Integrated Exploration and Visualization of Optimal Aircraft Conceptual Designs

M. Nunez1, J. Maginot2, M. Padulo3 and M. Guenov4 Cranfield University, Cranfield, MK43 0AL, UK

[Abstract] The objective of this paper is to present a novel methodology for enabling the visualization and exploration of data obtained during aircraft multiobjective optimization at conceptual design stage. The aim is to enhance the designer’s insight into the optimization problem at hand. To achieve this goal, the present methodology integrates suitable means for visualization and exploration of the optimal solutions. By combining different graphical interfaces, the designer is provided with diverse perspectives of the multidimensional data under study. Additionally, the method provides a posteriori exploration of further optimal solutions. This is made possible by means of appropriate approximation techniques which are integrated within the visualization environment and allow obtaining such information at no additional computational cost. The application of the present methodology to an aircraft sizing test case shows promising results in helping the designer understand the complex high-dimensional data involved.

Nomenclature

ρ = ambient air density γ = minimum second segment climb angle CDo = zero-lift coefficient A = aircraft wing aspect ratio CPI = carpet plot interface D = aircraft drag e = aircraft Oswald’s efficiency factor F = objective function vector fi = i-th objective function G = constraint vector gi = i-th constraint L = number of constraint functions M = number of objective functions MDO = multidisciplinary design optimization MDVI = multidimensional data visualization interface MOO = multiobjective optimization MTOW = maximum take-off weight N = number of input variables Ne = aircraft number of engines OSI = objective space interface P = projection matrix PCP = parallel coordinates plot S = design space 1 PhD Student, Aerospace Engineering Department, Bldg. 83. 2 Research Officer, Aerospace Engineering Department, Bldg. 83. 3 PhD Student, Aerospace Engineering Department, Bldg. 83. 4 Professor, Aerospace Engineering Department, Bldg. 83.

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50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br>17th4 - 7 May 2009, Palm Springs, California

AIAA 2009-2204

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

S = aircraft reference area SMP = scatter matrix plot SOM = self organizing map T = aircraft thrust V = aircraft speed W = aircraft weight x = design vector xi = i-th design variable

I. Introduction

ircraft multidisciplinary design optimization (MDO) takes into account simultaneously the contributions of different disciplines, which often exhibit strong coupling. Most of the MDO research effort so far has been

focused on handling the interactions between analysis tools, including the development of novel numerical techniques. However, the large multidimensional datasets resulting from such an approach are often too complex to be analyzed and completely understood by the designers who are accustomed to traditional design visualization tools. There is an apparent need for a suitable visualization methodology to make the results of a complex design optimization procedure fully readable and meaningful to the designer. The development of such a methodology is the subject of this paper.

A

The next section presents the formulation of the optimization problem and reviews multidimensional visualization techniques used in the field. Section III presents the novel integrated visualization methodology. Section IV focuses on the application of the method to a mid-range civil aircraft sizing test case. Finally, conclusions are drawn in section V.

II. Background

A. Optimization The aim of conceptual design is to develop the overall aircraft configuration arrangement for a given set of

requirements and specifications. Since a large number of alternative design concepts are evaluated, it follows that during this phase an iterative process, including continuous analysis, changes and improvements of the design layouts, is required to achieve a consistent design. As the design solutions advance step by step towards more detailed design, it becomes more difficult and expensive to introduce changes. Therefore, it is fundamental to obtain a good design at conceptual stage, so that the following design phases will not require major revisions. In this context the design of an aircraft frequently results in optimizing multiple criteria for a large number of design variables with the requirement to meet a set of constraints. For example, the designer may wish to maximize the aircraft range and minimize its maximum take-off weight according to constraints related to performance characteristics, regulations, safety and maintenance considerations.

Such design problems, referred to as multiobjective optimization (MOO) problems, are very complex and require specific numerical methods to obtain solutions.

