Instationary heat constrained trajectory optimization of a ... · transformed into a large scale state and control constrained, but purely ODE constrained optimal control problem,

Instationary heat constrained trajectory

optimization of a hypersonic space vehicle

by ODE-PDE constrained optimal control

K. Chudej1, H. J. Pesch1, M. Wachter2, G. Sachs3, and F. Le Bras4

1 Lehrstuhl fur Ingenieurmathematik, Universitat Bayreuth, Bayreuth, Germany2 German Institute of Science and Technology, Singapore3 Lehrstuhl fur Flugmechanik und Flugregelung, Technische Universitat Munchen,

Munchen, Germany4 Laboratoire de Recherches Balistiques et Aerodynamiques, Delegation Generale

pour l’Armement, Vernon, formerly: Ecole Polytechnique Paris, France

dedicated to Professor Angelo Miele on the occasion of his 85th birthday

Summary. During ascent and reentry of a hypersonic space vehicle into the atmo-sphere of any heavenly body, the space vehicle is subjected, among others, to extremeaerothermic loads. Therefore an efficient, sophisticated and lightweight thermal pro-tection system is determinative for the success of the entire mission. For a deeperunderstanding of the conductive, convective and radiative heating effects througha thermal protection system, a mathematical model is investigated which is givenby an optimal control problem subject to not only the usual dynamic equations ofmotion and suitable control and state variable inequality constraints, but also to aninstationary quasi-linear heat equation with nonlinear boundary conditions. By thismodel the temperature of the heat shield can be limited in certain critical regions.

The resulting ODE-PDE constrained optimal control problem is solved by asecond-order semi-discretization in space of the quasi-linear parabolic partial dif-ferential equation yielding a large scale nonlinear ODE constrained optimal controlproblem with additional state constraints for the heat load. Numerical results ob-tained by a direct collocation method are presented, which also include those foractive cooling of the engine by the liquid hydrogen fuel. The aerothermic load andthe fuel loss due to engine cooling can be considerably reduced by optimization.

1 Introduction

In hypersonic flight regimes, i. e. with Mach number greater than about five,the aerothermic heating constitutes one of the major problems requiring newmaterials and new heat protection systems that can withstand temperaturesup to 2000 K and simultaneously are lightweight and low maintenance. There-fore, advanced mathematical models are required, to compute and optimize

2 K. Chudej, H. J. Pesch, M. Wachter, G. Sachs, and F. Le Bras

the thermal load in hypersonic flight regimes. For that, one necessitates a so-phisticated model which couples a usual flight path optimization problem withan instationary heating constraint. This will lead to an optimal control prob-lem subject to a coupled system of ordinary differential equations (ODEs),which describes the equations of motion, and, in addition, a parabolic par-tial differential equation (PDE), which describes the heat conduction throughthe thermal protection system. One can refer the coupled nonlinear ODE-PDEsystem to as a nonlinear partial differential algebraic equation system (PDAE)(with zero differential time index as well as MOL index and differential spaceindex to be infinity [19]).

Among the different concepts for future space transportation systems, weconcentrate in this paper on the German Sanger II-concept [17]. This conceptis concerned with a two-stage winged aircraft, consisting of a rocket-propelledorbital stage and an airbreathing carrier being able to start and land hori-zontally.5 Optimal flight paths for two-stage vehicle of Sanger-type have beencomputed and analysed, e. g., by [2, 4, 5, 14, 30]. In this paper however, weconcentrate on the lower stage only which may be considered for interconti-nental hypersonic flights, too.

Thermal protection systems (TPSs) are a need for hypersonic flights andconsist of different insulated layers of suitable materials and thickness. Fora realistic simulation of the heat transfer, a detailed mathematical modelis required, that includes unsteady effects and also material properties; see,e. g. [8, 34]. Layered TPSs have been investigated in [13, 16, 36].

In this paper, we concentrate on the reduction of the extreme heating loadsby simultaneously optimizing a ODE-PDE system consisting of a sophisticatedmodel for hypersonic flight regimes and a quasi-linear heat equation, both forheat conduction and transport, with nonlinear boundary conditions includ-ing radiation effects. Simultaneous flight-path optimization and aerothermicheating have been investigated also in, e. g., [8, 9, 10, 13, 15, 20, 25, 28, 35, 36].

