Conference PaperPDF Available

Aerostructural design optimization of a 100-passenger regional jet with surrogate-based mission analysis

August 2013

DOI:10.2514/6.2013-4372

Conference: 2013 Aviation Technology, Integration, and Operations Conference

Authors:

Rhea P. Liem

Imperial College London

Charles Mader

University of Michigan

Edmund Lee

University of Toronto

Joaquim R. R. A. Martins

University of Michigan

In this paper we present a coupled aerostructural optimization procedure for the design of a fuelefficient regional aircraft configuration. A detailed mission analysis is performed on an optimized flight mission profile to accurately compute the mission range, fuel burn, and flight time. The mission analysis procedure is designed to allow flexible mission profiles including those with a variety of cruise, climb and descent segments in the profile. The direct operating cost (DOC) is computed based on the mission characteristics (fuel weight, range, and time), and is then used as the objective function in the optimization problem. We use a coupled aerostructural solver comprised of a high-fidelity structural solver and medium-fidelity aerodynamic solver to solve for the static aeroelastic shape of the lifting surfaces. Due to the large computational cost associated with these solvers, "kriging with a trend" surrogate models are employed to approximate the aerodynamic force and moment coefficients required in the mission analysis. This approach is demonstrated in two DOC minimization cases: a mission profile optimization with a fixed geometry, and an aerostructural optimization with fixed, previously optimized mission profiles for a 100-passenger regional jet aircraft.

C D distribution at 35 000 ft and tail rotation angle 0 @BULLET (for pressure drag).

…

: Fuel fraction values

…

A 3-view drawing of the baseline 100-passenger regional jet.

…

+10

Thickness and pressure distributions of a representative wing-tail structure, showing the geometry and the internal structural layout.

…

Figures - uploaded by Joaquim R. R. A. Martins

Content may be subject to copyright.

Content uploaded by Joaquim R. R. A. Martins

Content may be subject to copyright.

Aerostructural design optimization of a 100-passenger regional

jet with surrogate-based mission analysis

Rhea P. Liem∗, Charles A. Mader†, Edmund Lee†

University of Toronto Institute for Aerospace Studies, Toronto, ON, Canada

Joaquim R. R. A. Martins‡

University of Michigan, Department of Aerospace Engineering, Ann Arbor, MI, USA

In this paper we present a coupled aerostructural optimization procedure for the design of a fuel-

efﬁcient regional aircraft conﬁguration. A detailed mission analysis is performed on an optimized

ﬂight mission proﬁle to accurately compute the mission range, fuel burn, and ﬂight time. The mis-

sion analysis procedure is designed to allow ﬂexible mission proﬁles including those with a variety of

cruise, climb and descent segments in the proﬁle. The direct operating cost (DOC) is computed based

on the mission characteristics (fuel weight, range, and time), and is then used as the objective func-

tion in the optimization problem. We use a coupled aerostructural solver comprised of a high-ﬁdelity

structural solver and medium-ﬁdelity aerodynamic solver to solve for the static aeroelastic shape of

the lifting surfaces. Due to the large computational cost associated with these solvers, “kriging with

a trend” surrogate models are employed to approximate the aerodynamic force and moment coef-

ﬁcients required in the mission analysis. This approach is demonstrated in two DOC minimization

cases: a mission proﬁle optimization with a ﬁxed geometry, and an aerostructural optimization with

ﬁxed, previously optimized mission proﬁles for a 100-passenger regional jet aircraft.

I. Introduction

Aircraft fuel consumption is expected to keep increasing in the next few decades as air trafﬁc continues to grow [1].

With fuel prices consistently increasing over the past few years [2], total aviation fuel use now contributes signiﬁcantly

to the total aircraft operating costs and prices. In addition, fuel consumption is a direct surrogate for CO2emissions,

which is an important greenhouse gas, and also an indirect indicator for other aircraft emissions, such as NOxand

H2O [3]. Due to these growing environmental concerns and the challenging fuel economics of aircraft, the appropriate

fuel economy improvement goals for future aircraft need to be prudently selected from the available technological

improvement options and policy scenarios. To assist such decision making and policy analysis processes, optimization

techniques are often employed. However, to perform the optimization properly, ﬁrst and foremost the right objective

function needs to be carefully selected. Several objective functions have been proposed in previous work, such as

maximum takeoff weight (MTOW), direct operating cost (DOC) [4], and block fuel (the minimum fuel mass for

the ﬁxed range) [5]. Minimizing MTOW pushes both the acquisition costs and the operational costs lower, as fuel

weight is incorporated in the objective [6]. Wakayama [7] and Liebeck [8] successfully demonstrate this optimization

strategy to design the BWB concept on a coupled system. Kenway et al. [9] demonstrate and compare high-ﬁdelity

aerostructural optimization problems with fuel burn and MTOW as the objective functions. Kennedy and Martins [10]

use as a design objective a weighted combination of the takeoff gross-weight (TOGW), as a rough surrogate for the

overall aircraft acquisition cost, and fuel burn, which contributes signiﬁcantly to the aircraft’s DOC. By assigning

more weight on the fuel burn objective, they obtain optimum designs with increasing importance on the aerodynamic

performance of the aircraft.

In this work, DOC is used as the objective function since it captures many aspects of aircraft operation, such as

fuel burn, fuel price, ﬂight range, block time, and MTOW. The effect of those factors on the net change in DOC is

not straightforward, due to their interaction with each other. For example, increasing the cruise speed consequently

reduces block time, which reduces DOC, but increases fuel burn, which causes DOC to go up. The right tradeoff

is thus required for a net decrease in DOC. Fuel price determines how the fuel cost component weighs in the DOC

computation. Higher fuel price calls for a conﬁguration with reduced fuel burn, producing a more aerodynamically

efﬁcient aircraft with lower drag.

To compute DOC, it is critical to accurately and efﬁciently estimate the total fuel burn, mission range, and time

of aircraft missions. An accurate calculation of the mission performance requires modeling the aircraft operation as

∗Ph.D. Candidate, AIAA Student Member

†Research associate, AIAA Member

‡Associate Professor, AIAA Associate Fellow

1 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

2013 Aviation Technology, Integration, and Operations Conference

August 12-14, 2013, Los Angeles, CA AIAA 2013-4372

realistic as possible, including the mission proﬁle, aircraft weights and balance, and aircraft engine. To analyze the

aircraft performance, it is important to consider both aerodynamic and structural disciplines simultaneously, to account

for the interaction and tradeoffs between the two disciplines, for which we use a multidisciplinary design optimization

(MDO) framework. An MDO framework is an efﬁcient technique to assist the design of engineering systems, as it

takes into account the coupling in the system and automatically performs the optimal interdisciplinary trade-offs [11].

The classical Breguet range equation is commonly used to compute the amount of fuel burn during ﬂight [3,12,13].

