adjoint-based aerothermodynamic shape design of hypersonic

Adjoint-Based Aerothermodynamic Shape Design of

Hypersonic Vehicles in Non-Equilibrium Flows

Sean R. Copeland∗, Francisco Palacios†, and Juan J. Alonso‡

Stanford University, Stanford, CA, 94305, U.S.A.

In this paper, a new, continuous formulation of the adjoint equations is derived for asteady, continuum, inviscid gas mixture in thermochemical non-equilibrium for force-basedobjective functions. These adjoint equations, when solved in conjunction with the governingequations, provide sensitivity information that can be used in a gradient-based optimiza-tion framework for shape design. The governing and adjoint equations are implemented inan unstructured, three-dimensional CFD solver, enabling efficient, optimal design of com-plex geometries with aerothermodynamic considerations in the presence of high-enthalpy,chemically reactive gas mixtures. Surface gradients calculated using the adjoint formula-tion provided herein are validated against finite-difference gradients for blunt-body andsimple wing configurations.

Nomenclature

A Flux Jacobianc Speed of sound~d Force projection vectorE Energy per unit mass~f Force vector~F Vector of convective fluxesH Enthalpy per unit mass¯I Identity matrixJ Objective functionJ Objective function Lagrangiann Number of species~n Normal vectorP Pressureq Energy exchange source termQ Vector of source termsR Governing equation residualS Control surfaceT TemperatureU Vector of conserved variables~u Velocityw Pressure weighting term in the MSW methodw Chemical production source term~x Spatial coordinatesY Mass fractionα Angle of attackβ Angle of sideslip

∗Ph.D. Candidate, Department of Aeronautics & Astronautics, AIAA Student Member.†Engineering Research Associate, Department of Aeronautics & Astronautics, AIAA Senior Member.‡Associate Professor, Department of Aeronautics & Astronautics, AIAA Associate Fellow.

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American Institute of Aeronautics and Astronautics

ε Modified Steger-Warming pressure switch parameterε0 Lax-Friedrich artificial dissipation parameterΓ Domain boundaryΛ Diagonal matrix of eigenvectors~Φ Adjoint convective fluxΨ Vector of adjoint variablesρ DensityΩ Volumetric domain

Subscriptsi Node iij Interface between nodes i & jj Node jp Polyatomic speciess Speciessym Symmetry∞ Far field

Superscriptsc Convectiveel Electron/electronicLF Lax-FriedrichMSW Modified Steger-Warmingt− r Translational/Rotationalt : v Translational-rotational/Vibrational-electronic exchangev Vibrationalv − e Vibrational-electronic Formation

AcronymsAD Automatic differentiationCFD Computational fluid dynamicsDOF Degree of freedomFVM Finite volume methodGNC Guidance, navigation and controlOML Outer mold lineRRHO Rigid-rotator harmonic-oscillator

I. Introduction

The hypersonic aerothermodynamic environment is influenced by unique physical phenomena. Strongshockwaves in hypervelocity flow fields can lead to localized regions of chemical and thermodynamic non-equilibrium that strongly affect surface quantities including pressure, shear stress, and energy flux. Thesequantities are integrated over the surface to predict aerodynamic performance metrics (lift, drag, stability,etc.) and surface thermal conditions that are fed to other discipline-specific analysis tools that are requiredfor the overall vehicle design.

The hypersonic vehicle design process requires the synthesis of aerothermodynamic, structural, TPSmaterial response, GNC, trajectory, and payload analysis tools in a tightly-coupled design environment.Because of the computational expense associated with high-fidelity analysis tools, an initial vehicle config-uration, entry envelope, and design trajectory is typically established using low-fidelity, engineering-levelmethods.1–3 These can include hypersonic Newtonian aerodynamic theory, correlation-based stagnation lineheating rate predictions, and 3-DOF trajectory simulations. The vehicle design is matured via the intro-duction and coupling of higher-fidelity tools and the design is iterated until closure is reached, satisfying allsubsystem and component-level constraints. This process essentially defines an envelope in design parameter

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space using low fidelity tools, then systematically refines and narrows the boundaries of the design envelopeusing higher fidelity tools. This procedure is predicated under the assumption that the optimal design pointis located within the initial, and each subsequent iteration’s design parameter envelope. In general, this con-dition is not guaranteed, given the limitations of the physical modeling of the low-fidelity tools. Therefore,to guarantee optimal vehicle configurations, new high-fidelity strategies must be introduced earlier in thedesign process.4–6 This work proposes such a strategy via the formulation of the adjoint problem for thenon-equilibrium flow environment.

The adjoint method is one approach to efficiently acquire gradient information in high-dimensional spaceswhen function evaluations are expensive.7–11 This approach formulates an alternate system of PDEs andboundary conditions that, when solved in conjunction with the governing equations, provide sensitivity in-formation that can be used in a gradient-based optimization framework. A solution to the adjoint systemcomes at roughly the same cost as solving the governing equations and provides objective function sensitiv-ities regardless of the dimensionality of the design space. This is contrasted with other gradient acquisitionmethods (e.g. finite-difference, complex step) that scale linearly with the design space dimensionality. Thischaracteristic makes adjoint-based methods popular for optimal shape design of aerospace vehicles usingcomputational fluid dynamics. A rich body of literature exists for adjoint-based design,12,13 error estima-tion,14,15 and uncertainty quantification16 for a variety of steady, unsteady, inviscid, viscous, and turbulentproblems.