In addition, we investigate active cooling of the engine by the liquid hy-drogen fuel. This is especially suited because of its large heat capacity andlow temperatures; see also, e. g., [3, 11, 27, 29]. Moreover, it improves thecombustion of the fuel by increasing its activation energy. Active cooling is acommon design in liquid-fuel rocket propulsion.

This paper is an expanded version of [6] and is based on the report [18]and the PhD thesis [34]. In contrast to [34], we treat the problem as an opti-

5 Angelo Miele published some critical work on the NASP (National Aero-SpacePlane), which was supposed to be a single-stage configuration powered by fourpowerplants operating in sequence in different Mach number regimes (turbojet,ramjet, scramjet, rocket). His analysis was done under a grant from NASA-LRC.Miele’s results indicated that the NASP was not feasible as a single-stage con-figuration and this is why he urged NASA to look instead to at least a double-stage configuration. The associated publication can be found in the Acta of theAcademy of Sciences of Turin, a dusty academy founded by Lagrange; see [22].See also [23] and [24].

Instationary heat constrained trajectory optimization 3

mal control problem for a coupled ODE-PDE system. In terms of numericalmethods for PDEs, in [34] a cell oriented energy preserving method was usedwith lumped parameters for heat capacity and heat conductivity, called knotmodel approach; see [8]. This approach is equivalent to solving an initial-multipoint-boundary-value problem. In this paper, we treat the coefficients astemperature dependant. Significant technical improvements may be achievedby both approaches with respect to efficiency, costs, and weight of the TPS.

The paper is organized as follows. In Section 2 a two-dimensional flightpath optimal control problem for minimum fuel consumption is, first of all,presented and numerically solved by a direct collocation method. Its optimalsolution will constitute the reference trajectory. Then, three modified optimalcontrol problems are investigated. Two of which take into account that thecold hydrogen fuel can be used to cool down the structure and the turbo-ramjet engine to their individual demands and reused for powering the engineafterwards. In contrast, the third model variant simulates that the fuel isreleased after it has been flowed through the cooling devices.

Finally, a much more complex optimal control problem including an insta-tionary quasi-linear heat equation in one, resp. two spatial dimensions withnonlinear boundary conditions and a state constraint is considered, in orderto limit the temperature in critical reagions more realistically, for example atthe stagnation point or around the engine. The scheme used for discretizingthe partial differential equation involved is analyzed and associated numericalresults are discussed. This is the content of Section 3.

From a mathematical point of view, the resulting problem constitutes anoptimal control problem subject to a coupled system of ODEs and a PDE witha multiplicity of control and state variable inequality constraints. The couplingis done not only through coefficients of the PDE depending on state variablesof the ODEs, but also through boundary conditions of the parabolic PDE.The controls occur in the ODE system only. Hence, this problem constitutesa new type of ODE-PDE constrained optimization problem where the PDE issimultaneously controlled in a distributed and boundary-controlled manner.However, the controls exert its influence indirectly only via the state variablesof the ODE subproblem. Besides the typical state and control constraints inflight dynamics, the PDE subproblem additionally exhibits a state constraintwhich closes the dependency between ODE and PDE subproblem. This typeof optimal control problem not discussed so far in the literature has given riseto a subsequent paper, in which a detailed analysis of an abstract prototypeproblem of similar kind is presented; cf. [26].

For an overview of the theory of optimal control subject to PDEs we referto the new book of Troltzsch [33]. State constrained PDE optimal controlproblems and also quasi-linear problems are subject to actual research. Somefirst references on state constrained PDE optimal control problems can befound in [33], too.

Because of the complexity of the final model we pursue here the so-called direct approach despite its known lack of reliability; cp. [12]. By semi-


discretization in the spatial variables (i. e. using the vertical method of lines)the state and control constrained ODE-PDE optimal control problem is firstlytransformed into a large scale state and control constrained, but purely ODEconstrained optimal control problem, which itself is solved by a direct collo-cation method.

The paper is completed with some numerical results for the ODE-PDEconstrained optimization in Section 4.