This widely used range equation was derived and published independently in 1920 by Cofﬁn [14] and later in 1923

by Breguet [15]. This equation has since become a basic model describing the physics of aircraft, encompassing

the three dominant disciplines within an aircraft system: engine (by the thrust speciﬁc fuel consumption or TSFC),

aerodynamics (by the lift to drag ratio, L/D), and structural technologies (by the structural weight). This equation,

however, is only applicable under the assumption that TSFC, L/D, and ﬂight speed are constant. One important

implication is that the takeoff, climb, and descent segments are not properly modeled by this equation [16]. Simple

fuel fractions are typically used to compute the amount of fuel burned in ﬂight segments other than cruise. A fuel

fraction is deﬁned as the ratio of the aircraft total weight at the end of a ﬂight segment to the weight at the start of

the same segment. See, for example, Roskam [13] for values of suggested fuel-fractions corresponding to several

mission phases for various aircraft types. A combined factor is also sometimes used. For example, the fuel burn for

taxi, takeoff, approach and landing can be collectively approximated by taking a fraction of takeoff weight; typically

a factor of 0.007 is used for the total of those phases [17].

Lee and Chatterji [18] present the approximation functions for total fuel burn in climb, cruise, and descent phases.

To compute fuel burn during climb, they apply a climb fuel increment factor, which is deﬁned as the additional fuel

required to climb the same distance as it is for cruise, normalized with respect to the takeoff weight [17]. This

climb fuel increment factor is expressed as a quadratic polynomial function of cruise altitude and cruise speed with

different sets of coefﬁcients, depending on the aircraft type. Henderson et al. [19] present an object-oriented aircraft

conceptual design toolbox, pyACDT, which analyzes a given mission proﬁle to estimate the mission fuel burn and

point performance parameters. The Breguet range equation is used to calculate the cruise range. This toolbox uses a

potential ﬂow panel method for its aerodynamic module. The Program for Aircraft Synthesis Studies (PASS), created

by Desktop Aeronautics, Inc., is a conceptual design tool which evaluates all aspects of mission performance [20].

This software package can incorporate several analyses, including linear aerodynamic models for lift and inviscid

drag, sonic boom prediction for supersonic cases, weight and CG estimation, and full mission analysis. These rapid

analyses are coupled with optimization tools (gradient or non-gradient based) to perform aircraft design optimizations.

The fuel burn computations mentioned above are done with simpliﬁcatons of the aircraft performance and mission

proﬁle. This simpliﬁcation might lead to inaccurate prediction of the total aircraft fuel consumption. For example,

the constant L/D, TSFC, and ﬂight speed assumed in the Breguet range equation do not reﬂect the actual aircraft

operation, as their values vary across the ﬂight operating points in the mission proﬁle. This assumption in turn leads

to inaccurate calculation of fuel burn, mission range and time, and thus an inaccurate DOC. Moreover, most fuel burn

computation focuses more on the cruise portion, which is critical for long range missions, but not necessarily so for

shorter range missions. For shorter range missions, the climb segments will contribute signiﬁcantly to the total fuel

consumption as well. Though desirable, performing a detailed mission analysis could be computationally expensive

due to the thousands of performance evaluation needed to model all ﬂight operating points in a mission. Solving an

optimization problem with such expensive analyses can quickly become computationally prohibitive. Analytical and

empirical models are sometimes used to reduce the computational time [21], in the expense of accuracy. Liem et

al. [22] develop a mission analysis procedure to compute aircraft fuel burn using kriging surrogate models to approx-

imate the aerodynamic performance. The use of surrogate models help reducing the computational cost signiﬁcantly.

However, the detailed analysis is still only limited to the cruise segment.

The main objective of this work is to develop an integrated toolbox to perform a detailed mission analysis that

models all the climb, cruise, and descent segments. An engine model is employed to compute the TSFC value at

each ﬂight operating point, and a weight and balance model is used to estimate the weight and centre of gravity (CG)

location of each component. Given a mission proﬁle, aircraft geometry and engine speciﬁcations, this toolbox will

compute the total fuel burn and DOC of the mission. To make this procedure more computationally tractable, “kriging

with a trend” surrogate models are used to approximate the aerodynamic force and moment coefﬁcients required in

the mission computation.

To demonstrate the capability of this mission analysis module, we perform two DOC minimization cases of a

100-passenger regional jet aircraft. In the ﬁrst case, we optimize mission proﬁle parameters with a ﬁxed aircraft

geometry, to obtain the mission proﬁles to be used in the aerostructural optimization. In the second case, we perform an

aerostructural optimization to obtain the optimized aircraft geometry with ﬁxed mission proﬁles, using those obtained

in the ﬁrst optimization case.

2 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

The paper is organized as follows. In Section II, we describe the solution methods we use for our mission anal-

ysis module. The descriptions of the aircraft conﬁguration, engine model, and aerostructural solver are presented in

Section III.Section IV presents the optimization problem formulations. We then present and discuss the results in

Section V, and close with conclusion and future work in Section VI.

II. Mission analysis methods

In this section, we discuss the methods used to compute the mission characteristics, including fuel weight Wfuel,

range R, and mission time t. Each of these characteristics can be computed by numerically integrating a given mission

proﬁle. We need to provide the mission proﬁle parameters, such as altitude and Mach number for cruise segments,

ﬂight speed (Mach number or the knots indicated airspeed, KIAS), initial and ﬁnal altitudes for climb and descent

segments. The mission analysis also requires the initial takeoff weight and the ﬁnal zero fuel weight (ZFW), which

are determined by the weights and balance model in this work, for each mission. From this information, we then solve

iteratively for the weight, range and time spent in each of the mission segments. Once the overall system solution is

obtained, we can determine the characteristics for the entire mission.

In previous work on mission analysis, such as that of Henderson et al. [19], the mission is analyzed in a sequential

fashion, analysing each segment in succession. In such a case, the previous segment’s end weight is used as the

starting point for the current segment, requiring that the segments be analyzed in sequence. In the current work, we

have employed an all-at-once approach to solving the mission. The mission is broken up into several segments. Each

segment is limited to one phase of ﬂight, be that takeoff, climb, cruise, descent, or landing. By deﬁning the various

endpoint weights as states of the system, we are able to formulate a set of residual equations for each segment and

assemble a set of nonlinear system residuals, R, that is set to zero to solve the mission. This formulation of the mission

allows each of the segments to be analyzed independently, based only on the states of the system.

A. Mission segments

In order to understand the formulation of the residual equations, we must ﬁrst develop the equations for each of the

segment types. In this work we have equations for climb, cruise, and descent. Startup, taxi, takeoff, and landing are

handled through fuel fractions. The equations of numerical integration for each of these segment types is described

below.

