[American Institute of Aeronautics and Astronautics 35th AIAA Fluid Dynamics Conference and Exhibit...

American Institute of Aeronautics and Astronautics

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The Application of MDO Technologies to the Design of a High Performance Small Jet Aircraft

- Lessons learned and some practical concerns -

Keizo Takenaka* Mitsubishi Heavy Industries, Ltd., 10, Oye-cho, Minato-ku, Nagoya, 455-8515, Japan

Shigeru Obayashi† Tohoku University, Sendai, 980-8577, Japan

Kazuhiro Nakahashi‡, Kisa Matsushima§ Tohoku University, Sendai, 980-8579, Japan

In Japan, a 5 year R&D project has been in progress toward the development of an environmentally friendly high performance small jet aircraft under auspice from NEDO (New Energy Development Organization) since 2003, in which new technical features have been investigated including advanced aerodynamics, new materials, and human centered cockpit. In this paper, we focus on aerodynamics and Multi-Disciplinary Optimization (MDO) R&D activities and applications to actual design phase. The MDO among aerodynamics, structures, and aeroelasticity of the wing of a small jet aircraft was performed using high-fidelity evaluation models. The objective functions were minimizations of a block fuel and the maximum takeoff weight. As a result, several non-dominated solutions were obtained and one solution was found to have one percent improvement in the block fuel. However, through the study, we found there still remain some concerns to obtain “practically enough” optimized solution. These concerns are such as airfoil shape representation, definition of design space, efficiency of the evaluation, etc.. Several investigations to overcome these concerns are also presented in this paper. On the other hand, we have been developing the detail aerodynamic optimization modules simultaneously. These are engine-airframe integration and 2D high lift device design using unstructured adjoint method. We introduce overview of these modules and the whole design process utilizing the systems with these modules.

I. Introduction n Japan, a 5 year R&D project has been in progress toward the development of an environmentally friendly high performance small jet aircraft under auspice from NEDO1 since 2003, in which new technical features have been

investigated including advanced aerodynamics, new materials, and human centered cockpit by industry-government--university cooperation2. In this paper, we focus on aerodynamics and MDO R&D activities and applications to actual design phase.

Recently, MDO studies for aircraft design are conducted by industries and universities and have a growing recognition of its importance. As the technology for high-speed design (cruise configuration design) has matured, the development of the aircraft requires the application of MDO to be superior to the competitors. Conventionally, an aircraft design proceeds mainly with aerodynamic design and partly with some design revision required from other design process, such as structures and aeroelastics. Based on such aerodynamics oriented design process, it is difficult to realize an optimum design of an aircraft in the tight program schedule. MDO is expected to innovate the aircraft design process. * Research Engineer, Nagoya Aerospace Systems, Mitsubishi Heavy Industries, Ltd. † Professor, Institute of fluid science, Tohoku University, associate fellow AIAA ‡ Professor, Department of Aerospace Engineering, Tohoku University, associate fellow AIAA § Associate Professor, Department of Aerospace Engineering, Tohoku University, AIAA senior member

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35th AIAA Fluid Dynamics Conference and Exhibit6 - 9 June 2005, Toronto, Ontario Canada

AIAA 2005-4797

Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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At Mitsubishi Heavy Industries, Ltd. (MHI), which is the prime contractor of the project, have developed the high-fidelity MDO systems as a R&D study between MHI and Tohoku University.

Our goal is to develop the high-fidelity MDO systems applicable to not only the conceptual aircraft design phase, but also the preliminary aircraft design phase in commercial aircraft design. It is not easy task. Optimized solutions and those histories to the solutions in MDO give us useful trade-off information, sensitivities of design parameters to the objective functions, and seem to meet for conceptual design. However, to obtain “practically enough” optimized solution compared with conventional engineers’ knowledge and 3D CAD based design is not easy, even if we use high-fidelity evaluation models such as Navier-Stokes CFD analysis.

We developed the proto-type high-fidelity MDO systems for cruise wing design and evaluated it. From the evaluations, we obtained satisfactory achievement, however, simultaneously we have learned several concerns to overcome. Most important concern is the reasonable definition of the design space. This includes, airfoil shape representation, mesh generation, definition of design variables domain and so on. Several investigations to overcome these are also presented in this paper.

On the other hand, we have been developing the detail aerodynamic optimization systems simultaneously. These are engine-airframe integration and 2D High lift device design using unstructured adjoint method under practical aero, structure, equipment and manufacture constraints. We introduce overview of these modules and the whole design process utilizing the systems with hese modules.

II. Present MDO system The objective of the system is to optimize the three-dimensional wing shape for the transonic jet aircraft using

evolutionary multi-objective optimization with high-fidelity simulations. In the present study, high-fidelity simulation tools, such as Reynolds-averaged Navier-Stokes solver for aerodynamics, MSC NASTRANTM 7, versatile and high-fidelity commercial software, for structures and aeroelasticity were coupled together for MDO.

A. Objective functions In this system, minimization of the block fuel at a required target range derived from aerodynamics and

structures was selected as an objective function. In addition, two more objective functions were considered — minimization of the maximum takeoff weight and minimization of the difference in the drag coefficient between two Mach numbers, which are cruise Mach and target Maximum operating Mach number (MMO), to prevent decrease MMO.

