Multiobjective Optimization of Earth-Entry Vehicle Heat Shields

Joshua E. Johnson∗ and Mark J. Lewis†

University of Maryland, College Park, Maryland 20742

and

Ryan P. Starkey‡

University of Colorado, Boulder, Colorado 80309

DOI: 10.2514/1.42565

A differential evolutionary algorithm has been executed to optimize the hypersonic aerodynamic and stagnation-

point heat transfer performance of Earth-entry heat shields for manned and unmanned missions. Objective

functions comprise maximizing cross range, minimizing heat flux, and minimizing heat load. Each considered heat-

shield geometry is composed of an axial profile tailored to fit a base cross section. Axial profiles consist of spherical

segments, spherically blunted cones, and power law geometries. Heat-shield cross sections include oblate and prolate

ellipses, rounded-edge parallelograms, and blendings of the two. Aerothermodynamicmodels are based onmodified

Newtonian impact theory with semiempirical correlations for convection and radiation. Entry velocities of 11 and

15 km=s are used to simulate atmospheric entry conditions at lunar and Mars return conditions, respectively.

Results indicate that skip trajectories allow for vehicles with a low lift-to-drag ratio of 0.25 to achieve 1000 km cross

range, a factor of 4 increase in capability over direct entry. For 11 km=s entry with a 6g deceleration limit, the

spherical segment provides optimal performance. For 15 km=s entry with a 12g deceleration limit, the spherically

blunted cone produces an 8% lower heat flux when compared with spherical segments with similar aerodynamic

characteristics. This result is attributed to the blunted cone’s smaller shock standoff, which reduces the heat load

generated by thermal radiation, the dominant heat transfer mode.

Nomenclature

A = coefficient of power lawBC = ballistic coefficient, kg=m2

b = exponent of power lawC = aerodynamic coefficientD = drag, Nd = diameter, mE = total emitted power density, J=�m3 � s�e = eccentricitygw = ratio of wall enthalpy to total enthalpyg1, g2, g3 = coefficients and exponents [Eq. (6)]H = exponent [Eq. (7)]h = length along y direction, mht = altitude, kmj = semimajor axis length of a blunt body, mjmax = maximum mesh points in x directionk = semiminor axis length of a blunt body, mkmax = maximum mesh points in � directionL = lift, Nl = length along x direction, mM1 = freestream Mach numberm = mass, kgm1 = number of sides of the superellipsen̂ = normal unit vector, away from surfacencrew = crew number, person

nmax = peak deceleration load, Earth gn1, n2, n3 = superelliptic parameterspxrs = cross range, kmQ = heat load, kJ=cm2

q = heat flux,W=cm2

q1 = freestream dynamic pressure, Par = base radius, mrn = nose radius of blunted cone, mrs = radius of curvature of spherical segment, mS = area of base cross section, m2

t = time, std = total mission duration, daysVPR = pressurized volume, m3

v1, v2 = superellipse parametersV1 = freestream velocity, m=sx, y, z = coordinate values, m� = angle of attack, �

� = sideslip angle, �

� = trajectory flight-path angle; positive pointing awayfrom planet, �

�so = shock-standoff distance, m" = edge tangency angle, �

�v = volumetric efficiency�c = half-cone angle, �

�s = half-spherical segment angle, �

� = density, kg=m3

�2=�1 = normal-shock density ratio� = sweep angle, rad�b = bank angle, �

Subscripts

A = axial force, Nb = basecg = center of gravityconv = convectiveD = drag, NEI = entry interface, km (for Earth, ht is 122 km)eff = effectiveEV = entry vehiclef = final

Received 4December 2008; revision received 4August 2011; accepted forpublication 4August 2011. Copyright © 2011 by theUniversity ofMaryland.Published by the American Institute of Aeronautics and Astronautics, Inc.,with permission. Copies of this paper may be made for personal or internaluse, on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 0022-4650/12 and $10.00 in correspondence with the CCC.

∗Graduate Research Assistant, Department of Aerospace Engineering,3181 Glenn L. Martin Hall; currently Analyst, Johns Hopkins University,Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, Maryland20723; Joshua.Johnson@jhuapl.edu. Member AIAA.

†Professor, Department of Aerospace Engineering, 3181 Glenn L. MartinHall; lewis@umd.edu. Fellow AIAA.

‡Assistant Professor, Aerospace Engineering Sciences Department, 429UCB; rstarkey@colorado.edu. Associate Fellow AIAA.

JOURNAL OF SPACECRAFT AND ROCKETS

Vol. 49, No. 1, January–February 2012

38

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65

H = horizontal component relative to orbital planeHS = heat shieldL = lift, Nl, m, n = rolling, pitching, yawing moments, N �mmax = point of maximum heat transferN = normal force, Np = pressurerad = radiatives = stagnation pointsl = sea leveltot = totalV = vertical component relative to orbital planeY = side force, N� = derivative with respect to �, =rad� = derivative with respect to �, =rad

I. Introduction

V EHICLES returning to Earth from the moon or Mars mustsurvive an extreme hypersonic environment during atmos-

pheric entry. This involves entering Earth’s atmosphere at super-orbital velocities ranging from 10 to 15 km=s with correspondingMach numbers from 30 to 50 [1] while withstanding 3000 K�temperatures. The forward heat shield, which protects the vehicle, isthe primary contributor to the vehicle’s aerothermodynamic perfor-mance, i.e., the aerodynamic forces, moments, and heat transfer [2].The rest of the vehicle experiences significantly lower heatfluxes andregions of extremely low pressure due to flow separation.

Optimizing aerodynamic and heat transfer performance areconflicting objectives. To reduce surface heating and maximizethermal energy transferred to the surrounding environment, vehiclesdesigned for hypervelocity entry have typically been blunt-bodydesigns that limit entry aerodynamic performance [3]. Conse-quently, an aerothermodynamic balance must be achieved to satisfymission requirements without exceeding material technologyconstraints.

In this work, aerothermodynamic performance is represented bycrossrange capability and stagnation-point heating. Crossrangecapability enables missions that require diverting from the vehicle’sorbital ground track. It also allows an entry vehicle to execute coursecorrections to counter offnominal atmospheric conditions and switchlanding sites [4]. Stagnation-point heat transfer, although it may notbe the point of maximum heating, is a good relative measure of theexpected high heating over the heat shield. From an overallperspective, both are directly associated with mission requirementsand material constraints. This work seeks to find optimal blunt-bodyheat-shield designs from the standpoints of cross range andstagnation-point heat transfer.A complete trajectorymodelwith low-order aerothermodynamic models is applied to balance the need forfidelity with the desire to have practical computational times,allowing the optimizer to consider a wide range of heat-shieldgeometries.

Previous work has primarily emphasized heat shields based onspherical and conic shapes: the �25� spherical segment (SS) fromProject Apollo [5–8], and the 70� spherically blunted cone (SC) fromthe Mars Viking [9–11] missions. Although these shapes have beenused over the past 40 years, it is unknown whether either providesoptimal aerothermodynamic performance for lunar and Mars returnmissions. Properly broadening the design space would allow theoptimizer to determine which geometric features improve perform-ance. Recent work completed by the authors [12] introduced non-spherical designs consisting of elliptical and polygonal base crosssections. Single design-condition optimization indicated improve-ments in aerodynamic performance by generating oblate heat shieldswith parallelogram cross sections, increasing the hypersonic lift-to-drag ratio significantly. In general, a larger lift-to-drag ratio increasescrossrange capability [4]. For both spherical and nonspherical heat-shield designs, the present work extends the recent work byincorporating entry trajectory analysis, thus allowing the optimizer toaccount for peak heat flux, total heat load, cross range, anddeceleration loads.

To produce optimal tradeoff relationships between performanceparameters, this work uses a population-based multiobjectiveoptimization scheme, in which a differential evolutionary algorithmis employed to optimize cross range and heat transfer simultaneously.The optimized trajectories may be located on the bounds of thefeasible design space and likewise be sensitive to small deviations inbank angle. Thus, the trajectories themselves may not be flyableduring a real mission but are reported to recognize optimal perfor-mance boundaries for each heat-shield/trajectory configuration.Contributions of viscous shear forces and turbulence are not con-sidered. The overall level of analysis presented in this work isappropriate for assessment of the trade space at the conceptual designlevel for heat-shield geometries.

