Jan 18, 2013 - generated by ignition overpressure from the shuttle's solid rocket booster during launch. Results are presented for attenuating shocks traveling ...
AIAA JOURNAL Vol. 51, No. 3, March 2013
Method for Optimally Controlling Unsteady Shock Strength in One Dimension Nathan D. Moshman,∗ Garth V. Hobson,† and Sivaguru S. Sritharan‡ Naval Postgraduate School, Monterey, California 93943
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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 derived and used in an adjoint-based solution procedure for the optimal control. A novel 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 a, b f H J J~ K L Lhv mH2 Ov
= = = = = = = = = =
ϵ P Q T U u V x z γ ∂Ω ρ ρE ρe ρu Ω Ωs
= = = = = = = = = = = = = = = = =
Subscripts
Jacobian matrix weighting constants functional of final time Hamiltonian cost functional augmented cost functional final time penalty running cost functional (Lagrangian) latent heat of vaporization of water at 100°C energy equivalent mass distribution of water vapor produced by control action positive small constant gas pressure target pressure at final time final time three-component state vector gas velocity three-component adjoint vector one-dimensional spatial vector in Ω control variable gas constant spatial domain boundary gas density gas total energy gas internal energy gas momentum spatial domain in one dimension spatial interval upstream of shock where P > Q
i, j, k m
= =
variable number spatial index
Superscripts l n
= = =
control algorithm iteration index temporal index optimal quantity
I.
O
Introduction
PTIMAL control of fluid dynamics has undergone rapid developments during the past two decades [1–3], and parallel developments in optimal aerodynamic shape optimization also have seen exciting advances [4–9]. Optimal control theory of hyperbolic systems of conservation laws for applications of gas dynamics with shock waves is addressed in [10,11]. In this paper, we will solve an optimal control problem for a one-dimensional hyperbolic system of conservation laws that arise in an important rocket launching problem. In the present work, the cost functional to be minimized will penalize the magnitude of the jump in pressure across the front of an unsteady shock wave after a finite simulation time. The control variable is a distributed field that removes energy from the gas upstream of the moving shock front and changes in both space and 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 chamber quickly rises several orders of magnitude. This drives hot combustion products toward the nozzle and out to the open atmosphere at supersonic speeds. An IOP wave is a spherical blast wave, which originates from the exit plane of the nozzle and propagates spherically outward. Overpressures that the body of the rocket experiences are of the order 2:1 [12]. The region just outside the nozzle will experience further compression due to the displacement of gas along the blast wave’s direction of propagation and overpressures can approach 10:1. The portions of the IOP wave that become incident on the rocket body or launch platform components must have an overpressure below a known threshold to avoid costly damage. The current technique used by NASA and other launch providers is to spray water into the region around the nozzle prior to ignition. This forces the IOP wave to propagate through water prior to
Presented as Paper 2011-3847 at the 20th AIAA Computational Fluid Dynamics Conference, Honolulu, Hawaii, 27–30 June 2011; received 29 February 2012; revision received 19 September 2012; accepted for publication 10 October 2012; published online 18 January 2013. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 1533-385X/13 and $10.00 in correspondence with the CCC. *Research Assistant Professor, Mechanical and Aerospace Engineering Department, 700 Dyer Road. Member AIAA. † Professor, Mechanical and Aerospace Engineering Department, Mail Code: MAE/Hg. Member AIAA. ‡ Director, Center for Decision, Risk, Controls and Signals Intelligence, 1 University Circle. 606
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becoming incident on the rocket body or platform components. Through several dissipative mechanisms this causes a sufficient decrease 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 water arrangement in the nozzle region and its effects on the maximum IOP strength [13,14]. The first work on parametrically optimizing IOP attenuation with respect to water injection strategy is in [13]. The goal of this work is to develop a computational tool that can directly calculate a distributed optimal control for attenuating a range of blast waves to a desired minimal overpressure.
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II.
