Method for Optimally Controlling Unsteady Shock Strength ...

Method for Optimally Controlling Unsteady ShockStrength in One Dimension

Nathan D. Moshman,∗ Garth V. Hobson,† and Sivaguru S. Sritharan‡

Naval Postgraduate School, Monterey, California 93943

DOI: 10.2514/1.J051924

This paper presents a new formulation and computational solution of an optimal control problem concerning

unsteady shock wave attenuation. The adjoint system of equations for the unsteady Euler system in one dimension is

derivedandused in anadjoint-based solutionprocedure for the optimal control.Anovel algorithm is used to satisfy all

necessary first-order optimality conditions while locally minimizing an appropriate cost functional. Distributed

control solutions with certain physical constraints are calculated for attenuating blast waves similar to those

generated by ignition overpressure from the shuttle’s solid rocket booster during launch. Results are presented for

attenuating shocks traveling at Mach 1.5 and 3.5 down to 85%, 80%, and 75% of the uncontrolled wave’s driving

pressure. The control solutions give insight into the magnitude and location of energy dissipation necessary to

decrease a given blast wave’s overpressure to a set target level over a given spatial domain while using only as much

control as needed. The solution procedure is sufficiently flexible such that it can be used to solve other optimal control

problems constrained by partial differential equations that admit discontinuities and have fixed initial data and free

final data at a free final time.

Nomenclature

A = Jacobian matrixa, b = weighting constantsf = functional of final timeH = HamiltonianJ = cost functional~J = augmented cost functionalK = final time penaltyL = running cost functional (Lagrangian)Lhv = latent heat of vaporization of water at 100°CmH2Ov = energy equivalent mass distribution of water vapor

produced by control actionϵ = positive small constantP = gas pressureQ = target pressure at final timeT = final timeU = three-component state vectoru = gas velocityV = three-component adjoint vectorx = one-dimensional spatial vector in Ωz = control variableγ = gas constant∂Ω = spatial domain boundaryρ = gas densityρE = gas total energyρe = gas internal energyρu = gas momentumΩ = spatial domain in one dimensionΩs = spatial interval upstream of shock where P > Q

Subscripts

i, j, k = variable numberm = spatial index

Superscripts

l = control algorithm iteration indexn = temporal index� = optimal quantity

I. Introduction

O PTIMAL control of fluid dynamics has undergone rapiddevelopments during the past two decades [1–3], and parallel

developments in optimal aerodynamic shape optimization also haveseen exciting advances [4–9]. Optimal control theory of hyperbolicsystems of conservation laws for applications of gas dynamics withshock waves is addressed in [10,11]. In this paper, we will solve anoptimal control problem for a one-dimensional hyperbolic system ofconservation laws that arise in an important rocket launchingproblem. In the present work, the cost functional to beminimizedwillpenalize the magnitude of the jump in pressure across the front of anunsteady shock wave after a finite simulation time. The controlvariable is a distributed field that removes energy from the gasupstream of the moving shock front and changes in both spaceand time.Ignition overpressure (IOP) is a phenomenon present at the start of

an ignition sequence in launch vehicles using solid grain propellants.When the grain is ignited the pressure inside the combustion chamberquickly rises several orders ofmagnitude. This drives hot combustionproducts toward the nozzle and out to the open atmosphere atsupersonic speeds. An IOP wave is a spherical blast wave, whichoriginates from the exit plane of the nozzle and propagatesspherically outward. Overpressures that the body of the rocketexperiences are of the order 2:1 [12]. The region just outside thenozzle will experience further compression due to the displacementof gas along the blast wave’s direction of propagation andoverpressures can approach 10:1. The portions of the IOP wave thatbecome incident on the rocket body or launch platform componentsmust have an overpressure below a known threshold to avoid costlydamage. The current technique used by NASA and other launchproviders is to spray water into the region around the nozzle prior toignition. This forces the IOPwave to propagate throughwater prior to

Presented as Paper 2011-3847 at the 20th AIAA Computational FluidDynamics Conference, Honolulu, Hawaii, 27–30 June 2011; received 29February 2012; revision received 19 September 2012; accepted forpublication 10 October 2012; published online 18 January 2013. Thismaterial is declared a work of the U.S. Government and is not subject tocopyright protection in theUnitedStates.Copies of this papermaybemade forpersonal or internal use, on condition that the copier pay the $10.00 per-copyfee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,MA 01923; include the code 1533-385X/13 and $10.00 in correspondencewith the CCC.

*Research Assistant Professor, Mechanical and Aerospace EngineeringDepartment, 700 Dyer Road. Member AIAA.