B. Geometry definition First, the planform was fixed. The front and rear spar

positions were fixed in the structural shape based on the initial aerodynamic geometry. The wing structural model was substituted with shell elements. The design variables were related to airfoil, twist, and wing dihedral. The airfoil was defined at three spanwise cross-sections using the modified PARSEC3 with nine design variables (xup, zup, zxxup , xlo, zlo, zxxlo , αTE,βTE, and rLElo/rLEup )per cross-section as shown in Fig. 1. The twists were defined at six spanwise locations, and then wing dihedrals were defined at kink and tip locations. The twist center was set on the trailing edge in the present study. The entire wing shape was thus defined using 35 design variables. The detail of design variables is summarized in Table 1. In the present study, the geometry of each individual was generated by the unstructured dynamic mesh method4,5 using displacement from the initial geometry.

C. Constraints The five constraints were considered in the optimizer. The first three were geometrical constraints, while the last

two were constraints for flight condition as follows: 1. The distribution of the parameter Δy to describe leading-edge geometry was constrained in the spanwise direction to prevent abrupt stall characteristics. Here, Δy denotes an airfoil upper surface ordinate at 6% chord from the leading edge minus the ordinate at 0.15% chord. 2. Rear spar heights were greater than required for housing of the control surfaces.

Figure 1. Illustration of the modified PARSEC airfoil

shape defined by nine design variables


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3. The lower and upper surfaces of the spars changed monotonically in the spanwise direction. 4. The lift coefficients increased monotonically with increasing Mach number to satisfy target MLD (Lift Divergence Mach number). 5. The evaluated fuel for the given range was less than the wing fuel volume.

Table 1. Detail of design variables.

D. Optimizer Evolutionary algorithms (EAs), in particular genetic algorithms (GAs), are based on the theory of evolution,

where a biological population evolves over generations to adapt to an environment by selection, crossover, and mutation. Fitness, individuals, and genes in the evolutionary theory correspond to the objective function, design candidates, and design variables in design optimization problems, respectively. GAs search for optima from multiple points in the design space simultaneously and stochastically. GAs can prevent the search from settling in a local optimum. Moreover, GAs do not require computing gradients of the objective function. These features lead to the following advantages of GAs coupled with CFD:

1. GAs has the capability of finding global optimal solutions. 2. GAs have be processed in parallel. 3. High-fidelity CFD codes can be adapted to GAs easily without any modification.

Figure 2. Flowchart of the present MDO system


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4. GAs are not sensitive to any noise that might be present in the computation. Adaptive Range Multi-Objective Genetic Algorithm (ARMOGA)6 is an efficient multi-objective evolutionary

algorithm (MOEA) designed for aerodynamic optimization and multidisciplinary design optimization problems. ARMOGA has range adaptation based on population statistics, and thus the population is re-initialized every N generation so that the search region adapts toward more promising regions.

E. Evaluation Method The optimizer generates eight individuals per generation, and evaluates aerodynamic and structural properties of

each design candidate as follows; 1. Structural optimization is performed to Jig shape to realize minimum wing weight with constraints of strength and flutter requirements using NASTRAN. And then, weights of wing box and carried fuel are calculated. 2. Static aeroelastic analysis is performed at three flight conditions to determine the aeroelastic deformed shapes (1G shape) using the Euler solver and NASTRAN. 3. Aerodynamic evaluations are performed for the 1G shapes using the Navier-Stokes solver. 4. Flight envelope analysis is performed using the properties obtained as above to evaluate the objective functions. Using the objective functions, the optimizer generates new individuals for the next generation via genetic operations, such as selection, crossover, and mutation. The conceptual flowchart for the present MDO system is shown in Fig. 2. In the present study, MSC

NASTRANTM which is a high-fidelity commercial software is employed for the structural and aeroelastic evaluations. Besides, the unstructured mesh solver named as TAS-Code (Tohoku university Aerodynamic Simulation code)8, 9 is used to evaluate aerodynamic performance using Euler and Navier-Stokes equations.

F. Structural Optimization In the present MDO system, structural

optimization of a wing box is performed to realize minimum weight with constraints of strength and flutter requirements. Given the wing outer mold line for each individual, the finite element model of wing box is generated automatically by the FEM generator for the structural optimization. The wing box model mainly consists of shell elements representing skin, spar and rib and other wing components, such as control surfaces and subsystems etc., are modeled using concentrated mass elements.

In the present structural optimization, strength and flutter characteristics are evaluated with using NASTRAN. For the strength evaluation, the static load is calculated from the pressure distribution on the wing, which is computed by the Euler solver, assuming the 4.5G up-gust condition and then, static analysis is conducted where the static load acts on the wing box structure to obtain the stress on each element. For the flutter evaluation, doublet-lattice method is used to compute the unsteady aerodynamic forces on the wing and p-k method is used as a flutter solution to obtain the critical flutter velocity.