Unfortunately, progress in adjoint-based methods with application to the non-equilibrium aerothermo-dynamic environment has lagged, primarily due to the significant leap in the complexity of the physicalmodeling. Attempts have been made17,18 to utilize discrete adjoint (where the adjoint problem is formu-lated after the linearization and discretization of the direct governing equations) strategies, coupled withautomatic differentiation19 tools, to acquire the desired sensitivity information. This methodology avoidsthe complications associated with an analytic derivation of the adjoint equations and boundary conditionsand is capable of providing numerically exact gradient information in a flexible, automatic fashion, throughthe utilization of AD tools. However, because the discrete adjoint is formulated after the discretization ofthe governing equations, the adjoint system must be solved using the same numerical method as the directproblem. These numerical schemes are highly specialized for non-equilibrium flow solvers,20–23 and there isno guarantee that these methods are well-suited to solving the corresponding adjoint problem. The currentliterature indicates the resulting adjoint linear system can be very numerically stiff, and the resulting dis-crete adjoint solution can exhibit non-physical oscillations, failing to capture the behavior of the continuousobjective function when in the presence of strong shock waves on misaligned grid topologies24 (as is the casewhen solving hypersonic flow problems on unstructured meshes).

In response, this paper pursues an analytic derivation of the continuous adjoint system of equationsfor a continuum, inviscid, multi-component gas mixture in thermochemical non-equilibrium for the purposeof acquiring high-fidelity gradient information at acceptable cost that can be used for aerodynamic shapedesign, sensitivity analysis, and uncertainty quantification of hypersonic vehicles.

The paper is organized in the following manner. Section II outlines the mathematical formulation of thedirect and adjoint problems. Section III details the numerical implementation and solution procedure forthe direct and adjoint systems of II. This section also describes the procedure for manipulating the surfacegeometry and volume meshes for the finite-difference calculations shown the gradient validation of SectionIV. Lastly, Section V summarizes the findings and emphasizes the utility of the method.

II. Formulation

This section details the physical modeling of the system and the mathematical formulation of the gov-erning equations, the objective function, and the adjoint system of equations.

A. Direct Problem

For this work, we require a system of governing equations that model a steady, continuum, inviscid, multi-species gas mixture in thermochemical non-equilibrium. For this work, a single-fluid, two-temperature,multi-species model has been selected that governs the transport of species mass, mixture momentum, andmixture energy within the computational domain. This model is expressed in conservation-law form as,

R(U) ≡ ∇ · ~F c(U)−Q(U) = 0, (1)

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where,

U =

ρ1

...

ρns

ρ~u

ρE

ρEv−e

, ~F c =

ρ1~u...

ρns~u

ρ~u⊗ ~u+ P ¯I

ρ~uH

ρ~uEv−e

, ~Q =

w1

...

wns

¯0

0

qt:v +∑s wsE

v−es

.

Each species conservation equation tracks the finite-rate production and destruction of the chemical con-stituent through ws, which is calculated using an Arrhenius model with rate-coefficients determined inaccordance with Park’s 1990 model.25 The two-temperature approach presented above assumes thermalequilibrium between the translational-rotational and vibrational-electronic energy states within the gas mix-ture. The two energy equations are coupled via the energy exchange source term, qt:v, that is modeled usinga Landau-Teller vibrational relaxation model.26 The total energy per unit volume of the gas mixture is thesum over all species and energy states, illustrated in Eq. (2), where each mode of energy storage is calculatedassuming a rigid-rotator harmonic-oscillator (RRHO) thermodynamic model. Additional details regardingthe chemical system and the gas dynamic model can be found in Appendices A & B.

ρE =∑s

ρsEs =∑s

ρs

(Et−rs + Evs + Eels + Es +

1

2usiusi

). (2)

H = E +P

ρ(3)

B. Objective Function Definition

The adjoint problem establishes a mathematical framework for determining the sensitivity of a specifiedobjective function to high-dimensional design spaces in an efficient manner. To begin, we choose an objectivefunction of interest. For this work, we focus on integrated projected forces on a control surface, S,

J =

∫S

j(~f(U))ds =

∫S

(~d · P~n)ds, (4)

where ~d is a force projection vector. By changing the value of the force projection vector, different objectivefunctions pertaining to aerodynamic performance and stability can be evaluated, for example,

~d =

1C∞

(cos(α) cos(β), sin(α) cos(β), sin(β)) CD - Drag1C∞

(− sin(α), cos(α), 0) CL - Lift1C∞

(0, 0, 1) CL

CD- L/D

1C∞Lref

(−(y − y0), (x− x0), 0) Cmz - Z-moment

(5)

The analysis is formulated as follows,

Find J(U) =∫Sj(~f(U))ds,

such that R(U) = 0.(6)

By satisfying the governing equations, R(U) = 0, this equality-constrained analysis can be transformed toan unconstrained problem via the addition of an inner product of the adjoint variables with the governingequations to form the Lagrangian,

J =

∫S

j(~f(U))ds−∫

Ω

ΨTR(U)dΩ, (7)

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where we have introduced the adjoint variables Ψ as Lagrange multipliers to the linear system,

Ψ =

ψρ1...