†Professor, Mechanical and Aerospace Engineering Department, MailCode: MAE/Hg. Member AIAA.

‡Director, Center for Decision, Risk, Controls and Signals Intelligence, 1University Circle.

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becoming incident on the rocket body or platform components.Through several dissipative mechanisms this causes a sufficientdecrease to the pressure jump across the shock to prevent damage.The implementation of the water suppression system is ad hoc,specific to the shuttle, and it has not been reconsidered in decades.The latest work from NASA was a parametric study of waterarrangement in the nozzle region and its effects on themaximum IOPstrength [13,14]. The first work on parametrically optimizing IOPattenuationwith respect towater injection strategy is in [13]. The goalof this work is to develop a computational tool that can directlycalculate a distributed optimal control for attenuating a range of blastwaves to a desired minimal overpressure.

II. Computational Fluid Dynamics

Data on the shuttle grain and chamber pressure [15] were given asinput to Cequel, an open-source chemical equilibrium solverdeveloped by NASA. Cequel [16] uses chemical properties from anextensive database to minimize the Gibbs free energy of the givencombustion products and calculate equilibrium conditions. The outputgives the temperature and gas velocity at the nozzle exit plane for agiven pressure ratio. The computational fluid dynamics simulationdomain has boundaries at the nozzle exit plane, the rocket body andthe rest free space. Initially, ambient conditions inside the domainare present. The two-dimensional (2-D) ignition sequence simulationwas performed using the ESI-Fastran commercial software [17].The ESI-Fastran solver uses well-known and robust finite volumemethodologies for a range of physical models involving compress-ible flows.The spatial discretization scheme used was Van Leer’s flux vector

splitting [18] and was extended to second-order accuracy by a Barthlimiter [19]. The Barth limiter enforces monotonicity and, therefore,is appropriate for solutions with strong discontinuities. The timeintegration was fully implicit with a tolerance of 10−4 in the residualsover a maximum of 20 subiterations.Constant mass-flow boundary conditions equivalent to the steady-

state exit plane of the rocket nozzle on the shuttle were used in thebottom center of the domain on the right face of the step as shown inFig. 1. The Mach number is depicted in three snapshots in Figs. 1a,1c, and 1e, and the pressure is depicted in Figs. 1b, 1d, and 1f. The lastframe shows a snapshot of the flow 10 ms after ignition. The bottom

left edge of the domain represents the rocket body, whereas in thebottom right edge is the centerline of the normal to the nozzle exitplane and a symmetry boundary. All other edges are nonreflectingboundaries.Flow conditions over time were recorded at two locations marked

in Fig. 1a. Point 1 is near the rocket body 2.5 m above the nozzle andpoint 2 is 1.5m along the symmetry boundary and the plane normal tothe nozzle exit. The conditions at the recorded locations are used asthe boundary conditions in the one-dimensional Euler equationscalculation used for the optimal control calculation. Figure 2a showsthe flowconditions over time atmonitor point 1 (MP1) near the rocketbody and Fig. 2b shows the flow conditions over time for monitorpoint 2 (MP2) directly downstream of the nozzle. The transverseMach number in both cases is about 0.5. Neglecting this motioncorresponds to a 10% loss of total temperature. This is an acceptablesimplification because the purpose is to be able to control a range ofblast waves, not tomost accurately predict a specific flowfield.Hottergas will vaporize water more rapidly, thereby, extracting energy fromthe gas more quickly and yielding a more effective control action.Hence, no unfair advantage is gained by driving the inlet boundarycondition with a slightly cooler gas.The single-phase control calculation is meant to give insight into

a two-phase control calculation where water droplet size andplacement determines shock attenuation. The conservation of mass,momentum, and energy are obeyed for both the gas and the liquid andan additional equation relates the volume fraction of either phase tothe movement of the interface. A sink term for vaporization appearsin the gas balance laws. The formulation in this way yields sevenbalance laws for one dimension with a vaporization source term.Several interaction mechanisms between the water droplets and theIOP wave are present and not fully understood. The dominantdissipativemechanism for shocks betweenMach 1 and 2 is the loss ofenergy of the gas through the vaporization of the water droplets[14,20]. Experimental data from droplet-shock interactions in thisregime show that the other dissipative mechanisms, e.g., dropletvelocity, drag on droplets, sensible heating of droplets, chemicalreactions, changes in specific heat, etc., are less significant to IOPattenuation. In a two-phase calculation, the control action would takethe form of a liquid mass source, and the effect of vaporization mostcritical to IOP attenuation will be to take energy out of the gas phase.