II. Methodology

A. Mission Profile

To simulate Earth entry for lunar return, an initial entry velocity of11 km=s is applied [13]. For Mars return, a fast 180 day returnrequires entry velocities up to 14:7 km=s [14], and an initial entryvelocity of 15 km=s is applied. Although the hypersonic aero-dynamics at these two velocities are similar for a given heat-shielddesign, their heating environments are greatly different. Whileconvection typically dominates in heat load for a vehicle entering at11 km=s, radiation is projected to be the primary heat transfer modefor 15 km=s. The mission profile for the Orion crew explorationvehicle (CEV)with an overall duration of 18 days, a crew of four, anda pressurized volume of 5 m3=person is applied [15]. The Earth-entry simulation begins at the atmospheric entry interface, at analtitude of 122 km, and terminates after the freestreamMach numberbecomes less thanfive and accounts for the hypersonic aerodynamicsonly. For blunt-bodied capsules (low L=D), whether the trajectoryends at M1 < 5 does not strongly affect the values of the threeoptimization parameters in this analysis: cross range pxrs, peakstagnation-point heat flux qs;max, and total stagnation-point heat loadQs;tot. This mission profile is used for both lunar and Mars return.

B. Heat-Shield Geometries

The heat-shield geometry is defined by two contours: the basecross section of the heat shield and the axial profile that is swept aboutthe central axis and is tailored to fit the base cross section. Foroptimization, the ideal equation defining the geometry wouldproduce a wide, continuous range of cross sections. One suchequation is available using the superformula [16] of the superellipse.It defines the cross-section radius for 0 � � � 2�:

r��� ������cos�

14m1��v1

����n2

�����sin�

14m1��v2

����n3�1=n1

(1)

in which m1 corresponds to the number of sides of a polygonv1 � v2 � 1, n1 and n2 are modifiers, and n3 � n2 to render sharp orrounded-edge polygons. Increasing n2 transforms a polygon(n2 < 2) into an ellipse (n2 � 2) and then into a rounded-edgeconcave polygon (n2 > 2). Example cross sections are shown inFig. 1, and ranges of values for parametersm1, n1, and n2 are listed inTable 1. This work uses n2 � 2:0 to avoid regions of high heat fluxgenerated by corner flow that may be present around concavegeometries. Note that an ellipse is generated by Eq. (1) with n2 � 2independent of n1 and the chosen type of polygon. For any n2, oblate(e < 0) and prolate eccentricity (e > 0) are considered. Once thecross section is chosen, the heat shield’s axial profile, which is theshape that protrudes from its base, is selected. Three axial profiles areconsidered: the SS, the SC, and the power law (PL). To generate aheat-shield geometry, the axial profile is rendered and then swept360� about the central body axis using Eq. (1), based on the selectedcross-section contour.

Figures 2 and 3 show the SS and SC geometries, respectively. ThePL formula y� Axb offers axial profiles with a wide range ofbluntness being controlled by coefficientA and exponentb, as shown

JOHNSON, LEWIS, AND STARKEY 39

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65

in Figs. 4a and 4b, respectively. The fixed-body coordinate systemfor this work is included in Fig. 2. In the fixed-body coordinatesystem, it is convention to combine the three orthogonal forcevectorsCA, CN , and CY into the two aerodynamic force vectors CL and CD.This results invertical and horizontal components of lift (with respectto the z axis) represented by LV and LH, respectively, for the caseswith � ≠ 0. In calculating rolling moment stability, the sign of LVdetermines the direction of the positive rolling moment shown inFig. 2. This allows Cl;cg;� to be negative for all statically roll stableconfigurations. The moment coefficients are defined as the ratio ofthemoment to q1Sj, in which j is the length of the semimajor axis ofthe base cross section. Additional details of both the static stabilityand the heat-shield shapes in this work are given in [17].

C. Entry Vehicle Mass Estimation

Mass estimation and scaling of the entry vehicle are based on themission profile, heat-shield geometry, and dimension requirementsfor incorporating the crew or payload. Vehicle scaling is necessarydue to the wide range of heat shields in the design space. High �s, forexample, can allow the heat shield to encompass part of or the entirepressurized volume. If the entry vehicle geometry is assumed to havethe same base height hb (top to bottom) as the Orion with 5 m, thenheat shields with base cross sections of high eccentricity would haveover 15 timesOrion’s volume.To scale the entry vehicle, a heat shieldis categorized into one of four cases based on lHS. This differentiatesthe procedures applied for estimating the pressurized volume basedon the heat shield’s geometry. The scaled vehicle must also satisfyrequired crew seating dimensions. It is assumed that the seat dimen-sions required for a suited astronaut consist of a top-to-bottom heightof 1.4 m, a width of 0.7 m, and a depth of 1.1 m.

Once the pressurized volume is closely matched and seatdimensions are satisfied, the entry vehiclemass is estimated based onthe following empirical correlation [18]:

mEV � 592�ncrewtdVPR�0:346 (2)

Based on themission profile, Orion’s estimatedmass of 7330 kg iswithin 1% of the landingmass circa 2006 reported in the early designphase [15]. The entry vehicle masses for this analysis are nearlyconstant since heat-shield scaling is designed to produce a pres-surized volume that meets the mission requirements as closely aspossible. Details on the method of calculation are documented in[19]. This mass estimate is independent of heat load, whichdetermines the heat-shield material’s thickness in a detailed designanalysis. Several new heat-shield designs are considered; thus, theheat load for a given vehicle and flight path is unknown a priori. Therequired iterative process, which would increase the computationtime by a few factors, has not been integrated into the optimizationsetup. Some of the new heat-shield designs would requireunshrouded launch due to their size. The presence of a shroud is anaerodynamic requirement on some launch vehicle designs, but therehave been reentry vehicles launched unshrouded on successful testflights including the five aerothermodynamic elastic structuralsystems environmental tests from 1963–1965 [20]. In this program,the aerothermodynamic structural test vehiclewas attached to the topof a Thor launch vehicle. For unshrouded launch, there could be areasof the reentry vehicle that are protected by the heat shield duringreentry, but not during the launch phase, and may require additionalmaterial and weight for thermal protection.

Uniform density has been assumed here to calculate the center ofgravity location of the heat shield, and the prescribed Xcg has beenarbitrarily modified to equal 75% of the uniform density value.Bringing the Xcg forward increases the feasible design space byallowing more slender blunt bodies with higher L=D to belongitudinally statically stable. Using the uniform density assump-tion is a limitation, as it does not account for the specific placement ofsubsystems within the reentry vehicle geometry. In reentry vehicledesign, the placement of the center of gravity is important andrepresents a significant challenge. In some cases, ballasting orredistributing mass may be needed to achieve the desired placement.In reality, reentry vehicle conceptual design may generate asignificantly different center of gravity than the uniform densityassumption used in this heat-shield shape analysis.

Fig. 1 Examples of base cross sections.

Table 1 Superformula parameters forrounded-edge polygonsa (n3 � n2)

m1 n1

4 1.005 1.756 2.307 3.208 4.009 5.5010 7.00

a1:3 � n2 � 2:0

Yawing

x

z

φ θs

rs

y

Rolling

CL,V > 0

CL,V < 0 Pitching

Fig. 2 SS, �s � 60�, n2 � n3 � 2.

rn d

l

θc

Fig. 3 SC axial profile, rn=d� 0:25, �c � 60�.

40 JOHNSON, LEWIS, AND STARKEY

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65

D. Trajectory Modeling and Validation

The University of Maryland Parallel Trajectory OptimizationProgram (UPTOP) is applied for a three-degree-of-freedom (3-DOF)entry trajectory analysis [21,22]. Its capability covers not onlytrajectory optimization but also heat-shield shape optimization,which is discussed in Sec. II.F. UPTOP uses a fourth-order Runge–Kutta routine to propagate the point-mass equations of motion forrigid-body flight in a vertical plane [21–23]. The time step isnormally set to 1 s. A rotating, ellipsoidal Earthmodel is appliedwitha second harmonic gravity model J2 based on the WGS-84Geocentric Equipotential Ellipsoid model [24]. The U.S. 1976Standard Atmosphere [25] is applied for ht < 85 km, and theNRLMSISEE-00 Atmosphere [26] is applied for ht 85 km.

Trajectories generated with UPTOP are compared with those ofthe benchmark Program to Optimize Simulated Trajectories (POST)[27] in Fig. 5.UPTOP is capable of optimizingmultistage trajectorieswhere the vehicle may have multiple engines and fuel tanks. Thebenchmark case [28] for the optimal space shuttle transport ascenttrajectory through space shuttle main engine cutoff is provided inFig. 5a to demonstrateUPTOP’s comparable optimization capability.The optimal pitch and altitude profiles generated by UPTOP closelymatch POST’s optimal profiles. Additionally, the results from POSTas calculated by UPTOP match the POST profiles. Validation forEarth entry from lunar return atVE � 11 km=s is given in Fig. 5b. Forthe given bank angle profile, which rotates the lift vector, the skiptrajectory generated in UPTOP matched POST’s and illustratesUPTOP’s suitability for high-velocity entry applications. The 3-DOFentry analysis assumes that the reentry vehicles are dynamicallystable, and a higher-order analysis would be required to determinewhere in the design space this is and is not the case.