The present structural optimization is based on the following optimality criteria:

• Strength optimization

constFi

i =σ

0 5 10 15 20 25

Iteration

Min

Str

engt

h Fa

ctor

γm

in

Cri

tical

Flu

tter

Ver

ocity

[m/s

EA

S]

γmin

Critical VF

0.2 100

γmin=1.0

VF requried

(a) Strength factor/Critical flutter velocity

0 5 10 15 20 25

Iteration

Win

g B

ox S

truc

tura

l Wei

ght [

kg

20

(b) Wing box structural weight

Figure 3. Convergence histories of structural optimization for the initial geometry


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• Flutter optimization

constt

V

i

F =∂

∂

where σ denotes the stress for i-th shell element of the wing box, F is the allowable stress and FV is the critical flutter velocity in the several flight conditions.

In the structural optimization, thickness of shell elements is resized iteratively until the weight change is converged sufficiently under the strength and flutter constraints.

The resizing formula is as follows: • Strength optimization

( )i

oldinew

it

tminγ

=

• Flutter optimization

etti

F

i

F

oldi

newi

tV

tV

tt

arg∂

∂∂

∂

=

where minγ denotes the minimum strength factor

and the several constraints are considered as follows:

F

shear

tensile

ecompressiv

<

σσ

σ

FrequiredF VV >

mintti > Fig.3 shows the convergence history of structural

optimization for the initial geometry.

G. Static Aeroelastic Analysis In this module, static analysis is performed

iteratively, where the aerodynamic load calculated form Euler computation for the deformed wing shape is applied to the wing structure model using NASTRAN, until the displacement change is converged sufficiently. FLEXCFD, which is the MHI in-house interface code, is applied for automatic data communication between aerodynamics and structures and a moving mesh method4, 5 is applied to obtain a deformed aerodynamic unstructured mesh. The flowchart of this module is shown in Fig. 4. The displacement change computed by the present method usually converges after three iterations.

H. Aerodynamics Evaluation In the present study, TAS-Code was employed for aerodynamic evaluation. The three-dimensional Reynolds-

averaged Navier-Stokes (RANS) equations were computed with a finite volume cell-vertex scheme. The

Figure 5. Visualization between unstructured surface mesh for aerodynamic CFD model and wing box shell

elements for structural FEM model

Figure 4. Flowchart of static aeroelastic analysis


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unstructured hybrid mesh method10 was applied to capture the boundary layer accurately and efficiently. Spalart-Allmaras one-equation turbulence model11 was employed. Fig. 5 shows the unstructured CFD mesh and the wing box element for the structural FEM model. Taking advantage of the parallel search in EAs, the present optimization was parallelized on vector-parallel machines (NEC SX-5 and SX-7). The master processing element (PE) managed ARMOGA, while the slave PEs computed aerostructural evaluation processes. Slave processes did not require synchronization.

I. Flight Envelope Analysis Finally, the Block Fuel Module was executed to evaluate three objective functions as block fuel, maximum

takeoff weight, and drag divergence, and to check the constraints for flight conditions shown in Fig. 2. In this module, the wing box weight for structural-optimized shape and aerodynamic performance were used as input. As all eight individuals were evaluated.

III. Optimization Results The population size was set to eight, and then roughly 70 Euler and 90 RANS computations were performed in

one generation. It took roughly one and nine hours of CPU time on NEC SX-5 or SX-7 per PE for single Euler and RANS computations, respectively. The population was re-initialized every five generations for the range adaptation. First, evolutionary computation was performed for 17 generations. Then, the evolutionary operation was restarted using eight non-dominated solutions extracted from all solution of 17 generations, and two more generations were computed. A total evolutionary computation of 19 generations was carried out. It took several months and the evolution may not converge yet. However, the results were satisfactory because several non-dominated solutions achieved significant improvements over the initial design. Furthermore, a sufficient number of solutions were searched such that the sensitivity of the design space around the initial design could be analyzed. This will provide useful information for designers.

All solutions evaluated are shown in Fig. 6, and Fig. 7 shows all solutions projected on a two-dimensional plane between two objectives, the block fuel and the drag divergence. As this figure shows that the non-dominated front was generated, there was a tradeoff between the block fuel and the drag divergence. All solution projected on two-dimensional planes between other combinations was shown in Fig. 8. As the non-dominated solutions did not comprise Pareto front, this figure showed that there was no global tradeoff between these combinations of the objective functions.

A. Comparison between Initial and Strong Non-dominated Solution Geometries Although the wing box weight tends to increase as compared with that of the initial geometry, the block fuel can

also be reduced. Thus, the aerodynamic performance can redeem the penalty due to the structural weight. An individual on the non-dominated front shown in Fig. 7 was selected, indicated as ‘optimized’, and then the optimized geometry was compared with the initial geometry.

Fig. 9 shows a comparison of polar curves. Although the corresponding the drag minimization was not considered here, CD was reduced. CD of the optimized geometry was found to be reduced by 5.5 counts. Due to the drag reduction, the block fuel of the optimized geometry was decreased by over one percent even with its structural weight penalty.

