2D Euler shape design on non-regular flows using

adjoint Rankine-Hugoniot relations

Antonio Baeza∗

Madrid Institute for Advanced Studies in Mathematics, Madrid, 28049, Spain

Carlos Castro†

Universidad Politecnica de Madrid, Madrid, 28040, Spain

Francisco Palacios‡

Madrid Institute for Advanced Studies in Mathematics, Madrid, 28049, Spain

Enrique Zuazua§

Universidad Autonoma de Madrid, Madrid, 28049, Spain

Optimal Shape Design (OSD) aims at finding the minimum of a functional by controllingthe Partial Differential Equation (PDE) modeling the flow using surface (domain bound-aries) deformation techniques. As a solution to the enormous computational resourcesrequired for classical shape optimization of functionals of aerodynamic interest, one of thebest strategies is to use methods inspired in control theory. To do this, one assumes that agiven aerodynamic surface is an element that produces lift or drag by controlling or mod-ifying the flow. One of the key ingredients in the application of control methods relies onthe use of the adjoint system techniques to simplify the computation of gradients.

In this paper we will restrict our attention to optimal shape design in 2D systems gov-erned by the Euler equations with discontinuities in the flow variables (an isolated normalshock wave). We first review some facts on control theory applied to optimal shape design,and recall the 2D Euler equations (including Rankine-Hugoniot relations). We then studythe adjoint formulation, providing a detailed exposition of how the derivatives of func-tionals may be obtained when a discontinuity appears. Further on, adjoint equations willbe discretized and analyzed and some novel numerical experiments with adjoint Rankine-Hugoniot relations will be shown. Finally, we expose some conclusions about the viabilityof a rigorous approach to the continuous Euler adjoint systems with discontinuities in theflow variables.

I. Introduction

In last decades, OSD has evolved very close to the Computed Fluid Dynamics (CFD) developments. By theeighties, advances in computer hardware and algorithms had made feasible to develop accurate and efficient

analysis tools for inviscid flows.1 On the other hand, some of the groundbreaking works in Control Theoryare due to J.-L. Lions.2 Several years later O. Pironneau investigated the problem of optimum shape designfor elliptic equations using control theory.3 In the late eighties A. Jameson4 was the first to apply thesetechniques to the Euler and Navier-Stokes equations in the field of aeronautical applications. At the beginningof the XXI century new techniques as the Reduced Gradient Formulation5 and the Systematic Approach6

made a significant simplification to the continuous adjoint implementation on unstructured meshes.∗Research Scientist, iMath Project.†Assistant Professor, Departamento de Matematicas e Informatica, ETSI Caminos, Canales y Puertos.‡Director of Technological Innovation, IMDEA-Mathematics, AIAA Member.§Professor, Departamento de Matematicas, Facultad de Ciencias. Director, IMDEA-Mathematics.

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In gradient-based optimization techniques, the goal is to minimize a suitable cost or objective function(drag coefficient, deviation from a prescribed surface pressure distribution, etc.) with respect to a set ofdesign variables (defining, for example, an airfoil profile or aircraft surface). Minimization is achieved bymeans of an iterative process which requires the computation of the gradients or sensitivity derivatives ofthe cost function with respect to the design variables.

Dealing with gradients computation of an objective function defined in smooth inviscid flows, the pertur-bation of the flow field variables can be calculated from a linearization of the governing equation (or usinga more elaborate technique like the adjoint state). However, this is not valid in the neighborhood of flowdiscontinuities. Several options had been proposed by some authors, Iollo and Salas,7 Giles and Pierce8 orMatsuzawa and Hafez.9 Currently, most existing works ignore the shock motion sensitivity supposing thatshocks are smeared using numerical dissipation. However, this paper is intended to clarify that accuratetreatment of shock waves could be essential.

Aerodynamical applications of optimal shape design10 in systems governed by PDEs are formulated ona fluid domain Ω, containing a compressible fluid, usually air, delimited by disconnected boundaries dividedinto a “far field” Γ∞ and one or more solid wall boundaries S, usually airplane surfaces (see Fig. 1).

This kind of design problems are aimed at minimizing a functional J of the flow defined on the boundariesS, Γ∞, or in a subdomain ω ⊂ Ω. From now on we will restrict ourselves to the analysis of optimizationproblems involving functionals defined on the solid wall S, whose value depends on the flow variables Uobtained of the corresponding fluid flow equations. In this context, the generic optimization problem can besuccinctly stated as follows: find Smin ∈ Sad such that

J(Smin) = minS∈Sad

J(S), (1)

where Smin is the sought optimal surface, belonging to the set Sad of admissible boundary geometries and

J(S) =∫

S

j(U) ds, (2)

is the objective function, whose evaluation is subject to the resolution of the flow equations to obtain U .

Figure 1. Classical optimal design problem.