2 Trajectory optimization problems with active cooling

The well-known optimal control problem for a minimum fuel, resp. maximumfinal mass range flight of a hypersonic space vehicle over a spherical rota-tional earth under the assumptions of a point mass model and, for the sakeof simplicity, for a two-dimensional flight over the equator is given by

maxα(t),δT(t)

m(tf) (1)

subject to

v =1

m

(

T (v, h; α, δT) cos(α + σT) − D(v, h; α))

−g(h) sin γ + ω2E r(h) sin γ , (2)

γ =1

m v(T (v, h; α, δT) sin(α + σT) + L(v, h; α))

+ cos γ( v

r(h)−

g(h)

v+

ω2E r(h)

v

)

+ 2 ωE , (3)

h = v sin γ , (4)

ζ = v cos γ , (5)

m = −

βT(v, h; α, δT) (no active cooling,resp. active cooling I) ,

maxβT(v, h; α, δT), βC(v, h; α, δT) (active cooling II) ,βT(v, h; α, δT) + βC(v, h; α, δT) (active cooling III) .

(6)

A derivation of the equations of motion can be found, e. g. in Miele [21]and Vinh et. al. [37].

The state variables are velocity v, flight path angle γ, altitude h, pathlength ζ and vehicle mass m. The control variables are angle of attack α andthrottle setting δT. Note that due to the hypersonic flight also the angle ofattack exerts an influence on the instantaneous fuel consumption (6). Finally,the flight time interval is [0, tf ] with the final time tf unspecified in case ofno active cooling or active cooling I or II, resp. specified in case of activecooling III.

The approximations used here for the data fields of lift L =CL(v, h; α) (ρ(h)/2) v2 S, drag D = CD(v, h; α) (ρ(h)/2) v2 S, and thrust


T (v, h; α, δT), which are sufficiently often differentiable in the relevantaltitude-Mach number flight envelope (0 km < h < 40 km, 0 ≤ M ≤ 7.2), canbe found in [34]; see also [7, 36]. The model includes the possiblity of overfu-eled combustion. In [34], approximations are given for all relevant aerodynamicquantities as functions of air temperature Θair and air density using the lawsof thermodynamics. Air temperature Θair and air density themselves are ap-proximated as functions of the altitude h. The functions g(h) = g0 (rE/r(h))

2

and r(h) = rE + h in the Eqs. (2)–(6) denote the acceleration of gravity andthe distance of the vehicle from the geocenter.

Furthermore, the optimal trajectory must obey certain control and statevariable inequality constraints

−1.5

180π ≤ α ≤

20

180π , 0 ≤ δT ≤ 1 , (7)

0 ≤ n(v, h, m; α) ≤ 2 , 10 [kPa] ≤ q(v, h) ≤ 50 [kPa] (8)

limiting the angle of attack α, the throttle setting δT, the load fac-tor n(v, h, m; α) = L(v, h; α)/(m g0), and the dynamic pressure q(v, h) =((h)/2) v2.

The optimal control problem is completed by the following initial and finalconditions: v(0) = 150 [m/s], γ(0) = 0 [rad], h(0) = 500 [m], ζ(0) = 0 [km],m(0) = 244 000 [kg], v(tf) = 150 [m/s], γ(tf) = 0 [rad], h(tf) = 500 [m], andζ(tf) = 9 000 [km].

In the present paper, we will consider four different models for describingthe fuel consumption. The first model (1)–(6, no active cooling) takes intoaccount only the fuel consumption βT(v, h; α, δT) of the turbo-ramjet engine.It will be referred to as reference trajectory optimization problem. Its optimalsolution is shown in Figs. 1 and 2 and yields a fuel consumption of 60 614 [kg]and a final time of tf = 7 023 [s].

