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9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization 4-6 September 2002, Atlanta, Georgia

AIAA 2002-5405

INTEGRATED WING DESIGN WITH THREE DISCIPLINES1

G. Shi, G. Renaud and X. Yang Structures, Materials and Propulsion Laboratory Institute for Aerospace Research National Research Council Canada Ottawa, Canada K1A 0R6 F. Zhang and S. Chen Aerodynamics Laboratory Institute for Aerospace Research National Research Council Canada Ottawa, Canada K1A 0R6

ABSTRACT A wing design process, coupling the three disciplines of structures, aerodynamics and aeroelasticity, was successfully performed. This paper presents the multidisciplinary design procedure with emphasis on structural modeling and optimization. The linking and coupling techniques for the multidisciplinary integration is summarized and the methodology for automatic conversion of the aerodynamic wing shape and pressure distribution to the structural finite element model is presented. Furthermore, the automatic model generation used for structural optimization in the integrated design loop is described. The results obtained show that the developed techniques work well and that more complicated MDO operations can be undertaken. INTRODUCTION

linkage of distributed design teams are two important issues that must be addressed in MDO. The Institute for Aerospace Research of the National Research Council of Canada launched a project for developing MDO strategies for aerospace systems integrating structures, aerodynamics and aeroelasticity. At this stage, the research activities focused on the development of linking and coupling techniques. This paper presents the initial results of a preliminary wing design by integration of the three disciplines. In order to provide more information on this research work, this paper focuses on the structural aspects and coupling techniques. Another paper provides more details on aerodynamic modeling and optimazition1. MULTIDISCIPLINARY DESIGN CYCLE

Multidisciplinary Design and Optimization (MDO) has gained wide acceptance in the aerospace industry. The increasing interest in this methodology is due to the complexity of aerospace systems, which requires efficient coordination of various disciplinary analysis capabilities and effective communication among potentially geographically separated teams. The design departments in aerospace industry are often strongly segregated by disciplines, such as structures, aerodynamics and aeroelasticity. Each department is only responsible for specific aspects of the engineering work required for designing an aircraft. In each discipline, specific discipline-driven design techniques are developed and used, for instance, FEM for structural analysis and CFD for aerodynamic analysis. In order to develop an integrated aircraft design approach, the coupling of multiple disciplines and the

The objective of the multidisciplinary design process presented in this paper was to find the lightest wing box design for certain flight conditions that satisfies predefined geometry, stress, flutter, and displacement requirements. The loads acting on the wing box were determined from CFD analyses and depend on the structural weight. Figure 1 illustrates the M6 swept wing geometry and flight parameters used in the design process. The integrated design process included three coupled disciplines. First, the aerodynamics discipline evaluated the pressure distribution on the surface of the wing. Second, the structures discipline calculated the wing box stresses and deformations resulting from the air pressure. Third, the aeroelasticity discipline

Copyright 2002 by NRC. Published by AIAA with permission.

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Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

predicted the flutter speed with data from the other two disciplines.

Step 1: Structural and aeroelastic analysis and optimization A finite element model of the wing box was created after each aerodynamics analysis to include the changes in pressure loads acting on the structural members. Furthermore, the wing shape changes were taken into account when aerodynamics optimization was present. The wing box structural displacements and stresses caused by the aerodynamics loads were calculated through a finite element procedure. This analysis was coupled to an optimization algorithm that interpreted the structural response and modified the wing box design to minimize the weight under predetermined constraints. This procedure was repeated until the lightest structure with acceptable stress and deformation under the aerodynamics loads was determined. Aeroelasticity analyses were performed in parallel to the static structural analysis. No optimization work was involved in this discipline and only flutter speed calculations were carried out. The flutter speed was used as a constraint in the structural optimization. The objective of this constraint was to ensure that the flight speed did not reach the critical flutter speed. Step 2: Aerodynamics analysis/optimization Assuming that the wing shape was fixed (Case 1), the aerodynamics analysis was limited to determining the air pressure needed to maintain flight. To accomplish this, the angle of attack was adjusted at each aerodynamic step to ensure that the total lift was equal to the aircraft weight. In the present case, the aerodynamic analysis was conducted using the CFD solver KTRAN2. In case 2, a drag minimization was carried out for the ONERA M6 isolated wing, represented by seven sections in the spanwise direction. For simplicity, the original M6 wing planform was not altered and the optimization procedure only modified the section shapes. The free stream Mach number was set to 0.84 and the lift was kept equal to the weight of the airplane. The angle of attack was allowed to vary during the course of the optimization process. To obtain a realistic wing, geometry constraints forced the maximum wing thickness to be greater than or equal to 8% of its corresponding chord and the trailing edge angle to lie between 5 and 20 degrees, at each section. More details on aerodynamic analysis and optimization can be found in the reference1.