Let us consider an arbitrary (but small) perturbation of the boundary S which, without loss of generality,can be parameterized by an infinitesimal deformation of size δS along a outward pointing unitary normalvector ~nS :

S′ = ~x + δS (~x)~nS (~x) , ~x ∈ S . (3)

Assuming a regular flow solution U , the variation of the functional J under the deformation can beevaluated as11

δJ (S) =∫

δS

j (U) ds +∫

S

j′ (U) δU ds, (4)

where the first term, which stems from the displacement of the boundary, takes the form∫

δS

j (U) ds =∫

S

(∂nj − κj) δS ds, (5)

where ∂nj = ~nS · ~∇j is the normal derivative of j (U) and κ is the curvaturea of S. The second term in (4)is the contribution due to infinitesimal changes in the flow solution induced by the deformation.

aFor a plane curve given parametrically as f (η) = (x(η), y(η)) the curvature is defined as κ =˛˛ xy−yx

(x2+y2)3/2

˛˛, where the dot

denotes differentiation with respect to η.

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On the other hand, non-regular solutions of the flow variables are the most common in aeronautical appli-cations (in transonic and supersonic flow regime). If the body immersed in the fluid has an aerodynamicallysharp shape,12 then the discontinuity usually touches the body surface.13 In these cases a discontinuity (sonicshock wave) along a regular curve Σ must be considered (see Fig. 2) and the Rankine-Hugoniot relationsmust be added to the Euler equations to correctly account for the presence of the shock.

Figure 2. Optimal design problem with a shock wave Σ.

When taking flow discontinuities into account the previous computation of the derivative of the functionalfails and has to be modified by including the effect due to the sensitivity of the shock location with respectto shape deformations.14 Let xb = Σ ∩ S and x∞ = Σ ∩ Γ∞, that we assume to be unique points. Then theexpression for δJ is

δJ (U) =∫

δS

j (U) ds +∫

S\xbj′ (U) δU ds− [j(U)]xb

δΣ(xb)− [j(U)]xb(~nS · ~nΣ) δS(xb). (6)

The most expensive terms (in terms of time and required computational resources) are those which involvethe computation of δU and δΣ. These can be obtained by solving the linearized flow equations (togetherwith the linearized Rankine-Hugoniot conditions) once per each independent deformation (design variable).If the design space is large, as is the case in real applications, the computational cost is prohibitive. It isthen convenient to switch to the control theory approach, which reduces significantly the computational costof getting the gradients, using the adjoint or dual formulation of the shape design problem.

II. 2D Euler equations and Rankine-Hugoniot relations

Ideal fluids are governed by the Euler equations,13,15 which express the conservation of mass, momentum(with null viscosity) and energy. In the aeronautical framework, these equations are considered in a domainΩ delimited by disconnected boundaries divided into “far field” Γ∞ and solid wall boundaries S. The mostcommon way to pose the Euler equations is in conservative form:

∂tU + ~∇ · ~F = 0, in Ω, (7)

where UT = (ρ, ρvx, ρvy, ρE) are the conservative variables and ~F = (Fx, Fy) is the convective flux vector

Fx =

ρvx

ρv2x + P

ρvxvy

ρvxH

, Fy =

ρvy

ρvxvy

ρv2y + P

ρvyH

, (8)

where ρ is the fluid density, ~v = (vx, vy) is the flow speed in a Cartesian system of reference, E is the totalenergy, P the system pressure and H the enthalpy. The system of equations (7) must be completed by anequation of state which defines the thermodynamic properties of the fluid. For a perfect gas:

P = (γ − 1) ρ

[E − 1

2|~v|2

], (9)

where γ ≈ 1.4 for standard air conditions, and the identity ρH = ρE + P holds.

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On the other hand, the Euler equations (7) have to be completed with the following boundary conditions

~v · ~nS = 0, on S, (10)

where ~nS is an inward-pointing unit vector normal to S, and at the “far field” boundary Γ∞ boundaryconditions are specified for incoming waves, while outgoing waves are determined by the solution inside thefluid domain.16

Inviscid flows described by the Euler equations can develop discontinuities (shocks or contact discontinu-ities) due to the intersection of flow characteristics. If the shock wave is perpendicular to the flow directionit is called a normal shock. When this occurs, the Rankine-Hugoniot conditions relate the flow variables onboth sides of the discontinuity. For a shock located at Σ which propagates with speed s, these relations are

[~F · ~nΣ

]Σ− s [U ]Σ = 0, (11)

where ~nΣ = (nΣx, nΣy) is the unit vector normal to the curve Σ pointing in the same direction as the shockspeed s, and [A]Σ represents the jump of A across the discontinuity curve Σ, that is to say, [A]Σ = A+−A−.Using the Euler equations, the Rankine-Hugoniot relations can be written as

[ρ~v · ~nΣ]Σ − s [ρ]Σ = 0,

[(ρ~v · ~nΣ)vx + PnΣx]Σ − s [ρvx]Σ = 0,

[(ρ~v · ~nΣ)vy + PnΣy]Σ − s [ρvy]Σ = 0,

[Hρ~v · ~nΣ]Σ − s [ρE]Σ = 0.