Computational Fluid Dynamics
Data on the shuttle grain and chamber pressure [15] were given as input to Cequel, an open-source chemical equilibrium solver developed by NASA. Cequel [16] uses chemical properties from an extensive database to minimize the Gibbs free energy of the given combustion products and calculate equilibrium conditions. The output gives the temperature and gas velocity at the nozzle exit plane for a given pressure ratio. The computational fluid dynamics simulation domain has boundaries at the nozzle exit plane, the rocket body and the rest free space. Initially, ambient conditions inside the domain are present. The two-dimensional (2-D) ignition sequence simulation was performed using the ESI-Fastran commercial software [17]. The ESI-Fastran solver uses well-known and robust finite volume methodologies for a range of physical models involving compressible flows. The spatial discretization scheme used was Van Leer’s flux vector splitting [18] and was extended to second-order accuracy by a Barth limiter [19]. The Barth limiter enforces monotonicity and, therefore, is appropriate for solutions with strong discontinuities. The time integration was fully implicit with a tolerance of 10−4 in the residuals over a maximum of 20 subiterations. Constant mass-flow boundary conditions equivalent to the steadystate exit plane of the rocket nozzle on the shuttle were used in the bottom center of the domain on the right face of the step as shown in Fig. 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 last frame shows a snapshot of the flow 10 ms after ignition. The bottom
607
left edge of the domain represents the rocket body, whereas in the bottom right edge is the centerline of the normal to the nozzle exit plane and a symmetry boundary. All other edges are nonreflecting boundaries. 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 and point 2 is 1.5 m along the symmetry boundary and the plane normal to the nozzle exit. The conditions at the recorded locations are used as the boundary conditions in the one-dimensional Euler equations calculation used for the optimal control calculation. Figure 2a shows the flow conditions over time at monitor point 1 (MP1) near the rocket body and Fig. 2b shows the flow conditions over time for monitor point 2 (MP2) directly downstream of the nozzle. The transverse Mach number in both cases is about 0.5. Neglecting this motion corresponds to a 10% loss of total temperature. This is an acceptable simplification because the purpose is to be able to control a range of blast waves, not to most accurately predict a specific flowfield. Hotter gas will vaporize water more rapidly, thereby, extracting energy from the gas more quickly and yielding a more effective control action. Hence, no unfair advantage is gained by driving the inlet boundary condition 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 and placement determines shock attenuation. The conservation of mass, momentum, and energy are obeyed for both the gas and the liquid and an additional equation relates the volume fraction of either phase to the movement of the interface. A sink term for vaporization appears in the gas balance laws. The formulation in this way yields seven balance laws for one dimension with a vaporization source term. Several interaction mechanisms between the water droplets and the IOP wave are present and not fully understood. The dominant dissipative mechanism for shocks between Mach 1 and 2 is the loss of energy of the gas through the vaporization of the water droplets [14,20]. Experimental data from droplet-shock interactions in this regime show that the other dissipative mechanisms, e.g., droplet velocity, drag on droplets, sensible heating of droplets, chemical reactions, changes in specific heat, etc., are less significant to IOP attenuation. In a two-phase calculation, the control action would take the form of a liquid mass source, and the effect of vaporization most critical 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, and Δt 1 μs): Mach number at (1, 4, and 10 ms) (a, c, and e); pressure at (1, 4, and 10 ms) (b, d, and f).
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Fig. 2 Flow conditions over time for monitor points 1 and 2 shown in Fig. 1a (Δx 1 cm, Δt 1 μs).
To most simply replicate the dominant dissipative mechanism with a single-phase calculation the control will act as an energy sink with no corresponding mass or momentum sinks or sources, as shown in Eq. (2): U ρx; t; ρux; t; ρEx; tT
Ux; 0 given
0 1 0 1 0 1 ∂t ρu 0 ∂@ A ∂ @ 2 ρu ρu P A @ 0 A ∂t ∂x uρE P ρE zx; t
(1)
(2)
The control action is assumed to act instantaneously, but in a real twophase interaction the droplets take some amount of time to extract energy from the shock. As Jourdan et al. [14] showed, the more droplet surface area exposed to the gas, the greater the shock attenuation for a fixed amount of time. This is because the rate of vaporization is related to the total droplet surface area exposed. For a fixed amount of water smaller droplets expose more surface area than larger ones. Therefore, when interpreting the following single-phase results in a two-phase context, it is most appropriate to think of the smallest practical droplets that a typical atomizer can produce, a diameter of about 10 μm. The restriction that the instantaneous distributed control action, z, be only a sink, and, therefore, either negative 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 no new degrees of freedom to the calculation. The only place where the effect of water is replicated is in the energy sink and not in an equation of state that has been customized to gas and water vapor near the critical point (e.g., [21]). In addition, γ 1.4 for air and will not be allowed to change as it might in modeling a mixture of gas and water vapor: P ρeγ − 1
III.