1. General mathematical formulation A typical MOO problem can be formulated as follows:

{ }{ }

{ }

1 2

1 2

1 2

min ( ) ( ), ( ), ... , ( )

, , ... ,

. . ( ) ( ), ( ), ... , ( ) ,

MS

N

L

f f f

x x x

s t g g g

∈=

=

= ≤

xF x x x x

x

G x x x x 0

(1)

where F(x) is the vector of the M objectives to be minimized with respect to the design vector x in the N-dimensional design space S, subject to L constraints. A constraint gi is said to be active at a point x* of the design space S if a strict equality holds at this point, that is, gi (x) = 0.

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2. Pareto solutions and approximation Solving a MOO where the objectives are conflicting usually does not have a unique solution but a set of non-

dominated solutions also known as Pareto solutions. A feasible design point is said to be Pareto optimal if no other feasible design can improve some of the objectives without simultaneously being detrimental to others. For real-life engineering problems, the Pareto frontier cannot be described analytically. Numerical methods are thus required to obtain discrete Pareto solutions, which turns out to be computationally demanding. Research has been carried out to locally approximate the Pareto frontier by reusing gradient information obtained during the optimization procedure. These methods allow to derive new approximate Pareto solutions in the objective space and to obtain their corresponding design vectors in the design space1,2. These approximations based on Taylor expansion on the Pareto frontier are defined as follows:

*

1

( 1, ..., ) fn

pp p i f

ii

dff f f p n M

df=

= + Δ = +∑ (2)

( )*

1 , 1

1 ( 1, ..., ) 2

f fn np p

p p i j k fjkii j k

dff f f H f f p n M

df= =

= + Δ + Δ Δ = +∑ ∑ (3)

where p

i

dfdf

and 2

( ) ppjk

j k

d fH

df df= are the first and second order derivatives on the Pareto surface, nf is the dimension of

the largest family of linearly independent vectors P∇fi, where P is the projection matrix onto the hyper-plane normal to all gradients of active constraints. The derivatives can be computed from the usual gradients of objectives and active constraints obtained during the optimization process. Thus further information in the vicinity of a Pareto point can be obtained at no extra computational cost.

In this paper we will refer to the term optimal family of solutions to identify those designs belonging to the same local Pareto frontier and characterized by the same set of active constraints (including the active bounds of the variables).

B. Visualization A major requirement for an effective visualization technique is to be able to translate numerical datasets into

simple and meaningful graphical representations in order to facilitate data analysis and understanding. Previous efforts in this field have been based on the application of multidimensional visualization techniques in MOO, so that both evaluation and exploration of the Pareto frontier could be performed. In some cases, well-known methods have been implemented3,4, while in others ad hoc methodologies have been developed5,6. A brief summary is presented below.

1. Multidimensional data visualization methods Among all the multidimensional visualization methods, scatter matrix plots (SMP), parallel coordinates plots

(PCP) and self-organizing maps (SOMs) are widely used in MDO because of their capabilities to represent large multidimensional datasets. Scatter matrix plots allow for the analysis of relationships between every couple of variables of a multidimensional dataset7. Parallel coordinates plots are based on the idea of visualizing simultaneously all the values of all variables for each sample on the same graph by means of a set of vertical parallel axes8. SOM is an efficient technique for visualizing multidimensional data9. It consists of neurons positioned on low dimensional regular grids (component maps) and described by a multidimensional prototype vector in the high dimensional space. Through training, the map is organized to best describe the set of training data and allows projecting a high dimensional space onto two dimensional component maps. SOMs provide an appropriate technique for identifying similarities and clustering whereas the SMP and the PCP allow users to easily identify relationships between variables.

In the context of the above mentioned ad hoc methods, Stump10 proposes a data visualization interface suitable for design by shopping paradigm. In particular, the technique of preference shading enables the designer to focus on the regions of the design space corresponding to different preference weights on performance parameters.

