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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 August 2015

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NASA STI Program . . . in Profile Since its founding, NASA has been dedicated to the advancement of aeronautics and space science. The NASA scientific and technical information (STI) program plays a key part in helping NASA maintain this important role. The NASA STI Program operates under the auspices of the Agency Chief Information Ocer. It collects, organizes, provides for archiving, and disseminates NASA’s STI. The NASA STI Program provides access to the NTRS Registered and its public interface, the NASA Technical Reports Server, thus providing one of the largest collections of aeronautical and space science STI in the world. Results are published in both non-NASA channels and by NASA in the NASA STI Report Series, which includes the following report types: • TECHNICAL PUBLICATION. Reports of completed research or a major significant phase of research that present the results of NASA programs and include extensive data or theoretical analysis. Includes compilations of significant scientific and technical data and information deemed to be of continuing reference value. NASA counterpart of peer-reviewed formal professional papers, but having less stringent limitations on manuscript length and extent of graphic presentations. • TECHNICAL MEMORANDUM. Scientific and technical findings that are preliminary or of specialized interest, e.g., quick release reports, working papers, and bibliographies that contain minimal annotation. Does not contain extensive analysis. • CONTRACTOR REPORT. Scientific and technical findings by NASA-sponsored contractors and grantees. • CONFERENCE PUBLICATION. Collected papers from scientific and technical conferences, symposia, seminars, or other meetings sponsored or co-sponsored by NASA. • SPECIAL PUBLICATION. Scientific, technical, or historical information from NASA programs, projects, and missions, often concerned with subjects having substantial public interest. • TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientific and technical material pertinent to NASA’s mission. Specialized services also include organizing and publishing research results, distributing specialized research announcements and feeds, providing information desk and personal search support, and enabling data exchange services. For more information about the NASA STI Program, see the following: • Access the NASA STI program home page at http://www.sti.nasa.gov • E-mail your question to help@sti.nasa.gov • Phone the NASA STI Help Desk at 757-864-9658 • Write to: NASA STI Information Desk Mail Stop 148 NASA Langley Research Center Hampton, VA 23681–2199

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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

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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

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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 Introduction 3 Methodology 2 4 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 5 Summary and Conclusions 6 Directions of Future Work A DAKOTA Driver Script B DAKOTA Input Files C Optimization Problem Definition 10 11 13 16 18 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

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1 Nomeclature Acronyms/ Description Initialisms 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 Description Symbols Acore nozzle exit core area Units 2 m A˜ normalized nozzle exit core area 1 core Aexit Nozzle exit area ai weighting coecient e error F objective function 2 m 1 1 1 F˜ pseudo-objective function 1 feval function evaluations Hexit nozzle exit height 1 m i number of algorithm iterations 1 Lnozzle nozzle length m L˜ nondimensional nozzle length 1 nozzle Lref reference nozzle length m M Mach number 1 Mcenterline nozzle exit centerline Mach number 1 M˜ scaled nozzle exit centerline Mach number 1 centerline Mdesign targeted nozzle exit design Mach number 1 MM OC nozzle exit Mach number from Method of Characteristics 1 M˜ scaled nozzle exit Mach number from Method of Characteristics 1 M OC P penalty function p˜ scaled pressure p0 stagnation pressure 2 1 1 Pa

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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˜ nondimensional nozzle throat radius of curvature 1 c 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 Description Symbols variable placeholder Superscripts Description * optimized condition 3

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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

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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 1 at NASA Langley Research Center. The IMOCND code 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 2 clustering parameters near the walls and near the throat. Clustering was enforced to achieve sucient resolution in the wall boundary layer as well as in areas where rapid area changes occur. 3,4 The nozzle flow properties were simulated using the VULCAN-CFD code, developed and maintained at NASA Langley Research Center. VULCAN-CFD uses a 5

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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 5 frozen with non-vititated air. The flow fluxes were computed using LDFSS and were recombined with a 3rd order MUSCL interpolation strategy. The flow was 6 integrated temporally using the ILU scheme with local time stepping and an increasing CFL number schedule. Turbulence was modeled using the two-equation 7 Menter k-! model with a turbulent Prandtl number of 0.9. Wall matching func- 8 tions 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 Laboratory- 9 developed and -maintained code DAKOTA. 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

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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

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Table 1. DAKOTA and application parallelism use cases (table reproduced from 9 information in DAKOTA user manual 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 10,11,12 AHSTF supersonic tunnel. 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 13,14 be present in the gas flowing from the facility plenum, 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

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Dominated Solutions Nozzle Length Core Size Pareto Optimal (Non-Dominated Solutions) Optimum design (closest design to ideal solution) Ideal Solution (Not achievable) Unique solution for a given design variable combination 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 15 will become thicker with longer nozzles, 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˜ , M˜ ) = a A + a L (1) c M OC ˜ ˜ 1 core 2 nozzle 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 P that ai 2 [0, 1] and ai = 1. Because the objective function in this case is to be i 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˜ = A /A core core exit andL˜ = L /L . For more information on the mathematical optimization nozzle nozzle ref definition in this study, refer to appendix C. 9