1. Startup, taxi, takeoff, and landing

The amount of fuel burn during startup, taxi, takeoff, and landing are computed using the fuel fraction method, where a

constant ratio is used to deﬁne the weight at the end of the segment, given the initial weight for that particular segment.

This method is expressed as

Wf= (1 −ζ)Wi,(1)

where Wiand Wfrefer to the segment’s initial and ﬁnal weight, respectively. The fuel fraction values, ζ, used in this

work are listed in Table 1, following those suggested by Roskam [13], Raymer [23], and Sadraey [24].

Table 1: Fuel fraction values

Segment Startup Taxi Takeoff Landing Taxi & Shutdown

Fuel fraction,ζ0.01 0.01 0.005 0.003 0.008

2. Cruise, climb and descent segments

The fuel burn computation for the climb, cruise, and descent segments are derived from the range equation. To compute

fuel burn, we ﬁrst deﬁne the thrust speciﬁc fuel consumption, cT, as:

cT=Weight of fuel burned per unit time (N/s)

Unit thrust (N).(2)

We use an engine model to estimate the TSFC value, which is a property of the aircraft engine. The engine model used

in this work is described in Section III.E. The rate of reduction of aircraft weight can thus be computed as,

dt =−cTT, (3)

where Wand Tdenote aircraft weight and thrust, respectively.

3 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

For the cruise segment, the equation for range, R, can be expressed as a deﬁnite integral of ﬂight speed, V, over a

certain time interval,

R=Ztf

V dt. (4)

Substituting (3) to (4) lets us deﬁne the ﬂight range given the speciﬁc range, i.e., the range per unit weight of fuel

(−V/cTT), using the following expression,

R=ZWf

−V

cTTdW , (5)

where Wiand Wfrefer to the weights at the start and the end of a cruise segment, respectively. Similarly, by rear-

ranging (3), the cruise time can be computed as,

t=ZWf

−1

cTTdW . (6)

The range equation for the climb and descent segments are derived similarly. We use the following relation between

ﬂight speed and rate of climb, RC,

RC =dh

dt =Vsin γ, (7)

where hand γdenote ﬂight altitude and ﬂight path angle, respectively. We approximate RC based on the equation of

motion given below,

Tav cos (φT+α)−D−Wsin γ=W

dt ,(8)

where Tav denotes the available thrust, Ddenotes drag, and gis the gravitational acceleration. The thrust inclination

angle is denoted by φT(typicaly assumed to be zero [25]), and αrefers to the angle of attack. With small angle

approximations, we get the simpliﬁed equation as shown below,

Tav −D−W γ =W

dt (9)

Similarly, the rate of climb can be approximated as RC ≈V γ. Rearranging and substituting (9) into this approxima-

tion gives us:

RC =(Tav −D)V

W1 + V

dh (10)

For each point in the numerical integration we solve for a value of γthat is consistent with the differential between

the thrust and the drag to determine the actual rate of climb RC. Returning to (7), we can see that the time to climb is

simply,

t=Zhf

RC dh (11)

where hiand hfare the initial and ﬁnal altitudes for the climb segment. Note that for climb and descent we integrate

over altitude, since this is the independently speciﬁed variable for these segments.

Substituting (3) into (7), we can obtain an expression for the fuel burn with respect to altitude,

Wfuel =Zhf

−cTT

RC dh (12)

ﬁnally if we modify (4) to account for the climb angle so that,

dt =Vcos γ(13)

and substitute this equation into (7), we get the following expression for climb range,

R=Zhf

Vcos γ

RC dh. (14)

4 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

In this mission analysis module, the integrations to ﬁnd the mission range, time, and fuel are solved numerically, by

dividing each segment to a number of intervals.

Referring back to (10) we note that there is an undeﬁned derivative dV /dh. This leads to three separate climb

formulations, one for accelerated climb, one for constant velocity climb, and one for constant Mach climb. The

constant velocity refers to the constant indicated airspeed (IAS) (the airspeed when the aircraft is at sea level under

International Standard Atmospheric conditions) and it needs to be converted into the true airspeed (TAS) before we

evaluate the term dV/dh. For the accelerated and constant velocity climb segments, we use (15) to compute dV /dh,

where the second term goes to zero for the constant velocity case.

dh =∂VTAS

∂h +∂VTAS

∂VIAS

∂h .(15)

For constant Mach climb, dV

dh =Mda

dh,(16)

where ais the speed of sound. We compute the derivative terms ∂VTAS/∂h,∂VTAS/∂VIAS, and da/dh, using ﬁnite

differencing.

B. Residual equations

Having deﬁned the equations of numerical integration for the various segments, we now need to develop the residual

equations for the nonlinear system. We have deﬁned the states as the endpoint weights of each segment. This causes

duplicate values of the weight at each node of the mission. Therefore the primary residual equations are that the end

point weights of two adjacent segments must be equal,

Wfj−Wij+1 = 0,(17)

where j= 1, ..., Nseg denotes the segment index. The boundary conditions for the problem are formulated in a similar

manner, i.e. that the initial weight of the ﬁrst segment match the initial weight WTO and that the ﬁnal weight of the

last segment match the ﬁnal weight WZF, giving:

Wi1−WTO = 0 (18)

and

WfNseg −WZF = 0.(19)

For the climb and descent segments, this single residual equations is sufﬁcient as each segment adds only a single

degree of freedom to the system (Wiis independent, Wi+1 is determined by the integration). However, for each cruise

segment, both endpoints are state variables, therefore we need a second residual equation for each of these segments.

For this second equation, we constrain the ratio of the various cruise segments to be ﬁxed. In this case we constrain

that all cruise segments be equal, but other ratios are possible. To implement this constraint, the range of the ﬁrst

cruise segment is calculated and used as a reference range, then for each other cruise segment we have the equation:

(Rref −Ri)/Rref = 0.(20)

Note that we have scaled this residual by the reference range to improve the conditioning of the Jacobian of the system.

We then use a line search stabilized Newton’s method to solve the nonlinear system, where the Jacobian for the

Newton iteration is formed using ﬁnite differencing. This forces the weights of the various segments to be consistent

with each other, providing a valid, continuous mission proﬁle.

C. Aerodynamic data

To facilitate the efﬁcient numerical integration of the individual ﬂight segments, we use a surrogate model to approx-

imate the required aerodynamic coefﬁcients, CL,CD, and Cm. To generate the model, we sample a small number of

points (we use 28 in this work) in the four dimensional space of Mach, altitude, angle of attack, and tail rotation angle,

and compute the aerodynamic coefﬁcients at these points. To avoid having ill-conditioned kriging equations, these

design variables are normalized to [0,1] intervals, before the kriging models are constructed and later on used. From

this data, three surrogate models are generated, one for each coefﬁcient. In our case, all of the moment coefﬁcients

are computed about a ﬁxed reference point. Therefore we account for the variation in center of gravity location for the

different mission segments using the following equation,

∆M=W·g·(xcg −xref),(21)

5 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

where xref refers to the location of reference point used to compute pitch moment. This leads to a correction on the

pitching moment coefﬁcient,

∆Cm=∆M

2ρSrefV2·MAC ,(22)

where ρ,Sref, and MAC denote the atmospheric density, reference wing area, and the mean aerodynamic chord of the

wing, respectively. The velocity Vused in this equation refers to the true airspeed.