Fig. 1 Simulated shuttle IOP (domain size � 10 × 5 m,Δx � 1 cm, andΔt � 1 μs):Mach number at (1, 4, and 10ms) (a, c, and e); pressure at (1, 4, and10 ms) (b, d, and f).

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To most simply replicate the dominant dissipative mechanism with asingle-phase calculation the control will act as an energy sink with nocorresponding mass or momentum sinks or sources, as shown inEq. (2):

U � �ρ�x; t�; ρu�x; t�; ρE�x; t��T U�x; 0� given (1)

∂∂t

0@ ∂tρuρE

1A� ∂

∂x

0@ ρu

ρu2 � Pu�ρE� P�

1A �

0@ 0

0

z�x; t�

1A (2)

The control action is assumed to act instantaneously, but in a real two-phase interaction the droplets take some amount of time to extractenergy from the shock. As Jourdan et al. [14] showed, the moredroplet surface area exposed to the gas, the greater the shockattenuation for a fixed amount of time. This is because the rate ofvaporization is related to the total droplet surface area exposed. For afixed amount of water smaller droplets exposemore surface area thanlarger ones. Therefore, when interpreting the following single-phaseresults in a two-phase context, it is most appropriate to think of thesmallest practical droplets that a typical atomizer can produce, adiameter of about 10 μm. The restriction that the instantaneousdistributed control action, z, be only a sink, and, therefore, eithernegative or zero is imposed.Equation 3 relates the internal energy to the total energy of the gas:

ρE � ρe� ρu2

2(3)

The ideal gas equation of state [Eq. (4)]was chosen because it adds nonew degrees of freedom to the calculation. The only place where theeffect ofwater is replicated is in the energy sink and not in an equationof state that has been customized to gas and water vapor near thecritical point (e.g., [21]). In addition, γ � 1.4 for air and will not beallowed to change as it might in modeling a mixture of gas and watervapor:

P � ρe�γ − 1� (4)

A conservative Godunov-type method, second-order accurate inspace, proposed in [22,23], was implemented to solve thecompressible flow dynamics in one dimension under a distributedcontrol action. This method assumes that the solution in each cell ispiecewise linear and projects an intermediate solution on anonuniform grid based on the maximum characteristic speeds fromthe interpolated solutions at each cell interface. The familiarGodunov integration [24] on the uniform grid is then accurate to thesecond-order because of the intermediate finer grid. This numericmethod has been implemented previously in the literature for asingle-phase calculation [25] as well as a two-phase calculation [26]based on a model presented in [27].

III. Optimal Control System

In any optimal control problem there is a cost functional tominimize. Previous work on unsteady compressible flow control [7]was aimed at actuation near a boundary to suppress instabilities or thedevelopment of turbulence. In this work, the cost functional mustreflect that a decrease in the maximum jump in pressure (theoverpressure at the shock front) is most desirable at some finalmoment in time. It should also penalize control action but to a lesserdegree. Therefore, let J be the cost functional:

J � a2

ZT

0

ZΩz�x; t�2 dx dt� b

2

ZΩs�P�x; T� −Q�x��2� dx (5)

Here x represents a spatial vector, which is one-dimensional in thiscalculation.T is the final time, which is not fixed; z�x; t� is the controlaction; P�x; T� is the pressure at the final time; Q�x� is the desiredfinal pressure; and a and b are weighting constants. The larger b iscompared to a themore significant the final time penalty compared tothe penalty for using the control action. In Eq. (6), L denotes therunning penalty for control effort, whereas the final time penalty,denoted as K, penalizes the height of the pressure profile above atargetQ.Ω represents the entire one-dimensional simulation domain,whereas Ωs is the interval behind the shock where the pressuredistribution at the final time is above the target state, as shown inFig. 3:

L�z�·; t�� � a2

ZΩz�x; t�2 dx

K�U�·; T�� � b2

ZΩs�P�x; T� −Q�x��2� dx (6)

Fig. 2 Flow conditions over time for monitor points 1 and 2 shown in Fig. 1a (Δx � 1 cm, Δt � 1 μs).

Fig. 3 Illustration of iterative procedure.