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Of the optimization methods available in DAKOTA, three were chosen to demon- 16 strate the capability of the automated framework. The first was the DIRECT 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 17,18 as well as the dimensionality of the problem. The second was the COBYLA 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 9 a Quasi-Newton method that is distinct from a classical Newton method in that 19,20,21,22 the approximate inverse Hessian at a point is updated using the BFGS 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

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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˜⇤ M˜⇤ M˜ A˜ [m2] L˜ [m] f i c MOC centerline core nozzle eval 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

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(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

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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

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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 VUL- CAN 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

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[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

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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

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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

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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

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continuous_design = 2 descriptors ’MOCMach’ initial_point [MOC0] lower_bounds [MOCl] upper_bounds [MOCu] 10.0 discrete_state_set_real = 5 ’ThroatROC’ 7.5 5.0 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 interface, fork asynchronous evaluation_concurrency = 5 3000 3000 3000 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

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"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

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Appendix C Optimization Problem Definition The constrained optimization problem in this study was defined mathematically as Minimize: F (R˜ , M˜ ) = a A + a L (C1a) c M OC Subject to: M˜ centerline = 1 (C1b) ˜ ˜ 1 core 2 nozzle 5  R˜  10 (C1c) c 0.93  M˜ M OC  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˜ , M˜ ) + P (M˜ ) (C2) c M OC centerline 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 h i2 P (M˜ ) = R M 1 (C3) centerline ˜ centerline 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 23 functions, refer to Vanderplaats. Hence, the mathematical definition of the optimization problem using penalty functions takes the form 21

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Minimize: F˜ = F (R˜ , M˜ ) + P (M˜ ) (C4a) c M OC centerline Subject to: 5  R˜  10 (C4b) c 0.93  M˜ M OC  1.07 (C4c) with F and P respectively defined by eqs. (C1a) and (C3). 22

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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. Wall! Solution Domain! ! Wall Symmetry! Symmetry ! Wall ! Nozzle Crossflow Plane! 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 + 8 order to achieve y values suitable for the use of a wall matching function. 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

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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

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CFL log10(L2(Res)) Mdot Error (%) 6 4 2 log10(Residual) 0 25 20 1 10 15 0 10 10 CFL Number Mdot Error (%) -1 10 5 -2 10 2000 4000 6000 8000 Cycle Figure D3. Typical errors, residuals, and CFL number evolution during a VULCAN- CFD 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

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1 0.8 0.6 Y scaled 0.4 0.2 0 0.05 0.1 X scaled 1 0.8 0.6 Mach scaled 0.4 Y scaled 0.2 Mach scaled 0.15 0.2 0.25 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

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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 24 further degraded in terms of flow uniformity and core size. 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 max min e,avg = (D1) mean 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

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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] p˜ [1] 0 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 M p 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 1 1.000 7.500 0.144 2 1.048 7.500 -0.237 3 0.952 7.500 0.142 4 1.048 9.167 -0.238 5 1.048 5.833 -0.247 6 1.063 5.833 -0.246 7 1.032 5.833 -0.241 8 1.000 9.167 0.149 9 1.000 5.833 0.138 10 1.053 5.833 -0.247 11 1.042 5.833 -0.247 12 1.048 9.722 -0.237 13 1.048 8.611 -0.234 14 0.968 7.500 0.142 15 0.937 7.500 0.141 16 1.042 6.389 -0.246 17 1.042 5.278 -0.250 28 f2 0.965 1.001 0.919 1.010 1.010 1.025 0.997 0.965 0.965 1.015 1.005 1.010 1.003 0.934 0.904 1.005 1.005

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18 1.063 7.500 -0.241 19 1.032 7.500 -0.237 20 0.968 9.167 0.147 21 0.968 5.833 0.137 22 1.044 5.278 -0.250 23 1.041 5.278 -0.250 24 1.048 6.389 -0.244 25 1.048 5.278 -0.250 26 1.069 5.833 -0.246 27 1.058 5.833 -0.246 28 0.952 9.167 0.147 29 0.952 5.833 0.137 30 1.041 5.278 -0.250 31 1.040 5.278 -0.250 32 1.053 6.389 -0.240 33 1.053 5.278 -0.250 34 1.037 5.833 -0.247 35 1.026 5.833 -0.247 36 0.937 9.167 0.146 37 0.937 5.833 0.136 38 1.040 5.278 -0.250 39 1.040 5.278 -0.250 40 1.037 6.389 -0.246 41 1.037 5.278 -0.250 42 1.069 7.500 -0.241 43 1.058 7.500 -0.242 44 1.000 6.389 0.142 45 1.000 5.278 0.137 46 1.039 5.278 -0.250 47 1.035 5.278 -0.250 48 1.026 6.389 -0.196 49 1.026 5.278 0.082 50 1.063 9.167 -0.237 51 1.032 9.167 -0.233 52 1.000 8.056 0.146 53 1.000 6.944 0.143 54 1.036 5.278 -0.250 55 1.035 5.278 -0.251 56 1.058 6.389 -0.244 57 1.058 5.278 -0.250 58 1.063 9.722 -0.236 59 1.032 9.722 -0.232 60 1.000 9.722 0.150 61 1.000 8.611 0.147 62 1.038 5.278 -0.250 63 1.036 5.278 -0.250 29 1.026 0.996 0.935 0.935 1.007 1.003 1.028 1.010 1.030 1.020 0.921 0.921 1.004 1.003 0.998 1.015 1.000 1.001 0.905 0.906 1.003 1.003 1.000 1.000 1.031 1.021 0.965 0.965 1.002 0.998 0.990 0.990 1.025 0.995 0.966 0.963 0.999 0.998 1.026 1.020 1.025 0.995 0.970 0.965 1.001 0.999