E. Aerothermodynamics

The primary physical mechanisms that contribute to the hyper-sonic aerothermodynamics, i.e., the aerodynamic forces, moments,and heat transfer [2], in this work consist of 1) the surface pressuredistribution, 2) the velocity gradient along the heat shield, and 3) theradiating shock layer. The local bow shock strength, imposed on thevehicle by freestream conditions, strongly affects both the surfacepressure distribution and the resulting heat transfer along the heatshield. The surface pressure distribution produces a velocity gradientalong the heat shield and sets the velocity at the edge of the boundarylayer. The resulting boundary-layer velocity gradient results inconvective heat transfer at the surface of the heat shield. Thethickness of the high-temperature shock layer influences the thermalradiative heat flux. Low-order methods have been applied todetermine the effects of these physical mechanisms on the aero-thermodynamic performance.

1. Aerodynamics

Modified Newtonian flow theory is applied to produce thehypersonic surface pressure distribution about a blunt-body heat-shield design. Thismethod assumes that the freestreamflow’s normalmomentum is destroyed upon impact with the vehicle and that thepressure is negligible on the portion of the vehicle not facing thefreestream [29]. The expression for themodifiedNewtonian pressurecoefficient is

Cp � Cp;max�V1 � n̂=V1�2 (3)

for V1 � n̂ < 0, and Cp � 0 for V1 � n̂ 0, which represents theregion in aerodynamic shadow. The maximum pressure coefficient

Fig. 4 PL axial profile.

Fig. 5 Trajectory validation of UPTOP results with POST.

JOHNSON, LEWIS, AND STARKEY 41

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65

occurs at the stagnation point, and it is calculated according to theRayleigh pitot tube formula [29]. Figure 6 presents Newtonianresults plotted against wind-tunnel data of the Apollo commandmodule [30]; note the excellent agreement. An in-depth validation ofmodified Newtonian flow is provided in [17] and indicates thatresults are generally within 10% for aerodynamic coefficients forhigh Earth-entry velocities from both low Earth orbit and the moon.

2. Convective Heat Transfer

The properties of the high-temperature shock layer affect bothconvection and radiation due to the presence of dissociated andpartially ionized air. Additionally, the heat-shield geometry directlyaffects the surface pressure distribution, and thus the velocitygradient along the heat shield at the edge of the boundary layer. Asmaller local radius of curvature increases the velocity gradient,thereby increasing the local convective heat flux [3]. Low-ordercorrelations based on empirical data account for these upstreameffects with a local radius-of-curvature term. The stagnation-pointconvective heat flux has been correlated [31] to nose radius andfreestream conditions:

qs;conv � �1:83 � 108�r0:5n �1 gw��0:51 V31 (4)

in which this work assumes gw � 1.

3. Radiative Heat Transfer

For a given set of freestream conditions and a shock layer withemitted power densityE, qs;rad will be greater for the heat shield withthe larger �so [3]. Empirical results [32] indicate that the �so for asphere is directly proportional to its radius as a function of thenormal-shock density ratio. Kaattari’s semiempirical method [33,34]is used to estimate�so for zero and nonzero �. This solution appliesrelationships for shock-standoff and shock surface inclinations nearthe sonic point for elliptical bodies at specific heat ratios from 1.0 to1.4 for a range of high normal-shock density ratios. For the presentwork, it is assumed that the effective radius (reff) for a given heatshield is equal to the radius of that particular sphere that maintains anequivalent shock-standoff distance at the stagnation point. Ried et al.[32] offers an empirical curve fit that renders an acceptableapproximation:

�so=reff ��

��2=�1� 1�2

��2=�1� ����������������������������2�2=�1� 1

p 1

�1(5)

In this way, the �so is incorporated in the low-order method bymeans of reff . As a result, this effective radius term is different fromthe radius-of-curvature term applied for convection. Two qs;radcorrelations are applied over a range of freestream velocities. ForV1 < 9000 m=s, the correlation applies the following form:

qs;rad � reffg1�3:28084 � 104V1�g2��1�sl

�g3

(6)

in which g1 � 372:6, g2 � 8:5, and g3 � 1:6 from [35] forV1 < 7620 m=s, and g1 � 25:34, g2 � 12:5, and g3 � 1:78 from[36] for velocities 7620 to 9000 m=s. For velocities above9000 m=s, Tauber and Sutton [37] apply

qs;rad � 4:736 � 104rHeff�1:221 f�V1� (7)

in which

H � 1:072 � 106V1:881 �0:3251

and

f�V1�

��3:93206793�1012V4

1�1:61370008�107V312:43598601�103V2

1�16:1078691V139;494:8753 9000�V1�11;500m=s1:00233100�1012V4

1�4:89774670�108V318:42982517�104V2

1�6:25525796V117;168:3333 11;500<V1�16;000m=s

Thermochemical equilibrium is assumed. The curve-fit equation forf�V1� has a high number of significant figures in order to have lessthan 2% error with the published tabulated values [37].

The low-order heat transfer methods have been validated againstApollo 4 and FIRE II flight data. In comparison with the flightinvestigation of the reentry environment (FIRE) II calorimeter data[38] shown in Fig. 7, the low-order code overpredicts the peak totalheat flux (radiative and convective) by 7%. Table 2 indicatespredictions of maximum total heat transfer within 9.2% of reportedvalues. Further descriptions of the method and validation areprovided in [12]. All of these heat transfer correlations are limited tothe stagnation area, and while this is appropriate for this level ofanalysis, this work does not account for acreage heating, which could

Fig. 6 Modified Newtonian validation with Apollo wind-tunneldata [30].

Fig. 7 FIRE II qs;tot with flight data [38].

Table 2 Apollo 4 comparison of total heat transfer [12]

Parameters Apollo 4, [8] Results, % error

qmax;tot,W=cm2 480 470 (2:1%)

Qmax;tot, kJ=cm2 42.6 38.7 (9:2%)

42 JOHNSON, LEWIS, AND STARKEY

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65

have a significant impact on thermal protection system (TPS) designand requirements.

F. Optimization Setup

UPTOP’s flexible framework allows for an external code toprovide updated aerothermodynamics for a heat-shield designthroughout the trajectory calculation. The aerothermodynamicmodels have been integrated into the UPTOP setup to perform heat-shield shape optimization.

1. Optimization Method

Previous work [12] applied the modified method of feasibledirections gradient-based method to optimize over the geometricdesign space (without trajectory analysis) for a single objective.There were numerous local optima; over 200 runs were required tolocate the global optima for four objective functions. For the presentwork, a more robust and global search algorithm is required toaccount for both the additional complexity of multiobjective opti-mization and the broader design space with trajectory analysis.UPTOP applies a differential evolutionary scheme (DES) [21,39] forthis optimization. As an evolutionary algorithm,DESbases its searchfor an optimum on nature’s evolutionary principles [40]. It begins byrandomly selecting an initial population of designs, and 100s ofiterations are required to settle on an optimal solution. Each heat-shield design, known as an individual in a population of designs, isevolved throughout each iteration with other individuals based onmutation intensity and crossover parameters. Details on DES and itsparameters are provided in [39,41].

2. Objective Functions

Three objective functions are applied in this work: minimizingstagnation-point heat load Qs;tot, minimizing peak stagnation-pointheat flux qs;max, and maximizing cross range pxrs. These objectivefunctions have been selected on the basis of 1) relevance to missionrequirements, 2) connection to low-performing or restrictivecapabilities of existing blunt-body designs, and 3) the availability ofaccurate physical models suitable for optimization purposes.

The peak heat flux of the trajectory determines whichmaterials arecapable of surviving the selected entry conditions. Minimizing heatload reduces the heat shield’s thickness and mass indirectly.Minimizing them together requires the capability to fly hundreds ofentry trajectories and to calculate the heat flux along all of thosetrajectories. As a result, low-order computational models of theaerothermodynamics are implemented to balance the need forfidelity with the desire to have practical computational times. Heattransfer is tracked at the stagnation point. There ismore confidence inwell-validated stagnation-point correlations than in low-orderestimates of the maximum heat flux, especially when applied to awide range of geometries in extreme hypersonic conditions. Theyalso cost less computational time. In many cases, the stagnation-point heat flux is not representative of the heat flux experienced overthe entire heat shield, which is commonly lower, but its value isusually on the order of the maximum heat flux.