Optimal Control System
In any optimal control problem there is a cost functional to minimize. Previous work on unsteady compressible flow control [7] was aimed at actuation near a boundary to suppress instabilities or the development of turbulence. In this work, the cost functional must reflect that a decrease in the maximum jump in pressure (the overpressure at the shock front) is most desirable at some final moment in time. It should also penalize control action but to a lesser degree. Therefore, let J be the cost functional: Z Z Z a T b zx; t2 dx dt Px; T − Qx2 dx (5) J 2 0 Ω 2 Ωs Here x represents a spatial vector, which is one-dimensional in this calculation. T is the final time, which is not fixed; zx; t is the control action; Px; T is the pressure at the final time; Qx is the desired final pressure; and a and b are weighting constants. The larger b is compared to a the more significant the final time penalty compared to the penalty for using the control action. In Eq. (6), L denotes the running penalty for control effort, whereas the final time penalty, denoted as K, penalizes the height of the pressure profile above a target Q. Ω represents the entire one-dimensional simulation domain, whereas Ωs is the interval behind the shock where the pressure distribution at the final time is above the target state, as shown in Fig. 3: Z a zx; t2 dx Lz·; t 2 Ω Z b KU·; T Px; T − Qx2 dx (6) 2 Ωs
(4)
A conservative Godunov-type method, second-order accurate in space, proposed in [22,23], was implemented to solve the compressible flow dynamics in one dimension under a distributed control action. This method assumes that the solution in each cell is piecewise linear and projects an intermediate solution on a nonuniform grid based on the maximum characteristic speeds from the interpolated solutions at each cell interface. The familiar Godunov integration [24] on the uniform grid is then accurate to the second-order because of the intermediate finer grid. This numeric method has been implemented previously in the literature for a single-phase calculation [25] as well as a two-phase calculation [26] based on a model presented in [27].
Fig. 3
Illustration of iterative procedure.
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MOSHMAN, HOBSON, AND SRITHARAN
The control action will take the form of an internal energy sink. It only appears on the right-hand side of the energy balance equation of the 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 neighborhood of the shock front and does not have vanishing properties as ε → 0. A slight increase in the amplitude behind the shock perturbs the speed and, therefore, also the location of the shock front. This causes small perturbations to induce variations on the order of the jump across the shock. The presented method of solution avoids this issue, as will be demonstrated later. Because only decreasing the amplitude of a shock wave is desired it is apparent that any realistic control action will only slow the shock wave down. The target state Qx and final time penalty will be constructed in such a way that all variations of the solution will occur upstream of the shock front only. Matching the simulated pressure profile under control action with the target final state near the shock front will occur by allowing the final time to be free. Henceforth, it can be assumed that all variations are taken in smooth regions of the flow and that the solution procedure will not depend 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 the conservative vector: ρx; 0 1 kg∕m3 ux; 0 0 m∕s Px; 0 105 Pa ρux; 0 0 kg:m∕s ρex; 0 250000 J ρEx; 0 250000 J
(7)