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64 1.063 6.389 -0.244 65 1.063 5.278 -0.249 66 1.037 7.500 -0.237 67 1.026 7.500 0.118 68 1.039 5.463 0.136 69 1.039 5.093 -0.249 70 1.071 5.833 -0.246 71 1.067 5.833 -0.246 72 1.048 8.056 -0.241 73 1.048 6.944 -0.244 E.2 COBYLA Method %eval MachMOC ThroatMOC f1 1 1.071 7.500 -0.241 2 1.357 7.500 -0.200 3 1.071 8.500 -0.239 4 1.033 6.509 -0.239 5 1.034 7.018 -0.238 6 1.039 6.769 -0.042 7 1.029 7.267 -0.214 8 1.036 7.142 -0.238 9 1.016 7.013 0.144 10 1.035 7.143 -0.238 11 1.040 6.960 -0.244 12 1.040 6.991 -0.244 13 1.040 7.023 -0.243 14 1.042 6.978 -0.244 15 1.042 6.977 -0.244 16 1.041 6.984 -0.244 17 1.039 6.990 -0.244 18 1.040 6.999 -0.244 19 1.040 7.007 -0.244 20 1.040 7.015 -0.243 21 1.039 7.007 -0.245 22 1.040 7.015 -0.243 23 1.040 7.011 -0.244 24 1.040 7.015 -0.243 25 1.040 7.013 -0.245 26 1.040 7.015 -0.243 27 1.041 7.013 -0.244 28 1.041 7.013 -0.243 29 1.040 7.012 -0.244 30 1.040 7.013 -0.244 31 1.040 7.013 -0.245 32 1.040 7.013 -0.245 30 1.025 1.025 1.001 0.991 0.956 1.002 1.031 1.028 1.010 1.007 f2 1.034 1.295 1.032 0.996 0.994 0.988 0.991 0.997 0.977 0.995 1.000 1.000 1.012 1.002 1.002 1.001 0.999 1.000 1.000 1.012 0.999 1.012 1.000 1.012 1.000 1.012 1.000 1.013 1.000 1.000 1.000 1.000

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33 1.040 7.014 -0.245 34 1.040 7.014 -0.243 35 1.040 7.014 -0.245 36 1.040 7.014 -0.243 37 1.040 7.014 -0.245 38 1.040 7.014 -0.245 39 1.040 7.014 -0.245 40 1.040 7.014 -0.245 41 1.040 7.014 -0.245 42 1.040 7.014 -0.245 43 1.040 7.014 -0.245 44 1.040 7.014 -0.243 E.3 Quasi-Newton Method %eval MachMOC ThroatMOC f1 1 1.071 5.000 -0.253 2 1.042 7.105 -0.244 3 1.020 9.433 0.150 4 1.031 8.269 -0.228 5 1.036 7.687 -0.236 6 1.039 7.396 -0.243 7 1.040 7.251 -0.244 31 1.000 1.012 1.000 1.012 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.012 f2 1.032 1.002 0.984 0.994 1.001 1.002 1.002

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REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704–0188 The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE 01-08-2015 Technical Memorandum 4. TITLE AND SUBTITLE An Automated DAKOTA and VULCAN-CFD Framework with Application to Supersonic Facility Nozzle Flowpath Optimization 6. AUTHOR(S) Axdahl, Erik L. 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) NASA Langley Research Center Hampton, Virginia 23681-2199 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) National Aeronautics and Space Administration Washington, DC 20546-0001 12. DISTRIBUTION/AVAILABILITY STATEMENT Unclassified-Unlimited Subject Category 66 Availability: NASA STI Program (757) 864-9658 13. SUPPLEMENTARY NOTES An electronic version can be found at http://ntrs.nasa.gov. 14. ABSTRACT 3. DATES COVERED (From - To) 10/2013–9/2014 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 110076.02.07.05 8. PERFORMING ORGANIZATION REPORT NUMBER L–20571 10. SPONSOR/MONITOR’S ACRONYM(S) NASA 11. SPONSOR/MONITOR’S REPORT NUMBER(S) NASA/TM–2015–218796 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 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT b. ABSTRACT c. THIS PAGE U U U UU 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON OF STI Information Desk (email: help@sti.nasa.gov) PAGES 19b. TELEPHONE NUMBER (Include area code) 36 (757) 864-9658 Standard Form 298 (Rev. 8/98) Prescribed by ANSI Std. Z39.18