Crossrange capability can enable additional landing options.Existing designs such as the �25� SS and the 70� SC have lowcrossrange performance due to trajectory design and low L=D. For avehicle flying a direct entry trajectory, lunar return with a hypersonicL=D� 0:30, the maximum cross range is limited to �200 kmassuming a 5g constraint [1]. To increase crossrange capability, bothskip trajectories, which have been shown to increase cross rangesignificantly [1], and higher L=D designs are considered feasible.

3. Multiobjective Optimization

In single-objective optimization, the one optimal or nondominatedsolution is better than all other solutions. In multiobjective optimi-zation, two ormore objective functions are optimized simultaneouslyto produce a set of optimal or nondominated solutions known as thePareto frontier. When two objective functions are optimizedsimultaneously, a Pareto frontier has the form of a curve that

represents the optimal tradeoff between the two objectives. Shown inFig. 8, the results of minimizing Qs;tot and qs;max are providedtogether. This Pareto frontier is composed of those solutions in thefeasible population that are not dominated with respect to bothobjective functions; each point on the frontier represents an optimalsolution. In general, the Pareto frontier is a set of nondominatedsolutions, in which one solution is better than another with respect toat least one objective, but not all objectives [40].

Multiobjective optimization is used to optimize conflictingobjectives. Since an increase in cross range produces a larger heatload, maximizing cross range and minimizing heat load areconflicting objectives. Nonoptimal results may produce higher heatloads than necessary for a desired cross range. Minimizing heat loadandminimizing peak heatflux are also conflicting objectives. For thiswork, optimal solutions are provided in the form of Pareto frontiersbetween two objectives to highlight performance tradeoffs andprovide comparisons among axial profiles. For entry velocities of 11and 15 km=s, optimization is performed using two objectivefunction sets: 1) maximizing pxrs and minimizing Qs;tot and2) minimizing qs;max and Qs;tot.

The authors conducted a parametric study of the effects ofpopulation size, crossover probability, and mutation intensity on thePareto frontier for maximizing pxrs and minimizing Qs;tot in [42]. Itwas found that themost comprehensivePareto frontier for this designspace is produced with a crossover probability of 0.8 and a mutationintensity that is randomly varied for each generation between 0 and 1.An initial population of 390 individuals is applied since diminishingreturns were observed with increasing population. Twelve AMD2.2 GHz Opteron 248 processors were used in this analysis, and thetime required per run was approximately 10 h.

4. Design Variables

For each axial profile, the design variables along with their sideconstraints are listed in Table 3. For �s, the lower limit of 5� providesa blunt body that has a large but finite radius of curvature. For both �sand �c, the upper limit 89� removes numerical issues present if theupper limit is set to 90�. For the blunted cone, the upper limit of rn=dis chosen to provide overlap and continuity between the blunted coneand SS design spaces. For the PL, Newtonian impact theory mayhave an accuracy issue given the quick slope changes shown inFig. 4b for b� 0:1 or smaller; a lower limit value of 0.2 for b hasbeen chosen arbitrarily. The upper limit of b excludes the PL profileswith sharper nose regions. The arbitrary upper limit for A has beenchosen based on its similarity to the SS near this value. In generatingthe base cross section, the lower limit of n2 produces slightlyrounded-edge polygons. A maximum eccentricity of �0:968 waschosen to limit the axes ratio j=k to four. Additional reasons for thechosen side constraints that are geometry-related have been detailedin [12]. A mesh convergence study was completed in [19] to reduce

Fig. 8 Multiobjective function population with Pareto frontier, SS,

VEI � 15 km=s.

JOHNSON, LEWIS, AND STARKEY 43

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65

computational time; for optimization, the geometry’s mesh isjmax � 45, kmax � 101.

Awide range of �E including the Apollo [8] missions’ �E �6:5�is allowed. Tomodify the vehicle’s flight path through rotation of thelift vector, a bank angle profile with five control points is available.The optimizer can modify the five bank angles �b;0 through �b;4, aswell as the three intermediate times at which the bank angles areinitiated. Connecting the control points �t; �b� produces the �bprofile. A limit of 0 to 180� lowers the size of the design space; angles181 through 359� are not necessary since longitude and latitudeconstraints are not considered.

5. Design Constraints

Boundaries for the feasible design space are provided in Table 4.These constraints account for trajectory design limits, theorylimitations, and static longitudinal, directional, and roll stabilities.Aerodynamic and geometric constraints for this optimization aredetailed in [12]. The requirement for roll static stability changes signwhenCL;V changes sign, as explained in [17], thus requiring a changein the constraint. Note that, for heat shields with " < 15�, an aftbodywith an angle greater than 15� is assumed to allow heat shields withlow �s (for example, �s � 10�) to be feasible for j�j � 15�. Sinceguidance laws are not considered in the trajectory analysis, optimaltrajectories with similar entry interface characteristics but with lesscomplex �b profiles tend to generate longer duration trajectories. Acompromisewas chosen to be 2 h, tf � 7200 s, which is greater thantwice the upper limit to the estimated tf for the Orion CEV.

An arbitrary maximum altitude for skip trajectories has been set to3000 km. A final altitude ht;f no greater than 75 km has been chosenarbitrarily to ensure that the vehicle’s trajectory ends within theatmosphere. This allows for higher L=D vehicle designs to befeasible since their optimal trajectories may result in deceleration athigher altitudes in the atmosphere. ForVEI � 11 km=s, a peak g loadof 6g was chosen since it is the maximum allowable accelerationlevel for a deconditioned astronaut in a reclined position [43]. It isalso lower than the 7g that Apollo 10 experienced [44]. Preliminaryanalysis indicated that, for a 15 km=s entry, this optimization setupwould be overconstrained with a 6g upper limit. This limit wasincreased to 12g based on previous work that indicates a pilot cansustain 12g for up to 60 s and still continue to perform the assignedtasks [45]. Although this is not expected to conform with futurestandards for manned Mars return, the results provide a sense of theheating environment when entering at high hyperbolic velocities.Results for VEI � 15 km=s can be applied at least toward mostunmanned missions.

III. Results

Optimization has been performed using three types of heat-shieldaxial profiles: the SS, the SC, and the PL. A heat shield with aparallelogram-form base cross section (m1 � 4) provides the largesthypersonic L=D [12]. Since greater L=D increases crossrangecapability, this analysis focuses optimization on base cross sectionsof parallelogram and elliptical forms, and blendings of the two. Forinitial entry velocities of 11 and 15 km=s, Pareto frontiers areprovided for two multiobjective function sets: 1) minimizing heatloadQs;tot and maximizing cross range pxrs, and 2) minimizing heatload Qs;tot and heat flux qs;max. Design variable distributions areprovided for selected Pareto frontiers, and specific designs listed inTable 5 are discussed. Axial profiles and base cross sections for thespecific designs selected from the Pareto frontiers are shown inFigs. 9 and 10, respectively. Note that the base cross section fordesign C is circular.

A. Optimal Configurations for VEI � 11 Kilometers per Second

1. Minimizing Qs;tot and Maximizing pxrs

The lowest possible heat load is expected to increase with crossrange pxrs provided that down range is relatively constant orincreasing. A Pareto frontier is given for each type of axial profile inFig. 11a for cross ranges up to 1500 km and heat loads from 11 to33 kJ=cm2. For optimal designs with pxrs of 500, 1000, and1500 km, the values of Qs;tot are 14.7, 22.6, and 29:4 kJ=cm2,respectively. The optimizer produced similar Pareto frontiers for allthree axial profiles. The PL Pareto frontier is expected to be the leastaccurate since an artificial effective nose radius is applied. All threefrontiers closelymatch forpxrs > 750 km; close inspection indicatesthat, for this region, the SC and PL profiles are disguised SSs.