To compute the drag at each point in the mission, we use a Newton method to search for the angle of attack (α) and

tail rotation angle (η) that satisfy lift and trim conditions simultaneously. The general equation for Newton’s search

algorithm is

x(k)=x(k−1) −J−1fx(k),(23)

where kis the iteration index. The function fis to be driven to zero and Jis the corresponding Jacobian. In this search

procedure, the values for x,f, and Jare

x=α

η,f=CL−CLtarget

Cm+ ∆Cm,J="∂CL

∂α

∂CL

∂η

∂Cm

∂α

∂Cm

∂η #(24)

Once the values of αand ηare determined for a given Mach and altitude, the drag surrogate model can be evaluated

to determine the drag for that point. This drag computation is conducted at every point of numerical integration for each

segment. Thus, the number of evaluations required is the product of the number of missions, the number of Newton

iterations required to solve each mission system, the number of segments per mission, the number of intervals per

segment, and the number of Newton iterations to compute the drag for each point. This can easily lead to thousands

of potential evaluations per mission, which would be extremely expensive to compute directly. However, by using

surrogate models for the aerodynamic models, the cost of this evaluation can be reduce to mere seconds.

D. Kriging models

The kriging approximation technique is a statistical interpolation method that is constructed by minimizing the mean

squared error of the approximation, subjected to the unbiasedness constraint [26,27]. We refer the reader to previous

work for further details on the derivation and formulation of kriging models [26,28,29]. To build a kriging surrogate

model, we need Nssample points, x, and function values evaluated at those sample points, ys. A kriging predictor, ˆy,

at an untried point x0can then be expressed as,

ˆy(x0) = F(x0)β+r(xs,x0)TR−1(ys−F(xs)β)(25)

where Rdenotes the correlation matrix of size Ns×Ns,F(x)is a matrix of basis functions for the global model

(whose entry values might depend on the location of x0), and βdenotes the corresponding basis function coefﬁcients.

r0(xs,x0)denotes the correlation vector between x0and sample points. The ﬁrst term of (25) is a global model

representing the “function mean”, and the second term is a zero-mean stochastic realization with variance σ2and

non-zero covariance:

Cov yxi, y xj=σ2RRxi,xj,(26)

where Rxi,xjis a correlation function between any two sample points xiand xj. The stochastic term represents

localized deviations from the global model, which allows kriging models to perform interpolation between sampled

data points.

For the global model, a low-order polynomial (similar to the polynomial model in a response surface), is commonly

used. Typically, a constant global model is deemed sufﬁcient and thus used in the kriging approximation [26,30]. In

which case, all entries in F(xs)are a scalar 1and its coefﬁcient vector, β, is reduced to a scalar value β. Such a kriging

model is typically referred to as ordinary kriging. In a “kriging with a trend” model, on the other hand, the global

model is modeled by an analytical expression, which takes different values in space, to be the trend component [31].

The stochastic term is modeled as a Gaussian process, where the Gaussian exponential correlation is used for the

spatial correlation function. Other correlation functions, such as exponential and cubic spline functions, can also be

used. The formulation for the Gaussian correlation function is shown below,

Rxi,xj= exp "−

Ndv

k=1

θkxi

k−xj

k

2#,(27)

where Ndv is the problem dimension, θkis the unknown correlation parameter associated with the k-th design variable

used to ﬁt the model, and xi

kand xj

kare the k-th components of sample points xiand xj. This property of the

6 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

correlation function ensures that the approximation error of kriging is zero at sample points and increases with distance.

A vector of correlation parameters (denoted as θ) determines the “strength” of correlation in each direction. These

kriging hyperparameters are also referred to as distance weights. Large distance weight values correspond to weak

spatial correlation, whereas small values correspond to strong spatial correlation [32]. These kriging hyperparameters

are found via a maximum likelihood estimation (MLE) optimization, with the assumption that the approximation

errors are normally distributed and that the process variance is stationary, i.e., it is independent of the locations in the

design space. The gradient-free Hooke–Jeeves pattern search algorithm [33,34] is used to solve the MLE optimization

problem, as suggested by Lophaven et al. [35], which was also used in other previous work [28,36,37].

In this work, the kriging models are used to approximate the aerodynamic force and moment coefﬁcients across

the entire range of ﬂight conditions in a ﬂight mission, i.e. from takeoff to landing. We thus need to ensure that the

design space of the kriging models cover the operating points of the entire mission. The range of values for the four

kriging variables are shown in Table 2.

Table 2: Flight condition value ranges

Kriging variables Lower bound Upper bound

Mach 0.2 0.9

Altitude 0 ft 45 000 ft

Angle of attack (α)−5.0◦12.0◦

Tail rotation angle (η)−10.0◦12.0◦

Figure 1: CDdistribution at 35 000 ft and tail rotation angle 0◦(for pressure drag).

Figure 1 shows the CDdistribution (for pressure drag only) of a regional jet, at 35 000 ft altitude and tail angle

of 0◦. These CDvalues are obtained from a medium-ﬁdelity aerodynamic solver, TriPan, which uses the panel

method. From this CDdistribution, we observe a high drag gradient, especially in the high Mach number and high

angle of attack region. Ordinary kriging model is unable to capture this high drag gradient well, as shown in Figure 2a.

In general, accuracy can be improved by increasing the number of samples in the kriging model construction, but doing

so will signiﬁcantly increase the computational cost and time, which defeats the purpose of using surrogate models in

the ﬁrst place. Instead, we introduce a trend in the kriging model by using a nonconstant global model for CDkriging

approximation. The basis functions are selected to follow the trend of CD, especially in the high drag gradient region

(high Mach, high α). We thus select the following equation as the global model for CDkriging approximation,

f(x) = β0+φ(M, α)·β1.(28)

Here, xis a vector of [M, h, α, η], where M,h,α, and ηrefer to Mach number, altitude, angle of attack, and tail

rotation angle, respectively. The function φ(M, α)is given below,

φ(M, α) = (1

1−M2if α≤1.0

α2

1−M2if α≥1.0.(29)

The constant numerator for α≤1.0is used to remove the quadratic proﬁle (with respect to α) in low αregion, to be

consistent with the CDdistribution obtained from the aerodynamic solver. The basis function vector, F(x), and the

7 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

coefﬁcient vector, β, are thus expressed as follows,

F(x) = [1, φ (M, α)] (30)

β= [β0, β1]T.(31)

Thus at an evaluation point x0= [M0, h0, α0, η0], the kriging equation can be expressed as

ˆy(x0) = β0+β1φ(M0, α0) + r(xs,x0)TR−1[ys−β0−β1φ(M0, α0)] .(32)

The kriging model construction procedure will determine the basis function coefﬁcients, β0and β1, by the mean

square error minimization, subject to the unbiasedness constraint.