The inlet boundary condition is explicitly given by the IOP simulation data shown in Figs. 2a and 2b when the flow is supersonic. If the flow behind the shock front is subsonic, a nonreflection of the u-c characteristic is imposed [29]. In addition, the monitor point data were chosen where the flow was nearest to one dimensional; however, the 2-D data did have transverse motion. The data will still give a plausible one-dimensional blast wave with the inlet boundary condition set in this manner, and the goal of the calculation, controlling a range of blast waves, can be achieved. At the outlet boundary, the flow remains stationary because the final time will always be such that the shock wave will not have enough time to propagate 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 of the system and derive necessary conditions using the Pontryagin minimum principle and the calculus of variations. The pseudoHamiltonian for this system is Z ∂U dx HU; V; z; t LU; z V · ∂t Ω R ∂ρu ∂ρE Ω a2 z2 V 1 ∂ρ (8) ∂t V 2 ∂t V 3 ∂t dx where (V 1 , V 2 , V 3 ) is the adjoint vector. Writing the Euler equations in nonconservative form in this basis defines the Jacobian matrix:
B Aij U @
02
ρu γ−3 ρ2 2 3 −γρuρE γ − 1 ρu ρ2 ρ3
1 − ρu ρ γ − 3
γρE 3 ρu2 ρ − 2 γ − 1 ρ2
d ~ Jz εδzjε0 0 dε
(9)
(11)
Grouping the terms of like variational multipliers gives the optimal system. Integrating by parts until all derivatives are on the adjoint vector yields the following linear system of partial differential equations [10]: ∂V j ∂Aij dAij ∂Uk ∂V Aij i − · Vi ∂t ∂x ∂x dUk ∂x ∂ ∂L Aij vi ; δuj j∂Ω 0 (12) ∂x ∂Ui The nontrivial elements of the matrix dAij ∕dUk · ∂Uk ∕∂x are given below. ∂Ω denotes the boundary of the spatial domain. The righthand side of Eq. (12) is zero in this formulation because the running cost does not explicitly depend on the state vector: 2 ∂ ∂Uk u ∂ρ u ∂ρu γ − 3 − A21 · ∂Uk ∂x ρ ∂x ρ ∂x
(13)
∂ ∂Uk u ∂ρ 1 ∂ρu − γ − 3 A22 · ∂Uk ρ ∂x ρ ∂x ∂x
(14)
∂ ∂Uk u 2γE − 3γ − 1u2 A · ∂Uk 31 ∂x ρ ∂ρ 1 ∂ρu u ∂ρE 3γ − 1u2 − γE −γ ∂x ρ ∂x ρ ∂x ∂ ∂Uk 1 ∂ρ 3γ − 1u2 − γE A · ∂Uk 32 ∂x ρ ∂x u ∂ρu γ ∂ρE − 3γ − 1 ρ ∂x ρ ∂x ∂ ∂Uk γu ∂ρ γ ∂ρu − A · ∂Uk 33 ∂x ρ ∂x ρ ∂x
(15)
(16)
(17)
Because the final state is not fixed there is a necessary condition on the adjoint vector at the final time: V i x; T
∂ KUx; T ∂Ui
(18)
Therefore, in this basis, all three derivatives are nonzero, because the pressure is a function of all three conserved quantities: V 1 x; T bPx; T − Qx
∂Uj ∂Ui Aij U zi δi3 ∂t ∂x 0
one-dimensional Euler system using the Lagrange multiplier method each conservation law is multiplied by an adjoint variable and these ~ Then, terms are added to J. This is the augmented cost functional J. from expanding J~ in a Taylor series, the first-order necessary condition will be
∂P x; T ∂ρ
ux; T 2 γ − 1 2 ∂P x; T V 2 x; T bPx; T − Qx ∂ρu bPx; T − Qx
1 0 γ − 1 C A γρu ρ
(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 perturbed such that z z εδz where ε > 0 is a small constant. The variation in the control will cause variational terms in each of the other free variables of the system whose duality pairing must necessarily vanish at an optimal solution. To incorporate the constraints of the
−bPx; T − Qxux; T γ − 1 V 3 x; T bPx; T − Qx bPx; T − Qxγ − 1
∂P x; T ∂ρE (19)
The time derivative of the entire adjoint vector for all space at a discrete 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 in time will be of size km × 1, and the matrices in Eq. (12) will be of size km × km. A single component of the adjoint vector, e.g., V 1 x; t, will be size m × 1 at each time step. 00 11 V 1 x1 ; tn BB C B B V 1 x2 ; tn C CC BB CC .. B@ AC . B C B C B V 1 xm ; tn C B0 C 1 B V 2 x2 ; tn C B C B B V 2 x2 ; tn C C BB CC CC B .. ∂V lk ∂B B AC . ↔ B@ (20) C ∂t B V x ; tn C ∂t 2 m B C B C .. B C B C . n B 0 V k xk ; t 1 C B C B V k x2 ; tn C CC BB CC BB .. BB C . C AC B@ @ A n V k xm ; t All of the matrices in Eq. (12) have a diagonal-block structure. The Jacobian matrix for all of space at tn is shown in Eq. (21): Akν Un ↔ 1 0 1 1 0 0 n n x ;t x ;t A A 1k 1 C B 11 1 C CC B .. .. BB A ··· ··· @ AC B@ . . C B B A11 xm ;tn A1k xm ;tn C C B C B .. .. .. .. C B . . . . C B C B .. .. .. .. C B . . . . C B0 1 1 0 n C B Ak1 x1 ;tn x ;t A kk 1 C B C BB C C B . . .. .. B@ A ··· ··· @ AC A @ Ak1 xm ;tn Akk xm ;tn (21) The adjoint partial differential equation (PDE) is given in discrete form with an explicit integration in Eqs. (22) and (23): 1 IV kn−1 − V nk RUn V ni 0 Δt
RU A · DU
Dmxm
−2 B −1 1 B B B 2Δx B @
(23)
1 1 ..
