

add to unity. The values of a in a multi-objective function are generally defined such

that ai 2 [0, 1] and P

i ai = 1. Because the objective function in this case is to be

minimized, the coecient of a1 is negative due to the need to maximize the core area

and the coecient of a2 is positive due to the need to minimize nozzle length. For

this study, an equal weighting to both objectives was chosen such that a1 = a2 = 0.5.

For any given value of a1 and a2, the optimum value of the objective function F

yields a design on the Pareto optimal front of the design space, shown notionally

in fig. 5, which allows design trades to be done using di↵erent objective function

weightings. In this study, the core area and nozzle length were non-dimensionalized

in order to make them the same order of magnitude. Therefore, A ̃core = Acore/Aexit

and L ̃nozzle = Lnozzle/Lref . For more information on the mathematical optimization

definition in this study, refer to appendix C.

9
Of the optimization methods available in DAKOTA, three were chosen to demonstrate the capability of the automated framework. The first was the DIRECT16

method, which is a zeroth order method designed for engineering applications that

scales well with the number of design variables in the parameter space. DIRECT is

also guaranteed to find the global minimum within a user-defined spatial tolerance.

The number of simultaneous design points the method may evaluate is dependent

both on the number of regions where potential global minima have been identified

as well as the dimensionality of the problem. The second was the COBYLA17,18

method, which traverses the design space by producing successive linear approximations of the objective function and constraints seprately in an n + 1 dimensional simplex. The COBYLA method may only evaluate a single design point at

a time and converges according to a fixed step reduction schedule. The third was

a Quasi-Newton method9 that is distinct from a classical Newton method in that

the approximate inverse Hessian at a point is updated using the BFGS19,20,21,22

method using the local Jacobian. This makes the method quasi-second order where

local Jacobians are computed numerically—rather than analytically—in this study

by perturbing the locally sampled point at each iteration. As the calculation of numerical Jacobians increases the computational resources required by the algorithm

also increases, although a CFD code that incorporates an adjoint capability may be

able to mitigate this. As with the COBYLA method, the Quasi-Newton method

may only evaluate a single design point at a time, although numerical Jacobians

may be computed concurrently at the design point. Furthermore, a vulnerability

of both the COBYLA and Quasi-Newton methods is that they may converge to a

local minima instead of the global minima.

4 Results and Discussion

The optimization algorithms described in section 3.4 were run to convergence to the

constrained optimum design variable values. For more information on numerical

results and analysis using VULCAN-CFD, refer to appendix D for a run at the optimum location as determined by the Quasi-Newton method that is representative

of each iteration of the process. A list of objective function values for each algorithm

at each iteration are summarized in appendix E. Optimum designs as determined

by each optimization algorithm are summarized in table 2. The number of function

evaluations represents the number of simulations required to converge the algorithm.

The number of iterations represents the number of candidate design points evaluated

in the convergence of each algorithm. For each algorithm’s converged optimum, the

number of function evaluations and total number of algorithmic iterations shows

that the Quasi-Newton method is the most ecient of the three. The reason is because the method takes both numerical Jacobians and approximate inverse Hessians

into account, thereby improving the convergence characteristics of the method over

zeroth and first order methods. While the COBYLA and Quasi-Newton methods

both converge to approximately the same design point, the Quasi-Newton method is

more ecient in terms of function evaluations by over a factor of two. Another notable aspect of these results is that the DIRECT method converged to the shorter

10
Table 2. Results for di↵erent optimization algorithms are shown. Design variables

with asterisks indicate the optimum design variable combination. The listed Mach

numbers are scaled against the nozzle design Mach number.

Algorithm R ̃ ⇤

c M ̃ ⇤

MOC M ̃ centerline A ̃ core [m2] L ̃nozzle [m] feval i

DIRECT 5.278 1.039 1.002 0.778 0.278 73 73

COBYLA 7.014 1.040 1.012 0.778 0.293 44 44

Quasi-Newton 7.105 1.041 1.001 0.778 0.290 21 7

length nozzle while achieving nozzle core area and exit centerline Mach number

similar to those determined using the Quasi-Newton method. This suggests that

the COBYLA and Quasi-Newton methods converged into a local minima while the

DIRECT method identified the location of the global optimum.

Graphical visualizations of each algorithm’s design space exploration are shown

in fig. 6. While the sequence of iterations should not be inferred by any of the

plots, each gives indications of the methodology and the eciency of each method.