ψρns

ψρu

ψρv

ψρE

ψρEv−e

=

ψρ1...

ψρns

~φ

ψρE

ψρEv−e

. (8)

C. Adjoint Problem

Figure 1. Notional computational domain,Ω, and control surface deformation.

Our objective is to obtain gradient information to the specifiedobjective function, J , with respect to local normal perturba-tions to the control surface, δS. With the appropriate defini-tion of the adjoint problem and its boundary conditions, andby utilizing calculus of variations in conjunction with differen-tial geometry, it is possible to acquire this gradient informationwith a single solution of the governing equations. To accom-plish this, we take the first variation of the Lagrangian and seekto eliminate its dependence on variations in the fluid state, δU .

1. Lagrangian Variation

To determine the effect of local normal control surface pertur-bations, δS, on the Lagrangian, a first variation is applied toEq. (7).

δJ = δJ −∫

Ω

ΨT δR(U)dΩ. (9)

By inserting the definition of R, and applying the chain rule, the final term in Eq. (9) can be written as,∫Ω

ΨT δR(U)dΩ =

∫Ω

ΨT δ(∇ · ~F c −Q

)dΩ (10)

=

∫Ω

ΨT

(∇ · ~Ac − ∂Q

∂U

)δUdΩ, (11)

where ~Ac = ∂ ~F c

∂U = (∂F c

x

∂U ,∂F c

y

∂U ,∂F c

z

∂U ), is the Jacobian of the convective fluxes. The domain integral over Ωcan be split into domain and boundary components using integration by parts and the divergence rule,∫

Ω

ΨT (∇ · ~A)δUdΩ =

∫Ω

∇ · (ΨT ~AδU)dΩ−∫

Ω

∇ΨT · ~AδUdΩ

=

∫Γ

(ΨT ~AδU) · ~nds−∫

Ω

∇ΨT · ~AδUdΩ.

(12)

2. Objective Function Variation

The first variation of the objective function is determined by applying the chain rule to the definition of thelocal objective function, j, and the boundary, S,

δJ =

∫S

δjds+

∫δS

jds (13)

=

∫S

δ(~d · P~n)ds+

∫δS

(~d · P~n)ds. (14)

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Noting that the force projection vector, ~d, is constant, and by using the differential geometry definitions ofEqs. (50), the variation in the objective function can be simplified to the following expression,

δJ =

∫S

(~d · ~n)δPds+

∫S

(~d · ∇P )δSds. (15)

3. Adjoint Equations

By combining the results from Eqs. (10), (12), and (15), the variation of the Lagrangian can be written asthe sum of a domain integral and boundary integrals,

δJ =

∫S

(~d · ~n)δPds+

∫S

(~d · ∇P )δSds−∫

Γ

(ΨT ~AδU) · nds−∫

Ω

[−∇ΨT · ~Ac −ΨT ∂Q

∂U

]δUdΩ. (16)

The adjoint equations, boundary conditions, and surface sensitivities are determined by strategically selectingthe value of the integrands of the domain and boundary integrals of Eq. (16) to eliminate dependencies onvariations in flow quantities (δU and δP ). In the domain, Ω, this dependency can be eliminated by settingthe integrand of the final term to zero,

−∇ΨT · ~Ac −ΨT ∂Q

∂U= 0. (17)

These are the adjoint equations for the non-equilibrium problem.

4. Adjoint Boundary Conditions & Surface Sensitivity

The corresponding boundary conditions and surface sensitivities to the adjoint linear system of Eq. (17) aredetermined by manipulating the boundary integrals of Eq. (16). By inserting the definition of the convectiveflux Jacobian and enforcing flow tangency on the control surface, S, we can write,

ΨT ( ~Ac · ~n)∣∣∣(~u·~n=0)

=

(φ · ~n) ∂P∂ρr(φ · ~n) ∂P∂ρu + (

∑s ψρsYs + (~φ · ~u) +HψρE + Ev−eψρEv−e)nx

(φ · ~n) ∂P∂ρv + (∑s ψρsYs + (~φ · ~u) +HψρE + Ev−eψρEv−e)ny

(φ · ~n) ∂P∂ρw + (∑s ψρsYs + (~φ · ~u) +HψρE + Ev−eψρEv−e)nz

(φ · ~n) ∂P∂ρE(φ · ~n) ∂P

∂ρEv−e

T

. (18)

Finally, we take the inner product of Eq. (18) and δU to explicitly write the form of the adjoint boundaryintegral on the control surface,∫

S

ψT ( ~Ac · ~n)δUds =

∫S

(φ · ~n)δP + ρϑδ~u · ~n ds (19)