. −1
C C C C . C 0 1 A 2 −2
..
The necessary condition on the optimal control solution comes from maximizing the Hamiltonian for an unconstrained control. The unconstrained condition is justified because there is no restriction on control magnitude in regions where control is allowed. The integral is true over any domain Ω, and at any moment in time it should be true pointwise for all t ∈ 0; T: Z ∂H az x; t V 3 x; t dx 0 (26) U ; V ; z ; t ∂z Ω In summary, Eqs. (7), (9–12), (19), (25), and (26) give the complete set of first-order necessary conditions for the optimal system.
IV.
∂A ∂A ∂U − · ∂x ∂U ∂x
2 0 .. .
d HU x; T ; V x; T ; z x; T ; T KUx; T 0 dt Z a zx; T 2 bPx; T − Qx Ω2 1 0 ∂P ∂ρ ∂ρ x; T ∂t x; T C B C B ∂P ∂ρu B ∂ρu x; T ∂t x; T C C dx 0 ×B (25) B ∂P C C B ∂ρE x; T ∂ρE x; T ∂t A @ ∂P x; T ∂t
(22)
The matrix D is made up of discrete spatial derivative block matrices, central differencing in the domain interior, upwind differencing at the outlet and downwind at the inlet. A single block is shown in Eq. (24): 0
With careful direction of the memory there is more potential in the assembly of R for speed optimization than will be shown in Sec. V. The final part of the adjoint time step is the time integration, which boils down to matrix addition and matrix-vector multiplication for the explicit scheme. These operations are known to be adaptable to parallelization 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 dissipation will recover the nonlinear PDE in the limit of vanishing viscosity. A small numerical viscosity can stabilize the adjoint solution. These ideas have been extended to fluid dynamics systems [30] and are implemented in the current work in a manner which maintains consistency 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 the Hamiltonian in order for the first-order variation of the cost functional to vanish. For a free final time the necessary condition for the optimal final time T is given by the transversality condition, which is obtained by grouping the terms and multiplying the δt variation after expanding Eq. (11):
(24)
Each time step of the adjoint solution has four parts. Prior to time integration, the matrix R must be assembled. Some of the matrices, which makeup R, are known in closed form and require no discretized derivative. The three-component system requires assembling A and ∂A∕∂U from the known state data. The second part of the solution requires assembling the matrix and the vectors that have discrete derivatives ∂A∕∂x, ∂V∕∂x, and ∂U∕∂x. These two parts can be done in parallel. The third part, calculating R, requires sharing memory between the processes and does not lend itself well to parallelization.
Solution Procedure
Let the left-hand side of the transversality condition in Eq. (25) be defined as a functional fT with the final time as the independent variable. Then fT 0. T can be solved iteratively with the Newton–Raphson root finding method. The derivative df∕dt and the discrete form of f and its derivative are given in [31]: fT l T l1 T l − df l dt T
(27)
The superscript l denotes an iterate of the inner loop of the solution procedure in Fig. 4. As fT l → 0 then T l → T . The condition required by Eq. (26) is similarly iteratively satisfied by the update in Eq. (28): δzl1 xi ; tn −V l3 xi ; tn ∕a zl1 xi ; tn zl xi ; tn δzl1 xi ; tn
(28)
Physical control constraints are then imposed. Because the calculation concerns shock attenuation extracting energy from the gas using a sink is of interest. Consequently, zl ≤ 0 for all space and time is enforced. It has been shown that the adjoint variables will travel along the characteristics of the flow in the opposite direction. This means that the calculation will suggest putting control ahead of the shock wave, which is not relevant because physical droplet-shock
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magnitude of the rate of energy extraction from the gas in the control solution. In addition, it is desirable to have b much larger than a so that minimizing to total cost J is dominated by minimizing K, the overpressure of the blast wave above the target state.