2. Carpet plots Prior to the application of computational tools in aircraft conceptual design, optimization methods were based on

the development of a set of parallel layouts, each one characterized by different combinations of the design

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parameters. Design optimization was carried out with the help of carpet and matrix plots11-13 by estimating the impact of parametric variations on the aircraft layout and criteria such as mission, weight and cost. A typical carpet plot provides a means of visualizing performance requirements (e.g. cruise speed, second segment climb rate, take-off and landing field length distances) as a function of parametric variables such as thrust-to-weight-ratio (T/W) and wing-loading (W/S). A point on the carpet plot represents a particular aircraft design and provides the designer with information on how performance constraints are satisfied.

3. Recent developments in MOO visualization The hyper-space diagonal counting (HSDC) method presented in Agrawal et al.5 is intended to visualize

intuitively the Pareto frontier for large-scale MOO problems. HSDC exploits Cantor’s findings in set theory which enable the representation of multidimensional Pareto surfaces in a 2-D or 3-D graph without any loss of information.

Several authors have stressed the importance of representing the physical layout of the aircraft to simultaneously visualize the geometrical parameters characterizing each design solution. This has been achieved either by using simple parametric CAD models or schematics of the aircraft under consideration. These representations allow the designer to immediately understand the impact on the aircraft geometry (e.g. number of engines and their location, fuselage geometry, tail and wing plants, etc). However, such visualization might only be useful for experienced designers dealing with conventional aircraft design. By comparing the key features of a specific layout with similar existent aircraft, they can assess whether it corresponds to a feasible design. Furthermore, such representations can only display limited information with regard to aerodynamics, performance, civil/military regulations, design-constraints satisfaction and so forth. Finally, it does not allow appreciating the subtle, but still important differences between similar design solutions.

III. Integrated Exploration and Visualization of Pareto frontier

A. Methodology The proposed approach is aimed at enhancing the visualization of design solutions in an MOO framework. It is

based on the authors’ belief that in order to convey the full meaning of design optimization data to the designer a combination of visualization techniques have to be used together and also that the interpretation of such data should not require the designer to be proficient in numerical optimization methods. Depending on the analysis to be carried out, suitable methods have to be selected with respect to key features of the datasets to be investigated in the attempt to guarantee their full readability. The present methodology is based on the development of a matrix (Table 1) which identifies the suitable visualization techniques to be used in the context of common data analysis scenarios occurring during aircraft design optimization.

Technique Scenario

Carpet Plots SMP PCP Objective Space

Visualization of a single Pareto point Performance analysis Visualization of input, objective

and constraint values Objective value comparison Objective value comparison and visualization of optimal families of designs

Visualization of a reduced number of Pareto points (<10)

Performance comparison between design points

Trend and correlation analysis

Identification of design points sharing common features

Objective value comparison and visualization of optimal families of designs

Visualization of a large number of Pareto points (>10)

Analysis/Visualization of the entire solutions-set distribution

Trend and correlation analysis

Identification of design points sharing common features

Objective value comparison and visualization of optimal families of designs

Visualization of Pareto frontier approximation

Performance visualization of approximated Pareto frontier

Gradient information for local sensitivity analysis

Identification of common features shared between the approximated solutions

Constraint satisfaction study

Performance requirements check Study of how well constraints

are satisfied Objective value comparison

Active constraints study

Comparison of designs characterized by constraint activation

Analysis of designs characterized by constraint activation

Visualization of optimal families of designs

Table 1: Methodology matrix for visualization techniques

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A visualization tool implementing this methodology has also been developed: the Integrated Exploration and Visualization Interface (IEVI). It integrates three graphical interfaces whose synergic coupling aims at providing the designer with improved insight into the data for various analysis scenarios. The three graphical interfaces are described in detail in the following sub-sections. Each interface is focused on the representation of a particular perspective of the aircraft multiobjective optimization at conceptual design stage. The simultaneous update and interactive application of such interfaces aims to further improve the visualization of the complex information produced by the design optimization tools. In addition, the method is intended to provide a posteriori exploration of additional optimal solutions. This is made possible by means of the approximation technique described above, which is integrated into the visualization environment and do not entail any additional computational cost. A typical use of the visualization tool is given in Figure 1 while Table 1 identifies the set of suitable visualization techniques to be used for each common data analysis scenario occurring in the aircraft design optimization (green boxes in Fig. 1).