Figure 12 shows the design variable distribution for the SS Paretofrontier shown in Fig. 11a. The transformation variable n2 is nearlyconstant at 2.0, indicating an elliptical cross section rather than aparallelogram form is optimal for this set of pxrs. These resultsindicate that for low L=D designs, an elliptical cross section is betterdue to its larger drag area (CDS�D=q1), resulting in a lowerBC fora given mEV. For pxrs � 700 km, nondominated or optimal designshave highly oblate e�0:968, which is the lowest allowed value.With an increased heat-shield radius of curvature, this design allowsfor less convective heat transfer.Higher e also increases the drag area,thus decreasingBC. The nondominated heat-shield geometry is heldconstant by the optimizer untilpxrs � 700 km, at which point there isa jump in �s from 6.8 to 18�. In general, j�j is increasing throughoutthis portion of the Pareto frontier, indicating higher jL=Dj is requiredto produce additional cross range. At pxrs � 700 km, j�j � 16�, andthe geometric constraint j�j � j"� 1�j is active. The parametricanalysis in [17] (figure 19) indicates that, for specific combinations offixed � and e (i.e., �� 20�, e� 0), an increase in �s decreases themagnitude jL=Dj. Design A is in a similar part of the design space.Since the j�j is fixed at 16� unless �s > 15� and the optimizer isattempting to increase jL=Dj, it determines that a decrease in e alongwith an increase �s that allows for j�j> 16� would result in asufficient and incremental increase in jL=Dj. The increase in �s to18� limits the reduction in radius of curvature, thus concurrentlyminimizing Qs;tot. These necessary adjustments to �s, e, and �produce the sudden rise in Qs;tot on the Pareto frontier atpxrs � 700 km. Consequently, drag area is traded off with jL=Dj as

Table 3 Design variables with side constraints

Axial profile Profile-specific design variables Common design variables

SS 5:0� � �s � 89:0� 30� � � � 30�

0:968 � e � 0:96815:5� � �E � 0:05�

0� � �b;0 � 180�

SC 55:0� � �c � 89:0�

0:15 � rn=d � 2:001:30 � n2 � 2:00

t0 � 5 s � t1 � 7190 s0� � �b;1 � 180�

0� � �b;2 � 180�

PL 0:900 � A � 10:0000:200 � b � 0:650

t1 � 5 s � t2 � 7190 st2 � 5 s � t3 � 7190 s

0� � �b;3 � 180�

0� � �b;4 � 180�

Table 4 Trajectory and aerodynamic constraints

Optimization constraints

Trajectory Aerodynamic/geometric

tf � 7200 s M1;f � 5ht � 3000 km Cm;cg;� � 0:001ht;f � 75 km Cn;cg;� 0:001

nmax � 6g, VEI � 11 km=s sign�CL;V�Cl;cg;� � 0:01nmax � 12g, VEI � 15 km=s j�j � j"� 1�j

44 JOHNSON, LEWIS, AND STARKEY

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65

pxrs increases. Similar behavior is noted for the other axial profilesand not shown here.

Forpxrs � 250 km, optimal solutions use direct entry trajectories.To increase pxrs, the banked lift vector must turn the vehicle further.This is achieved by steepening �E to travel deeper in the atmospherefor a givenV1, increasing q1 and thus, the lift dedicated for turning.As shown in Fig. 12b,�b:0 is adjusted, pointing the lift vector slightlyupward, to keep the vehicle slightly higher in the atmosphere forminimizing Qs;tot. The change in behavior of �E and �b at pxrs �250 km indicates a switch from direct entry trajectories to skiptrajectories. This crossrange limit for direct entry trajectories isconsistent with the work of Putnam et al. [1]. For a skip trajectory, asteeper �E is used to dissipate sufficient energy to avoid skipping outof the atmosphere. Minimizing heat load restricts low j�Ej andtrajectory duration, while maximizing pxrs and the deceleration limitrestricts high j�Ej. For skip trajectories, larger pxrs requires �b toapproach 90�. Since there is a smaller proportion of lift dedicated toproducing the skip as �b approaches 90�, the vehicle requires asteeper �E, as indicated by Fig. 12b.

As seen in Fig. 11a, designs A, B, and C are optimal forpxrs � 520, 1010, and 1500 km, respectively. Design A representsthe SS geometry applied for pxrs � 700 km. Its heat-shield axialprofile is illustrated in Fig. 9, and its base cross section is illustrated inFig. 10a. The results shown in Table 5 indicate that, by usingdesign B, a 54% increase in Qs;tot is required to achieve doubledesign A’s cross range. Not only does qs;max affect Qs;tot but so doesthe change in individual contributions from convection and radiation.By halving the radius of curvature of design B from 6.3 to 3.15 musing �c � 60:4�, e�0:682, and rn=d� 0:615, the qs;max isapproximately unaffected with qs;conv � 140 W=cm2 and qs;rad�160 W=cm2. However, Qs;conv increases by 40% while Qs;rad

decreases by 20%. As a result,Qs;tot increases by 20%. For a designsimilar to the Viking’s SC, �c � 70�, e�0:682, rn=d� 0:25, and

Table 5 Specific optimal designs from two multiobjective function optimizations, m1 � 4a

Minimizing Qs;tot and maximizing pxrs Minimizing Qs;tot and qs;max

VE � 11 km=s (Fig. 11a) VE � 15 km=s(Fig. 14a)

VE � 11 km=s(Fig. 11b)

VE � 15 km=s (Fig. 14b)

Design variables A B C E F D G H

SS SC PL SS SC SC SS SCAxial profile �s � 6:80� �c � 60:4�

rn=d� 1:26b� 0:34A� 5:25

�s � 8:1� �c � 84:3�

rn=d� 1:29�c � 84:4�

rn=d� 2:00�s � 10:2� �c � 84:3�

rn=d� 1:30Base cross section n2 � 1:99

e�0:968n2 � 2:00e�0:682

n2 � 1:96e�0:003

n2 � 1:98e�0:968

n2 � 2:00e�0:968

n2 � 2:00e�0:968

n2 � 2:00e�0:968

n2 � 2:00e�0:968

Trajectory�E�t0; �b;0�. . .�tf; �b;f�

��13:7�6:01�

(0 s, 59.0�)(1440, 76.0�)

��23:8�6:14�

(0 s, 75.9�)(1530, 55.1�)

�� 30:0�

6:29�(0 s, 97.9�)

(1540, 135.7�)

�� 14:8�

6:60�(0 s, 43.7�)

(248.4, 161.5�)(1450, 137.4�)

��15:9�6:44�

(0 s, 143.5�)(190, 40.7�)(870, 84.5�)(1214, 85.8�)

��15:8�5:37�

(0 s, 144.0�)(267, 66.1�)

�� 9:00�

6:41�(0 s, 1.93�)(219, 6.14�)

��15:9�6:18�

(0 s, 177.0�)(197, 84.0�)

Parameters Skip trajectories Direct trajectories

nmax, g 6.0 5.9 6.0 12.0 11.8 5.9 11.9 11.7Qs;tot, kJ=cm

2

�Qs;conv; Qs;rad�14.7

(7.1, 7.6)22.6

(14.9, 7.7)29.4

(19.8, 9.6)82.4

(13.6, 68.8)65.2

(18.5, 46.7)12.4

(7.4, 5.0)76.7

(12.9, 63.8)63.6

(16.3, 47.3)qs;max, W=cm

2

�qs;conv; qs;rad�250

(50, 200)300

(100, 200)380

(130, 250)1930

(150, 1780)1400

(200, 1200)160

(60, 100)1500

(140, 1360)1100

(180, 920)pxrs, km 520 1010 1500 990 1000 120 10 100CD 1.62 1.32 1.17 1.57 1.56 1.57 1.60 1.56jL=Dj 0.22 0.36 0.50b 0.22b 0.24 0.24 0.12b 0.24Cm;cg;�; =rad 0:18 0:15 0:10 0:20 0:19 0:19 0:27 0:20BC, kg=m2 130 220 350 130 100 110 120 100hb;HS, m 3.4 5.0 5.0 3.5 4.0 3.8 3.6 4.0S, m2 36.9 27.1 19.0 38.1 49.0 45.8 41.0 49.0�v;HS 38.8% 60.6% 58.5% 43.0% 41.2% 40.5% 49.1% 41.3%tf , s 1440 1530 1540 1450 1214 267 219 197

amEV � 7800 kgbThe sign of L=D is negative for the listed �

Fig. 9 Axial profile designs from Table 5. Fig. 10 Specific base cross sections from Table 5.

JOHNSON, LEWIS, AND STARKEY 45

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65

the qs;max increase is approximately 27% (to 380 W=cm2); at thesame time, the Qs;tot increase is 68% over that for design B’s heat-shield geometry. The results in Fig. 12a indicate that the optimal heat-shield geometry for skip trajectories atVEI � 11 km=s is the SS withnonzero eccentricity. Design C doubles the Qs;tot to triple the pxrsover that of designA. Table 5 showspxrs, qs;max, andQs;tot increasingwith BC as expected.