Next, the mechanism of the drag reduction was investigated. Fig.10 shows a comparison of the pressure distributions at the 35.0% spanwise location. Then, the variation in the leading-edge bluntness works to depress the shock wave on the upper wing surface, i.e., to reduce the wave drag. Fig.11 shows a comparison of the shock wave visualized by the shock function. The shock wave of the optimized geometry was weaker than the initial geometry in the vicinity of the 35.0% spanwise location as shown in Fig. 17 indicating wave drag reduction.

Therefore, these figures show that the shape near the 35.0% spanwise location, i.e., the shape in the vicinity of the kink is effective to reduce the drag in this case.

For further detail investigation on the present results and data mining technique such as Self-Organizing Map (SOM) and Analysis of Variance (ANOVA), see also the previous reports12,13.


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Figure 9. Comparison of the CL-CD curves between initial and optimized geometries

under transonic flight condition

Figure 10. Comparison of Cp distributions between

initial and optimized geometries under transonic cruising flight condition at 35% semi span locations

Figure 11. Comparison of shock wave visualizations

colored by entropy under the transonic cruising flight condition between initial (left) and optimized

(right) geometries. CL is constant

Figure 6. All solutions plotted in three-dimensional space of all objective functions

Figure 7. All solutions on two-dimensional plane between block fuel and drag divergence

Figure 8. All solutions on two-dimensional plane between block fuel and maximum takeoff weight


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IV. Investigations toward the practically enough optimization Several solutions achieved significant improvements in block fuel, aerodynamic drag and structural weight over

the baseline design in the previous optimization study. Furthermore, from the data mining technique, useful information for much improvement was derived. However, comparing another results obtained by an optimization similar to the previous optimization study with the conventional engineer’s knowledge and 3D CAD based design (Engineer’s design) from the same baseline, interesting phenomena was observed. Fig.12 shows the all solutions projected on a two-dimensional plane between two objectives, the wing box weight and the cruise drag. As this figure shows that the non-dominated front was generated, there was a tradeoff and a non-dominated solution achieved significant improvement in both objectives. All solution projected on two-dimensional planes between drag divergence and the cruise drag was shown in Fig. 13. This figure showed that solutions comprise Parete fronto, however, Engineer’s design result was not included in all solutions. From careful investigations, three major possible factors for degrading non-dominated solutions were concerned. These are airfoil shape definition by PARSEC method, the geometry modification by the unstructured dynamic mesh method and inadequate definition of design variables domain.

All factors are related to the reasonable definition of the design space. We investigated on the improvement for these concerns. Investigations are described as follows.

A. Airfoil shape definition The adequacy of the design space of an optimization problem strongly affects whether we can get practically

enough optimized solution or not. In the wing optimization problem, the most dominant factor is the way of definition of the wing shape. Though the way of definition of the airfoil shape is the most basic technique, there is still no general way.

In the previous optimization problem, we applied the PARSEC method. If the PARSEC method cannot express the optimized airfoil shape, we cannot get the true optimized solution. This means the optimized solution is not included in the design space. In the previous problem, the high-speed drag of the solutions from the optimization was bigger than that of the airfoil which was designed by the conventional way based on the experience and the knowledge of the engineer using 3D CAD (this airfoil is called as "original airfoil"). It was assumed that the PARSEC method did not contain the true optimized solution in its design space.

Therefore, we investigated on the improvement of the airfoil shape definition. We tried the three ways of definition listed as follows, and evaluated the accuracy of each way.

1. PARSEC 2. PARSEC + least squares method 3. NURBS

CD@Cruise

CD

dive

rgen

ce

10cts.

Nondom_cNondom_b

Nondom_a

Baseline

Engineer's design

Optimum Direction

Figure 13. All solutions on two-dimensional plane

between cruise drag and drag divergence

Wing-box weight

CD

@cr

uise

5cts.

5% weight

Nondom_c

Nondom_b

Baseline

Nondom_a

Pareto Fronto

Optimum Direction

Figure 12. All solutions on two-dimensional plane

between cruise drag and wing-box weight


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For the above three definitions, parameter estimation for the minimizing the geometry difference between the original airfoil and the estimated airfoil was conducted using ARMOGA. The objective function is as follows;

∑=

−=n

iii YorgYcalObj

1)(

In these problems, the ARMOGA generates 8 individuals at each generation and continues until the generation reaches 400.

1. PARSEC

The 10 design variables used in this method is shown in Fig.1. The airfoil result is shown in Fig.14. From Fig.14, though the airfoil, which was estimated by this method, seemed well matched with the original

airfoil, there was small difference at the trailing edge (it is shown as the blue circle in the figure).

2. PARSEC + least squares method In this method, the airfoil is divided into two parts (thickness distribution + camber line). The former is expressed

by the PARSEC-5 (expressed by 5design variables) and the latter is expressed by the 5-th order least square method. The resultant shape is shown in Fig.15 and the Root-Mean-Square (R.M.S.) of the difference between the original airfoil and that of estimated by this method is shown in Fig.16.

In Fig.16, it is found that the error at the leading edge region (0%c-25%c) is the biggest of all. 3. NURBS

In this method, the airfoil is expressed by the 4-th order NURBS (Non-Uniform Rational B-Spline). NURBS is the way of expressing the complex curve developed in the CAD and its high flexibility makes this method general.