The comparison between ordinary kriging and kriging with speciﬁed basis functions (trend) for CDapproximation

is shown in Figure 2. From these ﬁgures, we can observe that the basis functions we specify for the kriging global

model signiﬁcantly improve the accuracy of kriging models in the high drag gradient area. This accuracy improvement

will in turn improve the accuracy of mission analysis with the same number of samples. The implementation of kriging

with speciﬁed basis functions only requires an additional 0.01 s in the construction time, from the 0.2 s required to

construct an ordinary kriging model. This additional computational time is considered very minimal.

(a) Ordinary kriging (b) Kriging with speciﬁed trend

Figure 2: Comparing CDapproximations (at 35 000 ft and 0◦tail rotation angle) using ordinary kriging and kriging with speciﬁed trend surrogate models (for

pressure drag).

For CLand CM, the proﬁles do not exhibit a high gradient in the design space, thus ordinary kriging models are

deemed sufﬁcient to approximate these quantities. In this work, 28 samples are used to construct the kriging models.

III. Optimization problem description

In this section, we describe the aircraft conﬁguration, as well as its wing structural model, that we use in the

optimization problems. The coupled aerostructural solver is also described. We then brieﬂy talk about the DOC com-

putation, engine model, weight and balance components of the analysis. Finally we discuss the software architecture

for the mission analysis module.

A. Aircraft conﬁguration

The aircraft being considered is a 100-passenger regional jet. This aircraft is conﬁgured with low swept-back wings,

two aft engines mounted on the fuselage, and a T-tail, similar to the other regional aircraft such as the Bombardier

CRJ1000, McDonnell Douglas DC-9, and Fokker 100. The fuselage is designed to accommodate a ﬁve-abreast seating,

which gives an overall fuselage length of 108.6 ft with a width of 10.8 ft. The wing for this initial conﬁguration is a

winglet design, with a sweep of 30◦at the leading edge, giving a wing area of 989 ft2and a span of 84.9 ft. A 3-view

of this conﬁguration is shown in Figure 3.

B. Wing structural model

Figure 4 illustrates the three-dimensional wing-tail structure with the internal structure layout that we use in this work.

The thickness and pressure distributions are also shown. A 3D wing and tail ﬁnite element structural geometry is

constructed using shell elements. The wing box structure consists of a front spar at 15% chord, rear spar at 65% chord,

and a subspar to support the landing gear structure. The thickness of this baseline wing is 14% t/c for the entire

8 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

Figure 3: A 3-view drawing of the baseline 100-passenger regional jet.

Figure 4: Thickness and pressure distributions of a representative wing-tail structure, showing the geometry and the internal structural layout.

9 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

centerbody section, 10%t/c for the wingtip and winglet, and linearly varying in between. Structural components

that are deﬁned include ribs, spars, skins, and stringers, with the ribs being oriented ﬂow-wise in the centerbody and

winglet, and perpendicular to the rear spar in the main wing. A ﬁxed leading edge structure is added for better load

transfer between the aerodynamic forces and structure; however, this structure is not sized and not accounted for in

the total weight. Top and bottom blade stringers are also included in the model, oriented perpendicularly to the ribs.

Since buckling is not considered at the moment, the required stringer thicknesses would not be captured. Therefore,

the stringer thickness are linked to the skin thickness for a better weight prediction.

The minimum thickness for all components is set at 0.080 in, except for the ribs which is set at slightly higher,

since they are not sized by buckling or the loading from the control and high-lift device attachments. Also, adjacency

thickness constraints are enforced that limit skin thickness variations between neighbouring patches. Aluminium 7075

is used for the top skin and top stringer, and aluminium 2024 is used for the rest of the structure. The material properties

for both of these alloys are summarized in Table 3. For stress constraint considerations, the limit is the lesser of the

ultimate stress with a safety factor of 1.5or the yield stress; the lower value for each material is highlighted in the

table.

Table 3: Material properties [38] of aluminium 2024 and 7075, including fatigue data [39] at 60 000 cycles.

2024-T4 7075-T73

ρ0.100 0.102 lb/in3

E10 600 10 400 ksi

ν0.330

σy40.056.0 ksi

σult 62.067.0ksi

σfat. 12.25 8.4 ksi

C. Aerostructural solver

The coupled aerostructural solution is obtained from a high-ﬁdelity structural solver, TACS, and a medium-ﬁdelity

aerodynamic solver, TriPan. Both solvers are developed by Kennedy and Martins [40]. The Toolbox for the Analy-

sis of Composite Structures (TACS) is a parallel ﬁnite-element solver purposely built for the analysis and optimization

of composite structures. TACS is used to solve the wing-tail structure. TriPan calculates the aerodynamic forces

and moments of inviscid, incompressible, external lifting ﬂows on unstructured grid using surface pressure integra-

tion, with constant source and double singularity elements. The surface of the body is discretized with quadrilateral

and triangular panels. TriPan also computes the aerodynamic loading at various ﬂying shapes. This aerodynamic

solver is also used to determine the lift, drag, and moment coefﬁcients (CL,CD, and Cm) used to construct the surro-

gate models for the mission analysis. Both the structural and aerodynamic solvers are equipped with an accurate and

efﬁcient analytic gradient computation routine, allowing a gradient-based optimizer to be used. Through the optimiza-

tion, the optimum thickness distribution of each structural member is determined. The ﬂuid-structure interface in the

aerostructural solver is done via a load displacement transfer procedure, where rigid links are used to extrapolate the

displacements from the structural surface to the outer-mold line (OML) of the aerodynamic surface [41,42]. A cou-

pled adjoint method that allows us to compute gradients of multidisciplinary functions of interest is one of the critical

components that make high-ﬁdelity aerostructural optimization tractable for large numbers of design variables [9,43].

D. DOC computation

The cost estimates for each component in DOC computation are derived based on empirical methods [44]. The com-

ponent breakdown, and the key contributing factors to each component, of the DOC computation are given in Table 4.

From the mission analysis procedure, we obtain the mission range, time, total amount of fuel burned (fuel weight).

MTOW is obtained from the weight and balance model. The technology year is used to compute the escalation factor

in the cost computation to the current year. A ﬁxed fuel price of $3 USD/gal is used in the computation.