It is possible for Qs;conv >Qs;rad, even though the peak qs;conv isless than the peak qs;rad. This situation occurs for designs B and Cwhen minimizing Qs;tot. Significant convective heat transfer occursthroughout the entire hypersonic trajectory, while radiative heattransfer contributes significantly only at the highest velocities forV1 7600 m=s. For V1 < 7600 m=s, the qs;rad is less than5 W=cm2.

2. Minimizing Qs;tot and qs;max

Pareto frontiers are provided in Fig. 11b for qs;max ranging from130 to 210 W=cm2, producing heat loads ranging from 11.8 to19:3 kJ=cm2. UPTOP could not generate a Pareto frontier for the PLform that followed behavior similar in form to the SS’s and the SC’s,although the effective nose radius of the geometry was notsignificantly different. As a result, the output data of the PL formwere found to be inconclusive. This may be due to the lack of amathematical relationship that is sufficiently accurate at determiningthe effective nose radius for a given design with a PL axial profile.The minimum Qs;tot decreases with increasing qs;max, as expected.The trajectory designvariable distribution for the SCgiven in Fig. 13,together with Fig. 11b, demonstrates that a shallower �E yields asmaller qs;max and largerQs;tot. The optimal geometric configurationsare similar to those with low Qs;tot in Fig. 11a that use a direct entry

approach, as suggested by the shallower �E and higher �b, than thosereported in Fig. 12b.

Both the SS and SC geometries are relatively constant throughoutthe Pareto frontiers. The SS geometry is �s � 6:83�, n2 � 2:00,e�0:968, and hb;HS � 3:2 m, and the blunted cone geometry islisted in Table 5 as design D. The two Pareto frontiers are within35 W=cm2 of each other and within the calculated correlationuncertainty. For altitudes from 66 to 72 km, stagnation-pointconvective and radiative heat transfer calculations were judgedagainst FIRE II flight data from [46] and high-order computationalresults from [37]. The uncertainty of peak qs;conv is �20 W=cm2,

Fig. 11 Pareto frontiers for Earth entry, VEI � 11 km=s.

Fig. 12 Design variable distribution for SS designs from Fig. 11a.

Fig. 13 Trajectory design variable distribution for SC designs from

Fig. 11b.

46 JOHNSON, LEWIS, AND STARKEY

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65

while the uncertainty of the peak qs;rad is �20 W=cm2 forVEI � 11 km=s, and�60 W=cm2 forVE � 15 km=s. Differences inqs within these uncertainties were not considered discriminators forthe selection of specific optimal designs.

Both configurations, the SS detailed above and SC design D, areequally optimal. Comparing the two geometries at the freestreamconditions of qs;max for design D, at ht � 73:1 km, and V1�10:5 km=s, the SS would produce a different combination of heatfluxes qs;conv � 40 W=cm2 and qs;rad � 140 W=cm2 (comparedwith 60 and 100, respectively, for design D, as shown in Table 5) butresult in nearly the same Qs;tot, although the radius of curvature is14.3 m for the SS and 7.5 m for SC. In this problem setup, a highlyeccentric base has an increase in drag area, which reduces BC, andthus indirectly reduces qs;max and Qs;tot since larger drag areaprovides deceleration at higher altitudes for a given mEV. However,the skip entry of design B has a 132% increase in time for which qsexceeds 5 W=cm2 over that for the direct entry of designD. This fact,together with a 0.77� steeper �E, makes radius of curvature assumegreater importance for skip entry.

B. Optimal Configurations for VEI � 15 Kilometers per Second

1. Minimizing Qs;tot and Maximizing pxrs

Entry at VEI � 15 km=s represents a kinetic energy level 85%greater than entry atVEI � 11 km=s, resulting in aQs;tot at least threetimes greater. Radiative heat transfer produces amajority ofQs;tot andcan be minimized by decreasing the radius of curvature to reduce�so, which is smaller for a SC than a SS for a given L=Djmax design.The different thermal environment is expected to result in differentoptimal configurations at 15 km=s than were found at 11 km=s.

Pareto frontiers are shown in Fig. 14a forpxrs � 2200 kmwithQs;tot

in the range 60–160 kJ=cm2 compared with 11–33 kJ=cm2 forVEI � 11 km=s. The Pareto frontier of the PL is composed ofeffective SS forms. For optimal designs with pxrs of 500, 1000, and1500 km, the values of Qs;tot are 64.5, 65.2, and 98:3 kJ=cm2,respectively.All three axial profiles follow similar behavior: theQs;tot

increases �12% from pxrs � 0 km to 1100 km.The significant difference in Qs;tot between the SS and SC Pareto

frontiers is caused primarily by differences in drag area. SS design Ehas a 27.8% lower drag area than blunted cone design F because theSS can maintain requirements with a smaller vehicle. Evidence ofthis is shown in Table 5 where design E has a 12.5% lower hb;HS thandesign F. Design E experiences qs;max at an altitude of 66.1 km, 2 kmdeeper than design F, and sees a 26% largerQs;tot than design F. Bothdesigns follow similar skip trajectories and experience qs;max atV1 � 13:5 km=s. The trajectory for design F is illustrated in Fig. 15.At 66.1 km, air density is 40% greater than at 68.1 km. Since thenormal-shock density ratio is relatively constant between the twoaltitudes with the same V1, the �so does not change significantly.This results in a 40% increase in power density E, and thus a 31%increase in qs;rad. This demonstrates the high sensitivity of heattransfer to density, and therefore altitude for travel at hyperbolicspeeds through the atmosphere.

A proper balance of radiative and convective heat loads isimportant for minimizing total heat load, but the optimal SS and SCgeometries produce nearly the same minimum Qs;tot. To isolate theeffects of trajectory design and geometry, the heat fluxes for bothdesigns are compared at each other’s qs;max freestream conditions.Table 6 provides a comparison of qs;max and Qs;tot generated at theqs;max freestream conditions of designs E and F, respectively. At theqs;max conditions of design E (ht � 66:1 km, V1 � 13:5 km=s),

Fig. 14 Pareto frontiers for Earth entry, VE � 15 km=s.

Fig. 15 Heat-shield skip trajectory of design F from Table 5.

Table 6 Max qs and Qtot comparison at qs;max freestream

conditions of designs E and F

Design E (SS) Design F (SC) Viking (SC) withe�0:968

FreestreamConditionsat qs;max

qs;max,W=cm2

�qs;conv; qs;rad�qs;max,W=cm

2

�qs;conv; qs;rad�qs;max, W=cm

2

�qs;conv; qs;rad�

Qs;tot;, kJ=cm2

�Qs;conv; Qs;rad�Qs;tot;, kJ=cm

2

�Qs;conv; Qs;rad�Qs;tot;, kJ=cm

2

�Qs;conv; Qs;rad�E: ht � 66:1 km,V1 � 13:5 km=s

1930(150, 1780)

82.4(13.6, 68.8)

1790(230, 1560)

——

——

F: ht � 68:1 km,V1 � 13:5 km=s

1580(120, 1460)

67.8For hb;HS � 4 m

1400(200, 1200)

65.2(18.5, 46.7)

1570(470, 1100)

86

JOHNSON, LEWIS, AND STARKEY 47

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65

design F would experience a 7% lower qs;max than design E yet resultin roughly the sameQs;tot as for design E. If design Ewere required tohave the same hb;HS as design F, hb;HS � 4 m, then design E wouldhave drag area similar to design F’s (since they have nearly the sameCD), resulting in an equivalent BC. They would then fly nearly thesame trajectories. At theqs;max conditions of design F (ht � 68:1 km,V1 � 13:5 km=s) and setting design E’s hb;HS equal to design F’s,design E would generate a 13% larger qs;max than design F, primarilygenerated by the 22% increase in qs;rad. The resultingQs;tot would beapproximately 4% greater than that of design F. At the qs;max

conditions of design F, a geometry similar to Viking, excepte�0:968, would produce a 12% larger qs;max and a 32% increaseinQs;tot over design F. This indicates thatQs;tot is sensitive to differentcombinations of radius of curvature and �so.

The design variable distributions for the blunted cone Paretofrontier are illustrated in Fig. 16; the trends in these distributions areconsistent with those discussed for VEI � 11 km=s (Fig. 12). Onlyskip trajectories were captured for the SC as its Pareto frontier beginsat pxrs � 200 km. The SS switches from direct entry trajectories toskip trajectories at pxrs � 170 km. As cross range is increased, thesudden changes in geometry are produced by the optimizer toincrementally increase L=D.