The NURBS function is shown as follows,

[ ]

( ) ( )( )

( ) ( ) ( )

+−−

+−

−=

≤≤≤≤

=

=

−++

+−

−+

+

+

−+

tNtt

tttNtt

tttN

ttttttt

tN

tttT

miimi

mimi

imi

imi

ii

iii

nm

1,11,1

,

1

11,

110

1

,01

Λ

By careful investigation, the number of the control points and its distribution were determined. The 26 design variables (control points) used in this method is shown in Fig.17.The resultant airfoil is shown in Fig.18. 4. Results

The pressure distributions under transonic Mach number and constant CL comparison of the original airfoil with airfoils estimated by each method is shown in Fig.19. The R.M.S. of the geometry differences among the original airfoil and all airfoils at each chord region and total values are shown in Fig.20 and Fig.21, respectively. It was found that there were large errors at the leading edge with the PARSEC and the PARSEC + least squares.

The PARSEC method uses the leading edge radius to express the shape around the leading edge, therefore the flexibility of the shape at this region is limited and this causes errors at the region. On the other hand, the NURBS can express the shape of the airfoil at whole region better than that of the PARSEC method. The R.M.S. of the geometry difference at the leading edge region is suppressed and this improves pressure distribution shown in Fig.19. The airfoil generated by the PARSEC cannot capture the suction peak at the leading edge, while the airfoil generated by the NURBS can express well. The R.M.S. difference of each method seems not large, however, pressure distribution in transonic regime is so sensitive that small geometry deviation causes large discrepancy.

Therefore, we adopted NURBS for the airfoil shape definition, though it leads the increase of the number of design variables and lost the designer friendly character, which the PARSEC variables have.


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Figure 17. Design variables (control points) used in NURBS

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-04

7.0E-04

0%c～25%c 25%c～75%c 75%c～100%cX/C [%]

Err R

MS

[/Cho

rd]

thicknesscamber

Figure 16. R.M.S. of geometry difference between original airfoil and the airfoil by

PARSEC + Least squares

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.50.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

X/C

Cp

OriginalPARSEC-10PARSEC + Least squaresNURBSCp*

Figure 19. Comparison of airfoil surface pressure distributions by each representation

(at transonic Mach number, constant CL)

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00X/C

Z/C

Thickness (Original)Camber (Original)Thickness (PARSEC-5)Camber (Leastsquares)

Figure 15. Airfoil representation result of PARSEC (thickness)+ Least squares (camber)

-0.10-0.08-0.06

-0.04-0.020.00

0.020.04

0.060.080.10

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

X/C

Z/C

OriginalNURBS

Figure 18. Airfoil representation result by NURBS

-0.10-0.08-0.06-0.04-0.02

0.000.020.040.060.080.10

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

X/C

Z/C

OriginalPARSEC-10

Figure 14. Airfoil representation result by PARSEC-10


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B. Geometry and computational mesh modification In the previous optimization problem, surface spline

function of the geometry deviation(ΔZ) was used for the modification of the wing shape (surface mesh) then, the volume mesh was modified by the unstructured dynamic mesh method. However, this process made the surface mesh distorted around the leading edge and highly limited the design space (Fig.22.). GAs does not necessary require the geometry sensitivity information, which is notably the merit compared with gradient based optimization methods which requires the geometry sensitivity and reversible shape modification process. Therefore, we developed the automatic and robust mesh generator to improve this problem. The flow chart of the mesh generation process is shown in the Fig.23 and Fig.24.

The automated mesh generation process is as follows; • Generate the wing geometry which satisfy the design

constraints. • Extract the intersection line between the wing and the

body. • Delete the wing geometry which is included in the

body and unite the wing and the body • Input mesh points distributions along created ridge

information • Generate unstructured surface mesh and volume mesh

automatically by the same methods adopted TAS-Mesh10

This tool is very robust and enables rich design space.

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

PARSEC-10 PARSEC + Leastsquares

NURBS

Err R

MS

[/Cho

rd]

0%c～25%c25%c～75%c75%c～100%c

Figure 20. Comparison of R.M.S. of geometry difference at each chord region among each

airfoil representation

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

PARSEC-10 PARSEC + Leastsquares

NURBS

Err R

MS

[/Cho

rd]

Figure 21. Comparison of R.M.S. of geometry difference among each airfoil representation

Figure 22. Example of distorted mesh around leading edge

Figure 23. Flowchart of the automated mesh

generation process


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C. Efficiency of optimization and definition of the design variables domain

The previous optimization problem uses the ARMOGA, which is one of the genetic algorithms as the optimizer and the viscous flow solver as the aerodynamics evaluation tool. GAs are very attractive to the engineering problems where discontinuities and multi-modalities may exist, because GAs do not utilize derivative information. Another merit of GAs is that they search the optimum point from a population of points, not single point. This feature is very promising to multi-objective problems. However, GAs require many objective function evaluations, which may be impractical when we rely on the high-fidelity analysis even with the current and near future supercomputer. One possible alternative is to construct a simple approximate model of the complicated CFD analysis solver. The approximate model expresses the relationship between the objective function (output) and the design variables (input) with simple equation. This model requires very little time to evaluate the objective function. It makes possible to save a lot of computation time and to explore more wide design space.