The other three models additionally include a complicated active coolingsystem of the engine modelled by an additional term βC(v, h; α, δT) where eachmajor component of the engine is cooled down to its individual demand, suchas walls and nozzle of the turbo combustion chamber, here to a temperatureof ΘE = 1 600 [K]. Furthermore, in two of those three model variants theinstantaneous amount of fuel for cooling must not exceed that for thrust, i. e.,

βC(v, h; α, δT) ≤ βT(v, h; α, δT) . (9)

Concerning the first mass model (6, no active cooling), this constraint, whenadditionally taken into account, allows that the entire amount of fuel for thecooling circuit can be reused for the thrust afterwards, or, in other words, theflight path has to be controlled in such a way, that the fuel needed for coolingmust not exceed that for thrust. The variant (1)–(6, active cooling I), (7) isreferred to as active cooling I. However, the optimal solution of this problemcoincides with the reference trajectory shown in Figs. 1 and 2, since the stateconstraint (9) becomes nowhere active.


velocity [m/s] flight path angle [deg]

altitude [m] range [m]

0 2000 4000 6000 80000

500

1000

1500

2000

0 2000 4000 6000 8000−10

−5

0

5

10

0 2000 4000 6000 80000

1

2

3

4x 10

4

0 2000 4000 6000 80000

2

4

6

8

10x 10

6

Fig. 1. Time histories of the state variables v, γ, h, and ζ for the reference trajec-tory optimization problem (no active cooling), identical to active cooling I and II(backflow of fuel from cooling devices)

Therefore, the optimal trajectory with (6, active cooling I), (7) replaced by(6, active cooling II) coincides with the reference trajectory, too. This variantwould assess the fuel for cooling only as consumption, if it really exceedsthat for thrust. It has been already investigated in [8, 34], where, however,the instantaneous fuel consumption βC exceeds βT at certain times and thesurplus had therefore to be released.

In addition, Figs. 3, 4 show the optimal trajectory for (1)–(6, active cool-ing III) when the fuel used for active cooling is not reused for the engines.When the terminal time is released to 7.500 [s], one obtains a fuel consumptionof 84.597 [kg]. This amount increases even to 91.504 [kg], if we screw down theterminal time to the optimal value of the reference trajectory.

All trajectores can be separated in three phases: ascent, cruise flight anddescent of the hypersonic vehicle; see Fig. 1. Within about 1 500 [s] the cruisingaltitude is reached followed by a strong acceleration with maximum thrust; seeFig. 2. The cruise proceeds with almost constant flight path angle, while thetwo controls, angle of attack and throttle setting, as well as the lift coefficientare also almost constant; see Figs. 1 and 2 again. The latter stays in thevicinity of the lift coefficient for minimum drag; cf. [34]. A slight periodicity


0 2000 4000 6000 80000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2000 4000 6000 80002.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5angle of attack [deg] throttle setting [−]

Fig. 2. Time histories of the control variables α and δT for the reference trajectoryoptimization problem (no active cooling), which is identical to active cooling I and II(backflow of fuel from cooling devices)

can be observed; see Fig. 2. This quasi-stationary flight phase covers about60 per cent of the entire flight time.

All numerical results have been obtained by the direct-collocation optimal-control software package Dircol of Stryk [31, 32].

So far we have not taken into account a goal-oriented reduction of thetemperature of the TPS in critical regions in the optimization process. Thiswill be the subject of the subsequent section.

3 Trajectory optimization problem with an instationary

heat constraint

In hypersonic flow the air stream impinges on the vehicle surface, and there-fore its kinetic energy is transformed into thermic energy which leads to airtemperatures of more than 2 000 K. In order to reduce the costs for the nec-essary thermal protection system, the reference problem is expanded so thata limitation of the surface temperature can be obtained by optimal control.For this purpose, the system of ordinary differential equations (2)–(6, no ac-tive cooling) for the dynamical behaviour of the vehicle is augmented by a


velocity [m/s] flight path angle [deg]

altitude [m] range [m]

0 2000 4000 6000 80000

2

4

6

8

10x 10

6

0 2000 4000 6000 80000

1

2

3

4x 10

4

0 2000 4000 6000 8000−10

−5

0

5

10

0 2000 4000 6000 80000

500

1000

1500

2000

Fig. 3. Time histories of the state variables v, γ, h, and ζ for the trajectory opti-mization problem with fixed final time tf = 7.500 [s], active cooling III (fuel fromcooling devices is released)

parabolic partial differential equation for instationary heat flow and an addi-tional state constraint for the temperature in critical regions of the surface.