2. Minimizing Qs;tot and qs;max

Pareto frontiers are shown in Fig. 14b for the SS and SC. Theresults for the PL form were inconclusive for the same reasons as forVEI � 11 km=s (III.A.2). For the two Pareto frontiers, the geometriesare relatively constant, listed as designs G and H in Table 5. Both theSS and blunted cone geometries are very similar to designs E and F,respectively. Similarly, the SS has a smaller hb;HS than the bluntedcone, thus having a smaller drag area and resulting in higherQs;tot andqs;max.

To isolate the effects of trajectory design and geometry, the heatfluxes for both designs are compared at each other’s qs;max freestreamconditions. Heat flux values are listed in Table 7. Design Gexperiences qs;max at ht � 68:7 km, V1 � 13:7 km=s, whiledesign H experiences qs;max at ht � 70:4 km, V1 � 13:6 km=s. Atthe qs;max freestream conditions of design G, design H wouldexperience a 26% increase in qs;max from its value at its own qs;max

conditions. At design G’s qs;max conditions, this blunted conedesign H experiences an 8% lower qs;max than design G. At the qs;max

conditions of design H, design G would experience a 21% decreasein qs;max from its value at its own qs;max conditions; at design H’sqs;max conditions, this SS design produces an 8% higher qs;max thandesign H. The SC generates a lower qs;max at both the qs;max

conditions of designs G and H in which qs;max is experienced.Differences in Qs;tot between both cases are negligible. The qs;rad ismore sensitive than Qs;tot for these optimal designs. Since thesegeometries are similar to E and F, the importance of balancingradiative and convective heat transfer also holds.

The trajectory design variable distributions are given in Fig. 17 forthe blunted cone. For this objective function set, the aims of both �and �b are the same sincepxrs is not being optimized. For the bluntedcone case, � is constant, and the optimizer varies �b to control howmuch lift is applied to counteract gravity. For the SS case (notshown), �b;0 is relatively constant at 0�, and � decreases withincreasing qs;max in order to lower the flight duration, and thusminimize Qs;tot.

IV. Conclusions

Optimization has produced optimal heat-shield configurations forEarth entry at VEI � 11 and 15 km=s using two objective functionsets: 1) maximizing pxrs and minimizing Qs;tot and 2) minimizingQs;tot and qs;max. The assumptions in this work are appropriate forassessment of the trade space at the conceptual design level for heat-shield shape optimization. ForVEI � 11 km=swith a 6g limit, the SSis the optimal axial profile formaximizingpxrs andminimizingQs;tot.Direct entry trajectories are best for pxrs � 250 km, and skiptrajectories are preferred for largerpxrs. For optimal designswithpxrsof 500, 1000, and 1500 km, the values of Qs;tot are 14.7, 22.6, and29:4 kJ=cm2, respectively. For pxrs > 750 km, the SC and PLsolutions are disguised SSs. The optimal designs for minimizing

Fig. 16 Design variable distribution for SC designs from Fig. 14a.

Fig. 17 Trajectory design variable distribution for SC designs from

Fig. 14b.

Table 7 Max qs comparison at qs;max freestream conditions

of designs G and H

Design G (SS) Design H (SC)

Freestream conditions at qs;max qs;max, W=cm2

�qs;conv; qs;rad�qs;max, W=cm

2

�qs;conv; qs;rad�G: ht � 68:7 km, V1 � 13:7 km=s 1500

(140, 1360)1390

(210, 1180)H: ht � 70:4 km, V1 � 13:6 km=s 1190

(130, 1060)1100

(180, 920)

48 JOHNSON, LEWIS, AND STARKEY

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65

Qs;tot and qs;max have direct entry trajectories, and the resulting SSand blunted cone geometries are equally optimal.

For VEI � 15 km=s with a 12g limit, neither the optimal bluntedcone nor SS is significantly better. Radiative heat transfer dominatesconvection in both heat flux and heat load. For optimal designs withpxrs of 500, 1000, and 1500km, thevalues ofQs;tot are 64.5, 65.2, and98:3 kJ=cm2, respectively. For maximizing pxrs and minimizingQs;tot, in general, the Qs;tot increases �12% from pxrs � 0 km to1100 km. For pxrs > 1200 km, although the optimal SS and bluntedcone designs produce nearly the same optimum Qs;tot with twodifferent sets of curvature and shock layer thicknesses, a properbalance of convective and radiative heat transfer is necessary tominimize Qs;tot.

As expected, it was observed that radiative heat transfer is moresensitive to air density for entry at hyperbolic speeds than for entry atlower velocities, including typical lunar return velocities of�10:5 km=s. The blunted cone designs have higher drag area sincethey are less volumetrically efficient, and thus decelerate 2 kmhigherin altitude with 40% less air density. This reduces qs;max andultimatelyQs;tot by 21%.TheSS can satisfy themission requirementswith a smaller-sized vehicle, which leads to the lower drag area andhigherQs;tot. Due to the high sensitivity ofQs;max to drag area, vehiclesizing and volumetric efficiency can have important roles indetermining the required capability of the TPS.

For entries at 11 and 15 km=s, a highly oblate eccentricity e�0:968 maximizes drag area, allowing deceleration at higheraltitudes, thus lowering both heat flux and heat load. As more pxrs isrequired, drag area is traded off with the need for larger L=D bydecreasing e. This behavior is consistent with previous work. Anelliptical cross section rather than a parallelogram form is optimal forL=D � 0:50, corresponding topxrs � 1500 andpxrs � 2200 km forVEI � 11 and 15 km=s, respectively. The parallelogram cross sectioncould be applied to increase L=D beyond the capability of theelliptical cross section, which is expected to increase pxrs anddecrease peak g limits. The TPS would be required to handle theexpected higher Qs;tot.

This work has identified and demonstrated an approach foroptimizing multiple objectives in a broad design space using adifferential evolutionary algorithm. With the proper setup, it can beused to automate the process of locating feasible solutions. Thisapproach can be used in a variety of disciplines to assess a broaddesign space and focus it to achieve the desired objectives.

Acknowledgments

This research was supported by the Space Vehicle TechnologyInstitute (SVTI), one of the NASAConstellation University InstituteProjects (CUIP), under grant NCC3-989, with joint sponsorshipfrom the Department of Defense. Appreciation is expressed toDarryll Pines, director of the SVTI at the University of Maryland,ClaudiaMeyer, programmanager of CUIP, and Jeffry Rybak, deputyprogram manager of CUIP of the NASA John H. Glenn ResearchCenter at Lewis Field, the support of whom is greatly appreciated.Gratitude is expressed to Falcon Rankins for software training andsupport of the University of Maryland Parallel TrajectoryOptimization Program code used for planetary entry trajectoryoptimization.

References

[1] Putnam, Z. R., Braun, R. D., Rohrschneider, R. R., and Dec, J. A.,“Entry System Options for Human Return from the Moon and Mars,”Journal of Spacecraft and Rockets, Vol. 44, No. 1, 2007, pp. 194–202.doi:10.2514/1.20351

[2] Bertin, J. J., Hypersonic Aerothermodynamics, AIAA EducationSeries, AIAA, Washington, DC, 1994, Chap. 3.

[3] Anderson, J. D., Hypersonic and High Temperature Gas Dynamics,AIAA, Reston, VA, 2000, Chap. 6.

[4] Griffin, M. D., and French, J. R., Space Vehicle Design, 2nd ed., AIAAEducation Series, AIAA, Reston, VA, 2004, Chap. 6.

[5] Crowder, R. S., and Moote, J. D., “Apollo Entry Aerodynamics,”Journal of Spacecraft and Rockets, Vol. 6, No. 3, 1969, pp. 302–307.doi:10.2514/3.29589

[6] Korkan, K. D., and Hanley, G. M., “Apollo Command ModuleAerothermodynamic Characteristics at Hyperbolic Earth EntryVelocities,” Journal of Spacecraft and Rockets, Vol. 3, No. 8, 1966,pp. 1274–1281.doi:10.2514/3.28639

[7] Lee, D. B., and Goodrich, W. D., “The AerothermodynamicEnvironment of the Apollo Command Module During SuperorbitalEntry,” NASATN D-6792, April 1972.

[8] Pavlosky, J., and L. Leger, “Apollo Experience Report—ThermalProtection Subsystem,” NASATN D-7564, Jan. 1974.