=

∫S

(φ · ~n)δPds+

∫S

ρϑ(−(∂n~u)δS · ~n+ ~u · ∇SδS)ds (20)

=

∫S

(φ · ~n)δPds−∫S

ρϑ(∂n~u) · ~nδSds+

∫S

ρϑ~u · ∇SδSds (21)

=

∫S

(φ · ~n)δPds−∫S

ρϑ(∂n~u) · ~nδSds−∫S

∇S · (ρϑ~u)δSds (22)

where we have used integration by parts on the last term and utilized Eq. (50c) to eliminate one of theintegrals, defining

ϑ =∑s

ψρsYs + (~φ · ~u) +HψρE + Ev−eψρEv−e . (23)

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We now combine the δP and δS terms,

δJ −∫S

ψT ~Ac · ~nδUds =

∫S

(~d · ~n)δPds+

∫S

(~d · ∇P )δSds−∫S

(φ · ~n)δPds+

∫S

ρϑ[(∂n~u)δS · ~n+ ~u · ∇SδS]ds

(24)

=

∫S

[(~d · ~n)− (φ · ~n)]δPds+

∫S

(~d · ∇P )δSds+

∫S

ρϑ[(∂n~u)δS · ~n+ ~u · ∇SδS]ds.

(25)

When written this way, it is clear that δP can be eliminated from δJ by setting the adjoint velocity equalto the force projection vector, ~φ = ~d, on S. The remaining δS-dependent terms are again manipulated usingEqns. (50) and can be expressed simply as,∫

S

[~d · ∇P +∇ · (ρϑ~u)]δSds =

∫S

∂J∂S

δSds. (26)

The expression for ∂J∂S is the surface sensitivity and is easily evaluated on unstructured mesh topologies when

written in this simplified form.

III. Numerical Implementation

The direct and adjoint problems defined in Section II were implemented in the SU2 open-source softwaresuite.27 This analysis package consists of C++ software modules, linked with python scripting and isspecifically architected for multi-physics analysis and design on unstructured mesh topologies.

A. Discretization of the Governing Equations

The discretized governing equations for the direct and adjoint problems are shown in Eqs. (27) & (28) and aresolved using a node-centered, edge-based, finite volume method (FVM). The direct problem convective fluxesare evaluated using a first-order upwind Modified Steger-Warming method28 with a pressure switch. Thisswitch modulates the numerical dissipation in the computational domain, asymptotically approaching theclassical Steger-Warming method in regions where ∇P is large and the Modified Steger-Warming methodas ∇P approaches zero. The adjoint problem uses centered Lax-Friedrich scheme29 with a user-specifiedartificial dissipation parameter, ε0. Both problems are integrated forward in time using the implicit Eulerscheme with local time-stepping. Numerical fluxes are calculated by applying the integral formulation ofthe governing equations to a dual grid control volume, Ωi, and performing exact integration on the controlvolume boundaries with cell-neighbors.

Un+1i − Uni

∆t|Ωi|+

mi∑j=1

~FMSWij · ~nijΓij −Qi|Ωi| = 0, (27)

Ψn+1i −Ψn

i

∆t|Ωi|+

mi∑j=1

~ΦLFij · ~nijΓij −∂Qi∂U|Ωi| = 0. (28)

Explicitly, the numerical fluxes are written as,

~FMSWij = ( ~Ac · ~n)+

ijUi + ( ~Ac · ~n)−jiUj , (29)

~ΦLFij = ( ~Aci )T

(Ψi + Ψj

2

)· ~nij + ε0λ(Ψi −Ψj), (30)

where the upwind projected Jacobians, ( ~Ac · ~n)±, are calculated using the eigenvalue/eigenvector decompo-sition at the ijth or jith state as defined by,

Uij = (1− w)Ui + wUj and Uji = wUi + (1− w)Uj (31)

( ~Ac · ~n)±ij = (P |~Λ± · ~n|P−1)ij , (32)

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and the pressure-weighting term is,

w =1

2

1

(ε∇P )2 + 1. (33)

Note that the adjoint system, as written in Eq. (17), is non-conservative and, in general, ~Φij 6= −~Φji. |Ωi|is the cell volume of Ωi and ~nij is the outward unit vector normal to the face associated with the grid edgeconnecting i and j. Γij is the area of the face between i and j and mi is the number of neighbors of thenode i. Convective fluxes, from node i to node j across dual grid interfaces, are reduced to one dimensionalproblems by projecting ~F and ~Φ onto ~nij .