(12)

If a steady problem is considered, the discontinuity velocity vanishes and then (12) is simplified to

[ρ~v · ~nΣ]Σ = 0,

[vx]Σ ρ~v · ~nΣ + [P ]Σ nΣx = 0,

[vy]Σ ρ~v · ~nΣ + [P ]Σ nΣy = 0,

[H]Σ = 0,

(13)

in this case, along the discontinuity, the following holds17

[ρ]Σ 6= 0, [P ]Σ 6= 0, [~v · ~nΣ]Σ 6= 0,[~v · ~tΣ

]Σ

= 0, (14)

where, to define ~nΣ = (nΣx, nΣy) we first define ~tΣ = (tΣx, tΣy) as a unitary tangent vector to the discon-tinuity beginning at the solid surface and pointing to the “far field” boundary, then ~nΣ = (nΣx, nΣy) is theπ/2 counter-clock-wise rotation of ~tΣ.

III. Continuous adjoint formulation for steady Euler equations

When developing an adjoint method to address optimal design problems in aeronautics, one of themain mathematical difficulties is the presence of discontinuities (sonic shock waves).8,9, 18–20 This is due, inparticular, to the intrinsic complexity of the adjoint system in the presence of shocks. Indeed, in the presenceof shock discontinuities, the formal linearization of the state equations, which can be rigorously justified forsome solutions, fails to be true and the adjoint system changes its nature. Indeed, when this occurs, thestate of the systems needs to be rather understood as a multi-body one in which both the state itself at bothsides of the shock and the location of the shock are considered as part of the state.

Thus, the sensitivity of the model needs to be analyzed with respect to perturbations of the solutionas well as to the location of the shock. The linearized flow equations turn out to be the classical ones onboth sides of the shock, plus an additional linear transport equation along the shock which stems from thelinearization of the Rankine-Hugoniot conditions. This allows defining the adjoint solution in a unique way.

III.A. Analytical formulation of the continuous adjoint method

The adjoint formulation is applied to an optimization problem defined in (1) where the objective is to evaluatethe variation of the functional (2) under shape changes of the surface S, where the flow governing equationsare the steady Euler equations.

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Assuming a flow discontinuity located along a smooth curve Σ that meets the boundary S at a pointx = xb and is oriented in such a way that it begins in xb, the variation of the functional δJ is written as:

δJ =∫

δS

j(U) ds +∫

S\xbj′(U)δU ds− [j(U)]xb

δΣ(xb)− [j(U)]xb(~nS · ~nΣ) δS(xb), (15)

where δΣ parameterizes an infinitesimal normal deformation of the discontinuity due to a change in thesurface S, and δU solves the linearized Euler equations

~∇ ·(

~A δU)

= 0, in Ω,

δ~v · ~nS = −δS ∂n~v · ~nS + (∂tgδS)~v · ~tS , on S,

(δW )+ = 0, on Γ∞,

(16)

with (δW )+ representing the incoming characteristics on the “far field” boundary which correspond tophysical boundary conditions in the Euler problem. ∂ ~F/∂U = ~A is the Jacobian matrix, ∂n = ~n · ~∇ and∂tg = ~t · ~∇ are the normal and tangential derivatives (respectively).

In this case, δS, which parameterizes infinitesimal deformations along the normal direction (3), is aninput datum to the design problem. In practice, δS has to be directly realized by means of the admissibledesign variables thus making impossible arbitrary deformations.21 Therefore, once the continuous analysishas been developed, allowing arbitrary deformations, a careful numerical interpretation is required to transferthose results to the context of the admissible design variables.

In (15), δΣ parameterizes an infinitesimal normal deformation of the discontinuity due to a change δSin the surface S. Note that the shock position Σ depends on S thought the resolution of the flow equations.The displacement of the shock due to the variation δΣ in the direction normal to the shock is another smoothcurve Σ′,

Σ′ = ~x + δΣ(~x)~nΣ (~x) , ~x ∈ Σ . (17)

Recall that δΣ is not a design parameter, but rather a dependent variable whose value is determinedby the simultaneous solution of the linearized Euler equations (16) and the linearized Rankine-Hugoniotconditions [

~A(δΣ ∂nU + δU)]Σ· ~nΣ +

[~F]Σ· δ~nΣ = 0. (18)

In order to eliminate δU and δΣ from (15), the adjoint problem is introduced through the Lagrange mul-tipliers ΨT = (ψ1, ψ2, ψ3, ψ4). Generally speaking, the method of Lagrange multipliers allows for calculatingthe minimum of a constrained multi-variate function, the constraints being in this case the Euler equations.