E. Engine model

The engine model is based on an engine map from data generated by GasTurb [45], a commercially available gas

turbine performance software commonly used within industry. Two sets of data are generated: (1) maximum climb,

with data at various Mach and altitude, and (2) cruise, with data at various Mach, altitude, and thrust setting. For each

set of engine data, three additional sets are generated for three different bypass ratios (BPR). These data allow the

mission analysis to determine the available thrust and the associated speciﬁc fuel consumption at a particular ﬂight

condition, by performing a high-dimensional linear interpolation.

10 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

Table 4: The dependency of DOC components on aircraft speciﬁcations and mission performance.

Components MTOW Takeoff

Thrust

Fuel

Weight

Block

Time Range

Crew 3 3

Attendent 3

Fuel 3

Oil 3

Airport Fees 3

Navigation Fees 3 3 3

Airframe Maint. 3 3

Engine Maint. 3 3

Insurance 3 3

Financing 3 3

Depreciation 3 3 3

Registry Taxes 3 3

F. Weight and balance

The total weight of the aircraft is decomposed into its various structural and system components, such as wing, fuse-

lage, furnishings, payload, and fuel, each with their own CG location estimate, as well as a forward and an aft CG limit

estimate. The addition of these component weights and moments gives an estimate of the entire aircraft’s weight, and

the nominal, forward, and aft CG locations. The nominal CG estimate can be used for the mission analysis during a

cruise ﬂight condition, while the CG limits can be used for the analysis of critical load cases. During the optimization,

the weight and CG location of these components can be individually updated, giving a more accurate picture of the

aircraft’s weight and balance in the mission optimization as the fuel is decremented in the integration.

G. Software architecture

The main computational part of our mission analysis module is implemented in Fortran, and wrapped in Python.

This combination has been proven to be very effective. Fortran offers a signiﬁcantly faster computational time as

compared to Python, and the object-oriented Python provides the more practical user interface (scripting), ease of

use of a class object, and plotting features. Using Python at scripting level has also facilitated the integration of the

different fortran modules (e.g., aerodynamic solver, mission analysis, kriging surrogate models, atmospheric module,

and engine model). The sensitivity computation in the fortan modules is done using Tapenade [46], which performs

algorithmic diffentiation (AD) directly on the modules in an integrated fashion. The optimization problem is solved

using SNOPT [47], a gradient-based sequential quadratic programming optimizer, which is used within pyOpt, an

object-oriented framework for nonlinear optimization [48]. The aerostructural optimization is performed on a mas-

sively parallel supercomputer [49]. In total, we use 96 processors to complete the aerostructural optimization, with the

breakdown shown in Table 5.

Table 5: Computational resources breakdown for the parallel aerostructural optimization.

Group Number of

Load Cases

Number of Processors

Aero Struct Total

Maneuver 4 6 ×4 6 ×4 48

Landing 5 10 10

Stability 2 8 8 16

Aerodynamics 28 21 21

Mission Analysis 2 1

Total 96

IV. Optimization problem formulation

In this section, we present the two DOC minimization cases performed in this work. The ﬁrst is a mission proﬁle

optimization where the aircraft geometry and engine are assumed ﬁxed, whereas the second one is an aerostructural

11 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

optimization with ﬁxed mission proﬁles. This mission proﬁle, obtained from the ﬁrst optimization case, is used for the

aerostructural optimization.

A. Mission proﬁle optimization

We demonstrate the use of the described mission analysis framework to minimize DOC by optimizing the mission

proﬁle. Two different missions are considered for this aircraft: a 2 000 nmi long range cruise (LRC) mission and

a500 nmi typical range cruise (TRC) mission, both with 100 passengers. The LRC mission ensures the aircraft

is capable of carrying the full fuel required for the maximum range, while the objective of the optimization is to

minimize the DOC based on the TRC mission. The LRC mission also has an impact on the DOC, as its fuel weight

contributes to the aircraft MTOW, which goes into the DOC computation.

The mission proﬁle for both the TRC and LRC missions are illustrated in Figure 5 (not drawn to scale). A single

cruise segment is assumed for the TRC mission, whereas the LRC mission is assumed to have two cruise segments.

The entire cruise segment for the LRC mission, from the initial cruise, through the step climb, to the ﬁnal cruise,

is done at a constant Mach number, with a single Mach number design variable. From 10 000 ft, the climb is done

at a constant knots indicated airspeed (KIAS), until it intercepts the desired cruise Mach number, at which point the

climb is done at a constant Mach number. The descent is also done in a similar fashion, with a constant Mach descent

followed by a constant KIAS descent. This procedure is implemented to reﬂect common operational procedures.

Startup

Taxi Takeoﬀ

Accelerated

climb

Constant

velocity climb

Constant

Mach climb

Cruise Constant

Mach descent

Constant

velocity descent Decelerated

descent

Landing Taxi

Cruise Mach

Cruise

altitude Fuel weight (from startup to post-landing taxi)

Thrust −Thrustreq ≥0(for all segments)

Range = 500 nm

(a) 500 nmi mission

Startup

Taxi Takeoﬀ

Accelerated

climb

Constant

velocity climb

Constant

Mach climb

Cruise 1

Cruise

climb

Cruise 2

Constant

Mach descent

Constant

velocity descent Decelerated

descent

Landing Taxi

Cruise Mach

Cruise

altitude 1

Cruise

altitude 2

Fuel weight (from startup to post-landing taxi)

Step climb ≥2 000 ft

Thrust −Thrustreq ≥0(for all segments)

Range = 2 000 nm

(b) 2 000 nmi mission

Figure 5: Mission proﬁle for the TRC and LRC missions (not drawn to scale).

The optimization problem formulation is presented in (33) below. The design variables and constraints are also

shown in Figure 5 (indicated in blue and red, respectively). Note that the altitudes shown here, denoted as h, refer

to the cruise altitude. The subscripts 1and 2in hLRC denote the ﬁrst and second cruise segment in the LRC mission,

respectively.

minimize DOC

w.r.t 2 fuel variables :WfuelTRC , WfuelLRC

2 cruise Mach numbers :MTRC, MLRC

3 cruise altitudes :hTRC, hLRC1, hLRC2

such that RTRC = 500 nmi

RLRC = 2 000 nmi

Thrust −Thrustreq ≥0.0

hLRC2−hLRC1≥2 000 ft

(33)

The ﬁrst two constraints are imposed to ensure that the mission proﬁle can ﬂy the range requirements for the

two missions. The thrust requirement constraint is imposed to all segment intervals (for the numerical integration

computation), and they all have to be positive. Lastly, the step climb altitude for the LRC mission has to be larger than

12 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

2 000 ft, to account for the trafﬁc separation rule for aircraft operating under IFR, as speciﬁed in FAR §91.179 [50].

The aircraft geometry and engine properties are ﬁxed for this optimization. The maximum thrust of the engine is set

at 14 000 lbf, and the bypass ratio (BPR) is set at 11.5.

