optimization of hypersonic aircraft using genetic algorithms

Applied Mathematics and Computation 242 (2014) 423–434

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Optimization of hypersonic aircraft using genetic algorithms

http://dx.doi.org/10.1016/j.amc.2014.05.1200096-3003/� 2014 Published by Elsevier Inc.

⇑ Corresponding author.E-mail address: [email protected] (R.J. Hartfield).

Vivek Ahuja, Roy J. Hartfield ⇑, Andrew SheltonDepartment of Aerospace Engineering, Auburn University, Auburn, AL 36830, United States

a r t i c l e i n f o a b s t r a c t

Keywords:HypersonicDesign optimizationGenetic algorithmIntegrated aero-propulsion

A methodology has been developed to take advantage of hypersonic aircraft design charac-teristics to allow design optimizations using computational fluid dynamics and geneticalgorithms. This methodology highlights the inherent advantages obtained from decou-pling and then adding via explicit formulations, the viscous, thermal, species-transportand combustion physics into the implicitly solved Euler formulations of fluid flow. Theresulting formulation is found to be robust and less sensitive to volume mesh refinement,thereby making it possible to use an automated mesh generation process. The mesh gen-erator also takes advantage of the inherent two-dimensionality of the fluid flow aroundhypersonic aircraft and uses block-structured rhombohedrum meshes that are easy tocontrol using an optimizer. A sample case study is presented in the form of the X-43 hyper-sonic aircraft which is then optimized for a Mach 6.5 flight using this new methodology.

� 2014 Published by Elsevier Inc.

1. Introduction

Hypersonic flight using airbreathing propulsion continues to be the elusive next logical step in aeronautical engineering.However, with the realization of the basics of Scramjet technology via unmanned technology demonstrator vehicles over thelast two decades and the advancement of technology in the fields of metallurgy and high enthalpy flow engines, the idea ofre-usable hypersonic air-breathing vehicles has begun to reach a tipping point [1–5]. Nevertheless, development of hyper-sonic speed vehicles continues to be an exercise in multidisciplinary optimization and multi-goal compromises in order to beviable [4].

The evaluation of the aero-propulsive performance of a typical hypersonic vehicle continues to be an area of research[1–3,5]. Until recently, it has not been possible to generate three-dimensional volume meshes without manual input for agiven geometry. The dependency of the turbulence flow conditions and viscous effects continue to force practitioners ofnumerical fluid mechanics to apply a lot of manual effort in order to achieve accurate results. All of these limitations havemade the idea of using an automated optimizer to design an aerospace vehicle using computational fluid dynamics veryunlikely.

However, there are fluid-dynamic regimes where the physics becomes relatively simple and the geometry becomes pre-dictable [4]. For these regimes it should be possible to do optimizer controlled design studies as shown by Ahuja et al. [4,6].Hypersonic flow is one such regime. At hypersonic Mach numbers, the geometry that can be used with current technologylevels and the physics of the flow are so well defined, that the geometries essentially revert to two-dimensions oraxisymmetric.

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Nomenclature

u local velocityq local densityp local pressureE total energy per unit volumeCP specific heat of the fluidT local temperaturel viscosity of the fluidk constant of fluid thermal conductivityM Mach numberRex local Reynolds numberqw heat transfer rate of the fluida angle of attackU free stream velocityq free stream atmospheric densitym friction correlation constantv local tangential velocity componenth ramp angleb shock angle created by the oblique shockM1 Mach number of the free-stream flowMf mass fraction of fuelDcT;i;j;k local diffusion coefficientRfi;j;k

rate of formation of the species at a cell

424 V. Ahuja et al. / Applied Mathematics and Computation 242 (2014) 423–434

Further, decoupling of the viscous flow formulations, the thermal properties, combustion and species-transport simplifythe fluid equations that need to be solved so that the solution sensitivity to the mesh can be dramatically reduced.