The first step of the procedure amounts to multiplying the linearized Euler 16 equations by Ψ and tointegrating them over the domain

0 =∫

Ω\ΣΨT ~∇ ·

(~A δU

)dΩ

= −∫

Ω\Σ~∇ΨT · ~A δU dΩ +

∫

S\xbΨT ~A δU · ~nS ds +

∫

Σ

[ΨT ~A δU

]Σ

~nΣ ds. (19)

Let us now analyze separately each of the terms of (19) so as to derive the adjoint system that will allowthe direct computation of the variation of the functional:

• The first term of (19) is a volume integral that vanishes provided Ψ satisfies the steady Euler adjointequation

− ~AT · ~∇Ψ = 0. (20)

• The second term of (19) is an integral over the solid surface S. Substituting the Jacobian matrix byits value the integral becomes

∫

S\xbΨT ~A δU · ~nS ds

=∫

S\xb(δ~v · ~nS) ϑds +

∫

S\xb(~ϕ · ~nS) δP ds

= −∫

S\xb

((∂n~v · ~nS)ϑ + ∂tg

((~v · ~tS

)ϑ))

δS ds +∫

S\xb(~ϕ · ~nS) δP ds. (21)

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where ~ϕ = (ψ2, ψ3) and ϑ = ρψ1 + ρ~vS · ~ϕ + ρHψ4.

• The third term of (19) is an integral over the discontinuity curve Σ that touches the solid surface atthe point xb and the far field boundary at x∞. This integral can be expanded by taking into accountthe conditions the adjoint variables must satisfy at the discontinuity,

∫

Σ

[ΨT ~AδU

]Σ· ~nΣ ds =

∫

Σ

[ΨT

]Σ

~AδU · ~nΣ ds +∫

Σ

ΨT

[~AδU

]Σ· ~nΣ ds, (22)

where f is the mean value computed as (f+ + f−)/ 2. From the linearized Rankine-Hugoniot conditions(18) we have

[~AδU

]Σ

= −δΣ[~A∂nU

]Σ· ~nΣ −

[~F]Σ· δ~nΣ = −δΣ

[∂n

(~F · ~nΣ

)]Σ

+ ∂tg(δΣ)[~F]Σ· ~tΣ. (23)

On the other hand, on Σ we can decompose the divergence operator in the Euler equation into itstangential and normal components as follows:

0 = ~∇ · ~F∣∣∣Σ

= ∂tg(~F · ~tΣ) + κΣ~F · ~nΣ + ∂n(~F · ~nΣ), (24)

where κΣ is the curvature of Σ, and this allows to simplify the normal derivative of ~F · ~nΣ in (23).Thus, taking into account the Rankine-Hugoniot condition we obtain

[~AδU

]Σ

= ∂tg

(δΣ

[~F · ~tΣ

]Σ

). (25)

We now replace this expression into (22) and integrate by parts,∫

Σ

[ΨT ~AδU

]Σ· ~nΣ ds = −

∫

Σ

∂tgΨT

[~F · ~tΣ

]Σ

δΣ ds−ΨT(xb)

[~F · ~tΣ

]xb

δΣ(xb)), (26)

where the continuity of the adjoint variables through the shock[ΨT

]Σ

= 0 has been used.

Having analyzed the first three terms in (19), this identity can be rewritten as∫

S\xb(~ϕ · ~nS) δP ds = −

∫

Σ

∂tgΨT

[~F · ~tΣ

]Σ

δΣ ds

−ΨT (xb)[~F · ~tΣ

]xb

δΣ (xb) +∫

S\xb

((∂n~v · ~nS)ϑ + ∂tg

((~v · ~tS

)ϑ))

δS ds. (27)

This equation will be used to eliminate the linearized variables from the variation of the functional J(15) upon identifying the corresponding terms in (27) and (15). Since the only unknown variation δU ofthe flow variables left in (27) corresponds to the pressure, only functionals which depend on the pressure Palone are allowed a priori.22 So in (15), we assume that j(U) = j(P ) . Luckily, functionals which dependsolely on the pressure are the most common in aerodynamic design applications with Euler equations (e.g.lift or drag coefficients). Thus, (15) can be written as

δJ(S) =∫

S

((∂nj(P )− κj(P )) δS ds

+∫

S\xb

((∂n~v · ~nS)ϑ + ∂tg

((~v · ~tS

)ϑ))

δS ds− [j(P )]xb(~nS · ~nΣ) δS (xb)

−∫

Σ

∂tgΨT

[~F · ~tΣ

]Σ

δΣ ds +(

ΨT (xb)[~F · ~tΣ

]xb

− [j(P )]xb

)δΣ(xb) . (28)