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The control action will take the form of an internal energy sink. Itonly appears on the right-hand side of the energy balance equation ofthe one-dimensional Euler system, as was shown in Eq. (2).It has been pointed out in the literature [10,28] that if the state

variables have shocks, a perturbation is not small in the neighborhoodof the shock front and does not have vanishing properties as ε → 0. Aslight increase in the amplitude behind the shock perturbs the speedand, therefore, also the location of the shock front. This causes smallperturbations to induce variations on the order of the jump across theshock. The presented method of solution avoids this issue, as will bedemonstrated later. Because only decreasing the amplitude of a shockwave is desired it is apparent that any realistic control actionwill onlyslow the shock wave down. The target state Q�x� and final timepenalty will be constructed in such a way that all variations of thesolution will occur upstream of the shock front only. Matching thesimulated pressure profile under control action with the target finalstate near the shock front will occur by allowing the final time to befree. Henceforth, it can be assumed that all variations are taken insmooth regions of the flow and that the solution procedure will notdepend on a shock location variable and corresponding adjoint state,and a more sophisticated variation is not required.Initial conditions are stationary, ambient air. Stating the density,

velocity, and pressure determines the internal and total energy and theconservative vector:

ρ�x; 0� � 1 kg∕m3 u�x; 0� � 0 m∕s P�x; 0� � 105 Pa

ρu�x; 0� � 0 kg:m∕s ρe�x; 0� � 250000 J

ρE�x; 0� � 250000 J (7)

The inlet boundary condition is explicitly given by the IOPsimulation data shown in Figs. 2a and 2bwhen the flow is supersonic.If the flow behind the shock front is subsonic, a nonreflection of theu-c characteristic is imposed [29]. In addition, the monitor point datawere chosen where the flow was nearest to one dimensional;however, the 2-D data did have transverse motion. The data will stillgive a plausible one-dimensional blast wave with the inlet boundarycondition set in this manner, and the goal of the calculation,controlling a range of blast waves, can be achieved. At the outletboundary, the flow remains stationary because the final time willalways be such that the shock wave will not have enough time topropagate though the entire domain and reach the outlet.To determine the optimal control z��x; t� that minimizes the cost

functional J it is necessary to define the pseudoHamiltonian ofthe system and derive necessary conditions using the Pontryaginminimum principle and the calculus of variations. ThepseudoHamiltonian for this system is

H�U;V; z; t� � L�U; z� �ZΩV ·

∂U∂t

dx

�RΩa2z2 � V1

∂ρ∂t � V2

∂�ρu�∂t � V3

∂�ρE�∂t dx (8)

where (V1, V2, V3) is the adjoint vector. Writing the Euler equationsin nonconservative form in this basis defines the Jacobian matrix:

∂Ui∂t� Aij�U�

∂Uj∂x� ziδi3 (9)

Aij�U� �

0B@

0 1 0�ρu�2ρ2

γ−32

− �ρu�ρ �γ− 3� �γ− 1�−γ�ρu��ρE�

ρ2��γ− 1� �ρu�3

ρ3γ�ρE�ρ − 3

2�γ− 1� �ρu�2

ρ2γ�ρu�ρ

1CA

(10)

To derive necessary conditions the optimal state must be defined�U��x; t�; P��x; t�; z��x; t�; T�� and the optimal control perturbedsuch that z � z� � εδzwhere ε > 0 is a small constant. The variationin the control will cause variational terms in each of the other freevariables of the system whose duality pairing must necessarilyvanish at an optimal solution. To incorporate the constraints of the

one-dimensional Euler system using the Lagrange multiplier methodeach conservation law is multiplied by an adjoint variable and theseterms are added to J. This is the augmented cost functional ~J. Then,from expanding ~J in a Taylor series, the first-order necessarycondition will be

d

dε~J�z� � εδz�jε�0 � 0 (11)

Grouping the terms of like variational multipliers gives the optimalsystem. Integrating by parts until all derivatives are on the adjointvector yields the following linear system of partial differentialequations [10]:

∂Vj∂t� Aij

∂Vi∂x��∂Aij∂x

−dAijdUk

·∂Uk∂x

�Vi

� ∂∂x��Aijv�i ; δuj��j∂Ω �

∂L∂Ui� 0 (12)

The nontrivial elements of the matrix dAij∕dUk · ∂Uk∕∂x are givenbelow. ∂Ω denotes the boundary of the spatial domain. The right-hand side of Eq. (12) is zero in this formulation because the runningcost does not explicitly depend on the state vector:

∂∂Uk

A21 ·∂Uk∂x� �γ − 3�

�−u2

ρ

∂ρ∂x� u

ρ

∂�ρu�∂x

�(13)

∂∂Uk

A22 ·∂Uk∂x� �γ − 3�

�u

ρ

∂ρ∂x

−1

ρ

∂�ρu�∂x

�(14)