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V.
Fig. 4
Block diagram of solution procedure.
interactions only take energy out of the gas behind the shock wave. Therefore, Eq. (29) gives the restrictions on the control. The shock speed S is estimated based on the location of the shock front at T l : zl1 xi ; tn
0 0
if zl1 xi ; tn > 0 if xi > S · tn
(29)
The overall algorithm, which is used to satisfy all of the above necessary optimal conditions, is shown in block-diagram form in 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 so that the physical units of overpressure are scaled to a meaningful
Results and Discussion
The plots shown in Fig. 5a are the distributions of the energyequivalent mass of water vapor produced by the optimal control solutions. By dimensional analysis, inspection of the energy balance equation indicates that the units of z x; t are watts. Integrating the control solution from 0; T gives an energy distribution in space. This energy can be directly equated to a required mass of water vapor that must be produced from liquid water droplet vaporization. The latent heat of vaporization of water is Lhv −2.26e6 J∕kg at 100°C. Equation 30 relates the optimal control solution z , distributed in space and time, to an energy equivalent distribution of the mass of water vapor produced over the entire simulation time interval mH2 0v : Z T 1 z x; t dt (30) mH2 0v x Lhv 0 The solution procedure in Fig. 4 assumes a given target state Qx at the final time T . To illustrate trends in the optimal control solution, prescribing a consistent and meaningful target state or sequence of target states is a necessity. Each target state is defined by locating the shock front of P0 x; T and setting Qx equal to a fraction, (0.85, 0.80, 0.75), times the value of P0 x; T upstream of the shock front and equal to P0 x; T downstream of the shock front. The results for controlling the blast data from MP1 are shown in Fig. 5. The solid black curve in Fig. 5a is the optimal water vapor mass distribution that yields a final time pressure profile closest to, and below, a target state, which has 85% of the pressure magnitude of the uncontrolled blast
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 target state as a dashed black curve and the optimal pressure profile as a solid black curve. The solid blue curves in Figs. 5a and 5b are the optimal control and pressure profiles, respectively, for a target with 80% of the magnitude of P0 x; T (dashed blue curve). The red curves 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 the cost functional J decreasing monotonically over solution iterations for a single target state. The bottom plot in Fig. 5c shows the optimal final 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 is decreased. The control curves are approximately linear, increasing from downstream to upstream of the direction of shock propagation. This indicates that the further upstream the water vapor can be produced the more effect it will have on diminishing the overall magnitude of pressure at the final time. The final time converges to a larger value than was used to obtain P0 x; T, because the speed of the shock is directly related to its amplitude. Hence, as the control action attenuates the amplitude, the shock will also slow down, requiring a longer simulation time for the shock front to match up with 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 water vapor mass distribution that yields a final time pressure profile closest to, while below, a target state, which has 85% of the magnitude of pressure of the uncontrolled blast wave P0 x; T behind the shock front. Figure 6b shows the target state as a dashed black curve and the optimal pressure profile as a solid black curve. The blue curves in Figs. 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 with 75% 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 the minimum value of J is increasing for the three separate targets. Although the final time penalty K roughly goes to zero in each case more control effort is needed to attenuate the pressure to a smaller amplitude and, hence, the increase in the minimum of J is due to L, which penalized the additional control effort. The argument is made that J is minimized because it monotonically decreases over the solution procedure, all necessary conditions of a minimum are satisfied, and that using less control would increase K, and, hence, J. Conversely, using more control would also increase J because L would 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 that the optimal final time increases over the three intervals with decreasing target amplitudes, because each of the waves will be moving slower and require a longer simulation time to match the target shock front. A comparison of the optimal water vapor distributions in Figs. 5a and 6a gives further insight into water as a control. Take, for example, the black curves representing a target state with an amplitude equal to 85% of P0 x; T. For MP1, this corresponds to an absolute pressure decrease of about 0.3 atm, whereas for MP2 it is about 1.2 atm. The absolute pressure decrease for MP2 is four times that of MP1, yet the maximum amount of water vapor required for MP2 is less than three times that of MP1. The reason is that the MP2 data have a hotter driving gas than MP1 and, therefore, cooling that gas with an internal energy sink has a greater effect on shock strength. In addition, the amount of space in the MP1 simulation is 1 m whereas that of MP2 is 2 m. In a 1 m domain, the calculated time required to optimally attenuate a blast driven by MP1 data to 75% of P0 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. 