Obviously, in shape design problems δS is known (design variables), but the dependence between δΣand δS (through the linearized Euler equation and R-H relations) is unknown. Thus the expression (28) isnot fully satisfactory since it does not allow to compute the variation of the functional J directly out of thesolutions of the state and adjoint equations, and the variation of S. But, let us, by now, maintain δΣ in our

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calculus. Also, in order to cancel the terms containing δP on solid surface S and “far field” Γ∞, the adjointequation is stated as follow

− ~AT · ~∇Ψ = 0, (29)

and the following adjoint boundary conditions must be imposed

~ϕ · ~nS = j′(P ), on S \ xb,ΨT

(~A · ~nΓ∞

)δU = 0, on Γ∞ \ x∞,

(30)

Using the expressions (28), (29), (30) we are able to solve any shape design problem with the Eulerequations. However, this strategy is unpractical because it needs to localize the discontinuity curve Σ andcompute its variation δΣ for each design variable. In order to face this difficulty, two different methods areproposed for computing the functional gradient using shock information.

1. Method 1.- Using adjoint Rankine-Hugoniot relations (shock localization).

• Step 1.- Find the discontinuity curve Σ.

• Step 2.- Impose adjoint Rankine-Hugoniot relations over the discontinuity Σ in order to cancelthe dependence of the functional J with respect to δΣ, where several simplifications are possibledealing with normal shock waves.

• Step 3.- Solve (29) using (30) with adjoint Rankine-Hugoniot relations and evaluate (28) (or asimplified expression).

2. Method 2.- Without using adjoint Rankine-Hugoniot relations.

• Step 1.- Ignore the sensibility of functional due to the displacement of the discontinuity Σ and setas if the flow was continuous across Σ.

• Step 2.- Compute the functional gradient without shock considerations.

• Step 3.- Use the term ∂J/∂Σ to find a correction for the computed gradient (e.g. introducingdesign variables which their main effect is a shock displacement).

III.A.1. Method 1.- Continuous adjoint system using adjoint Rankine-Hugoniot relations

If no simplification is made for the expression in (28), the variation of the functional J depends on thesurface shape S and the shape of the shock wave curve Σ. The computation of δΣ for all design variablesis unpractical. For that reason we transform them into boundary conditions on the shock for the adjointsystem

∂tgΨT

[~F · ~tΣ

]Σ

=([ρ~v ]Σ · ~tΣ

)(∂tgΨ1 + H∂tgΨ4)

+[ρ (~v)2 + 2P

]Σ

~tΣ · ∂tg ~ϕ = 0, on Σ,

ΨT (xb)[~F · ~tΣ

]xb

= [j(P )]xb, at xb.

(31)

This result has been derived in8 in the context of the quasi-1D Euler equations. Then the functionvariation is computed as

δJ(S) =∫

S

((∂nj(P )− κj(P )) δS ds

+∫

S\xb

((∂n~v · ~nS)ϑ + ∂tg

((~v · ~tS

)ϑ))

δS ds− [j(P )]xb(~nS · ~nΣ) δS (xb) . (32)

The direct application of (31) and (32) in a real design problem is complex because it is necessary to findthe shock curve, the value of the adjoint variables at both sides of the discontinuity and finally evaluate theexpression (31) coupled with the adjoint Euler equations.

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Another possibility consists of supposing normal shock wave. In this framework, the equations (32) and(31) are simplified to obtain

δJ(S) =∫

S

((∂nj(P )− κj(P )) δS ds +∫

S\xb

((∂n~v · ~nS) ϑ + ∂tg

((~v · ~tS

)ϑ))

δS ds, (33)

with the following relations over the discontinuity

~tΣ · ∂tg ~ϕ = 0, on Σ,

~ϕ · ~nS = j′(P ), at xb.(34)

Solving the continuous adjoint equations (29) and (30), coupled with (34) and evaluate (33) is mucheasier than using (31) and evaluate (32). The viability of this approach will be shown in Section IV.

III.A.2. Method 2.- Continuous adjoint system without using adjoint Rankine-Hugoniot relations

In order to eliminate the shock position depending terms of (28), a method which takes into account theanalytical information of those terms will be designed. Note that, those terms reflect the effect over thefunctional J when a infinitesimal perturbation in the shock position is made (by a solid surface S modificationor a “far field” boundary condition variation).