As shown in fig. 6(a), the DIRECT method sampled the design space at gradually

refining resolution, probing areas where potential global minima were located. The

method terminated once a defined search size tolerance was reached. The pattern

of the search indicates that the region where COBYLA and Quasi-Newton methods

converged was investigated by the DIRECT algorithm as potentially containing a

global minimum. The COBYLA algorithm’s search points are shown in fig. 6(b).

The method gradually approached the local optimum point by linearizing the design

space at successive points. The Quasi-Newton method search points are shown

in fig. 6(c). Here the method rapidly approached the local optimum design point

through the computation of the numerical Jacobian and approximation of the inverse

Hessian at each point.

As shown in these results, the Quasi-Newton method has diculty identifying

global minima when local minima are present or if the design space is not smooth or

continuous globally. For such problems, the converged solution will be sensitive to

the selection of the initial search point. This issue is not problematic with a method

such as DIRECT, which guarantees that a global minimum will be found within

some user-defined size tolerance. However, this comes at the cost of additional

runs relative to higher order methods such as Quasi-Newton. Because the DIRECT

algorithm scales well with the number of design variables, a sequential optimization

method may be employed where the DIRECT method is allowed to end in the

general vicinity of the global minimum with the ending point of DIRECT being the

starting point for a Quasi-Newton optimization method. To this end, DAKOTA is

able to handle sequential optimization with an arbitrary number of algorithms.

5 Summary and Conclusions

A method for integrating the high-fidelity VULCAN-CFD code in an automated

DAKOTA-driven framework was demonstrated with application to the design optimization of a supersonic facility nozzle at conditions representative of testing in

11
(a)

(b)

(c)

Figure 6. Design space exploration points using the (a) DIRECT, (b) COBYLA,

and (c) Quasi-Newton methods. Points indicate locations of iterations and colors indicate the pseudo-objective function value at that point. The color map is saturated

above -0.2 and below -0.24. 12
the NASA Langley AHSTF. The optimization framework was demonstrated for

the same facility stagnation conditions and design requirements but with di↵erent

optimization methods demonstrating zeroth and quasi-second order sampling characteristics. The benefit of utilizing this framework for facility nozzle design is in

the automated optimization without user input or decision making until the process

has completed, thereby speeding up the design process relative to manual processes

for the same level of design precision. Using the same framework, objective trades

can be done by selecting di↵erent values of the objective weightings in eq. (1) and

producing a Pareto front of designs. Once a suite of optimized designs are produced,

a single design may be selected by considering facility or manufacturing constraints

and maximizing for a particular objective such as nozzle core area. Alternatively,

the designer may choose a design on the Pareto front that is closest to the ideal

design according to the minimum L2-norm distance to the best possible objective

function values of all observed optima. This is depicted in fig. 5.

6 Directions of Future Work

The purpose of this study was to demonstrate the value added by integrating the

high-fidelity VULCAN-CFD code in an automated framework driven by DAKOTA.

For the present application in multi-objective optimization, further improvements

can be made to the methodology to achieve better nozzle designs. For example,

instead of using the centerline Mach number, the mean core Mach number could be

automatically extracted and used instead. Using that value of Mach number would

provide a more integrated estimate of the core Mach number instead of using a single

value. Furthermore, constraints may be applied to the statistical characterizations of

core flow quantities during the optimization process. This would allow the designer

to place upper bounds on the allowable departures from flow uniformity during the

optimization process according to accepted design criteria or requirements.

While the supersonic facility nozzle optimization problem was chosen as a proofof-concept due to the well-defined nature of the design space and the relatively few

number of design variables a↵ecting aerodynamic nozzle performance, elements of

the present framework may be applied to other problems of interest. The same

framework could be used to conduct studies in uncertainty quantification and sensitivity analyses to design more robust nozzles and characterize the e↵ect of uncertainties in tunnel plenum conditions. Uncertainties due to manufacturing could

also be identified, characterized, and integrated into the analysis. Furthermore, the

same HPC job structure and modification to some of the driver scripts will allow

the VULCAN-CFD and DAKOTA framework to be applied to other research areas

of relevance to high speed flowpaths.