The most widely used approximation model is polynomial-based model16-17, due to its simplicity and ease of use. However, this model is not suitable for representing multi-modalities and non-linearity that often appear in the aerodynamic problem.

Recently, the Kriging18-19 model, developed in the field of spatial and geostatistics, has gained popularity in this field. This model predicts the value of the unknown point using stochastic processes. Sample points are interpolated

with the Gaussian random function to estimate the trend of the stochastic processes. The model has a sufficient flexibility to represent the nonlinear and multimodal functions at the expense of computation time. However, the computation time to construct the Kriging model is still short compared to that of the direct evaluations. Another advantage of using the approximation methods is that it shows the relation between the objective function and the design variables. By applying the functional analysis of variance20 (ANOVA) to the approximation model, one can not only identify which design variables are important for the objective and the constraint functions quantitatively, but also change the design space adequately. This feature is very attractive to the large-scale complex engineering problems where it is difficult for human to define the domain of all design variables adequately beforehand. Moreover, based on the results of the ANOVA, design variables that do not have a significant influence to the objective function can be eliminated effectively21.

Therefore, we are now evaluating to adopt the Kriging-based Multi-Objective Genetic Algorithm (MOGA) as the alternative optimizer, and the ANOVA as a design space monitor. This method can drastically reduce the computational time required for the objective function evaluation and can change the domain of design variables automatically. The CFD analysis in the objective function is replaced with the Kriging model. However, it is possible to miss the global optimum in the searching space if we rely solely on the prediction value of the Kriging model, because the model includes uncertainty at the prediction point. For the robust exploration of the global optimum point, both the prediction value and its uncertainty should be considered at the same time. This concept is expressed in the criterion ‘expected improvement (EI)22’. EI indicates the probability of a point being optimum in the design space. By selecting the maximum EI point as the additional sample point, the improvement of the model and the robust exploration of the global optimum can be achieved at the same time. The Kriging model is constructed for each objective and constraint function. In the Kriging model for the objective function, the expected improvement is calculated, and in the Kriging model for the constraint, the probability of satisfying the constraint is calculated. Based on these values, an additional sample point for the balanced local and global search is selected using MOGA optimizer.

Figure 24. Automated wing-body mesh


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The present method is applied to a transonic airfoil design as the process validation. In the transonic region, the flow-field is drastically changed even with the small fluctuation. For the robust transonic airfoil design, the objective functions are defined as follows:

Minimize obj1: Cd@Cl=0.75 (Mach=0.70) obj2: Cd@Cl=0.67(Mach=0.74) subject to t/c > 11%

First objective function is for the design condition and the other is for the off-design condition. The design constraint is the maximum thickness of the airfoil. Geometry of airfoil is defined by using NURBS14 to represent free-form shape. 26 design variables are used to represent the airfoil as described previous section. The initial search region of each design variable is determined to avoid unrealistic airfoil geometry such as a fishtailed airfoil.

Sample points to fit the Kriging model are selected by using the Latin Hypercube Design (LHS)22 with the constraint evaluation. Once LHS selects a point (airfoil), the point is checked whether it satisfies the design constraint or not. If the point satisfies the constraint, the point is selected as sample point, if not, the point is rejected. Total 26 sample points are selected with 50 divisions in LHS. Design variable distribution of 26 sample points are shown in Fig. 25. Sample points are uniformly spread in the search region of all design variables except dv7. This means that airfoil with small value of dv7 doesn’t satisfy design constraint. In case of LHS without the constraints evaluation, only 13 points satisfy the design constraints with same divisions. It means that 37 sample points are selected from the infeasible design space. By using LHS coupled with the constraint evaluation, it is possible to select the sample points only in the feasible design space effectively. Aerodynamic drag of 26 sample airfoils are evaluated by using a Navier-Stokes evaluation. With the data obtained from the Navier-Stokes analysis, the Kriging parameter is determined by solving maximization problem. On the Kriging models, EIs of two objective functions are maximized by using MOGA to find the non-dominated solutions. In the present MOGA, number of populations and generations are 512 and 100, respectively. Fig. 26 shows the non-dominated solutions about EI after the first efficient global optimization(EGO) loop. EIs of objective functions are in the tread-off relation. Among these non-dominated solutions, three points are selected for the update of the Kriging models: i)the point whose EI of obj1 is the largest, ii)the point whose EI of obj2 is the largest, iii) mid point in the non-dominated solutions. Fig. 27 shows the non-dominated solutions obtained after the several EGO loops. The EIs are gradually decreased with EGO loops. It means that the accuracy near the Pareto front is improved by adding the additional sample points. Fig. 28 shows the initial sample points and the sample points selected by using EGO. Consequently, total 89 points are evaluated in this design process. The sample points selected by EGO show better characteristics than the initial sample points. It means that EGO algorithm selected the sample points for update of the Kriging models correctly.