Let us suppose that the shape of the discontinuity curve Σ is used as a design variable (analogous to Sfor the solid surface). Now the optimization problem can be stated as follows

J(Smin) = minS∈Sad

J(S, Σ(S)), (35)

where we maintain Σ = Σ(S) to emphasize the dependence between Σ and S. As before, Smin is the soughtoptimal surface, belonging to the set Sad of admissible solid boundary geometries. Then, it is possible toobtain a compete expression of the form

δJ(S,Σ) =∫

S\xb.GδS ds + Gxb

δS (xb) +∫

Σ

GshockδΣ ds + Gshockxb

δΣ(xb) , (36)

where G is the local gradient of J with respect to an infinitesimal movement of S in a normal direction ~nS

to the surface S, and Gshock is the local gradient of J with respect to an infinitesimal movement of Σ in anormal direction ~nΣ to the discontinuity surface Σ. As before, xb denotes the point in which the discontinuitytouches the solid surface.

Next, (36) is discretized using n nodes over the solid surface and m nodes over the discontinuity surface,the subindex b is used to denote the numerical node in which the shock touches the solid surface

δJ(S, Σ) =n 6=b∑

i=1

Gi(δS)i + G∗b(δS)b +m6=b∑

k=1

Gshockk (δΣ)k + Gshock∗

b (δΣ)b. (37)

Using the chain rule and keeping in mind that a deformation of Σ does not produce any deformation onthe solid surface S, equation (37) is written as

δJ(S, Σ) =n∑

i=1

(∂J

∂Si+

∂J

∂Σj

∂Σj

∂Si

)(δS)i +

m∑

k=1

∂J

∂Σk(δΣ)k, (38)

where the dependence of J with respect to infinitesimal variations of Σ could be very important (term of(38) multiplied by δΣ), so it is necessary to develop an optimization method which introduces this extrainformation. Comparing (38) with (37) and using (28) the effect of the introduction of the adjoint relationson the shock wave is shown. Next, detailed expressions for the functional sensitivity are obtained

Gi =

∂nj(Pi)− κij(Pi)+ (∂n~vi · ~nS,i)ϑi + ∂tg

((~vi · ~tS,i

)ϑi

), if i 6= b,

∂nj(Pi)− κj(Pi)− [j(Pi)]xi(~nS,i · ~nΣ,i) , if i = b,

(39)

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where for i = b a term which depends on the angle between the shock wave and the solid surface appears[j(P )]xb

(~nS · ~nΣ). This term can be easily computed by using a finite difference strategy with some selecteddesign variables over the solid surface (with influence over the shock), or by a direct evaluation. Notice, thatthis term is well evaluated when a discrete adjoint strategy or a finite difference method is used.

On the other hand, terms which depend on the shock wave displacement are computed as

Gshockk =

∂tgΨT

k

[~Fk · ~tΣ,k

]Σ

, if k 6= b,

ΨTk

[~Fk · ~tΣ,k

]xk

− [j(Pk)]xk, if k = b.

(40)

It is noteworthy that the shock displacement sensitivity Gshock does not appear in the discrete adjointmethod neither in the finite difference method because, in those methods, the shock position is not consideredas a design variable (just infinitesimal variations of the solid surface S shape are considered).

It is important to remark that when we are not considering the existence of a flow discontinuity, we arejust computing the value of the functional sensitivity over the surface due to the infinitesimal displacement ofS. However, the continuous adjoint method indicates that there is a very particular deformation of the solidsurface (which produce a shock movement) that could imply an important variation of the cost function.Now efforts must be focused in looking to develop a method that introduces this extra information providedby the equation (40), there are at least two ways of doing that.

1. Compute the functional gradient without shock considerations. In a second stage, use the term ∂J/∂Σfor finding a correction to the computed gradient. E.g. by using a inverse design problem, find theshape S which produces a shock deformation equivalent to Ψ

T[~F · ~tΣ

]Σ

which is the greatest descentdirection using the shock displacement. Finally, use it as a new design variable.

2. A simplified version of the first method consists in introducing this extra information by posing aninverse design problem: find the surface which produce a infinitesimal shock displacement over thesurface. As before, this shape function which moves the shock will be used in the optimization problemas a new design variable.

III.B. Discretization of the adjoint equations

The adjoint equations have been discretized with a standard edge-based finite volume formulation on thedual grid,23–25 obtained by applying the integral formulation of the adjoint equations to a dual grid controlvolume Ωj surrounding any given node j of the grid and performing an exact integration around the outerboundary of this control volume. Using the divergence theorem

|Ωj | dΨj

dt− ~AT

j

∫

Γj

Ψ~n dS = |Ωj |dΨj

dt−

mj∑

k=1

~fjk · ~njkSjk = 0, (41)

where ~ATj is the (transposed) Euler Jacobian evaluated at the node j, Γj is the boundary of Ωj , and |Ωj |

its area. For every neighbor node k of j, ~njk is the outward unit vector normal to the face of Γj associatedwith the grid edge connecting j and k and Sjk is its length, ~fjk is the numerical flux vector at the said face,Ψj is the value of Ψ at the node j (it has been assumed that Ψj is equal to its volume average over Ωj), andmj is the number of neighbors of the node j. The solution is advanced in time using a first order (forwardEuler) scheme:

Ψn+1j = Ψn

j −∆t

|Ωj |mj∑

k=1

~fnjk · ~njkSjk = 0, (42)

and a multistage Runge-Kutta was used in the computations. Next, we will review several alternativeschemes for defining the numerical flux vector.