B. Shape & Volume Deformation

Shape deformation in three dimensions is achieved using a Free-Form Deformation (FFD) strategy.30,31 Here,an initial box encapsulating the design surface is parameterized as a Bezier solid. A set of control pointsare defined on the surface of the box, the number of which depends on the order of the chosen Bernsteinpolynomials. The solid box is parameterized by the following expression

X(u, v, w) =

l∑i=0

m∑j=0

n∑k=0

Pi,j,kBli(u)Bmj (v)Bnk (w), (34)

where l, m, n are the degrees of the FFD function, u, v, w ∈ [0, 1] are the parametric coordinates, Pi,j,k arethe coordinates of the control point (i, j, k), and Bli(u), Bmj (v) and Bnk (w) are the Bernstein polynomials.The Cartesian coordinates of the points on the surface of the object are then transformed into parametriccoordinates within the Bezier box. Control points of the box become design variables, as they control theshape of the solid, and thus the shape of the surface grid inside. The box enclosing the geometry is thendeformed by modifying its control points, with all the points inside the box inheriting a smooth deformation.With FFD, arbitrary changes to the thickness, sweep, twist, etc. are possible for the design of any aerospacesystem. Once the deformation has been applied, the new Cartesian coordinates of the object of interest canbe recovered by simply evaluating the mapping inherent in Eq. 34.

(a) Surface deformation using FFD control point movement. (b) Volumetric mesh deformation.

Figure 2. Shape deformation within SU2.

Deformation of the volumetric grid is achieved by modeling the computational domain as an elasticsolid. Each cell within the grid is assigned a variable Young’s modulus corresponding to the volume ofthe cell. With this strategy, near-body cells undergo nearly rigid-motion, while larger cells further from

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the body accommodate the deformation of the surface. The linear elasticity equations are solved using aFinite-Element Method.

IV. Surface Gradient Validation

The derivation of the adjoint system of equations, boundary conditions, and surface sensitivity in SectionII enables adjoint-based gradient calculations of force-based objective functions in inviscid non-equilibriumflows. In this section, we validate the adjoint derivation by comparing the adjoint-based gradient calculationsfor drag coefficient, CD, to gradients calculated using a finite-difference method. Two validation cases arepresented: a sphere-cone blunt body, and a lifting surface.

A. RAM-C II Flight Test

1. Problem Description

During the late 1960s, a series of hypersonic flight tests32 were conducted by NASA to quantify electronnumber densities around blunt-body entry vehicles. The 9 sphere-cone RAM-C II test article is of particularinterest in that series due to its non-ablative, beryllium nose cap. This characteristic makes the vehicle anideal candidate for verification and validation of computational tools. Specifics of the RAM-C II geometryand experimental conditions are provided in Tab. (1) & (2).

Nose Radius 0.1524m

Cone Half Angle 9

Vehicle Length 1.295m

Table 1. RAM-C II vehicle geometry.

Case 5 Case 6 Case 7

H (km) 61 71 81

T∞ (K) 254 216 181

Re 19500 6280 1590

M∞ 23.9 25.9 28.3

Table 2. RAM-C II flight conditions.

For this work, we simulate the ‘Case 5’ conditions using a two-species Nitrogen gas chemistry modeldetailed in Appendix A. This model, while less accurate than a more sophisticated air gas chemistry model,exhibits all the appropriate physics necessary to demonstrate the adjoint methodology for non-equilibriumproblems. As our primary interest is in validating the gradients for a given gas model in thermochemicalnon-equilibrium, this is an acceptable simplification to reduce the overall computational expense that isintrinsic to the finite-difference analysis.

2. Computational Domain & Design Variable Description

The computational domain for the RAM-C II problem consists of a body-conformal, 63x65x3, hexahedralmesh. A 10 sector of the axisymmetric flow field is simulated with characteristic-based far field inflow andoutflow boundaries and flow-tangency enforced on the vehicle surface and side-walls of the sector.

The design variables for the surface perturbations consist of manipulations of FFD control points. Specifi-cally, we parameterize the box with 4th, 3rd, and 2nd order Bernstein polynomials in the i, j, and k dimensionsrespectively. Surface deformation is performed by manipulating the three control points at fixed i & j coor-dinates of the box. The cartesian displacement of the control points is specified by the user and is shown inTab. (2). The mesh and design variables for this test case are shown in Fig. (3).

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Design Var. FFD Nodes (i, j, k) (∆x, ∆y, ∆z)

0 (0,0,0), (0,0,1), (0,0,2) (-1, 0, 0), (-1, 0, 0), (-1, 0, 0)

1 (0,1,0), (0,1,1), (0,1,2) (-1, 0, 0), (-1, 0, 0), (-1, 0, 0)

2 (0,2,0), (0,2,1), (0,2,2) (-1, 0, 0), (-1, 0, 0), (-1, 0, 0)

3 (0,3,0), (0,3,1), (0,3,2) (-1, 1, 0), (-1, 1, 0.0875), (-1, 1, 0.1763)

4 (1,3,0), (1,3,1), (1,3,2) (-1, 0, 0), (0, 1, 0.0875), (-1, 0, 0.1763)

5 (2,3,0), (2,3,1), (2,3,2) (-1, 0, 0), (0, 1, 0.0875), (-1, 0, 0.1763)

6 (3,3,0), (3,3,1), (3,3,2) (-1, 0, 0), (0, 1, 0.0875), (-1, 0, 0.1763)

7 (4,3,0), (4,3,1), (4,3,2) (-1, 0, 0), (0, 1, 0.0875), (-1, 0, 0.1763)

Table 3. RAM-C II design variable definitions.