Fig. 29 shows the geometry and the pressure distributions of airfoil, which shows the best performances about both objective functions among 89 sample points. At both conditions, the designed airfoil shows the low drag characteristics.

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Figure 25. Distribution of initial sample points in the

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Figure 26. Non-dominated solutions about EI


14

This Kriging-based MOGA requires still many objective function evaluations with costly CFD solver. Therefore, we are trying the sequential evaluation process, where the initial Kriging model is constructed based on the aerodynamic evaluations using full-potential CFD solver as a low-fidelity model and the subsequent reconstructing of the Kriging model is conducted based on the evaluations using Euler/RANS CFD solver as a high-fidelity model, to reduce the computational time and not to sacrifice the quality of solutions simultaneously. Overall process of this approach is shown in Fig.30.

1. Initial Kriging models are constructed for all objective functions based on full-potential aerodynamic

evaluations (low-fidelity model) with N sample points uniformly spread in the design space. 2. MOGA operations

- Generation of initial population and evaluation of expected improvement (EI) - Selection parents - Crossover and mutation - Evaluation of new individuals in Kriging models - When the generation exceeds 100, the points which gives maximum EI is selected as an additional

sample points (Nadd)

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Figure 27. Non-domiated solutions of EIs

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Figure 28 Initial sample points and sample points

selected by EGO

Figure 29. Geometry and pressure distributions of

designed airfoil


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Wing-box weight

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5cnts

Full-potential base evaluation

Euler base evaluation

Sample B

Sample C

Sample A

Engineer's design

Optimum Direction

Figure 31. Additional sample points on two-dimensional plane between cruise drag and wing-

box weight

3. Reconstruction of Kriging models - Evaluation of additional sample points with full-potential CFD solver and Euler/RANS CFD solver

(high-fidelity model) - Collections in the objective function values if necessary - Construction of new Kriging models with N=N+Nadd sample points based on collected objective

function values 4. Go back to the No.2 process This routine is iterated until the termination criterion is reached. In our current study, termination criterion is the

maximum number of additional sample points. Fig.31 shows the additional sample points, projected on a two-dimensional plane between the wing box weight and the cruise drag, which are obtained by our current optimization similar to the previous optimization study. This optimization does not converged yet. However, as shown in Fig.31, the full-potential base evaluations give good agreement with the Euler base evaluations quantitatively, and there are significant improvements in the cruise drag and the wing box weight over the conventional engineer’s knowledge based design (engineer’s design). This result indicates that the present sequential evaluation process from low-fidelity model to high-fidelity model can reduce computational time of a practical aircraft MDO problem drastically without decrease of solution accuracy.

We believe that this approach will make it possible to overcome practical concerns of the prototype MDO system and to apply it to the practical aircraft design.

Selection of Initial Sample points uniformly spread in the design space

Construction of objective functions Kriging Model with initial sample points

Generation of Initial population And Evaluate the fitness of each

individual using the Kriging model

Selection of ParentsNGEN>100NGEN>100

Crossover and Mutation

Evaluate the fitness of new individuals

Evaluate the objective function values of

Additional sample points

N=N+Nadd:

Sample points

Evaluate the objective function values of

Additional sample points

Construction of objective functions Kriging Model

with N sample points

Collections in the objective function values if necessary

Yes No

:Operation in Kriging model

:Operation with Full-potential CFD code

:Operation with Euler/RANS CFD code

Figure 30. Sequential evaluation process using Kriging based MOGA optimizer


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V. Detail aerodynamic optimization modules and the whole design process

A. 2D High lift device optimization module For a successful design of an aircraft, the development of efficient high-lift devices is one of the important

requirements. Performance of high-lift devices has a strong and direct impact on the operating cost of the aircraft. From the point of view of jet engine noise, low drag at take-off and landing is also desirable.

We developed the 2-dimensional high lift device optimization module using a 2-dimensional unstructured Navier-Stokes code and its adjoint code23, 24. Adjoint method is independent of the number of the design variables and the adjoint code is developed by a discrete approach so that the Navier-Stokes flow solver and the adjoint code are exactly consistent with each other. Therefore, this module is very efficient and accurate. The developed design tool is applied to a high-lift device design. The flowchart of the optimization procedure is shown in Fig.32. As a design example, 3-element high-lift device configuration as shown in Fig.33 was optimized. The design objective is to maximize lift for a landing condition at angle-of-attack near Clmax. Design variables are totally 26, rigging parameter (gap, overlap, deflection) of slat and flap, airfoil geometry by Hicks-Henne shape function. 7% lift improvement was obtained as shown in Fig.34. Successful design results confirm validity and efficiency of the present design method. We extend the module to be able to evaluate at multi-design points to prevent peaky characteristics. Furthermore, we try to couple with the kinematics of high lift device so as to enable the optimization for takeoff and landing configuration at once.