It is the purpose of this effort to elucidate the aforementioned assertions via an optimization trade study of the X-43hypersonic airframe outer mold line. This paper shows the performance benefits of using an optimizer-controlled, integratedCFD-based aero-propulsive environment for cruise level hypersonic flight.

2. Mesh generation

In light of the need for efficiency in the mesh generation process during optimization, a block structured mesh design waschosen. On account of the two-dimensional/axisymmetric nature of the airframes chosen for this study, the individual cellswere derived from the two-dimensional structured blocks by extruding them. As a result, the rhombi two-dimensional cellswere in effect representative of a generalized rhombohedrum extrusion volume.

The surface patch interfaces of each block were not forced into conformity with the neighboring blocks. This allowed thegeneration of isolated structured mesh blocks with non-conformal patch interfaces. The onus of extracting the neighboringface sets for each interface cell face was then placed on the solver. This decision greatly simplified the mesh generationprocess.

The two-dimensional and axisymmetric nature of the mesh was chosen in light of the customized nature of this study.Most contemporary hypersonic airframes at their center-plane are essentially two-dimensional because they attempt to takeadvantage of an integrated aero-propulsive environment. The X-43 and X-51 designs fall into this category [1–3]. The otherdesigns tend to be closer to rocket powered missiles and invariably are circular in cross section. This simplification of overalldesign philosophies greatly assists in the application of CFD to the design optimization phases mainly as a result of muchsimplified mesh generation requirements.

Each structured mesh patch has four numerical perimeter edges, but is permitted to have zero length physical edges. Thissetup permits the use of simplified structured mesh mechanics. Each patch is defined with two numerical axes (I, J) and isrefined based on the distribution of points along these perimeter edges. Note that since the mesh patch is structured, twopairs of edges are refined along each of the numerical axes. This simplification for meshing effectively precludes any confor-mity at the patch interfaces except for simpler geometries.

The distribution of points in physical space is controlled by the use of successive ratio schemes. Both single and bi-direc-tional distribution schemes were applied depending on the mesh refinement required. This distribution scheme resulted insufficient mesh refinement control for each block so that the required regions of the geometry could be refined depending on

Fig. 1. Near-field and far-field refinement using block-structured rhombohedrum extrusion.

Fig. 2. Localized refinement using multiple patches with growth-transfer.

V. Ahuja et al. / Applied Mathematics and Computation 242 (2014) 423–434 425

the complexity of the physics involved. An example of this is shown in Fig. 1 for the center-plane of an X-43. The growthregions near the surface of the fuselage are highly refined compared with the cells further away.

It is also useful to note from Fig. 1 that the structured mesh block strategy chosen for this effort has an inherent disad-vantage: the refinement along one axis effectively refines its corresponding edge on the opposite side of the block. This hasunintended consequences as shown for the inlet and exhaust sections of the geometry. These regions are much more refinedin the far field than required. A solution to this kind of unwarranted increase in mesh cell count was found by the introduc-tion of more patches as shown in Fig. 2 for the inlet lip. With more patches applied, and growth transferred from one to inter-face to the other, the far-field refinement can be controlled. It is believed that the use of unstructured meshes using

Fig. 3. Ramp recirculation region mesh refinement using directional growth schemes.


tetrahedral or polyhedral cells with frozen patch interfaces will not encounter this limitation of the extrusion-based struc-tured rhombohedrum meshes.

One of the controlling parameters for the meshes in the optimization was for refined meshes near wall-boundaries. Forcecoefficients from pressure and skin-friction were evaluated using local pressure and velocities on the surface and necessi-tated greater refinement. As a result, it was invariably necessary to apply directional growth rates from wall boundaries intothe free-stream. This was achieved using successive ratio schemes along each numerical axis. An example of such a refine-ment is shown in Fig. 3 for the recirculation region downstream of a compression section of the main engine on an X-43design.

In addition to the physical mesh for each block, the solver requirements necessitated the presence of ‘‘ghost’’ cells alongeach of the four numerical patch perimeter edges. These ghost cells are extrusions of the patch perimeters and have celldepths dependent on the choice of the solver numerical schemes. The current effort uses an second-order numerical scheme,the ghost cell extrusion depth included two cells.