(a) X-Y plane projection of the mesh. (b) FFD box surrounding the surface geometry.

Figure 3. Computational domain and FFD design variable locations for the RAM-C II geometry.

3. Results

X (m)

Y (m

)

0 0.2 0.40

0.2

0.4

0.6lN2 (kg/m3)

0.00130.00120.00110.0010.00090.00080.00070.00060.00050.00040.00030.00020.0001

(a) N2 density contours along the Z = 0 plane.

X (m)

Y (m

)

0 0.2 0.40

0.2

0.4

0.6s N2

0.40.350.30.250.20.150.10.050-0.05-0.1-0.15-0.2

(b) Adjoint-N2 density contours.

Figure 4. Direct and adjoint solutions of the baseline configuration.

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To acquire the drag coefficient gradients, nine direct solutions and one adjoint solution were run. Allsolutions are converged O(10−10) in the density and adjoint-density residuals. Fig. (4) shows the directand adjoint solutions within the computational domain for ρN2 and corresponding adjoint variable ψρN2

.Closer examination of the adjoint solution reveals an area of sensitivity along the shoulder of the sphere-conebody, in the vicinity of design variable 3. This sensitivity region is confirmed upon examination of Fig. (5),indicating design variable 3 has the greatest influence on the calculated vehicle drag coefficient. For designcases with more complex geometries, this kind of information from the adjoint flow field is useful in specifyingdesign parameters that will have the most effect on the quantity of interest.

0 1 2 3 4 5 6 7

Design Variable #

0.010

0.005

0.000

0.005

0.010

0.015

0.020

CD/y i

FD

Adjoint

Figure 5. Gradient comparison between FD andadjoint surface sensitivity.

An examination of the drag coefficient gradient withrespect to the eight prescribed surface deformations showgood agreement between the finite-difference analysis andthe surface sensitivities calculated using the adjoint ap-proach. The finite difference step in the analysis is 5x10−3

and this factor is applied to the prescribed control pointmovements described previously. Some differences be-tween the approaches is expected, due to the error in thegradient calculation introduced by the finite-differencestep size and also due to the numerical methods used tosolve the adjoint equations (and consequently the surfacesensitivities). Specifically, we see the largest discrepanciesfor design variables 0 and 1, corresponding to the controlpoints nearest the stagnation line. Contributions to vehi-cle drag are largest in this area, but we would not expectnearly-normal surface deformation to have a significanteffect on the calculated drag force in this region. As aconsequence, step-size related errors are expected to belargest in this region. Moreover, the direct solution ex-hibits some carbuncle-like behavior along the stagnationline, and it is likely that the surface perturbations near the stagnation line are influencing the size and shapeof the phenomena, thus affecting the quality of the gradient information in this region.

B. Hypersonic NACA 0012 Wing

1. Problem Description

This test case is intended to demonstrate the adjoint methodology on a simple, yet representative geometry forlifting surfaces on hypersonic vehicles. The NACA 0012 airfoil is a canonical case for low speed aerodynamics,but its blunt leading edge and large thickness-to-chord ratio share similar characteristics to the aerodynamicsurfaces onboard existing hypersonic space planes. The airfoil is extruded in the y-direction to create athree-dimensional domain. Free-stream conditions for this test case are shown in Tab. (4).

P∞ (Pa) 20.0

T∞ (K) 273.15

M∞ 24.0

α 3

Table 4. NACA 0012 free-stream conditions.

2. Computational Domain & Design Variable Description

The computational mesh for the NACA 0012 airfoil consists of a 129x5x41 hexahedral O-mesh and a pro-jection of the mesh on the X-Z plane is shown in Fig. (6). The mesh is grown from the airfoil surface usinghyperbolic extrusion, then extruded again in the y-direction to create the three dimensional domain.

The FFD box for this case is parameterized with 4th, 1st, and 2nd order Bernstien polynomials in the i,j, and k dimensions respectively. Control point manipulation is constrained to the ±z axis for the upper and

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lower surfaces of the of the wing along the centerline, such that a positive deformation results in an increasein the local thickness.

(a) X-Y plane projection of the mesh. (b) FFD box surrounding the surface geometry.

Figure 6. Computational domain and FFD design variable locations for the RAM-C II geometry.

Design Var. FFD Nodes (i, j, k) (∆x, ∆y, ∆z)

0 (0,1,1) (0, 0, 1)

1 (1,1,1) (0, 0, 1)

2 (2,1,1) (0, 0, 1)

3 (3,1,1) (0, 0, 1)

4 (0,0,1) (0, 0, -1)

5 (1,0,1) (0, 0, -1)

5 (2,0,1) (0, 0, -1)

5 (3,0,1) (0, 0, -1)

Table 5. NACA 0012 design variable definitions.

3. Results

0 1 2 3 4 5 6 7

Design Variable #

0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

CD/y i

FD

Adjoint

Figure 7. Gradient comparison between FD andadjoint surface sensitivity.