13
References

[1] R. L. Ga↵ney, B. K. Stewart, and S. F. Harvin. “The Design of a High-Q,

Mach-5 Nozzle for the NASA Langley 8-Foot HTT”. In: 25th Aerodynamic

Measurement Technology and Ground Testing Conference. AIAA-2006-2954.

San Francisco, CA, June 2006.

[2] K. A. Ho↵mann and S. T. Chiang. Computational Fluid Dynamics. Fourth.

Vol. 1. Engineering Education System, 2000. isbn: 0962373109. url: http:

//www.worldcat.org/isbn/0962373109.

[3] J. White and J. Morrison. “A Pseudo-Temporal Multi-Grid Relaxation Scheme

for Solving the Parabolized Navier-Stokes Equations”. In: 14th Computational

Fluid Dynamics Conference. AIAA-1999-3360. Norfolk, VA, Nov. 1999.

[4] NASA Langley Research Center. VULCAN-CFD Ver. 6.3. Miscellaneous. Hampton, VA, June 2014. url: http://vulcan-cfd.larc.nasa.gov.

[5] J. R. Edwards. “A Low-Di↵usion Flux-Splitting Scheme for Navier-Stokes Calculations”. In: Computers & Fluids 26.6 (1997), pp. 635–659.

[6] D. Litton, J. Edwards, and J. White. “Algorithmic Enhancements to the VULCAN Navier-Stokes Solver”. In: 16th Computational Fluid Dynamics Conference. AIAA-2003-3979. Orlando, FL, June 2003.

[7] F. R. Menter. “Zonal Two Equation k-! Turbulence Models for Aerodynamic

Flows”. In: 24th Fluid Dynamics Conference. AIAA-1993-2906. Orlando, FL,

July 1993.

[8] D. Wilcox. “Wall Matching, a Rational Alternative to Wall Functions”. In:

27th Aerospace Sciences Meeting. AIAA-1989-0611. Reno, NV, Jan. 1989.

[9] B. Adams, L. Bauman, W. Bohnho↵, K. Dalbey, M. Ebeida, J. Eddy, M.

Eldred, P. Hough, K. Hu, J. Jakeman, L. Swiler, and D. Vigil. DAKOTA,

A Multilevel Parallel Object-Oriented Framework for Design Optimization,

Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis:

Version 5.4 Theory Manual. Technical Report SAND2011-9106. Updated 11

2013. Sandia National Laboratories, Dec. 2011.

[10] S. Thomas and R. Guy. “Increased Capabilities of the Langley Mach 7 Scramjet Test Facility”. In: 18th Joint Propulsion Conference. AIAA-1982-1240.

Cleveland, OH, June 1982.

[11] S. R. Thomas and R. W. Guy. Expanded Operation Capabilities of the Langley

Mach 7 Scramjet Test Facility. Technical Paper 2186. NASA, Oct. 1983.

[12] S. R. Thomas, R. T. Voland, and R. W. Guy. “Test Flow Calibration Study

of the Langley Arc-Heated Scramjet Test Facility”. In: 23rd Joint Propulsion

Conference. AIAA-1987-2165. San Diego, CA, 1987.

[13] W. MacDermott, D. Horn, and C. Fischer. “Flow Contamination and Flow

Quality in Arc Heaters used for Hypersonic Testing”. In: 17th Aerospace

Ground Testing Conference. AIAA-1992-4028. Nashville, TN, July 1992.

14
[14] K. Fischer and K. Rock. “Calculated E↵ects of Nitric Oxide Flow Contamination on Scramjet Performance”. In: 31st Joint Propulsion Conference and

Exhibit. AIAA-1995-2524. San Diego, CA, July 1995.

[15] J. Crown and W. Heybey. Supersonic Nozzle Design. Naval Ordnance Laboratory Memorandum 10594. U.S. Naval Ordnance Laboratory, Apr. 1950.

[16] D. Jones, C. Perttunen, and B. Stuckman. “Lipschitzian Optimization Without the Lipschitz Constant”. In: Journal of Optimization Theory and Application 79.1 (Oct. 1993), pp. 157–181.

[17] M. Powell. “Advances in Optimization and Numerical Analysis”. In: ed. by

S. Gomez and J.-P. Hennart. Kluwer Academic, 1994. Chap. A Direct Search

Optimization Method that Models the Objective and Constraint Functions by

Linear Interpolation, pp. 51–67.

[18] M. Powell. A View of Algorithms for Optimization without Derivatives. Apr.