∂∂Uk

A31 ·∂Uk∂x� u

ρ�2γE − 3�γ − 1�u2�

∂ρ∂x� 1

ρ�3�γ − 1�u2 − γE� ∂�ρu�

∂x− γ

u

ρ

∂�ρE�∂x

(15)

∂∂Uk

A32 ·∂Uk∂x� 1

ρ�3�γ − 1�u2 − γE� ∂ρ

∂x

− 3�γ − 1� uρ

∂�ρu�∂x� γ

ρ

∂�ρE�∂x

(16)

∂∂Uk

A33 ·∂Uk∂x� −

γu

ρ

∂ρ∂x� γ

ρ

∂�ρu�∂x

(17)

Because the final state is not fixed there is a necessary condition onthe adjoint vector at the final time:

V�i �x; T�� �∂

∂UiK�U�x; T��� (18)

Therefore, in this basis, all three derivatives are nonzero, because thepressure is a function of all three conserved quantities:

V�1�x; T�� � b�P�x; T�� −Q�x��∂P∂ρ�x; T��

� b�P�x; T�� −Q�x�� u�x; T��2

2�γ − 1�

V�2�x; T�� � b�P�x; T�� −Q�x��∂P

∂�ρu� �x; T��

� −b�P�x; T�� −Q�x��u�x; T���γ − 1�

V�3�x; T�� � b�P�x; T�� −Q�x��∂P

∂�ρE� �x; T��

� b�P�x; T�� −Q�x���γ − 1� (19)

The time derivative of the entire adjoint vector for all space at adiscrete time tn is shown in column-upon-column form in Eq. (20).Let m be the number of spatial grid points and k be the number of

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adjoint variables. Then the adjoint vector V at a discrete moment intimewill be of size km × 1, and thematrices in Eq. (12) will be of sizekm × km. A single component of the adjoint vector, e.g., V1�x; t�,will be size m × 1 at each time step.

∂Vlk∂t

↔∂∂t

0BBBBBBBBBBBBBBBBBBBBBBBBBBB@

0BBB@V1�x1; tn�V1�x2; tn�

..

.

V1�xm; tn�

1CCCA

0BBB@V2�x2; tn�V2�x2; tn�

..

.

V2�xm; tn�

1CCCA

..

.

0BBB@

Vk�xk; tn�Vk�x2; tn�

..

.

Vk�xm; tn�

1CCCA

1CCCCCCCCCCCCCCCCCCCCCCCCCCCA

(20)

All of the matrices in Eq. (12) have a diagonal-block structure. TheJacobian matrix for all of space at tn is shown in Eq. (21):

Akν�Un�↔0BBBBBBBBBBBBBBBB@

0B@A11�x1;tn�

. ..

A11�xm;tn�

1CA ··· ···

0B@A1k�x1;tn�

. ..

A1k�xm;tn�

1CA

..

.

..

.

. .. . .

.

. .. . .

.

..

.

..

.

0B@Ak1�x1;tn�

. ..

Ak1�xm;tn�

1CA ··· ···

0B@Akk�x1;tn�

. ..

Akk�xm;tn�

1CA

1CCCCCCCCCCCCCCCCA

(21)

The adjoint partial differential equation (PDE) is given in discreteform with an explicit integration in Eqs. (22) and (23):

1

ΔtI�Vn−1k − Vnk� � R�Un�Vni � 0 (22)

R�U� � A · DU� ∂A∂x

−∂A∂U

·∂U∂x

(23)

ThematrixD is made up of discrete spatial derivative blockmatrices,central differencing in the domain interior, upwind differencing at theoutlet and downwind at the inlet. A single block is shown in Eq. (24):

Dmxm �1

2Δx

0BBBBB@

−2 2

−1 0 1

. .. . .

. . ..

−1 0 1

2 −2

1CCCCCA

(24)

Each time step of the adjoint solution has four parts. Prior to timeintegration, the matrix R must be assembled. Some of the matrices,whichmakeupR, are known in closed form and require no discretizedderivative. The three-component system requires assembling A and∂A∕∂U from the known state data. The second part of the solutionrequires assembling the matrix and the vectors that have discretederivatives ∂A∕∂x, ∂V∕∂x, and ∂U∕∂x. These two parts can be donein parallel. The third part, calculating R, requires sharing memorybetween the processes and does not lend itself well to parallelization.