B. Aerostructural optimization problem

For the coupled aerostructural optimization, we consider the same 2 000 nmi LRC mission and a 500 nmi TRC mis-

sion, both with 100 passengers and fuel for a 100 nmi reserve. The DOC model computes the cost based on mission

characteristics such as total ﬂight time, total fuel burn, and cruise Mach number, as well as aircraft parameters such as

MTOW.

Figure 6 shows the XDSM (eXtended Design Structure Matrix) diagram [51] of the aerostructural optimization

problem. The black and shaded grey lines in the XDSM diagram represent process and data ﬂows, respectively. The

design variables for this optimization, x, include the global variables, xglobal, aerodynamic variables, xaero, structural

thickness variables, xstruct, and mission variables, xmission.Table 6 lists the design variables within each of these

groups, as well as the state variables (yaero and ystruct). At each optimization iteration, a multidisciplinary analysis

(MDA) performs coupled aerostructural analyses and evaluate some of the optimization constraints, i.e. thicknesses,

stresses, and static margin for stability. To build kriging models (in the surrogate model module), we need aerodynamic

samples. These samples are evaluated in the aerodynamic module at 28 ﬁxed sample locations. Once the kriging

models are built, they are then passed to the last module, which performs mission analysis at ﬁxed mission proﬁles.

xopt [M, h, α, η ]

samples

Mission

proﬁles

x∗

opt Optimization xglobal

xaero

xglobal

xstruct

xmission

MDA ˆ

ystruct

y∗

aero Static margin yaero Aerodynamic

analysis

yaero CL,CD,Cm

samples

y∗

struct Thickness

Stresses ystruct Structural

analysis Weights

Surrogate

models ˆ

CL,ˆ

CD,ˆ

Objective

Constraint

DOC

Thrust

Range

Mission

analysis

Figure 6: The XDSM diagram of the aerostructural optimization with surrogate-based mission analysis.

Table 6: List of design variables and state variables for the aerostructural optimization

xDesign variables

xglobal Wing twists

xaero Angles of attack, tail angles

xstruct Structural thicknesses

xmission Fuel weights for TRC and LRC missions

yState variables

yaero Pressure coefﬁcients

ystruct Displacements and weights

13 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

In addition to the coupled aerodynamic loading obtained from TriPan, other loads are also considered, including

load from self-weight, secondary weight from control surfaces and high-lift devices that are not part of the structural

model, as well as fuel weight loads. The inertia load is based on the weight of the structure at the corresponding load

condition. In addition, the secondary weights from the control surfaces and high-lift devices are also accounted for,

determined based on the method by Torenbeek [52]. There are weight estimates for each of the leading edge slats,

trailing edge ﬂaps, ailerons, as well as the ﬁxed portion of the leading edge and trailing edge. These weight estimates

are functions of the aircraft’s MTOW and the area of the device. These loads are applied to the leading edge spar and

the rear-most spar at each rib bay. Fuel loading is also applied to the wing box, with the load applied on the bottom

skin for a positive gloading, while for a negative gloading, the load is applied on the top skin only.

These loads are applied to several certiﬁcation maneuver conditions, and the stresses from these ﬂight conditions

are constrained to be within the lesser of the material yield stress σy, or the ultimate stress σult with a safety factor

of 1.5. In total, four ﬂying load cases with different loads being applied are considered: a pull up condition of 2.5g

at both MTOW and ZFW, and a pushover condition of −1.0gat MTOW. In addition, fatigue is also considered at

MTOW by using a reduced allowable stress obtained from Hangartner [39], assuming a service life of 60 000 cycles,

with a safety factor of 1.3. The maneuver ﬂight conditions and fatigue load case are analyzed at the optimized cruise

Mach number and 70% of the cruising altitude. For all ﬂight conditions, lift (L) and moment (m) are also constrained

to ensure the wings are generating sufﬁcient lift and that the aircraft is trimmed.

In addition to the maneuver load cases considered above, landing load cases are also considered to account for

the forces transferred from the landing gear to the wingbox. Five landing conditions are applied at maximum landing

weight (MLW) according to the requirements set out by FAR-25 [53], taking into consideration the level landing

conditions (§25.479) and side load conditions (§25.485). In order to determine the actual load that is to be applied to

the wing box structure, the landing gear is modeled as a separate beam structure, assumed to be connected to the wing

box structure at three points with the translational DOF constrained but not the rotational DOF. By applying the load

at the axle, the reaction forces at the three attachments points are determined, and that reaction forces are applied to

the actual wing box structure distributed over an area.

Aircraft stability is also enforced with two stability conditions. One with the aircraft at MTOW and one with the

aircraft at ZFW. Both cases are simulated at the cruise altitude and speed with an aft center of gravity. The static

margin Knof these two load cases are determined using

Kn=−CMα

CLα

,(34)

and are constrained to be Kn≥0.05. The approximation of the static margin constraint is also described in more detail

in previous work [54]. In total, ﬁve angle of attack variables, one for each ﬂight conditions, are added to provide the

required lift, and ﬁve tail angles are added to ensure the aircraft is trimmed. The shape and thickness design variables

are illustrated in Figure 7. To summarize, the optimization formulation is as follows:

minimize DOC

w.r.t 278 panel thicknesses :t

2 fuel variables :WfuelTRC , WfuelLRC

5 angles of attack :α

5 tail angles :η

5 wing twists :θ

such that ti≥0.080 in

RTRC = 500 nmi

RLRC = 2 000 nmi

Thrust −Thrustreq ≥0.0

Kn≥5%

Additional constraints in Table 7

(35)

The ﬁrst constraint deﬁnes the minimum gauge thickness bound of 0.080 in. The ranges as obtained from the

mission analyses are also constrained to ensure that the mission requirements are satisﬁed. The thrust requirement

constraint is imposed to all segment intervals, and they all have to be positive. The table also summarizes the additional

constraints for each load case to ensure that the aircraft can generate enough lift, is trimmed with mcg = 0, is

within allowable stresses, and has the correct fuel loading. The Kreisselmeier–Steinhauser (KS) constraint aggregation

technique [55,56] is used to combine the thousands of stress constraints into eight for each load case, four for each

wing and tail.

14 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

Table 7: Aerostructural speciﬁc constraints for each of the operating conditions considered.

Group Mach Alt (ft) Cm= 0 KS(σ)≤Weight Load Factor

Maneuver 0.65 25 900 fwd-CG min(σy, σult/1.5) MTOW 2.5

0.65 25 900 fwd-CG min(σy, σult/1.5) ZFW 2.5

0.65 25 900 fwd-CG min(σy, σult/1.5) MTOW -1.0

0.65 25 900 fwd-CG σfatigue MTOW 1.3

Landing (×5) – – – min(σy, σult/1.5) MLW −

Stability 0.65 37 000 aft-CG - MTOW 1.0

0.65 37 000 aft-CG - ZFW 1.0

Figure 7: Design variables considered for the aerostructural optimization.