B. Engine-Aircraft aerodynamic integration module Engine-Aircraft aerodynamic integration design is one of the most important items in aircraft design. The design

has many design parameters and constraints to be accounted for25, 26. Furthermore, due to the complexity of the geometry, it takes much time for the designers to modify the geometry for evaluation. To achieve airplane performance objectives within the tight program schedule, an efficient engine-aircraft aerodynamic integration tool has been developed adopting the surface mesh movement method for wing-pylon installation junctions27. Three-dimensional unstructured Euler solver is used for the flow analysis. The sensitivity analysis is made by a discrete adjoint method. Surface mesh points on installation junctions are moved directly on the curved body surface by using a spring analogy based on the geodesic distance27. This module enables us to cut the time-consuming geometry modification task using 3D CAD and enhances overall design efficiency and accuracy. As a design example, DLR-F6 wing-body-nacelle-pylon configuration as shown in Fig.35 was optimized. The design objective is to minimize drag and eliminate strong shock wave around the pylon to reduce buffet risk at climb condition (low CL). Design variables are inboard wing shape, pylon shape deformation by Hicks-Henne shape function, nacelle vertical movement and nacelle pitch angle. The total number of design variables is 82. Optimization results are shown in Fig.36 and Fig.37. Strong shock wave near inboard of the pylon, which leads to buffet risk, greatly suppressed and drag reduction due to the shock suppression was achieved by the optimization. Successful design results confirm validity and efficiency of the present design method.

C. The whole aero-structural aircraft design process We have developed wing-body aero-structural MDO system, 2D-high lift device optimization module and

engine-aircraft aerodynamic integration module as described above. Integrating the system with these modules, we developed the innovative aero-structural multi-disciplinary design process under realistic constraints such as, manufacturability, etc. Schematic chart of conventional and new innovative design process is shown in Fig.38. In new design process, 2D airfoil design can be omitted and the process greatly reduces design iterations between aerodynamics and structural design. Further more, the present wing-body MDO system will be extended to wing-body nacelle MDO system to enable simultaneous aero-structural design and preliminary engine-aircraft aero integration. Detail integration design is conducted engine-aircraft integration module. The system enables us fast and high quality aircraft design.


17

Figure 32. Flowchart of high lift device optimization

Figure34. Lift maximization optimization result

Figure 33. Unstructured mesh of a 3-element

Figure 35. Unstructured surface mesh of DLR-F6

configuration

-2.5

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x/c

Cp

initialdesign

Figure 37. Optimization results of DLR F6 dragminimization problem under transonic climbcondition ( inboard wing (η=0.3) airfoil and

Figure 36. Optimization results of DLR F6 drag minimization problem under transonic climb

condition (surface Mach number)


18

Conclusions The development and application of an MDO system was presented in this paper. In this approach, high-fidelity

aerodynamic and structural analysis code were linked to the MDO and applied to optimization of a commercial jet aircraft wing design. As a result, several non-dominated solutions were obtained and one solution was found to have one percent improvement in the block fuel.

However, through the careful investigations, three major possible factors for degrading non-dominated solutions were concerned. These are airfoil shape definition by PARSEC method, the geometry modification by the unstructured dynamic mesh method and inadequate definition of design variables domain. All factors are related to the reasonable definition of the design space. We investigated on the improvement for these concerns.

As a consequently, we adopted NURBS airfoil representation and developed automated geometry and computational mesh modification tool for first two concerns. For the last concern, we are now evaluating Kriging-based MOGA as the alternative optimizer and the ANOVA as a design space monitor This method can drastically reduce the computational time required for the objective function evaluation and can change the domain of design variables automatically. Furthermore, we are trying to develop variable fidelity sequential optimization process to reduce the computational time and not to sacrifice the quality of solutions simultaneously. Preliminary optimization study showed promising results for such approach.

On the other hand, we have been developing the detail aerodynamics optimization modules simultaneously. These are engine-airframe integration and 2D high lift device design using unstructured adjoint method.

Integrating the system with these modules, we developed the innovative aero-structural multi-disciplinary design process. In new design process, 2D airfoil design can be omitted and the process greatly reduces design iterations between aerodynamics and structural design. Further more, the present wing-body MDO system will be extended to wing-body nacelle MDO system to enable simultaneous aero-structural design and preliminary engine-aircraft aero integration. The system enables us fast and high quality aircraft design.

Acknowledgments The authors would like to express special thanks to Dr. Chiba of JAXA, Dr. Jeong, Dr. Kim and Mr. Salim of

Tohoku University and Mr. Morino and Mr. Hatanaka of MHI for their sincere cooperation to the present work. Also, we would like to express special thanks to Dr. Ito of the University of Alabama at Birmingham for his cooperation to the automatic mesh generation tool development.

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Conventional design process New design process utilize MDO system

Figure. 38 Schematic chart of the whole aircraft aero-structural design process


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24Kim, H. J., and Nakahashi, K., “Discrete Adjoint Method for Unstructured Navier-Stokes Solver,” AIAA 2005-0449, 2005. 25Dennis L. Berry, “The Boeing 777 Engine/Aircraft Integration Aerodynamic Design Process”, ICAS-94-6.4, 1994 26Paul E. Rubbert, Edward N. Tinoco, “The Impact of Computational Aerodynamics on Aircraft Design”, AIAA-83-2060,

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