For this test case, we again compare the adjoint-basedgradient to the finite-difference gradient for a drag-coefficient objective function. As in the RAM-C II testcase, N2 density and adjoint densities were convergedO(10−10) orders of magnitude to provide satisfactory ac-curacy in the comparison of the force coefficients. Directand adjoint solutions are shown in Fig. (8). A compari-son of the two gradients shows excellent agreement, withonly minor discrepancies at design variables 0 and 3 (cor-responding to the control points nearest the leading edgeof the wing on the upper and lower surfaces). These dif-ferences are within the expected variation between thegradient calculation methods.

At this point, it is worth emphasizing the utility ofthe adjoint approach again. The finite-difference gradientcalculation requires a baseline solution and 8 perturbedsolutions for each of the design variables. On the otherhand, the adjoint approach requires only one direct solu-tion and one adjoint solution that is of comparable com-putational expense as the direct problem. This benefit is

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X/c

Y/c

-0.5 0 0.5 1 1.5-1

-0.5

0

0.5

1

Mach

222018161412108642

(a) Mach contours around the NACA 0012 wing.

X/c

Y/c

-0.5 0 0.5 1-1

-0.5

0

0.5

1 lE

1E-077E-084E-081E-08-2E-08-5E-08-8E-08-1.1E-07-1.4E-07-1.7E-07-2E-07-2.3E-07-2.6E-07

(b) Adjoint energy field.

Figure 8. Direct and adjoint solutions of the NACA 0012 wing configuration.

compounded as the dimensionality of the design space increases since the expense of the adjoint problemremains fixed, while the finite-difference approach scales linearly with the number of design variables.

V. Conclusion

In this work, we have derived a continuous adjoint formulation for a steady, inviscid, continuum, multi-species fluid in thermochemical non-equilibrium. Using principles from differential geometry, the adjointboundary conditions are surface sensitivities are written concisely for easy implementation in computationalfluid dynamics codes. This formulation was implemented in a three-dimensional, unstructured CFD solver,and tested against finite-difference gradients for a blunt-body and lifting surface geometries. In both cases,the adjoint-based drag coefficient gradients showed excellent agreement with the finite-difference method. Us-ing the adjoint approach, high-fidelity gradient information can be efficiently acquired in the non-equilibriumaerothermodynamic environment, even in the presence of large numbers of design variables and for complexgeometries. These characteristics make the adjoint method a powerful tool for next-generation analysis anddesign in the hypersonic regime.

References

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11Giles, M., Duta, M., Muller, J., and Pierce, N., “Algorithm Developments for Discrete Adjoint Methods,” AIAA Journal ,Vol. 41, No. 2, 2003, pp. 198–205.

12Anderson, W. K. and Venkatakrishnan, V., “Aerodynamic Design Optimization on Unstructured Grids with a ContinuousAdjoint Formulation,” 35th AIAA Aerospace Science Meeting & Exhibit , AIAA Paper 97-0643, Reno, NV, 1997.

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14Flynt, B. T. and Mavriplis, D. J., “Discrete Adjoint Based Adaptive Error Control in Unsteady Flow Problems,” 50thAIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, AIAA Paper 2012-0078,Nashville, TN, January 2012.

15Duraisamy, K., Alonso, J., Palacios, F., and Chandrashekar, P., “Error Estimation for High Speed Flows Using Continuousand Discrete Adjoints,” 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition,AIAA Paper 2010-128, Orlando, FL, January 2010.

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24Bueno-Orovio, A., Castro, C., Duraisamy, K., Palacios, F., and Zuazua, E., “When the ‘exact’ discrete gradient is notthe best choice in optimal shape design?” 49th AIAA Aerospace Science Meeting including the New Horizons Forum andAerospace Exposition, AIAA Paper 2011-1298, Orlando, FL, 2011.

25Park, C., Nonequilibrium Hypersonic Aerothermodynamics, Wiley, New York, NY, 1990.26Lee, J. H., “Basic Governing Equations for the Flight Regimes of Aeroassisted Orbital Transfer Vehicles,” Thermal Design

of Aeroassisted Orbital Transfer Vehicles, edited by H. F. Nelson, Vol. 96, AIAA, New York, 1985, pp. 3–53.27Palacios, F., Alonso, J. J., Duraisamy, K., Colonno, M. R., Aranake, A. C., Campos, A., Copeland, S. R., Economon,

T. D., Lonkar, A. K., Lukaczyk, T. W., and Taylor, T. W. R., “Stanford University Unstructured (SU2): An Open SourceIntegrated Computational Environment for Multiphysics Simulation and Design,” 51st AIAA Aerospace Sciences MeetingIncluding the New Horizons Forum and Aerospace Exposition (Submitted), Grapevine, TX, 2013.

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AIAA Paper 1985-0247, Reno, NV, Jan. 1985.

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A. Chemical Model

A two-species Nitrogen chemical model is used with the following chemical reactions:

N2 +N2 ←→ 2N +N2, (35a)

N2 +N ←→ 3N. (35b)

(35c)

The extent of each of the chemical reactions can be determined using classical statistical mechanics33 andthe Law of Mass Action.