For the sake of simplicity and computability, we restrict ourselves to themost critical regions, the stagnation point and the engine. When neglect-ing the heat flow tangential to the layers of the TPS, it is sufficient to dealwith a spatially one-dimensional heat equation. This is, for example, suitablefor an investigation of the heating at the stagnation point. However, for theinvestigation in the neighborhood of the engine we have to deal with a spa-tially two-dimensional problem. In both cases, the PDE subsystem for theheat load Θ(x, t), x ∈ Ω ⊂ Rd, d = 1, 2, t ∈ (0, tf), Ω open, bounded andlieing in the vertical symmetry plane of the aircraft, is given by the followinginitial-boundary-value problem:

(h)(

cp(Θ) + c′p(Θ)Θ) ∂Θ

∂t+ ′(h) h cp(Θ)Θ

−∇ · (λ(Θ, p)∇Θ) = 0 for all (x, t) ∈ Ω × (0, tf) (10)

Θ(x, 0) = Θ0(x) = 300 [K] for all x ∈ Ω (11)

∂Θ

∂n(x, t) = qconv − qrad for all x ∈ ∂Ω and all t ∈ (0, tf) , (12)


0 2000 4000 6000 80000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2000 4000 6000 80002.5

3

3.5

4

4.5

5

5.5

6

6.5

7angle of attack [deg] throttle setting [−]

Fig. 4. Time histories of the control variables α and δT for the trajectory optimiza-tion problem with fixed final time tf = 7.500 [s], active cooling III (fuel from coolingdevices is released)

where the heat capacity cp depends on the temperature Θ and the heat con-ductivity λ on both the temperature Θ and the pressure p = (h)R Θair(h);R is the gas constant. Hence, the pressure itself is a function of the altitude h.Heat capacity and heat conductivity are given as follows:

cp = Rκ

κ − 1with κ =

∑5i=0 di Θi

∑1j=0 ej Θj

,

λ(Θ, p) =

∑3i=0 gi Θi if Θ ≤ 1 400 [K]

ξ1

∑3i=0 gi Θi + ξ2 Pλ(Θ, p) if 1 400 [K] < Θ < 1 600 [K]

Pλ(Θ, p) if 1 600 [K] ≤ Θ

.

Here, ξ1 = (1− tanh(100 (Θ− 1500))/2 and ξ2 = (1 + tanh(100 (Θ− 1500))/2denote transfer functions between the pressure independent regime below1 400 K and the pressure dependent regime above 1 600 K. Furthermore, Pλ isa polynomial approximation of the heat conductivity based on suitable datafields; see [34] for details.

A derivation of Eq. (10) is given in the Appendix, Part 1.In the nonlinear boundary condition (12), the quantities qconv and qrad

relate to the convective and radiative heat fluxes. If the outer normal n pointsinto the exterior environment of the vehicle, we use

qconv := qair(v, h, Θair; α; xL, Q) with Θair = τ(v, h, Θair; α) ,


qrad := ε σ(

Θ4 − Θ4air

)

,

where the air temperature Θair after the shock depends on the actual flightconditions. Futhermore, xL denotes the length of the laminar-to-turbulenttransition. It varies with the position Q on the vehicle surface. The constants εand σ denote emissivity and Stefan-Boltzmann constant.

If the outer normal n points into the interior of the vehicle we use

qconv := αq (Θ − Θint) ,

qrad := ε σ(

Θ4 − Θ4int

)

,

with the heat transfer coefficient αq = αq(Θint, p) and the interior tempera-ture Θint which relates to the inner surface of the TPS. This may be either apart of the vehicle structure or the inside air. Again all formulae can be foundin [34].

For the sake of simplicity, we do not have included a layered TPS whichwould require to include radiative fluxes at each boundary layer of the TPS,too, making (10)–(12) an initial-multipoint-boundary-value problem. By a so-called knot model approach this has been included in the optimization processin [34]. Actually this approach can be interpreted as a semi-discretization inspace where the step size equals the thickness of the layers.