3. Numerical simulation

The integrated aero-propulsive environment typically seen in hypersonic aircraft requires the use of a high fidelitynumerical simulation. This simulation must also be expedient in order to account for a large number of iterations duringthe design optimization study.

The use of generalized three-dimensional Navier–Stokes solvers was discounted because of computational intensity.Instead, for the present study, an Euler solver was judged to provide the appropriate balance between solution fidelityand expediency. The code was modified to allow the decoupled inclusion of viscous effects via semi-empirical formulations.Further, the mesh generation is reduced to two-dimensions for efficiency without dramatic compromise in fidelity for thisparticular application.

The Euler solver uses the robust Steger–Warming Flux-Vector-Splitting (FVS) model with limited upwind state extrapo-lation in the style of Van Leer’s MUSCL technique [6]. This model is explicit in time and the main numerical discretizationscheme used throughout the code is based on the first-order forward difference scheme as applied to the case of the stateand inviscid flux vectors and is given as:

DyDxDtðUnþ1

i;j � Uni;jÞ þ ½ðFiþ1

2;j� Fi�1

2;jÞDyþ ðGi;jþ1

2� Gj;i�1

2ÞDx�

n¼ 0; ð1Þ

where the inviscid flux vectors are obtained as:

U ¼

qquqvqet

8>>><>>>:

9>>>=>>>;;

F ¼

qu

qu2 þ ðc� 1Þ qet � q ðu2þv2Þ

2

� �

qvu

q et þ ðc� 1Þ et � ðu2þv2Þ

2

� �� u

8>>>>><>>>>>:

9>>>>>=>>>>>;;

G ¼

qvqvu

qv2 þ ðc� 1Þ qet � q ðu2þv2Þ

2

� �

q et þ ðc� 1Þ et � ðu2þv2Þ

2

� �� v

8>>>>><>>>>>:

9>>>>>=>>>>>;: ð2Þ

The Van-Leer Monotone Upwind Scheme for Conservation Laws (MUSCL) [6] was used to control the order of accuracy ofthe solver. This method can be described as:

ULiþ1

2j ¼ Ui;j þ 12 ðUi;j � Ui�1;jÞ;i;j

URiþ1

2j ¼ Uiþ1;j � 12 ðUiþ2;j � Uiþ1;jÞuiþ1;j

ULi�1

2j ¼ Ui�1;j þ 12 ðUi�1;j � Ui�2;jÞ;i�1;j

URi�1

2j ¼ Ui;j � 12 ðUiþ1;j � Ui;jÞui;j

ð3Þ

where, the limiters ;i;j;ui;j etc. are set up so that the system reduces to a first order system when these limiters are equal tozero and then to a second order system when they are unity. In practice, how a1 ever, it was found that the second orderapproach yielded comparatively better solutions for the external supersonic and hypersonic environments found for hyper-sonic aircraft.


The Euler formulations allow only for the evaluation of the compressible flow fields. The viscous effects, thermal conduc-tivity, combustion and species transport equations are not coupled with the formulations shown in Eqs. (1)–(3). These mod-els are thus decoupled and are added explicitly to the solver equations.

Specific heat values, viscosity, the specific heat ratio and the thermal conductivity coefficients are evaluated as a functionof temperature for all of the species involved. However, the species were simplified into two groups only: air and fuel. Thelatter was taken as hydrogen for which the flammability limits, thermal properties and combustion properties were wellestablished. The properties of air were evaluated using the following formulation [6]:

CP ¼ a0 þ a1T þ a2T2 þ a3T3 þ a4T4 þ a5T5 þ a6T6 þ a7T7; ð4Þ

where, a0 = 0.25020051, a1 = �5.1536879e�5, a2 = 6.551948e�8, a3 = �6.7178376e�12, a4 = �1.512825e�14,a5 = 7.6215767e�18, a6 = �1.4526772e�21, a7 = 1.011554e�25.