2007.

[19] C. Broyden. “The Convergence of a Class of Double Rank Minimization Algorithms, Parts I and II”. In: Journal of the Institute of Mathmatics and its

Applications 6 (1970), pp. 76–90, 222–231.

[20] R. Fletcher. “A New Approach to Variable Metric Algorithms”. In: The Computer Journal 13 (1970), pp. 317–322.

[21] D. Goldfarb. “A Family of Variable Metric Methods Derived by Variational

Means”. In: Mathematics of Computation 24 (1970), pp. 23–36.

[22] D. Shanno. “Conditioning of Quasi-Newton Methods for Function Minimization”. In: Mathematics of Computation 24 (1970), pp. 647–656.

[23] G. N. Vanderplaats. Numerical Optimization Techniques for Engineering Design with Applications. McGraw-Hill, 1984.

[24] S. P. Schneider. Method of Characteristics Design of a Supersonic Wind Tunnel Nozzle with Square Cross-Section. Contractor Report 194359. NASA, Jan.

1993.

15
Appendix A

DAKOTA Driver Script

A sample tcsh driver script controlling the execution of codes and flow of information in DAKOTA is shown.

#!/bin/tcsh

# See Advanced Simulation Code Interfaces chapter in Users Manual

# $argv[1] is params.in from Dakota

# $argv[2] is results.out returned to Dakota

# Extract run number

set num=‘pwd | awk -F. ’{print $NF}’‘

# Create VULCAN Input File

../../scripts/dprepro $argv[1] \

nozzle_unsplit_real_template.inp \

nozzle_unsplit_real.inp

rm nozzle_unsplit_real_template.inp

# Create MOC Input File

../../scripts/dprepro $argv[1] \

input_template.dat \

input.dat

echo "Generated MOC Input File"

# Create postprocess script

../../scripts/dprepro $argv[1] \

../../scripts/outflow_reduce_template.py \

../../scripts/outflow_reduce.py

echo "Generated post-processing script"

../../scripts/dprepro $argv[1] \

../../scripts/generate_grid_template \

../../scripts/generate_grid

echo "Generated grid generation script"

# Create PBS Script

sed ’’s/jobid/$num/g’’ runfile_template > runfile

rm runfile_template

chmod +x runfile

# Generate wall profile, grid, and split grid and input file

tcsh ../../scripts/generate_grid > grid_generation_output.out

16
# Submit job to PBS

qsub runfile

# Wait for job to complete by waiting for file "jobdone"

bash ../../scripts/waitforit

# Move to results directory and copy tecplot files

cd Recomp_files/Plot3d_files

cp ../../../../tecplot/* .

# Extract objective function data

tec360 -b tec_macro.mcr > tecplot_extract.out

python ../../../../scripts/outflow_reduce.py > objective.dat

# Copy objective function value to root

cp objective.dat ../../results.out

17
Appendix B

DAKOTA Input Files

The following is an example DAKOTA input file used for each optimization

method outlined in this paper. Bracketed items are placholders where a user would

input a numerical value in the actual input file.

# DAKOTA INPUT FILE - dakota.in

strategy,

tabular_graphics_data

pareto_set

method_pointer = ’QNewt’

weight_sets = 0.5 0.5

method,

id_method = ’QNewt’

optpp_q_newton

method,

id_method = ’C_DIRECT’

coliny_direct

min_boxsize_limit = 0.5 # 0.5 converges roughly

constraint_penalty = 1.0 # Default is 1000.0

method,

id_method = ’COBYLA’

coliny_cobyla

model,

single

variables,

# MOCMach = The design Mach number for the MOC code

# when generating a new nozzle profile. [1]

# ThroatROC = The throat radius of curvature normalized

# against throat height when creating new

# designs. [1]

# M_design = Nozzle design Mach number [1]

# exit_size = half-side width/height for square

# nozzle [inches]

# M = Longitudinal number of grid points [1]

# N = Transverse number of grid points [1]

18
continuous_design = 2

descriptors ’MOCMach’ ’ThroatROC’

initial_point [MOC0] 7.5

lower_bounds [MOCl] 5.0

upper_bounds [MOCu] 10.0

discrete_state_set_real = 5

descriptors ’M_design’ ’exit_size’ ’p0’ ’rho0’ ’vi’

set_values [M_design] [exit_size] [p0] [rho0] [vi]

discrete_state_set_integer = 5

descriptors ’M’ ’N’ ’ITC’ ’ITM’ ’ITF’

set_values 257 57 3000 3000 3000

interface,

fork

asynchronous

evaluation_concurrency = 5

analysis_driver = ’simulatorScript’

work_directory named ’cases/workdir’

directory_tag

directory_save

parameters_file = ’io/params.in’

results_file = ’io/results.out’

template_directory ’template’