Finally, the model is completed by a state constraint on the temperature:

Θ(x, t) ≤ Θmax(x) for all (x, t) ∈ Ω × (0, tf) . (13)

In order to afford the numerical computations, we simplify the upperbound Θmax to a constant and confine the domain Ω in (13) either to a spatialinterval (1D case), e. g., extending from the stagnation point inwards throughthe TPS by neglecting any fluxes tangentially to the TPS, or to a rectangle(2D case) in the vertical symmetry plane of the aircraft also pointing inwardsthrough the TPS, e. g., for investigating heating effects near the engine. In-deed, it is sufficient to limit the temperature only in the most critical regions.In the 1D case, the parabolic partial differential equation is discretized with re-spect to the space variable x and with step size ∆x using an implicit method.This is necessary because of the stiffness of the resulting ODE system. Wehave used the scheme:

(h)(

cp(Θ1) + c′p(Θ1)Θ1

)

Θ1

:=1

∆x

(

(

qair(v, h, Θair; α; xL, Q) − ε σ(

Θ41 − Θ4

air

)

)

−λ

(

Θ1 + Θ2

2

)

Θ1 − Θ2

∆x

)

− ′(h) h cp(Θ1)Θ1 , (14)


(h)(

cp(Θi) + c′p(Θi)Θi

)

Θi

:=1

∆x

(

λ

(

Θi−1 + Θi

2

)

Θi−1 − Θi

∆x− λ

(

Θi + Θi+1

2

)

Θi − Θi+1

∆x

)

−′(h) h cp(Θi)Θi , for i = 2, . . . , n − 1 , (15)

(h)(

cp(Θn) + c′p(Θn)Θn

)

Θn

:=1

∆x

(

λ

(

Θn−1 + Θn

2

)

Θn−1 − Θn

∆x

−(

αq (Θn − Θint) − ε σ(

Θ4n − Θ4

int

)

)

)

−′(h) h cp(Θn)Θn , (16)

where Θi = Θi(t) := Θ(xi, t).For the sake of brevity, we show, for the analogous discretization in 2D,

only the stencil of the discretization; see Fig. 5. A detailed derivation is givenin the Appendix, Part 3.

interior of vehicle

air

Θ11 Θ21 Θ31

Θ12 Θ22 Θ32

convection convectionradiation radiation

conduction conduction

conduction conduction

convection convectionradiation radiation

Fig. 5. Discretisation stencil (here n = 3, m = 2) for the 2D case.

Here, the notations are Θij = Θij(t) := Θ(xij , t) with xij := (xi, yj),i = 1, . . . , n, j = 1, . . . , m. In case of dealing with a layered TPS the verticalconduction arrows in Fig. 5 are to be replaced by convection and radiationarrows likewise as at the initial or terminal lines.

These discretization schemes can be shown to be of order two; see Ap-pendix, Parts 2 and 3.

The ODE part (2)–(6, no active cooling) is now augmented by either thesystem (14)–(16) or an equivalent system associated with the stencil of Fig. 5


including n, resp. n m additional discretized state variable inequality con-straints

Θi(t) ≤ Θmax for all t ∈ (0, tf) with i = 1, . . . , n

or

Θi,j(t) ≤ Θmax for all t ∈ (0, tf) with i = 1, . . . , n , j = 1, . . . , m .

Note that due to the quasi-linearity of the PDE the resulting ODE sys-tem (14)–(16) is nonlinear.

We now present the numerical results, first for the 1D case. Fig. 6 showsthe results for a limit temperature of Θmax = 1 000 [K] at the stagnationpoint. It exhibits a boundary arc on the first line which still can be seen onthe subsequent lines, however, with decaying maximum temperatures.

0 5000 10000 [s]0

500

1000

1500

2000velocity [m/s]

0 5000 10000 [s]0

0.5

1

1.5

2

2.5

3

3.5x 10

4 altitude [m]

0 5000 10000 [s]2

4

6

8

10flight path angle [deg]

0 5000 10000 [s]200

400

600

800

1000

1200temperature 1st line [K]

0 5000 10000 [s]

400

600

800

1000

1200temperature 2nd line [K]

0 5000 10000 [s]

400

600

800

1000

1200temperature 3rd line [K]

Fig. 6. Time histories of state variables v and h, and γ. Dashed lines: referencetrajectory optimization problem. Solid lines: temperature constrained optimizationproblem with Θmax,i = 1000 [K] at the stagnation point.