With careful direction of the memory there is more potential in theassembly of R for speed optimization than will be shown in Sec. V.The final part of the adjoint time step is the time integration, whichboils down tomatrix addition andmatrix-vectormultiplication for theexplicit scheme. These operations are known to be adaptable toparallelization in a straightforward way.For adjoint calculations of a scalar PDE with a discontinuity it has

been shown [28] that a relaxed system with second-order dissipationwill recover the nonlinear PDE in the limit of vanishing viscosity. Asmall numerical viscosity can stabilize the adjoint solution. Theseideas have been extended to fluid dynamics systems [30] and areimplemented in the current work in a manner which maintainsconsistency for the numerical adjoint solution.The transversality condition describes how the time rate of change

of the final time penalty must balance with the value of theHamiltonian in order for the first-order variation of the cost functionalto vanish. For a free final time the necessary condition for the optimalfinal time T� is given by the transversality condition, which isobtained by grouping the terms and multiplying the δt variation afterexpanding Eq. (11):

H�U��x; T��; V��x; T��; z��x; T��; T�� � d

dtK�U�x; T��� � 0

ZΩ

a

2z�x; T��2 � b�P�x; T�� −Q�x��

×

0BBBBBB@

∂P∂ρ �x; T��

∂ρ∂t �x; T���

∂P∂ρu �x; T��

∂ρu∂t �x; T���

∂P∂ρE �x; T��

∂ρE∂t �x; T���

∂P∂t �x; T��

1CCCCCCAdx � 0 (25)

The necessary condition on the optimal control solution comes frommaximizing the Hamiltonian for an unconstrained control. Theunconstrained condition is justified because there is no restriction oncontrol magnitude in regions where control is allowed. The integral istrue over any domain Ω, and at any moment in time it should be truepointwise for all t ∈ �0; T�:

∂H∂z�U�; V�; z�; t� �

ZΩaz��x; t� � V�3�x; t� dx � 0 (26)

In summary, Eqs. (7), (9–12), (19), (25), and (26) give the completeset of first-order necessary conditions for the optimal system.

IV. Solution Procedure

Let the left-hand side of the transversality condition in Eq. (25) bedefined as a functional f�T� with the final time as the independentvariable. Then f�T�� � 0. T� can be solved iteratively with theNewton–Raphson root finding method. The derivative df∕dt and thediscrete form of f and its derivative are given in [31]:

Tl�1 � Tl − f�Tl�dfdt �Tl�

(27)

The superscript l denotes an iterate of the inner loop of the solutionprocedure in Fig. 4. As f�Tl� → 0 then Tl → T�. The conditionrequired by Eq. (26) is similarly iteratively satisfied by the update inEq. (28):

δzl�1�xi; tn� � −Vl3�xi; tn�∕azl�1�xi; tn� � zl�xi; tn� � δzl�1�xi; tn� (28)

Physical control constraints are then imposed. Because thecalculation concerns shock attenuation extracting energy from thegas using a sink is of interest. Consequently, zl ≤ 0 for all space andtime is enforced. It has been shown that the adjoint variables willtravel along the characteristics of the flow in the opposite direction.This means that the calculation will suggest putting control ahead ofthe shockwave, which is not relevant because physical droplet-shock

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interactions only take energy out of the gas behind the shock wave.Therefore, Eq. (29) gives the restrictions on the control. The shockspeed S is estimated based on the location of the shock front at Tl:

zl�1�xi; tn� ��0 if zl�1�xi; tn� > 0

0 if xi > S · tn(29)

The overall algorithm, which is used to satisfy all of the abovenecessary optimal conditions, is shown in block-diagram formin Fig. 4.In each of the results, the weighting constants a and b from Eq. (5)

are 10−6 and 104, respectively. Their relative magnitudes are set sothat the physical units of overpressure are scaled to a meaningful

magnitude of the rate of energy extraction from the gas in the controlsolution. In addition, it is desirable to have b much larger than a sothat minimizing to total cost J is dominated by minimizing K, theoverpressure of the blast wave above the target state.

V. Results and Discussion

The plots shown in Fig. 5a are the distributions of the energy-equivalent mass of water vapor produced by the optimal controlsolutions. By dimensional analysis, inspection of the energy balanceequation indicates that the units of z��x; t� are watts. Integrating thecontrol solution from �0; T�� gives an energy distribution in space.This energy can be directly equated to a required mass of water vaporthat must be produced from liquid water droplet vaporization. Thelatent heat of vaporization of water isLhv � −2.26e6 J∕kg at 100°C.Equation 30 relates the optimal control solution z�, distributed inspace and time, to an energy equivalent distribution of the mass ofwater vapor produced over the entire simulation time intervalmH20v:

mH20v�x� �1

Lhv

ZT�

0

z��x; t� dt (30)

The solution procedure in Fig. 4 assumes a given target stateQ�x�at the final timeT�. To illustrate trends in the optimal control solution,prescribing a consistent and meaningful target state or sequence oftarget states is a necessity. Each target state is defined by locating theshock front of P0�x; T� and setting Q�x� equal to a fraction, (0.85,0.80, 0.75), times the value of P0�x; T� upstream of the shock frontand equal to P0�x; T� downstream of the shock front. The results forcontrolling the blast data from MP1 are shown in Fig. 5. The solidblack curve in Fig. 5a is the optimalwater vapormass distribution thatyields a final time pressure profile closest to, and below, a target state,which has 85% of the pressure magnitude of the uncontrolled blast

Fig. 4 Block diagram of solution procedure.