# Tag files

file_tag

# Save files

file_save

responses,

objective_functions = 2

nonlinear_equality_constraints = 1

descriptors "norm coreArea [1]"

"norm nozzleLength [1]"

19
"norm exitMach [1]"

# Optimization direction for objective functions

sense "max" "min"

# Nonlinear equality constraint target value (Mcenter/Mdesired)

targets 1.0

numerical_gradients

no_hessians

20
Appendix C

Optimization Problem Definition

The constrained optimization problem in this study was defined mathematically as

Minimize:

F(R ̃c, M ̃MOC) = a1A ̃core + a2L ̃nozzle (C1a)

Subject to:

M ̃centerline = 1 (C1b)

5  R ̃c  10 (C1c)

0.93  M ̃MOC  1.07 (C1d)

Equation (C1a) is the objective function, which is a linear combination of the response variables of normalized core area and non-dimensionalized nozzle length,

which are themselves functions of non-dimensionalized nozzle throat radius and the

MOC nozzle exit Mach number. Equation (C1b) is an equality constraint that defines the feasibility of designs as the optimization procedure advances. Here, the

constraint requires that designs have an exit Mach number as close to the target

design value as possible. Finally, eqs. (C1c) and (C1d) are side constraints that

come into play for optimization algorithms that require them such as DIRECT. The

bounds are defined such that certain optimization algorithms will only search within

a domain that satisfies the side constraints.

While some optimization algorithms handle equality and inequality constraints

directly and separately from the objective function (e.g., COBYLA), the DAKOTA

framework may alternatively use a pseudo-objective function that incorporates both

the objective function and a penalty function that handles the constraints indirectly.

For the optimization problem in this study, the pseudo-objective function takes the

form

F ̃ = F(R ̃c, M ̃MOC) + P(M ̃centerline) (C2)

The penalty function itself takes di↵erent forms depending on whether equality or

inequality constraints are included, but for this study it has the form

P(M ̃centerline) = R

h

M ̃centerline  1

i2

(C3)

The form of the penalty function ensures that its minimum is located at the equality constraint value. The penalty parameter R defines the strength of the penalty

function and was set to 1.0 in this study. The penalty parameter was kept at this

value in order to keep the penalty function value at the same order of magnitude

as the other objectives. For more information on the use and definition of penalty

functions, refer to Vanderplaats.23 Hence, the mathematical definition of the optimization problem using penalty functions takes the form

21
Minimize:

F ̃ = F(R ̃c, M ̃MOC) + P(M ̃centerline) (C4a)

Subject to:

5  R ̃c  10 (C4b)

0.93  M ̃MOC  1.07 (C4c)

with F and P respectively defined by eqs. (C1a) and (C3).

22
Appendix D

Representative VULCAN-CFD Solution

The discussion of the CFD solution in this appendix is representative of any given

solution during the optimization process. While the nozzle profiles will change from

iteration to iteration of the optimization algorithm, the methodology of grid generation, boundary conditions definition, and iteration to convergence is essentially

the same. The representative solution chosen here was that of the optimum point

as converged to by the Quasi-Newton method in table 2. For more information on

the VULCAN-CFD settings used to arrive at this solution, refer to section 3.1.

Solution Domain

Symmetry

Symmetry

Nozzle Crossflow Plane

Wall

Wall

Wall

Wall

Figure D1. A two-dimensional nozzle area cross-section showing the solution domain

as well as wall and symmetry boundary conditions.