Figure 1: Example of the procedure for the use of the visualization tool IEVI.

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B. Visualization Interfaces 1. Objective Space Interface (OSI) Currently the OSI is aimed at representing the Pareto frontier in a simple and conventional way for up to three

objectives, providing the value of the objective functions for each alternative optimal solution given by the optimizer. (It should be noted that the OSI can utilize the PCP to visualize more than three objectives. A future enhancement of the interface will be focused on the integration of additional multidimensional visualization techniques, e.g., star coordinates plots, self organizing maps, etc.)

Such a representation enables the user to immediately identify the most interesting and promising areas of the Pareto frontier. Apart from verifying formulation errors of the optimization process, valuable information can be rapidly obtained from it, for instance:

- The identification of the objective minimum and maximum values for the set of optimum points;

- The position of local Pareto regions, which correspond to different optimal families of solutions sharing the same features;

- The density of the design solutions in a specific area of the Pareto front; - The objective space location of the solutions characterized by one or more specific active constraints.

2. Carpet Plot Interface (CPI) In the CPI, carpet plots are used to visualize MOO results. Carpet plots provide a straightforward and physical representation of the optimization results. In the T/W-W/S space (or an equivalent parametric space), the users can immediately obtain information about the performance constraint satisfaction of the design under study. A simultaneous visualization of multiple Pareto points in the carpet plot would present not only a distinct location of the design points, but also a different arrangement of their respective sets of performance constraints, as shown in Figure 2. This is due to the dependence of the constraints on design parameters which are peculiar to each solution, but not explicitly represented in the T/W-W/S space.

Figure 2: Example of the carpet plots of two different design points, one in red and the other in blue. Hatching denotes inadmissible side of constraint curves.

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This graphical interface gives the designer an effective way to assess any design solution. The integration of carpet plots in the visualization framework allows the designer to evaluate the Pareto frontier with respect to a set of design constraints by using a traditional design tool without the burden of mathematical complexity.

As outlined in section II, approximated solutions can be obtained at no extra cost in the vicinity of any Pareto solution of interest. These approximated solutions can be plotted both in the OSI and CPI which gives the designer the possibility to explore other Pareto solutions positioned in a more desirable area of the T/W-W/S space. Such a posteriori analysis allows finding design solutions which are approximately Pareto optimal, but may help meeting additional criteria not considered in the initial MOO formulation.

3. Multidimensional Data Visualization Interface (MDVI) In the MDVI, the PCP is the current multidimensional visualization technique available to the designer. This

method is especially useful when exploring a small set of design solutions for identifying relationships among the design parameters, and for checking constraint satisfaction and activation. The PCP capabilities are especially useful for the visualization of high-dimensional data on a simple two-dimensional plot, representing all the variables on the graph at the same time. However, since the axes are plotted side by side, the i-th dimension is linked at most to two other dimensions. In an n-dimensional problem no information is visualized about the relationships among the i-th axis and the other (n-3) axes which are not by its sides. Therefore, it is evident the need of implementing the interface so that it is possible to permute the axes. This allows finding out different views of the problem and other possible relationships among the design parameters. Furthermore, the user would be provided with the Selective Ranges function, which can enable him/her to analyze only those solutions within an established range of values for any design parameter of interest.

An on-going effort is concentrated on integrating the SPM into the MDVI. The former is well suited for discovering or checking correlations among data or for comparing local relationships between couples of variables, constraints and objectives. The MDVI functionality will be enhanced by the simultaneous update between the PCP and SPM thus allowing the user to better explore the relationships among the parameters under consideration.

C. Operation of the Visualization Tool The analysis of the optimal solutions has to be carried out by considering simultaneously a suitable set of the

visualization interfaces described above. Table 1 facilitates choosing the most appropriate visualization methods according to the data analysis tasks and scenarios.