l ¼ T1:5

Tþ111

� �ð1:46� 10�6Þ

k ¼ T1:5

Tþ112

� �ð1:99� 103Þ

: ð5Þ

Species transport for the solver is built around the generalized Fickian advection–diffusion equation [6]:

@

@tðqi;j;kMfi;j;k

Þ þ r � ðqi;j;kv i;j;k��!Mi;j;kÞ ¼ �r � ð�qi;j;kDcT;i;j;k

rMfi;j;kÞ þ Rfi;j;k

: ð6Þ

The diffusion coefficients used in Eq. (6) are predetermined for various fuel–air mixtures and temperature combinations.The advection term of the equation used the local velocities in each cell. The Upwind style numerical discretization scheme isapplied for the application of the equation inside the FVS solver.

The solver models combustion effects as localized enthalpy increases based on the fuel–air concentration and the flam-mability limits in the cells downstream of the injection and ignition points. The species concentration data is used to eval-uate the flammability zones in the combustor [7]. The chemical reaction model is based on the reduction of the Gibbs freeenergy model for the evaluation of the enthalpy release at each cell under equilibrium conditions [7,8]. This enthalpy releaseis driven by the distribution of pressure, temperature and species mixture at each cell [8]. Given the high speed flow, the veryhigh velocities inside the combustion chamber allow a relatively good fidelity on the combustion model without detailedturbulence modeling results. The results are considered detailed enough for a preliminary level optimization. For detailedwork a full Navier–Stokes solver must be used to evaluate the results of this optimization study.

The skin-friction coefficient is also accounted using explicit, decoupled models. Based on the data presented by Koppen-wallner [9] for high speed flows, the skin friction drag coefficient was evaluated using the following formulation:

CD ¼0:045

R16ex

; CF ¼0:0375

Re162:

ð7Þ

In Eq. (7), the Reynold’s number is evaluated locally. An integration of the pressure and skin-friction coefficients providesthe net force vector components in the body-frame of the vehicle. All forces including the thrust, drag and aerodynamic lift-ing forces are resolved into the body-fixed frame of the vehicle.

Fig. 4. Mesh refinement study on the X-43; refined mesh (top) and coarse mesh (bottom).


4. Mesh refinement study

The choice of solver described previously greatly reduced the sensitivity of the computed aerodynamic loads on the meshrefinement. This was a crucial achievement in order for the overall scheme to be automated under the control of an opti-mizer. An example of the insensitivity of the numerical scheme to the mesh is given here.

A refinement study was conducted against baseline X-43 HXRV outer mold line configuration [10] with the scramjetengine pod closed and sealed with an inlet flow fence and nozzle flow fences (Fig. 4).

Two meshes were chosen for the study. The first mesh was coarse and had only 9375 volume cells. The second mesh washighly refined with 30,375 volume cells. Both meshes were run through the solver and the data extracted for the aerody-namic coefficients. The Mach number distribution is as shown in Fig. 4. The aerodynamic coefficients are plotted as a func-tion of angle of attack for the two meshes in Fig. 5. The force coefficients are seen to remain significantly free from volumemesh refinements. This result is intuitive of the nature of the solver chosen and its theoretical underpinnings, i.e. Euler for-mulation of the fluid dynamical equations. Fig. 5 suggests strongly why an automated environment can indeed be created forhypersonic airframe design optimization using numerical fluid dynamics.

5. Boundary conditions

The four patch perimeter edges for each structured volume block in the mesh represent the boundaries of that block. Thefaces of those perimeter cells are applied with one of five types of boundary conditions. These are slip walls, symmetryplanes, velocity inlets, pressure outlets and internal flow interfaces.

Fig. 5. Normal and axial force coefficients for the X-43 mesh refinement study.