III.B.1. Central Scheme with Artificial Dissipation

In the current work, we have developed a central scheme inspired by the standard JST scheme,26 followingthe adaptation to unstructured flow solvers presented in.27 In our scheme, the numerical flux is computed

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as

f centjk ≡ ~fjk · ~njk = AT

jk

(Ψj + Ψk

2

)+ djk, (43)

where ATjk ≡ ~AT

j ·~njk is the projected Jacobian and djk denotes the artificial dissipation. A simplified, fourthorder differences scheme has been chosen for the artificial dissipation

djk = ε(4)jk

(∇2Ψj −∇2Ψk

)Φjkλjk, (44)

where ∇2 denotes the undivided Laplacian

~∇2Ψj = −mjΨj +mj∑

k=1

Ψk, (45)

ε(4)jk are user-defined constants, and λjk is the local spectral radius

λjk = (|~ujk · ~njk|+ cjk)Sjk, (46)

where ~ujk = ~uj+~uk

2 and cjk = cj+ck

2 denote the flow speed and sound speed at the face, respectively. Φjk isintroduced to account for the stretching of the mesh cells and is defined as Φjk = 4 ΦjΦk

Φj+Φkwhere Φj = λj

4λjk.

III.B.2. Roe’s Upwind Scheme

In addition to the central scheme presented above, an upwind scheme based upon Roe’s flux differencesplitting scheme28,29 has been developed for the adjoint equations.

In our case, the aim is to use an upwind type formulas to evaluate a flow of the form ~AT · ~∇Ψ. Takinginto account that AT = − (

PT)−1 ΛPT , where AT = ~AT · ~n is the projected Jacobian matrix, Λ is diagonal

matrix of eigenvalues and P is the corresponding eigenvector matrix, the upwind flux is computed as

fupwjk =

12

(~AT

j · ~njk (Ψj + Ψk)− ∣∣ATjk

∣∣ (Ψk −Ψj))

=12

(AT

j (Ψj + Ψk) +(PT

)−1 |Λ|PT δΨ)

, (47)

where∣∣∣AT

jk

∣∣∣ = − (PT

)−1 |Λ|PT , the subindex jk denotes an avereage value in the interface. Note that

fupwjk 6= fupw

kj .

IV. Numerical experiments

The aim of this section is to investigate, with some numerical experiments, the significance of imposingthe adjoint Rankine-Hugoniot internal boundary conditions in order to introduce the influence of the shockdisplacement into the functional gradient computation.

The proposed problem consists in minimizing the wave drag using as initial geometry the classical NACA0012 airfoil. Gradients of the cost function are obtained with respect to variations of 50 Hicks-Henne sine“bump” functions,21 centered at various locations along the upper surfaces of the baseline airfoil. Thelocations of these geometry perturbations are ordered sequentially such that they start at the 25% of thechord (upper surface), proceed forward to the trailing edge until the 75% of the chord (upper surface), seeFig. 3 for an example of one “bump” functions applied to a NACA 0012 airfoil.

The drag objective function CD, on the surface S is defined as

J =∫

S

P

0.5v2∞ρ∞L~nS · ~d ds, ~d = (cos α, sin α) (48)

where ~nS is the inward unit vector normal to the boundary S, α is the airfoil angle of attack, L is thecharacteristic length of the airfoil and v∞, P∞ are the free-stream velocity and pressure, respectively.

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Figure 3. Effect of a Hicks-Henne sine “bump” function.

Figure 4. Iso-Mach lines and CP of a NACA 0012 (Mach 0.8, α = 0.0)

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IV.A. Symmetric configuration

In this subsection, a redesign of an airfoil profile NACA 0012 in transonic regime (Mach 0.8, α = 0.0) hasbeen selected as the baseline numerical test. In Fig. 4 the iso-Mach lines (left) and CP coefficient (right)are shown. In this configuration, the shock wave is orthogonal to the NACA 0012 surface and is located ona nearly flat zone (horizontal) of the airfoil profile.

According with the flow results exposed in the Fig. 4. A priori we can expect that the influence of a shockwave displacement on CD coefficient will be very small because the shock is located on a nearly horizontalplate and the influence of its specific position in this zone on the drag is negligible.

Next, the continuous adjoint formulation developed in this paper is applied. Instead of using the completeadjoint relation over the shock (31) a simplified version (34) is used. The crucial step of this method is todevelop an algorithm for detecting shock waves and subsequently impose the right adjoint Rankine-Hugoniotrelations.