1. Optimal Solutions Study IEVI is aimed at allowing the exploration and analysis of the Pareto frontier in a intuitive manner, where:

- the user can choose the graphical interface to interact with amongst OSI, CPI and MDVI; - the point to be analyzed can be intuitively selected; - diverse perspectives on the design under consideration can be enabled by a simultaneous update of the

visualization on the remaining interfaces; The above features are also applicable to multiple design points comparison.

2. Constraint Study A constraint behavior analysis is required to enhance the designer’s insight into the optimization outcome. This

can be achieved in two different ways:

- the user can click on a design point in the OSI, so that all the constraints are visualized in the PCP and the SPM. In the former graph, it would be possible to numerically check constraints satisfaction. Moreover, all active and inactive performance constraints will be represented in different colors, for example, in red and green, respectively. In the SPM plot, a local sensitivity analysis will be provided by conveying the effect of each variable on each constraint;

- the decision-maker can select a constraint in the PCP in order to visualize all the solutions for which it is

active in the OSI and CPI.

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Both approaches allow the user to gain a better understanding of the design and objectives spaces. Depending on the information the designer is interested in, it may be essential to discern:

- what solutions are characterized by the activation of one or more particular constraints; - what are the active constraints that feature each optimal family of solutions; - what constraints have an influence on a specific area of the Pareto frontier or of the Carpet Plot space; - what variables and what range of values are determinant for the activation of each constraint.

IV. Integrated Exploration and Visualization Example

A. Description of the test case The proposed methodology has been evaluated on a sizing test case which makes use of the analysis capabilities of FLOPS (Flight Optimization System), which is a NASA Langley’s code developed for the evaluation and optimisation of aircraft designs at conceptual and preliminary stage 16. It consists of nine primary modules: weights, aerodynamics, engine cycle analysis, propulsion data scaling and interpolation, mission performance, takeoff and landing, noise footprint, cost analysis and program control. These modules provide a means of evaluating advanced aircraft concepts, predicting their overall performance, weights, costs and environmental factors. Moreover, FLOPS has the capability to perform a detailed analysis for all mission segments and to estimate performance constraints such as approach speed, missed approach climb gradient, second segment gradient, landing and takeoff field length.

The input variables, constraints and objectives considered for the set-up of the test case are described in Table 2.

Wing Data: S Reference wing area [ft2]TR Taper ratio of the wing SWEEP Quarter-chord sweep angle of the wing [deg]TCA Wing thickness-chord ratioSPAN wing span [ft] DIH Wing dihedral (positive) or anhedral (negative) [deg]FCOMP Decimal fraction of amount of composites used in wing structureTails, Fins, Canards Data: SHT Horizontal tail theoretical area [ft2]SWPHT Horizontal tail 25% chord sweep angle [deg]ARHT Horizontal tail theoretical aspect ratioTRHT Horizontal tail theoretical taper ratioTCHT Thickness-chord ratio for the horizontal tailVertical Tail Data: SVT Vertical tail theoretical area (per tail) [ft2]SWPVT Vertical tail sweep angle at 25% chord [deg]ARVT Vertical tail theoretical aspect ratioTRVT Vertical tail theoretical taper ratioTCVT Thickness-chord ratio for the vertical tailPropulsion System Data: THRSO Rated thrust of baseline engine [lb]BPR Bypass ratio NEW Number of wing mounted enginesNEF Number of fuselage mounted enginesConfiguration Data: DESRNG Design range [NM] MTOW Maximum Take-Off Weight [lb]VCMN Cruise Mach number

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CH Maximum cruise altitude [ft]HTVC Modified horizontal tail volume coefficientVTVC Modified vertical tail volume coefficientHHT Decimal fraction of vertical tail span where horizontal tail is mountedVAPPR Maximum allowable landing approach velocity [kts]FLTO Maximum allowable takeoff field length [ft]WFC Wing fuel capacity [lb] FFC Fuselage fuel capacity [lb]Crew and Payload Data: NPF Number of first class passengersNPB Number of business class passengersNPT Number of tourist class passengersNSTU Number of flight attendantsFuselage Design Data: FPITCH Seat pitch for the first class passengers [in]NFABR Number of first class passengers abreastBPITCH Seat pitch for business class passengers, [in]NBABR Number of business class passengers abreastTPITCH Seat pitch for tourist class passengers [in]NTABR Number of tourist class passengers abreast