The CFD solution of the supersonic nozzle was carried out in a three dimensional simulation representing a quarter of the nozzle domain. This was because the

nozzle design has two planes of symmetry in the vertical and lateral axis directions

emanating from the center of each cross-flow plane moving downstream. Because

of this, two boundaries of the domain were defined to be the adiabatic walls of the

nozzle and the other two were defined to be symmetry boundary conditions. This is

shown schematically in fig. D1. The dimensions of the grid shown in fig. D2 has 257

points along the nozzle flow direction and 57 points in each dimension of the crossflow plane. The number of grid points chosen allows the VULCAN-CFD solution to

use a coarse-to-fine sequencing technique to a total of three levels (coarse, medium,

fine). The grid was clustered near the nozzle throat due to the rapid changes in the

nozzle profile shape in that region. The grid was also clustered near the walls in

order to achieve y+ values suitable for the use of a wall matching function.8

Convergence of the solution was achieved by local time stepping of the solution for 3000 iterations at each level of grid resolution. Figure D3 shows the

23
Figure D2. Nozzle grid for the quarter-domain is shown.

logarithmically-transformed residual, mass flow error, and CFL number as a function of iteration of the solution. Discontinuities in the mass flow error and residuals

indicate points of grid refinement. The CFL number was also chosen to gradually

increase to its maximum value (here chosen to be 25) over the coarse grid solution.

Once converged, the y+ values along the nozzle walls were verified to ensure they

were suitable for solution using wall matching functions as shown in fig. D4. Because the maximum value was 16.42, it was determined that the grid resolution was

suitable for the purposes of aerodynamic optimization.

The optimized nozzle profile and nozzle centerline longitudinal Mach number

evolution are shown in fig. D5. The classical features of a supersonic nozzle are

evident, with the flow beginning at subsonic conditions in the converging section

and transitioning to supersonic conditions past the nozzle throat. The nozzle was

also designed such that the nozzle profile would become parallel to the centerline at

the nozzle exit. Note that due to boundary layer growth, having a parallel nozzle

profile at the nozzle exit does not necessarily ensure that the flow is parallel to the

centerline at all points across the nozzle exit plane.

Nozzle exit contours of Mach number and logarithmically-transformed pressure

are shown in figs. D6 and D7, respectively. Both plots also feature a Mach contour

line that represents 99% of the design Mach number in order to better visualize where

the core is defined. While the entire region in excess of the 99% Mach number could

be defined as the core, here it was defined simply as the box that fits within that

contour as illustrated in fig. 2. The symmetry planes are defined by the location of

the axes and the maxima of the axes define the adiabatic walls. In the Mach contour

plot, the boundary layer is deformed into a “dog-bone” shape that is a result of the

24
Cycle

log10(Residual)

Mdot Error (%)

CFL Number

2000 4000 6000 8000

0

2

4

6

10-2

10-1

100

101

5

10

15

20

25

CFL

log10(L2(Res))

Mdot Error (%)

Figure D3. Typical errors, residuals, and CFL number evolution during a VULCANCFD nozzle flow simulation. Discontinuities indicate a refinement in the grid topology.

Figure D4. Reprentative y+ for a solved case is shown. The maximum value of

16.42 occurs in the throat region of the flow.

25
X scaled

Y scaled

Mach scaled

0 0.05 0.1 0.15 0.2 0.25

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

Y scaled

Mach scaled

Figure D5. Scaled nozzle profile and evolution of centerline scaled Mach number for

the length of the nozzle is shown. The Y-axis is scaled by nozzle exit height.

Figure D6. Scaled Mach contours in the nozzle core exit are shown for a quarter

domain. The intersection of the symmetry planes is at the origin. The core is defined

within the M ̃ = 0.99 contour, which is within 1% of the design Mach number. The

Y- and Z-axis directions are both scaled by nozzle exit height.

26
Figure D7. Logarithmically scaled pressures contours in the nozzle core exit are

shown for a quarter domain. Pre-scaled pressures were non-dimesnionalized by the

mean nozzle exit static pressure. The intersection of the symmetry planes is at the

origin. The core is defined within the M ̃ = 0.99 contour, which is within 1% of the

design Mach number. The Y- and Z-axis directions are both scaled by nozzle exit

height.

expansion of the nozzle wall in the Y-direction and the ensuing vorticity. Note that

if expansion were allowed in the Z-direction as well this contour may have been

further degraded in terms of flow uniformity and core size.24

Due to the variation of the flow properties across the nozzle exit, uniformity of

the flow properties was evaluated as a statistical measure using data points contained

within the nozzle exit area. Table D1 summarizes an a posteriori statistical analysis

of key nozzle parameters in the exit area. Furthermore, an average error for any

given variable (denoted by the placeholder ) was defined by

e,avg = max  min

mean

(D1)

that gave a metric to define the range of errors from the mean in the nozzle exit.