Figure 5. Adjoint 2D Euler solution and drag gradient (symmetric case)

In Fig. 5 the adjoint variable field (left) is shown (imposing and not imposing the Rankine-Hugoniotrelations). On the other hand, a most relevant result is shown in the right part of Fig. 5. In this case thesensitivity of the functional CD with respect to infinitesimal variations in the shape of the NACA 0012 ispresented (imposing or not, adjoint Rankine-Hugoniot relations on the shock). Results in both cases (withand without Rankine-Hugoniot relations) are almost equal.

To sum up, in this example internal conditions of Rankine-Hugoniot are naturally imposed in the casewhere the sensitivity of the functional with respect to variations in the position of the shock is negligible.That is to say, the term that multiplies to δΣ in (38) is zero, so (40) vanishes on the shock. This ratifies thefact that under certain circumstances the imposition of internal conditions is not necessary.

IV.B. Asymmetric 2D configuration

Now we take a step forward with an asymmetric case. As before, we are looking to redesign an airfoil profileNACA 0012 in transonic regime but now with an asymmetric flow field (Mach 0.8, α = 1.20). In fig. 6 theiso-Mach lines and CP are shown. In this case, due to the asymmetry of the configuration, the shock is notperpendicular to the x axis, so a displacement of the shock produces a significant variation in the functional.

In Fig. 7 (left) both adjoint solutions (with and without using Rankine-Hugoniot relations on the shock)are compared for the third adjoint variable. Also, in Fig. 7 (right) the computed surface gradient inboth cases is shown. In contrast to the symmetric case, in this configuration the imposition of the adjointRankine-Hugoniot relations has an important influence into the gradient computation.

In Fig. 8 the influence of the exact shock wave localization in order to impose adjoint Rankine-Hugoniotrelations is shown. Once the shock wave is located, the adjoint solution is computed (using adjoint Rankine-Hugoniot relations), and the drag surface gradient is evaluated. It is time to integrate the surface drag

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Figure 6. Iso-Mach lines and CP of a NACA 0012 (Mach 0.8, α = 1.2)

Figure 7. Adjoint solution (left) and drag surface sensitivity (right).

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gradient using 50 Hicks-Henne sine “bump” functions centered at various locations along the upper surfacesof the baseline airfoil, see Fig. 8 (right).

Figure 8. Influence of shock wave location, and CD gradient

Finally in Fig. 9 an excellent result is shown. In this case, we are computing the improvement that wouldproportionate the right use of the adjoint internal conditions in the functional minimization. The validationprocedure has been the following

1. Compute the functional gradients (with and without internal conditions).

2. Normalize the gradient value with respect to the Euclidean norm.

3. Provided a common advance step for both problems (with and without internal conditions).

Using the above procedure, if we do not use the internal conditions drag is reduced to 112 drag counts.On the other hand if we use the Rankine-Hugoniot adjoint relations we obtain a drag reduction value of 100drag counts, that approximately supposes an improvement in 10% which is a remarkable improvement. Stillbetter results are obtained for other functionals more sensible to the shock position.

Figure 9. Estimate improvement using internal boundary conditions

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The next step, is to state a complete optimization problem to compare the performance between imposingadjoint boundary conditions or not. In Fig. 10 a drag minimization problem is shown.

Figure 10. Complete optimization problem

We are looking for diminishing the drag of the NACA 0012 profile in transonic regime (Mach 0.8, α =1.20), as constraint we impose that the lift coefficient must be greater than 0.36. As we can see in thisexample, to impose the adjoint internal boundary conditions improves the optimization process in terms ofdrag minimization and lift maximization. Also, is remarkable that one iteration less is needed to obtain thebest result.

V. Conclusions

In this work the continuous adjoint methodology for the calculation of gradients of functionals of the flowfield (defined on the solid surface) has been developed dealing with discontinuities in the flow variables.

The continuous adjoint methodology derives the adjoint problem from the continuous formulation of theflow equations, and as such it constitutes a shortcut that allows to maintain the rigor throughout the wholeprocedure. However, when working with continuous schemes, it is often necessary to deal with the problemof discontinuities in solutions of the state equation. In this case, shocks must be treated as singularities onwhich the adjoint Rankine-Hugoniot conditions must be enforced. The enforcement of these conditions isdelicate and requires the numerical location of the shock.

Nevertheless, satisfactory results have been obtained without the imposition of these Rankine-Hugoniotconditions across the shock.8,30,31 On the other hand, in this paper a simplified version of the adjointRankine-Hugoniot relations is used and numerical test reveal the significance of using the functional sensi-tivity with respect to shock movements. Moreover some alternative methods are proposed to include extrainformation that is not provided by the classical finite difference method or discrete adjoint method whichdoes not consider the influence of the shock movement.

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