7 Mesh convergence results: Qx ∼ 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. 5b) is about 1.8 ms whereas that of MP2 is about 2.1 ms (red curve in Fig. 6b). It will take more space but not much more time to attenuate hot blast waves with larger overpressure (MP2) than it will with 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 decreased to keep the cfl number constant. Figure 7a is a closeup view of the shock front for the optimal pressure profile using MP2 data with a target state of 85% of P0 x; T. The simulation domain was only 1 m in contrast to the 2 m domain in Fig. 6. It can clearly be seen that the dissipative error near the shock front tends to zero, and the solution approaches an appropriate discontinuous jump at the shock front as the discretization gets smaller. Figure 7b shows the integral of the optimal final time pressure profile converging over increasing spatial resolution. Figures 7c and 7d show a closeup view of the distributed water vapor control distribution and optimal final time, respectively, over increasingly finer spatial and temporal discretization. The convergence criteria for each of the results was to stop when the maximum of the final pressure profile was below 6.65 · 105 Pa and the relative change to the final time between iterations was less than 5 · 10−5 . Figure 8 shows how the solution of the adjoint PDE was optimized with respect to run time in MATLAB. In all cases, an outer loop for the time index was used. The slowest implementation, shown in blue, uses a loop for the spatial index as well. The other three curves represent code with vectorized statements at each time step. The green curve gives the run time when taking advantage of the sparse structure of the matrices, as shown in Eq. (21). The red curve shows the run time when the code is run in parallel on a graphics processing
unit (GPU) using the MATLAB wrapper Jacket [32]. The solution of the adjoint PDE requires communication between the threads and that overhead makes the run time longer for smaller problems. It is only for the largest problem considered (500 spatial grid points and 5000 time steps) that the benefit of multiple processors outweighs the overhead of memory communication. For problems formulated in larger domains or with additional spatial dimensions taking
Fig. 8 Comparison of run time durations required to solve the adjoint partial differential equation over the time interval T ;0 with increasing problem size.
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advantage of the parallel execution on GPUs would be essential to making efficient calculations.
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VI.
Conclusions
A new iterative solution procedure was developed, which can calculate distributed optimal control solutions for the unsteady Euler equation in a single dimension. The algorithm allows discontinuities in the states and free final data and final time. This procedure has been successfully applied to single-phase, one-dimensional compressible gas dynamics with the goal of diminishing overpressure in unsteady shock waves. The control solutions with physical constraints 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 solutions generated are mesh convergent, and the adjoint partial differential equation has been run time optimized in MATLAB. The examples of optimal attenuation to blast waves typically encountered in the launch environment of the shuttle’s solid rocket boosters during an ignition are given. For a characterized dissipative mechanism (e.g., water droplets) the generated control solutions give insight as to the magnitude and spatial distribution of the energy-equivalent mass of water vapor that must be produced via droplet vaporization to achieve a given level of overpressure reduction. By comparing the results for the two inlet boundary conditions, monitor point 1 (MP1) and monitor point 2 (MP2), it is possible to see the scope of how the control action will affect the range of unsteady shock fronts that can be expected in an ignition overpressure (IOP) launch environment.
Acknowledgments Nathan D. Moshman thanks the Naval Postgraduate School and the Army Research Office for funding this research. In addition, Nathan D. Moshman would like to thank Chris Brophy, Frank Giraldo, and Wei Kang at the Naval Postgraduate School and Bruce Vu at NASA Kennedy Space Center.
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W. K. Anderson Associate Editor