The slip wall boundary is applied using the Neumann condition on the averaged surface normal of a cell face. For theinterface condition, the solver detects the presence of patch neighbors for the interface faces and evaluates the flux compo-nent from each neighbor depending on the area overlap encountered. Perimeter cell faces that were free are automaticallymarked as velocity inlets or pressure outlets depending on the relative orientation of the mean surface normal vector withthe marked free-stream direction.

6. Validation

The solver was validated over a range of simple geometries from one-dimensional shock tubes to two-dimensional obli-que shocks. The force coefficients were validated against the X-43 wind tunnel data [10,11]. A few of these results are pre-sented here for completeness.

The case of the oblique shock problem is designed to test the overall solver performance characteristics. The requiredpressure and density ratios are given from the basic shock theory equations as:

P2P1¼ 1þ ðM2

1ðsinðbÞÞ2 � 1Þ 2ccþ1

q2q1¼ ðcþ1ÞM2

1ðsinðbÞÞ2

ðc�1ÞM21ðsinðbÞÞ2þ2

ð8Þ

Sta

tic P

ress

ure

Sta

ticP

ress

ure

0

0.8

1

1.2

1.4

1.6

1.8

2

0

0.8

1

1.2

1.4

1.6

1.8

2

Sta

tic P

ress

ure

0

2

2

2

2

4D

4Dis

4Distanc

4stance

ce alon

along

6ng botto

6the bo

om Wa

ottom W

8all

NuAn

8Wall

NuAn

umericnalytic

umericnalytic

10

cal Solcal Sol

10

cal Solal Res

utionution

utionsults

12

12

Fig. 6. Comparison of solver results with theory (left: first order; right: second order).


The solver results are shown in Fig. 6. The pressure and density variations along the wall are in good comparison withtheory.

The NASA X-43 program consists of three autonomously controlled research flights at speeds up to Mach 10 to demon-strate, validate, and extend scramjet technology [10,11]. The airframe is integrated with a single airframe-integrated scram-jet as shown in Fig. 7. These vehicles are boosted using a modified Pegasus TM booster, air launched from the NASA DrydenFlight Research Center (DFRC) B-52 [10]. The desired test conditions in free flight are a dynamic pressure of 1000 lb persquare foot (psf). The research vehicle is boosted to approximately 95,000 feet for Mach 7, and 110,000 feet for Mach 10,corresponding to Reynolds numbers of 11 and 8 million respectively, based on vehicle length [10,11].

The X-43 model was tested in the wind tunnel with the scramjet engine pod fitted with an inlet flow fence and a nozzleflow fence at the combustor exit (Fig. 8) [11]. Fuel flow was switched off and no combustion modeling and thrust effectswere monitored. This geometry was tested at varying angles of attack and the vehicle normal and axial force componentswere evaluated experimentally.

For the initial solver validation efforts, the geometry was loaded and tested and the performance data compared withexperiments. The results are shown in Figs. 9 and 10. While the axial force prediction was lower than experiments, the trendswere captured accurately. Similar results were obtained for the normal forces as well.

Fig. 7. The X-43 hypersonic vehicle [1].

Fig. 8. X-43 engine pod experimental configuration [10].

Fig. 9. X-43 normal force coefficients.


7. Optimization

A binary-encoded genetic algorithm (GA) [12] was employed for the optimization. Genetic algorithms have proven to be agreat tool for optimizing aerospace systems. They have been used in a myriad of aerospace applications including rockets,missiles [13–15] etc. The GA was chosen over traditional gradient marching methods as it was found to be superior [14].

Fig. 11. The optimization design space.

Fig. 10. X-43 axial force coefficients.

Table 1Combustor geometry design space (all lengths in meters, angles in degrees).

Parameter Max Min

Xc1 0.40 0.10Xc1 0.20 0.10Xc1 0.20 0.10Xc1 0.15 0.05Ac1 25.00 �10.00Ac1 25.00 �25.00Ac1 25.00 �25.00Ac1 25.00 �65.00

Fig. 12. Goal residuals vs. generations.

Fig. 13. Inlet lip angle vs. generations.