Constant Variables NEW = 2 NEF = 0 NPF = 20 NPB = 20 NPT = 150 NSTU = 10 FPITCH = 34 in NFABR = 4 BPITCH = 30 in NBABR = 4 TPITCH = 28 in NTABR = 6 HTVC = 1 VTVC = 1 HHT = 0.02

Input Variables x ∈ [xlb,xub] SPAN = [90,130] ft DIH = [-4,4] deg FCOMP = [0,0.35] SHT = [200,250] ft2 SWPHT = [30,32] deg ARHT = [2.5,3.5] TRHT = [0.225,0.25] TCHT = [0.07,0.1] SVT = [200,250] ft2 SWPVT = [30,32] deg ARVT = [1.3,1.7] TRVT = [0.225,0.25] TCVT = [0.07,0.1] THRSO = [29200,32000] lb DESRNG = [2500,3000] NM S = [1500,1800] ft2 TR = [0.225,0.25] SWEEP = [30,32] deg TCA = [0.07,0.085] VCMN = [0.78,0.82] CH = [25000,35000] ft BPR = [5,6]

Table 2: Description of the test case set-up

The multi-objective optimization problem has been formulated as follows:

min ( )MTOW

Total Fuel⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭x

F x

Subject to:

1

2

( ) 9000 ;

( ) 140 ;

g FLTO ft

g VAPPR kts

= ≤

= ≤

x

x

1

min

1

3 22 2

( ) 0;1

e

e

TOe

TO e

TNT DgW N T L

γ⎛ ⎞⎡ ⎤ ⎡ ⎤= − ⋅ ⋅ +⎜ ⎟⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎝ ⎠

x ≥

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0

2 2

42

12( ) 0;

12

DTO TO

TO TO

TO

WC VT ST Wg

WW T WV A eS

ρ

ρ π

⎛ ⎞⎡ ⎤⎜ ⎟⋅ ⋅ ⋅ ⎢ ⎥ ⎛ ⎞⎡ ⎤ ⎣ ⎦⎜ ⎟= − ⋅ + ⋅⎜ ⎟⎢ ⎥ ⎜ ⎟⎡ ⎤⎣ ⎦ ⎝ ⎠⋅ ⋅ ⋅ ⋅ ⋅⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

x ≥

with:

;≤ ≤lb ubx x x

where the subscripts TO and 2 refer to the take-off and second segment conditions, the sub-subscript 1e denotes the thrust of a single engine, the constraints g3 and g4 represent the requirements on second segment with one engine inoperative and cruise speed respectively.

B. Data Analysis Solutions of the above MOO problem have been obtained using an optimization tool developed at Cranfield

University14,15. The default visualization mode of the optimization outcome is displayed in Figure 3, where all the design solutions are displayed simultaneously in each graphical interface. For this graphical mode the CPI is aimed at highlighting the region of the T/W-W/S space where the Pareto solutions are clustered, while the PCP provides a numerical data perspective only for the axes required by the user. Through these visual settings the designer is helped in understanding the overall features of the results set at hand, especially by highlighting the available ranges of values for each design parameter.

Figure 3: Default visualization of the optimization results on the Integrated Exploration and Visualization Interface (IEVI)

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The Pareto exploration and analysis can be conducted by interactively selecting the design of interest on any graphical interface (OSI, CPI and MDVI) thus updating the other interfaces visualization only with the data of the sample under study. Alternatively, the user can compare solutions by clicking on a set of Pareto points as it is shown in Figure 4, where the corresponding carpet plots and polylines are displayed in the CPI and PCP, respectively.

Figure 4: Results visualization in Interactive Mode for OSI and in Multiple Samples Mode for CPI and PCP. By clicking on the optimal design solutions of interest in the OSI, the user can compare these by analyzing further data perspectives of the selected samples on the other interfaces, which are updated simultaneously.