Fig. 5 MP1 results (dx � 1 cm, dt � 1 μs, T0 � 1.6 ms); a) optimal control solutions integrated over time; b) pressure profiles at optimal final time c,top) logarithm of cost functional iterates; c, bottom) final time iterates.

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wave P0�x; T� behind the shock front. Figure 5b shows the targetstate as a dashed black curve and the optimal pressure profile as asolid black curve. The solid blue curves in Figs. 5a and 5b are theoptimal control and pressure profiles, respectively, for a target with80% of the magnitude of P0�x; T� (dashed blue curve). The redcurves in Figs. 5a and 5b are the optimal control and pressure profiles,respectively, for a target with 75% of the magnitude of P0�x; T�(dashed red curve). The top plot in Fig. 5c shows the logarithm of thecost functional J decreasing monotonically over solution iterationsfor a single target state. The bottom plot in Fig. 5c shows the optimalfinal time T� converging to a larger value over solution iterations.As shown in Fig. 5a, an increasing amount of water vapor must be

produced in order that magnitude of the final time pressure profile isdecreased. The control curves are approximately linear, increasingfrom downstream to upstream of the direction of shock propagation.This indicates that the further upstream the water vapor can beproduced the more effect it will have on diminishing the overallmagnitude of pressure at the final time. The final time converges to alarger value than was used to obtain P0�x; T�, because the speed ofthe shock is directly related to its amplitude. Hence, as the controlaction attenuates the amplitude, the shock will also slow down,requiring a longer simulation time for the shock front to match upwith the target.Analogous results for controlling the blast data from MP2 are

shown in Fig. 6. The solid black curve in Fig. 6a is the optimal watervapormass distribution that yields a final time pressure profile closestto, while below, a target state, which has 85% of the magnitude ofpressure of the uncontrolled blast wave P0�x; T� behind the shockfront. Figure 6b shows the target state as a dashed black curve and theoptimal pressure profile as a solid black curve. The blue curves inFigs. 6a and 6b are the optimal control, target, and pressure profiles,respectively, for a target with 80% of the magnitude of P0�x; T�(dashed blue curve). The red curves in Figs. 6a and 6b are the optimal

control, target, and pressure profiles, respectively, for a target with75% of the magnitude of P0�x; T� (dashed red curve).The top plot in Fig. 6c shows the logarithm of the cost functional J

decreasing monotonically over each interval that a target state is set.Notice that theminimumvalue of J is increasing for the three separatetargets. Although the final time penalty K roughly goes to zero ineach case more control effort is needed to attenuate the pressure to asmaller amplitude and, hence, the increase in theminimumof J is dueto L, which penalized the additional control effort. The argument ismade that J isminimized because itmonotonically decreases over thesolution procedure, all necessary conditions of a minimum aresatisfied, and that using less control would increaseK, and, hence, J.Conversely, using more control would also increase J because Lwould increase, and K cannot decrease any further.The bottom plot in Fig. 6c shows the optimal final time T�

converging to a larger value over solution iterations. Also notice thatthe optimal final time increases over the three intervals withdecreasing target amplitudes, because each of the waves will bemoving slower and require a longer simulation time to match thetarget shock front.A comparison of the optimal water vapor distributions in Figs. 5a

and 6a gives further insight intowater as a control. Take, for example,the black curves representing a target statewith an amplitude equal to85% of P0�x; T�. For MP1, this corresponds to an absolute pressuredecrease of about 0.3 atm, whereas for MP2 it is about 1.2 atm. Theabsolute pressure decrease for MP2 is four times that of MP1, yet themaximum amount of water vapor required for MP2 is less than threetimes that of MP1. The reason is that the MP2 data have a hotterdriving gas thanMP1 and, therefore, cooling that gas with an internalenergy sink has a greater effect on shock strength. In addition, theamount of space in theMP1 simulation is 1 mwhereas that of MP2 is2 m. In a 1 m domain, the calculated time required to optimallyattenuate a blast driven byMP1 data to 75% ofP0�x; T� (red curve in

Fig. 6 MP2 results (dx � 1 cm,dt � 1 μs,T0 � 2 ms); a) optimal control solutions integrated over time; b) pressure profiles at optimal final time c, top)logarithm of cost functional iterates; c, bottom) final time iterates.