These average errors for each tracked flow parameter are summarized in table D2.

The results of tables D1 and D2 were for informational purposes only and were not

fed back into the design process. However, a di↵erent optimization procedure could

optimize on mean core Mach number instead of centerline Mach number and use

the ensuing statistical parameters to constrain the allowable errors in the nozzle exit

and use those values as constraints in the optimization algorithm.

27
Table D1. Statistical properties of nozzle core exit parameters. Mach number is

caled by the design value and all other properties are scaled by their mean value.

M [1]  ̃p [1] T ̃ [1]  ̃p0 [1]

Mean 1.003 1.000 1.000 1.000

Standard Deviation 0.003 0.015 0.005 0.005

Min 0.992 0.975 0.993 0.965

Max 1.009 1.036 1.018 1.002

Table D2. Average error of nozzle exit parameters

Mp T p0

eavg 1.71% 6.04% 2.55% 3.76%

Appendix E

DAKOTA Iteration Data

The following subsections summarize the points sampled in the design space.

Note that all function evaluations are not represented; only those that represent a

candidate design point are shown. The column labeled f1 represents the objective

function value while f2 represents the constraint value. MachMOC and ThroatROC

are scaled MOC Mach number and nondimensionalized throat radius of curvature,

respectively.

E.1 DIRECT Method

%eval MachMOC ThroatMOC f1 f2

1 1.000 7.500 0.144 0.965

2 1.048 7.500 -0.237 1.001

3 0.952 7.500 0.142 0.919

4 1.048 9.167 -0.238 1.010

5 1.048 5.833 -0.247 1.010

6 1.063 5.833 -0.246 1.025

7 1.032 5.833 -0.241 0.997

8 1.000 9.167 0.149 0.965

9 1.000 5.833 0.138 0.965

10 1.053 5.833 -0.247 1.015

11 1.042 5.833 -0.247 1.005

12 1.048 9.722 -0.237 1.010

13 1.048 8.611 -0.234 1.003

14 0.968 7.500 0.142 0.934

15 0.937 7.500 0.141 0.904

16 1.042 6.389 -0.246 1.005

17 1.042 5.278 -0.250 1.005

28
18 1.063 7.500 -0.241 1.026

19 1.032 7.500 -0.237 0.996

20 0.968 9.167 0.147 0.935

21 0.968 5.833 0.137 0.935

22 1.044 5.278 -0.250 1.007

23 1.041 5.278 -0.250 1.003

24 1.048 6.389 -0.244 1.028

25 1.048 5.278 -0.250 1.010

26 1.069 5.833 -0.246 1.030

27 1.058 5.833 -0.246 1.020

28 0.952 9.167 0.147 0.921

29 0.952 5.833 0.137 0.921

30 1.041 5.278 -0.250 1.004

31 1.040 5.278 -0.250 1.003

32 1.053 6.389 -0.240 0.998

33 1.053 5.278 -0.250 1.015

34 1.037 5.833 -0.247 1.000

35 1.026 5.833 -0.247 1.001

36 0.937 9.167 0.146 0.905

37 0.937 5.833 0.136 0.906

38 1.040 5.278 -0.250 1.003

39 1.040 5.278 -0.250 1.003

40 1.037 6.389 -0.246 1.000

41 1.037 5.278 -0.250 1.000

42 1.069 7.500 -0.241 1.031

43 1.058 7.500 -0.242 1.021

44 1.000 6.389 0.142 0.965

45 1.000 5.278 0.137 0.965

46 1.039 5.278 -0.250 1.002

47 1.035 5.278 -0.250 0.998

48 1.026 6.389 -0.196 0.990

49 1.026 5.278 0.082 0.990

50 1.063 9.167 -0.237 1.025

51 1.032 9.167 -0.233 0.995

52 1.000 8.056 0.146 0.966

53 1.000 6.944 0.143 0.963

54 1.036 5.278 -0.250 0.999

55 1.035 5.278 -0.251 0.998

56 1.058 6.389 -0.244 1.026

57 1.058 5.278 -0.250 1.020

58 1.063 9.722 -0.236 1.025

59 1.032 9.722 -0.232 0.995

60 1.000 9.722 0.150 0.970

61 1.000 8.611 0.147 0.965

62 1.038 5.278 -0.250 1.001

63 1.036 5.278 -0.250 0.999

29
64 1.063 6.389 -0.244 1.025

65 1.063 5.278 -0.249 1.025