In order to reduce the temperature at the stagnation point, one obviouslyhas to fly in lower altitudes at lower velocities. For a reduction of the stag-nation point temperature compared to the reference optimization problem


900 950 1000 1050 1100 1150 [K]0.98

1

1.02

1.04

1.06

1.08

1.1tmax/tmax0mB0/mB

Fig. 7. Maximum temperature ratio Θmax, ref/Θmax (upper curve) and ratio ofthe total fuel consumption mfuel, ref/mfuel (lower curve) versus maximum tempera-ture Θmax. The index ref refers to the reference optimization problem.

by 10 per cent, one has to increase, thanks to optimization, the total fuelconsumption by about 1 per cent only; see Fig. 7.

It has to be noticed that we had to restrict the relevant time interval for theheat equation to the interval [482 [s], 5478 [s]]. For larger intervals numericalresults could not be obtained anymore. However, this does not play a role,since the heat load obviously is less than its maximum outside of this interval.

Finally, we present the numerical results for the 2D case close to the frontpart of the fuel tank; see Fig. 8. The temperature constraints do not becomeactive here.

The temperatures also remain moderate at the lower surface; see Fig. 9.

4 Conclusions

A complex mathematical model has been presented in order to control theheating of thermal projection systems of hypersonic aircraft. The model is acoupled system of nonlinear ordinary and a quasilinear parabolic partial dif-ferential equation with non-linear boundary conditions. Altogether the math-ematical model presented here is an ODE-PDE control-and-state-constrainedoptimal control problem. By a semi-discretization in space using a finite vol-ume method, this problem is transformed into a large scale ODE control-and-state-constrained optimal control problem. Note that the resulting problemcannot be solved by standard software packages to higher accuracies. DirectODE constrained optimal control software as well as their incorporated SQPmethods come to their limits. Nevertheless, despite the coarse discretizationof the PDE, satisfactory results could be obtained, which show that detailedmodelling and sophisticated numerical methods can help to determine thenecessary dimensioning of thermal protection systems for hypersonic aircraft.


33

33.5

34[m]

0 1000 2000 3000 4000 5000 6000 7000 [s]

300

350

400

450

500

550

600

[K]

33

33.5

34[m]

0 1000 2000 3000 4000 5000 6000 7000 [s]

300

350

400

450

500

550

600

[K]

Fig. 8. Temperature profil Θ(x, ·, t) close to the front of the fuel tank on thesurface (left: Θ(x, 0, t)) and inside the structure (right: Θ(x, ∆y, t)) on time in-terval [482 [s], 5478 [s]]

−35

−30

−25

−20

−15

−10

−5 01000

20003000

40005000

60007000

8000 [s]

300

400

500

600

700

[K]

Fig. 9. Temperature profil Θ(x, 0, t) at the bottom side of the hull before the tankstarts on time interval [482 [s], 5478 [s]]

Moreover, the problem constitutes also a challenge for the growing field ofPDE, resp., PDAE constrained optimization in Applied Mathematics, since itcontains features which have, so far, not been theoretically studied in the con-text of optimal control theory. For an abstract twin problem of an equivalenttype, see [26].


5 Acknowledgement

We are indebted to Prof. Dr. Oskar von Stryk for providing us with his directoptimal control software package Dircol.

6 Appendix

A. Proof of Eq. (10): Let Θ(x, t) be a sufficiently smooth inhomogeneoustemperature distribution in Ω × R≥0. According to Fourier’s law, this givesrise to a conductive energy flux qcond = −λ(Θ)∇Θ. Considering the energydensity (h) cp(Θ)Θ in an arbitrary subset ω ⊂ Ω, there holds, because ofthe integral form of the law of conservation of energy,

d

dt

∫

ω

(h(t)) cp(Θ(x, t))Θ(x, t) dx = −

∫

∂ω

qcond(x, t) · n ds .

Using Gauss’ Theorem, this implies the differential equation

d

dt((h(t)) cp(Θ(x, t))Θ(x, t)) = ∇ · (λ(Θ(x, t))∇Θ(x, t))

which yields Eq. (10) ⋄

B. Proof of the consistency order of Eq. (15): Let Θ(x, t) be the suf-ficiently smooth exact solution of Eq. (10). Let L∆x denote the differenceoperator of Eq. (15) due to the spatial step size ∆x. For Eq. (15) being atleast of second consistency order, i. e., L∆x(Θ(x, t)) = O(∆x2), we only haveto show that this scheme is symmetric, i. e., that L∆x = L−∆x. This, however,can be easily seen. By Taylor expansion it can be shown that the scheme (15)is of order 2 indeed. Moreover, it is also stable, hence convergent, since it isthe 1D analog of the finite volume scheme discussed below.