Fig. 14. Normalized inlet lip length vs. generations.



A conventional GA optimization scheme was set up. A Pareto scheme was not initiated so that a global optimizationscheme was thus created with a single weighted goal used. The design goals were created mathematically in such a mannerthat the design constraints were always passed back to the GA in a form that required minimization. Creep mutation wasdeactivated for these runs and a single point crossover scheme was used for the analysis. The genetic diversity of the pop-ulation members of the GA generations was maintained by using unsteady conditions with mutation rates. The best per-formers from each generation were also preserved. Niches were discouraged from forming within the runs. A typical runconsisted of 100 generations created with 50 members per population.

Numerous GA based optimization runs were conducted for the X-43 HXRV under various design constraints. Since the X-43 airframe geometry depends on fuselage integrated pre-compression shocks to provide intake flow compression, the vehi-cle forward ventral surfaces are opened up for optimization. The geometry of the combustor has a large bearing on the thrustproduction capability especially in scramjet engines [15] and is also made part of the optimization design space as shown inFig. 11 and Table 1.

Fuel injection angles and locations were varied but the injection properties were held constant. Overall, a total of eightgeometry parameters were available for the GA to optimize. The primary goal of the optimization run included maximizationof thrust at cruise Mach number.

Fig. 15. Injection step angle vs. generations.

Fig. 16. X-43 optimization geometry outputs (top: worst performer; bottom: best performer).

Table 2Objective function results.

M = 6.5, Alt = 120,000 ft. L/D ratio Thrust/drag

Optimized design 1.29 3.19Worst performer 0.94 1.67


8. Analysis

The optimizer objective function is presented as a function of generations in Fig. 12. The GA is seen to have converged onthe optimized design within 20 generations. The objective function shows a consistent reduction.

Similar plots can be created for all of the individual geometric parameters from Table 1. A couple of these plots are shownin Figs. 13 and 14 for the inlet lip angle (Ac1 in Table 1 and Fig. 11) and the inlet lip length (Xc1 in Table 1 and Fig. 11). Inboth cases the trends are very similar. These trends are intuitive since the GA detects the better values early in the run. Thisis particularly true of parameters that prove crucial to the objective function. From Fig. 13 it is seen that the inlet-lip angledirecting the flow into the engine section just downstream of the pre-compressive ramp is fixed at 10� at Mach 6.5 free-stream flow at 120,000 feet altitude.

Fig. 15 shows the variation of the fuel injection angle during the optimization process. The GA is seen to quickly convergeon the design space upper limit of �62�. This suggests that the better injection angle for mixing inside the combustor is closeto streamline injection angles rather than normal injection angles.

The overall OML shapes of the best and worst performers of the optimization are shown in Fig. 16. The variation in theinternal fluid flow regions of the two engines is dramatic. The engine performances also reveal this difference as shown inTable 2. At Mach 6.5 and 120,000 feet, the lift to drag ratio is 1.29 for the optimized design while for the worst performer it is0.94. Equally important are the thrust to drag ratios. For the optimized vehicle it is 3.19 and that for the worst performer it is1.67. The diversity in design space is highlighted by these numbers as well as the combustor geometries shown in Fig. 16.

9. Conclusions

This methodology paper has demonstrated the feasibility of integrating computational fluid mechanics within a genetic-algorithm controlled environment in order to optimize design vehicles for hypersonic flight. The optimizer was able todeduce physically valid geometries from the design space for the test case of the X-43 outer mold line. This is encouragingin that with further development, this concept could be extended to generalized three-dimensional flow. The hybrid solverwas developed around the addition of explicit formulations for viscous flow, thermal properties, combustion and speciestransport to the implicitly solved Euler formulations. This solver was shown to have robustness and insensitivity to meshrefinement for supersonic flows. Further testing and validation of these models in three dimensions are needed before itis possible to include generalized three-dimensional geometry optimizations.

References

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optimization of hypersonic aircraft using genetic algorithms

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