66 1.037 7.500 -0.237 1.001

67 1.026 7.500 0.118 0.991

68 1.039 5.463 0.136 0.956

69 1.039 5.093 -0.249 1.002

70 1.071 5.833 -0.246 1.031

71 1.067 5.833 -0.246 1.028

72 1.048 8.056 -0.241 1.010

73 1.048 6.944 -0.244 1.007

E.2 COBYLA Method

%eval MachMOC ThroatMOC f1 f2

1 1.071 7.500 -0.241 1.034

2 1.357 7.500 -0.200 1.295

3 1.071 8.500 -0.239 1.032

4 1.033 6.509 -0.239 0.996

5 1.034 7.018 -0.238 0.994

6 1.039 6.769 -0.042 0.988

7 1.029 7.267 -0.214 0.991

8 1.036 7.142 -0.238 0.997

9 1.016 7.013 0.144 0.977

10 1.035 7.143 -0.238 0.995

11 1.040 6.960 -0.244 1.000

12 1.040 6.991 -0.244 1.000

13 1.040 7.023 -0.243 1.012

14 1.042 6.978 -0.244 1.002

15 1.042 6.977 -0.244 1.002

16 1.041 6.984 -0.244 1.001

17 1.039 6.990 -0.244 0.999

18 1.040 6.999 -0.244 1.000

19 1.040 7.007 -0.244 1.000

20 1.040 7.015 -0.243 1.012

21 1.039 7.007 -0.245 0.999

22 1.040 7.015 -0.243 1.012

23 1.040 7.011 -0.244 1.000

24 1.040 7.015 -0.243 1.012

25 1.040 7.013 -0.245 1.000

26 1.040 7.015 -0.243 1.012

27 1.041 7.013 -0.244 1.000

28 1.041 7.013 -0.243 1.013

29 1.040 7.012 -0.244 1.000

30 1.040 7.013 -0.244 1.000

31 1.040 7.013 -0.245 1.000

32 1.040 7.013 -0.245 1.000

30
33 1.040 7.014 -0.245 1.000

34 1.040 7.014 -0.243 1.012

35 1.040 7.014 -0.245 1.000

36 1.040 7.014 -0.243 1.012

37 1.040 7.014 -0.245 1.000

38 1.040 7.014 -0.245 1.000

39 1.040 7.014 -0.245 1.000

40 1.040 7.014 -0.245 1.000

41 1.040 7.014 -0.245 1.000

42 1.040 7.014 -0.245 1.000

43 1.040 7.014 -0.245 1.000

44 1.040 7.014 -0.243 1.012

E.3 Quasi-Newton Method

%eval MachMOC ThroatMOC f1 f2

1 1.071 5.000 -0.253 1.032

2 1.042 7.105 -0.244 1.002

3 1.020 9.433 0.150 0.984

4 1.031 8.269 -0.228 0.994

5 1.036 7.687 -0.236 1.001

6 1.039 7.396 -0.243 1.002

7 1.040 7.251 -0.244 1.002

31
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4. TITLE AND SUBTITLE

An Automated DAKOTA and VULCAN-CFD Framework with

Application to Supersonic Facility Nozzle Flowpath Optimization

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110076.02.07.05

6. AUTHOR(S)

Axdahl, Erik L.

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NASA Langley Research Center

Hampton, Virginia 23681-2199

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14. ABSTRACT

Removing human interaction from design processes by using automation may lead to gains in both productivity and design

precision. This memorandum describes e↵orts to incorporate high fidelity numerical analysis tools into an automated

framework and applying that framework to applications of practical interest. The purpose of this e↵ort was to integrate

VULCAN-CFD into an automated, DAKOTA-enabled framework with a proof-of-concept application being the optimization

of supersonic test facility nozzles. It was shown that the optimization framework could be deployed on a high performance

computing cluster with the flow of information handled e↵ectively to guide the optimization process. Furthermore, the

application of the framework to supersonic test facility nozzle flowpath design and optimization was demonstrated using

multiple optimization algorithms.

15. SUBJECT TERMS

DAKOTA; VULCAN-CFD; optimization; automation; nozzle

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