For each axis selected in the PCP, the horizontal green ticks show the minimum and maximum values of the set

of Pareto solutions obtained during the optimization. Furthermore, the designer can easily identify all the designs having similar features with respect to a particular set of parameters. This is made possible by specifying the range of values to satisfy for the parameters of interest. In Figure 5 the green horizontal ticks identify again the bounds of the Pareto set for each parameter, whereas the yellow ticks represent the ranges interactively chosen by the user for three different parameters. It should be noted that in this hypothetic data-analysis case the designer is interested in identifying all those solutions for which:

- the constraint on VAPPR is activated (rightmost axis in Fig 5); - the values of CH are within a desirable range; - the minimum optimal value for SW is obtained.

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Figure 5: PCP in Selective Ranges Mode.

Figure 6: Visualization of the Pareto solutions (in green in the OSI and CPI) obtained during the optimization procedure and the local approximations (in orange) of the design point under study (in sky blue). The green ticks displayed in the PCP represent the bounds of the Pareto set for each parameter.

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In Figure 6 and Figure 7 the approximation capabilities of the visualization tool have been used to obtain new approximated Pareto solutions in an attempt to explore a region of the Pareto front in the vicinity of a particular solution. The approximation is displayed with a series of orange points both in the OSI and in the CPI. This enables the designer to assess performance constraint satisfaction and also to understand how a particular area of the Pareto front is mapped to the (W/S – T/W) space. Additionally, the PCP of the approximated Pareto solutions can be visualized through a set of polylines of the same color. This provides extra information about the relationships among the optimization variables and allows the designer to identify key features of the designs in the vicinity of the solution under consideration. The user can easily assess the range of variations observed on the Pareto front and make comparisons between the variables and objectives changes derived with the approximation.

These approximation capabilities can be very useful for computationally and time demanding problems. The user is allowed to reduce the optimization efforts by obtaining only a reduced number of optimal points and, subsequently, to explore through a set of approximated solutions the Pareto frontier such points belong to.

Figure 7: Zoom-in on the CPI – carpet plot of the Pareto solution under study (sky blue point) including all Pareto solutions obtained during the optimization procedure (green) and some approximated Pareto solutions (orange).

V. Conclusions and Future work

Presented is a novel methodology aiming to enhance the decision making process associated with aircraft multiobjective optimization at conceptual design stage. It includes the integration of suitable visualization and exploration capabilities. The combination of different graphical interfaces and visualization techniques provides the designer with diverse perspectives on the data under study and bridges the gap between MOO and the more conventional design approaches. Ultimately, the method allows the designer to explore further optimal solutions

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with respect to other criteria not considered in the initial MOO formulation. This is made possible by means of proper approximation techniques integrated within the visualization environment and without any significant additional computational cost. The methodology has been tested with a sizing test case in order to evaluate and validate the method capabilities.

Future work will concentrate on enhancing the proposed methodology by:

- integrating the SPM in order to allow the user to obtain and display the local sensitivity on the Pareto surface of the point under study for a chosen set of input variables, constraints and objectives;

- integrating additional multidimensional visualization methods to display Pareto frontiers with more than

three objectives;

- integrating uncertainty visualization techniques in order to facilitate the results analysis of robust optimization problems.

Acknowledgments The authors gratefully thank L. Arnold McCullers at NASA Langley for supplying FLOPS.

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KS, 1979. 14 Fantini, P. “Improving the Effectiveness of Multi-Disciplinary Design Optimization at the Aircraft Conceptual Design

Phase,” Ph.D. Dissertation, Aerospace Engineering Department, Cranfield University, UK, 2007. 15 Fantini, P., Balachandran, L. K., and Guenov, M. D., “Computational Intelligence in Multi-Disciplinary Optimization at

Feasibility Design Stage,” First International Conference on Multidisciplinary Design Optimization and Applications, ASMDO, 2007.

16 McCullers, L. A., FLOPS User’s Guide, Release 7.40. Text file included with the FLOPS code.

American Institute of Aeronautics and Astronautics

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