The structural mesh for the wing and tail consists of 94 684,2nd order elements, with a total of 542 556 DOFs.

The TriPan mesh consists of 30 spanwise panels and 15 chordwise panels. An inital, structural-only, weight-

minimization optimization is performed to determine the initial thickness distribution of the structural panels. For

this optimization, the geometry is ﬁxed, and the structure is loaded with a ﬁxed aerodynamic load. The solution from

this optimization is then used as the initial starting thickness distribution for the aerostructural optimizations.

V. Optimization results

The results from the mission proﬁle optimization, as well as an aerostructural optimization that utilizes the opti-

mized mission proﬁle, are presented and discussed in this section. For both optimization cases, the numerical integra-

tions within the mission analysis are done by dividing each segment into 50 intervals.

A. Mission proﬁle optimization results

The results of the mission proﬁle optimization (33) are presented here. The optimized mission proﬁles are presented

in Table 8 and Table 9, respectively, and are illustrated in Figure 8. These ﬁgures display some mission segment

characteristics, including the range, time, altitude, ﬂight speed, and the amount of fuel burn. Note that “SL” refers to

sea-level altitude. For the TRC mission, the aircraft ﬂies at a relatively high Mach number of 0.81 for cruise, whereas

for the LRC mission, it ﬂies at a lower Mach number of 0.65, which is the lower bound set in the optimization. The

15 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

reduced Mach number contributes to a signiﬁcant fuel saving: 8.83 lb/nmi for the LRC mission versus 14.4 lb/nmi

for the TRC mission. Since the aircraft’s MTOW is determined based on the fuel required for the LRC mission, and

reducing the MTOW also reduces the DOC of the TRC mission, it is beneﬁcial to reduce the fuel required for the LRC

mission regardless of the ﬂight time needed. For the LRC mission, the cruise segments dominate the mission range,

time and fuel burn. For the TRC mission, though the cruise segment still dominates the mission range, time, and fuel

burn, the contribution from other segments, especially climb, is also signiﬁcant. This observation further emphasizes

the importance of considering the entire mission proﬁle, instead of only cruise, to have an accurate estimate of the fuel

burn of shorter range missions.

Table 8: Optimized TRC mission (500 nmi) details.

Segment Altitude Speed Range Time Fuel Burn

ft Mach/kt nm min lb

0 Start/WarmUp 0 - 0.0 - 805.64 (11.21%)

1 Taxi 0 - 0.0 - 797.59 (11.09%)

2 Takeoff 0 - 0.0 - 394.81 (5.49%)

3 Climb 1 500 →10 000 150 KIAS →250 KIAS 13.4 3.7 408.79 (5.69%)

4 Climb 10 000 →33 082 290 KIAS 138.9 19.7 1 670.22 (23.23%)

5 Climb 33 082 →34 185 M0.81 25.1 3.2 222.46 (3.09%)

6 Cruise 34 185 M0.81 234.4 30.1 1 857.46 (25.83%)

7 Descent 34 185 →33 082 M0.81 2.2 0.3 3.17 (0.04%)

8 Descent 33 082 →10 000 290 KIAS 66.1 10.0 133.81 (1.86%)

9 Descent 10 000 →1 500 250 KIAS →150 KIAS 19.9 5.2 81.75 (1.14%)

10 Landing 0 - 0.0 - 222.57 (3.10%)

11 Taxi 0 - 0.0 - 591.73 (8.23%)

TRC Mission Total 500.0 72.3 7 190.0

Table 9: Optimized LRC mission (2 000 nmi) details.

Segment Altitude Speed Range Time Fuel Burn

ft Mach/kt nm min lb

0 Start/WarmUp 0 - 0.0 - 910.42 (5.15%)

1 Taxi 0 - 0.0 - 901.31 (5.10%)

2 Takeoff 0 - 0.0 - 446.15 (2.53%)

3 Climb 1 500 →10 000 150 KIAS →250 KIAS 16.2 4.6 497.21 (2.81%)

4 Climb 10 000 →21 810 290 KIAS 40.3 6.6 665.50 (3.77%)

5 Climb 21 810 →35 000 M0.65 82.5 12.9 932.82 (5.28%)

6 Cruise 35 000 M0.65 873.4 139.8 6 313.45 (35.74%)

7 Climb 35 000 →37 000 M0.65 20.3 3.3 184.95 (1.05%)

8 Cruise 37 000 M0.65 873.4 140.6 5 769.91 (32.66%)

9 Descent 37 000 →21 810 M0.65 39.7 6.2 66.86 (0.38%)

10 Descent 21 810 →10 000 290 KIAS 34.2 5.7 82.51 (0.47%)

11 Descent 10 000 →1 500 250 KIAS →150 KIAS 19.9 5.2 81.75 (0.46%)

12 Landing 0 - 0.0 - 222.57 (1.26%)

13 Taxi 0 - 0.0 - 591.73 (3.35%)

LRC Mission Total 2 000.0 324.8 17 667.1

The optimized DOC (based on the TRC mission) is tabulated in Table 10, broken down to its component costs. The

component cost is sorted in a descending order. The highest contributor for DOC is fuel cost. By looking at Table 8,

relatively high percentages of fuel burn are observed in the startup, taxi, takeoff, and landing segments. In fact, the

total fuel burn from those segments amount to 39% of the total fuel burn for the 500 nmi mission. The fuel burn of

those segments are estimated using the fuel fraction method, with fuel fraction values taken from [13,23,24]. It is

possible that the fuel fraction values are on the high side, which results in more fuel burn than they actually should

be. Note that the mission range and time during those segments are not counted in getting the mission total values.

These “extra” fuel burn, without any additional mission time, might contribute to the fuel burn dominating the DOC

computation, as we observe.

Figure 9 shows the convergence history of the merit function, feasibility, and optimality, as deﬁned in SNOPT [47].

16 of 24

American Institute of Aeronautics and Astronautics

Downloaded by UNIVERSITY OF MICHIGAN on October 5, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4372

0: Startup

00:04

805.6lb

1: Taxi

00:10

797.6lb

2: Takeoﬀ

00:01

394.8lb

3: Climb

13.4nm / 00:03

1500 →10000ft

150kt →250kt

408.8lb

4: Climb

138.9nm / 00:19

10000 →33082ft

290kt

1670.2lb

5: Climb

25.1nm / 00:03

33082 →34185ft

M0.81

222.5lb

6: Cruise

234.4nm / 00:30

34185ft

M0.81

1857.5lb

7: Descent

2.2nm / 00:00

34185 →33082ft

M0.81

3.2lb

8: Descent

66.1nm / 00:10

33082 →10000ft

290kt

133.8lb

9: Descent

19.9nm / 00:05

10000 →1500ft

290kt →150kt

81.8lb

10: Landing