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Fig. 5b) is about 1.8 ms whereas that of MP2 is about 2.1 ms (redcurve in Fig. 6b). It will take more space but not much more time toattenuate hot blast waves with larger overpressure (MP2) than it willwith weaker, cooler blast waves (MP1).To demonstrate that the overall solution procedure is mesh

convergent the spatial discretization was decreased by integer factors(2, 3, 4, 5, and 6). The temporal discretization was similarly decreasedto keep the cfl number constant. Figure 7a is a closeup view of theshock front for the optimal pressure profile using MP2 data with atarget state of 85%ofP0�x; T�. The simulationdomainwas only1m incontrast to the 2 m domain in Fig. 6. It can clearly be seen that thedissipative error near the shock front tends to zero, and the solutionapproaches an appropriate discontinuous jump at the shock front as thediscretization gets smaller. Figure 7b shows the integral of the optimalfinal time pressure profile converging over increasing spatialresolution. Figures 7c and 7d show a closeup view of the distributedwater vapor control distribution and optimal final time, respectively,over increasingly finer spatial and temporal discretization. Theconvergence criteria for each of the results was to stop when themaximum of the final pressure profile was below 6.65 · 105 Pa andthe relative change to the final time between iterations was lessthan 5 · 10−5.Figure 8 shows how the solution of the adjoint PDEwas optimized

with respect to run time in MATLAB. In all cases, an outer loop forthe time indexwas used. The slowest implementation, shown in blue,uses a loop for the spatial index as well. The other three curvesrepresent code with vectorized statements at each time step. Thegreen curve gives the run time when taking advantage of the sparsestructure of the matrices, as shown in Eq. (21). The red curve showsthe run timewhen the code is run in parallel on a graphics processing

unit (GPU) using the MATLABwrapper Jacket [32]. The solution ofthe adjoint PDE requires communication between the threads andthat overhead makes the run time longer for smaller problems. It isonly for the largest problem considered (500 spatial grid points and5000 time steps) that the benefit of multiple processors outweighsthe overhead of memory communication. For problems formulatedin larger domains or with additional spatial dimensions taking

Fig. 7 Mesh convergence results: Q�x� ∼ 0.85 · P0�x;T� in a 1 m domain where T0 � 1 ms; a) shock front; b) integral of final pressure over space;c) distributed control integrated over time; d) optimal final time.

Fig. 8 Comparison of run time durations required to solve the adjointpartial differential equation over the time interval �T�;0�with increasingproblem size.

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advantage of the parallel execution on GPUs would be essential tomaking efficient calculations.

VI. Conclusions

A new iterative solution procedure was developed, which cancalculate distributed optimal control solutions for the unsteady Eulerequation in a single dimension. The algorithm allows discontinuitiesin the states and free final data and final time. This procedure has beensuccessfully applied to single-phase, one-dimensional compressiblegas dynamics with the goal of diminishing overpressure in unsteadyshock waves. The control solutions with physical constraints arepresented for attenuating shocks traveling at Mach 1.5 and 3.5 downto 85%, 80%, and 75% of the uncontrolled wave’s driving pressure.The solutions generated are mesh convergent, and the adjoint partialdifferential equation has been run time optimized in MATLAB. Theexamples of optimal attenuation to blast waves typically encounteredin the launch environment of the shuttle’s solid rocket boosters duringan ignition are given. For a characterized dissipativemechanism (e.g.,water droplets) the generated control solutions give insight as to themagnitude and spatial distribution of the energy-equivalent mass ofwater vapor thatmust be produced via droplet vaporization to achievea given level of overpressure reduction. By comparing the resultsfor the two inlet boundary conditions, monitor point 1 (MP1) andmonitor point 2 (MP2), it is possible to see the scope of how thecontrol action will affect the range of unsteady shock fronts that canbe expected in an ignition overpressure (IOP) launch environment.

Acknowledgments

Nathan D. Moshman thanks the Naval Postgraduate School andthe Army Research Office for funding this research. In addition,Nathan D. Moshman would like to thank Chris Brophy, FrankGiraldo, and Wei Kang at the Naval Postgraduate School and BruceVu at NASA Kennedy Space Center.

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W. K. AndersonAssociate Editor

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