C. Derivation of the 2D difference scheme as finite volume method,resp. energy balance method: Starting point is the investigation of the energybalance in a finite volume ωi,j of the discretized domain Ω:∫

ωi,j

∇ · (λ(Θ(xi, yj))∇Θ(xi, yj)) dω =

∫

∂ωi,j

λ(Θ(xi, yj))∇Θ(xi, yj) · n ds ,

where we suppress the dependence on t, since we are essentially interested ina spatial semi-discretization. First, we introduce the following notatations

xi+ 12

= xi+1−xi

2 , yj+ 12

=yj+1+yj

2 ,

∆xi = xi+1−xi−1

2 , ∆yj =yj+1−yj−1

2 ,∆xi+ 1

2= xi+1 − xi , ∆yj+ 1

2= yy+1 − yj ,

λi+ 12

j = λ(Θ(xi+ 12, yj)) , λi+ 1

2j = λ(Θ(xi, yj+ 1

2)) ,

fi,j =[

(h)(

cp(Θ) + c′p(Θ)Θ)

Θ + ′(h) h cp(Θ)Θ]

xi,yj.


Using the support values Θi,j in the center of each element ωi,j , one canthen approximate the line integrals by

∫

∂ωi,j east

λ∂Θ

∂xdy ≈ λi+ 1

2j

Θi+1 j − Θi j

∆xi+ 12

∆yj ,

∫

∂ωi,j west

λ∂Θ

∂xdy ≈ λi− 1

2j

Θi j − Θi−1 j

∆xi− 12

∆yj ,

∫

∂ωi,j nord

λ∂Θ

∂ydx ≈ λi j+ 1

2

Θi j+1 − Θi j

∆yi+ 12

∆xi ,

∫

∂ωi,j south

λ∂Θ

∂ydx ≈ λi j− 1

2

Θi j − Θi j−1

∆yj− 12

∆xi ,

and∫

ωi j

f dω ≈ fi,j ∆xi ∆yj

with f analogously defined as fi,j .Finally, the energy balance over the element ωi,j yields

[

λi+ 12

j

Θi+1 j − Θi j

∆xi+ 12

− λi− 12

j

Θi j − Θi−1 j

∆xi− 12

]

∆yj

+[

λi j+ 12

Θi j+1 − Θi j

∆yj+ 12

− λi j− 12

Θi j − Θi j−1

∆yj− 12

]

∆xi

= fi,j ∆xi ∆yj .

The evaluation of λ at the midpoints xi+ 12, resp. yj+ 1

2, requires an interpo-

lation, for example: Θi+ 12

j = [Θ(xi, yj) + Θ(xi+1, yj)] /2 + O(∆x). Herewith,

we obtain the final 2D scheme analog to (15),[

λ

(

Θi−1 j + Θi j

2

)

Θi−1 j − Θi j

∆xi− 12

− λ

(

Θi j + Θi+1 j

2

)

Θi j − Θi+1 j

∆xi+ 12

]

∆yj

+

[

λ

(

Θi j−1 + Θi j

2

)

Θi j−1 − Θi j

∆yj− 12

− λ

(

Θi j + Θi j+1

2

)

Θi j − Θi j+1

∆yj+ 12

]

∆xi

= fi,j ∆xi ∆yj .

This yields the main loop of the 2D discretization scheme. In a similar waythe boundary conditions are employed. In the case of Neumann conditions asgiven here, so-called ghost values outside the domain Ω need not be consideredas for Dirichlet conditions. This scheme is also at least of order 2 because ofits symmetry in ∆x as well as ∆y. It is known that finite volume schemes ofthis type are stable and thus convergent; see, e. g., [1]. Obviously, it is also aconservative scheme, since it preserves the energy not only locally, but alsoglobally, if ∪n,m

i=1,j=1 ωij = Ω. ⋄——————————————————————————


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