[9] Kirk, D. B., Intrieri, P. F., and Seiff, A., “Aerodynamic Behavior of theViking Entry Vehicle: Ground Test and Flight Results,” Journal of

Spacecraft and Rockets, Vol. 15, No. 4, 1978, pp. 208–212.doi:10.2514/3.57307

[10] Chrusciel, G. T., and Hull, L. D., “Theoretical Method for CalculatingAerodynamic Characteristics of Spherically-Blunted Cones,” AIAAPaper 1968-0674, June 1968.

[11] Sammonds, R. I., and Kruse, R. L., “Viking Entry VehicleAerodynamics at M� 2 in Air and Some Preliminary Test Data forFlight in CO2 atM � 12,” NASATN D-7974, June 1975.

[12] Johnson, J. E., Starkey, R. P., and Lewis, M. J., “AerothermodynamicOptimization of Reentry Heat Shield Shapes for a Crew ExplorationVehicle,” Journal of Spacecraft and Rockets, Vol. 44, No. 4, July–Aug. 2007, pp. 849–859.doi:10.2514/1.27219

[13] Whitmore, S. A., Banks, D. W., Anderson, B. M., and Jolley, P. R.,“Direct-Entry, Aerobraking, and Lifting Aerocapture for Human-RatedLunar Return Vehicles,” AIAA Paper 2006-1033, Jan. 2006.

[14] Wooster, P. D., Braun, R. D., Ahn, J., and Putnam, Z. R., “TrajectoryOptions for Human Mars Missions,” AIAA Paper 2006-6308,Aug. 2006.

[15] “Constellation Program: The Orion Crew Exploration Vehicle,”NASAJohnson Space Center, NASAfacts FS-2006-08-022-JSC, 2006.

[16] Gielis, J., “A Generic Geometric Transformation that Unifies a WideRange of Natural and Abstract Shapes,” American Journal of botany,Vol. 90, No. 3, 2003, pp. 333–338.doi:10.3732/ajb.90.3.333

[17] Johnson, J. E., Starkey, R. P., and Lewis, M. J., “Aerodynamic StabilityofReentryHeat Shield Shapes for aCrewExplorationVehicle,” Journalof Spacecraft and Rockets, Vol. 43, No. 4, July–Aug. 2006,pp. 721–730.doi:10.2514/1.20044

[18] Larson, W. J., and Pranke, L. K. (eds.), Human Spaceflight Mission

Analysis and Design, Space Technology Series, McGraw–Hill,New York, 1999, Chaps. 10, 12.

[19] Johnson, J. E., Starkey, R. P., and Lewis, M. J., “Multi-ObjectiveOptimization of Entry Vehicle Heat Shields with Reentry TrajectoryAnalysis,” 58th International Astronautical Congress, Hyderabad,India, International Astronautical Federation IAC Paper 07-C2.7.09,Paris, Sept. 2007.

[20] Darling, D., “ASSET (Aerothermodynamic Elastic Structural SystemsEnvironmental Tests),”TheComplete Book of Spaceflight: FromApollo

1 to Zero Gravity, Wiley, Hoboken, NJ, 2003.[21] Starkey, R. P., Rankins, F., and Pines, D. J., “Effects of Hypersonic

Cruise Trajectory Optimization Coupled with Airbreathing VehicleDesign,” AIAA Paper 2006-1036, Jan. 2006.

[22] Rankins, F., The University of Maryland Parallel Trajectory

Optimization Program User’s Manual, Univ. of Maryland, CollegePark, MD, 2007.

[23] Stevens,B. L., andLewis, F. L.,Aircraft Control and Simulation,Wiley,New York, 1992.

[24] “Department of Defense World Geodetic System 1984, Its Definitionand Relationships With Local Geodetic Systems,” NationalImagery and Mapping Agency TR 8350.2, 3rd ed., Reston, VA,July 1997.

[25] “U.S. Standard Atmosphere, 1976,” NASA TM-X-74335, U.S.Government Printing Office, Washington, D.C, Oct 1976.

[26] Picone, J.M., Hedin, A. E., Drob,D. P., andAikin,A. C., “NRLMSISE-00 Empirical Model of the Atmosphere: Statistical Comparisons andScientific Issues,” Journal ofGeophysical Research, Vol. 107,No.A12,Dec. 2002, pp. 1468–1483.doi:10.1029/2002JA009430

[27] Brauer, G. L., Cornick, D. E., Habeger, A. R., Petersen, F. M.,and Stevenson, R., “Program to Optimize Simulated Trajectories(POST): Volume 1 Formulation Manual,” NASA CR-132689,April 1975.

[28] Powell, R. W., Striepe, S. A., Desai, P. N., and Braun, R. D., Programto Optimize Simulated Trajectories (POST). Volume 2 Utilization

Manual, Version 5.2, NASA Langley Research Center, Oct. 1997.

JOHNSON, LEWIS, AND STARKEY 49

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65

[29] Rasmussen,M.,HypersonicFlow,Wiley,NewYork, 1994,Chaps. 8, 10.[30] Moseley, W. C., Jr., Graham, R. E., and Hughes, J. E., “Aerodynamic

Stability Characteristics of the Apollo Command Module,” NASATN D-4688, 1968.

[31] Tauber, M. E., and Menees, G. P., “Aerothermodynamics ofTransatmospheric Vehicles,” Journal of Aircraft, Vol. 24, No. 9,1987, pp. 594–602.doi:10.2514/3.45483

[32] Ried, R. C., Jr., Rochelle,W. C., andMilhoan, J. D., “RadiativeHeatingto the Apollo Command Module: Engineering Prediction and FlightMeasurement,” NASATM X-58091, April 1972.

[33] Kaattari, G. E., “AMethod for Predicting Shock Shapes and PresssureDistributions for a Wide Variety of Blunt Bodies at Zero Angle ofAttack,” NASATN D-4539, April 1968.

[34] Kaattari, G. E., “Shock Envelopes of Blunt Bodies at Large Angles ofAttack,” NASATN D-1980, Dec. 1963.

[35] Bertin, J., Hypersonic Aerothermodynamics, AIAA Education Series,AIAA, New York, 1993, p. 271.

[36] Lovelace, U. M., “Charts Depicting Kinematic and Heating Parametersfor a Ballistic Reentry at Speeds of 26,000 to 45,000 Feet Per Second,”NASATN D-968, Oct. 1961.

[37] Tauber, M. E., and Sutton, K., “Stagnation-Point Radiative HeatingRelations for Earth and Mars Entries,” Journal of Spacecraft and

Rockets, Vol. 28, No. 1, 1991, pp. 40–42.doi:10.2514/3.26206

[38] Cauchon, D. L., “Radiative Heating Results from the FIRE II FlightExperiment at a Reentry Velocity of 11.4 Kilometers per Second,”NASATM X-1402, July 1967.

[39] Madavan, N. K., “On Improving Efficiency of Differential Evolutionfor Aerodynamic Shape Optimization Applications,” AIAAPaper 2004-4622, Aug. 2004.

[40] Deb, K.,Multi-Objective OptimizationUsing Evolutionary Algorithms,Wiley, Chichester, England, U.K., 2001.

[41] Rai,M.M., “Robust Optimal AerodynamicDesignUsing EvolutionaryMethods and Neural Networks,” AIAA Paper 2004-778, Jan. 2004.

[42] Johnson, J. E., Lewis, M. J., and Starkey, R. P., “Coupled Entry HeatShield/Trajectory Optimization for Lunar Return,”AIAA AtmosphericFlight Mechanics Conference, AIAA Paper 2008-6557, Aug 2008.

[43] Jones, E., and Shiva, K. (eds.), “NASATechnical Standards Program,”NASASTD-3001, Vol. 2, https://standards.nasa.gov/documents/detail/3315785 [retained 2011].

[44] Graves, C. A., and Harpold, J. C., “Re-Entry Targeting Philosophy andFlight Results from Apollo 10 and 11,” AIAA Paper 1970-0028,Jan. 1970.

[45] Young, J. W., and Smith, R. E., “Trajectory Optimization for AnApollo-Type Vehicle Under Entry Conditions Encountered DuringLunar Return,” NASATR R-258, May 1967.

[46] Scott, C. D., Ried, R. C., Maraia, R. J., Li, C. P., and Derry, S. M., “AnAOTVAeroheating and Thermal Protection Study,” Thermal Design ofAeroassistedOrbital Transfer Vehicles, edited byH. F. Nelson, Vol. 96,of Progress in Astronautics and Aeronautics, AIAA, New York, 1995,pp. 309–337.

K. WursterAssociate Editor

50 JOHNSON, LEWIS, AND STARKEY

Dow

nloa

ded

by U

nive

rsity

of

Surr

ey o

n N

ovem

ber

1, 2

012

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/1

.425

65