Ri = −kfi∏

j=reactantsi

(ρjMj

)+ kbi

∏k=productsi

(ρkMk

), (36)

where the chemical rate constants kf and kb in Eqn. (36) are Arrhenius-type relations with parametersCfm , ηm, θm and Keq determined experimentally.?, 34

kfm = kfm(T ) = Cfm Tηm exp−θm/T , (37a)

kbm = kbm(T ) = kfm/Keq(T ), (37b)

Keqm = Keqm((T /Tref ) = Z) = exp(A1m +A2mZ +A3mZ

2 +A4mZ3 +A5mZ

4). (37c)

Each of the rate constants enumerated in Eqns. (37a-37c) are temperature dependent. The appropriatetemperature, T depends on the type of reaction and the collision partners. The chemical system is coupledto the flow equations via source terms, ws, that are determined by summing the appropriate extent ofreaction, Eqn. (36), corresponding to the chemical reactions Eqns. (35a & 35b) where the species appears aseither a product or a reactant.

ws = Ms

− ∑i:s∈productsi

Ri +∑

j:s∈reactantsj

Rj

. (38)

R1 = −kf1,m(ρN2

MN2

)(ρN2

MN2

)+ kb1,m

(ρNMN

)(ρNMN

)(ρN2

MN2

)(39a)

R2 = −kf1,m(ρN2

MN2

)(ρNMN

)+ kb1,m

(ρNMN

)(ρNMN

)(ρNMN

)(39b)

(39c)

wN2= MN2

(R1 +R2) , (40a)

wN = MN (−2R2 − 2R1) . (40b)

(40c)

B. Fluid Dynamics Model

Translational-rotational and vibrational energy per unit mass are defined,

et−rs =

(32 + ξs

2

)Ru

MsT, for s 6= el.

32Ru

MsT v−e, for s = el.

(41)

evs =

Ru

Ms

θvsexp(θvs/T

v−e)−1 , for polyatomic species.

0 for monatomic species & electrons.(42)

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eels =

Ru

Ms

∑∞i=1 gi,sθ

eli,s exp(−θeli,s/T

v−e)∑∞i=0 gi,s exp(−θeli,s/Tv−e)

, for s 6= el.

0 for s = el.(43)

where corresponding translational-rotational and vibrational-electronic temperatures can be determined byinverting Eqns. (?? & ??). Vibrational energy is modeled as a harmonic oscillator with characteristicvibrational temperature θvibs . Partial pressures are assumed to obey the ideal gas law with contributiononly from the translational-rotational states, and mixture pressure obeys Dalton’s law of partial pressures

Translational-vibrational exchanges are described by a Landau-Teller relation between species r andspecies s,

qt:v =

p∑s

ρsEv−e∗s (T )− Ev−es⟨

τsL−T

⟩ , (44)

with vibrational relaxation time,

⟨τsL−T

⟩=

∑rNr∑

rNr/τr,sL−T

, for r 6= e. (45)

Landau-Teller inter-species relaxation time (and constants),

τr,sL−T=

1

pexp

[Ar,s(T

−1/3 − 0.015µ1/4r,s )− 18.42

], p in atm,. (46)

Ar,s = 1.16× 10−3µ1/2r,s θ

4/3vs . (47)

µr,s =MrMs

Mr +Ms. (48)

C. Jacobian Matrices

~Ac·~n =

(δs,r − Ys)(~u · ~n) Ysnx Ysny Ysnz 0 0

−u(~u · ~n) ∂P∂ρr nx (~u · ~n) + unx + ∂P∂(ρu) nx uny + ∂P

∂(ρv) nx unz + ∂P∂(ρw) nx

∂P∂(ρE) nx

∂P∂(ρEv−e) nx

−v(~u · ~n) ∂P∂ρr ny vnx + ∂P∂(ρu) ny (~u · ~n) + vny + ∂P

∂(ρv) ny vnz + ∂P∂(ρw) ny

∂P∂(ρE) ny

∂P∂(ρEv−e) ny

−w(~u · ~n) ∂P∂ρr nz wnx + ∂P∂(ρu) nz wny + ∂P

∂(ρv) nz (~u · ~n) + wnz + ∂P∂(ρw) nz

∂P∂(ρE) nz

∂P∂(ρEv−e) nz

( ∂P∂ρr −H)(~u · ~n) ∂P∂(ρu) (~u · ~n) +Hnx

∂P∂(ρv) (~u · ~n) +Hny

∂P∂(ρw) (~u · ~n) +Hnz (1 + ∂P

∂(ρE) )(~u · ~n) ∂P∂(ρEv−e) (~u · ~n)

−ρEv−e

ρ (~u · ~n) ρEv−e

ρ nxρEv−e

ρ nyρEv−e

ρ nz 0 (~u · ~n)

(49)

D. Differential Geometry Formulas

∫δS

qds =

∫S

[∂nq − 2Hmq]δSds, (50a)

δ ~ns = −∇sδS, (50b)∫S

∇s · ~qds = 0, (50c)

∇ · ~q = ∂n(~q · ~ns)− 2Hm(~q · ~ns) +∇s · ~q. (50d)

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adjoint-based aerothermodynamic shape design of hypersonic

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