317.3 in

Aspect ratio: A = 3.8 Taper ratio: = 0.56 O Sweep angle: 25% = 26.7 Mach number: M = 0.84 Wing geo.: M6 swept wing Max thick. at root: 31.5 in

133.3 in

471.0 in

Figure 1. Wing characteristics. Two formulations were implemented for the integrated design process. The first case involved structural optimization with aerodynamics and aeroelasticity analysis. The second case involved both structural and aerodynamics optimization, with aeroelasticity analysis. Both cases included an initial step for determination of the initial weight and pressure distribution corresponding to the initial design. The actual process of system optimization comprised three steps: 1) Minimization of the wing box weight, under stress, deformation, and flutter constraints. Calculation of wing drag. If aerodynamics optimization is performed, the wing shape is modified to minimize the drag. Evaluation of process convergence.

2)

3)

These three steps were repeated until the process convergence met a certain tolerance criteria. The multidisciplinary design process is illustrated in Figure 2 and described in the next paragraphs.

Structural Optimization Aerodynamics Analysis/ Optimization

Pressure Geomerty

Structural & Aeroelasticity Analysis (Figure 5) Optimization

Converged? No Yes

No Yes Converged?

Weight

Figure 2. Multidisciplinary design loop. 2

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178.7 in

Step 3: Convergence evaluation. The total weight of the aircraft was used for convergence purposes. This weight value was calculated using the optimized wing box obtained in step 1. For the results presented in this paper, a relative weight change smaller than 0.01% was required between two subsequent iterations as the criterion of termination. WING BOX MODEL A medium-complexity wing box model was assumed, composed of two skins, three spars, eleven ribs, and four stringers. A schematic of the wing box, without its upper skin, is shown in Figure 3.

tensile, compressive, respectively.

and

shear

yield

stresses,

AUTOMATIC MODEL GENERATION (COUPLING TECHNIQUES) One of the most challenging aspects of multidisciplinary wing design and optimization is the sharing of disciplinary responses between the various analysis codes3. In the wing box case, all structural loads came from aerodynamic analysis. The quality of data transfer therefore directly affected the quality of the system analysis, and hence the final optimal design. The major difficulty came from the fact that the FE and CFD grids were different. The exchange of information from aerodynamics to structures was done during the finite element model generation. The data transmission was performed using ASCII files containing the linking information. Two types of coupling were considered in the present study. The first one concerned the outer wing geometry. It was assumed that the wing shape was controlled by aerodynamic performance and that it could be modified at any time outside the structures discipline operations. The structural geometric and finite element models therefore had to be updateable at the initialization phase of the structural optimization step. The second type of coupling concerned the aerodynamics pressure loads calculated by CFD that had to be converted to finite element nodal forces. Conceptually, the aerodynamics-to-structure coupling involved the first and fourth steps of the fivestep automatic model generation procedure illustrated in Figure 4. The first three steps were optional if no aerodynamics optimization was performed. In that case, since the geometry was assumed fixed, the geometry and finite element model could be saved and re-loaded when needed. However, since the applied loads changed after each CFD analysis, step 4) needed to be performed before the start of each structural optimization. It should be noted that all finite element pre-processing was accomplished using PATRAN4. 1) Outer wing geometry import The aerodynamics-discipline wing shape was stored in a predefined data file as a collection of points located in three dimensions. For the problem presented in this paper, 11 equally spaced sections defined by 204 points were used. A function written in PATRAN Command Language (PCL) read this file and imported the points into the PATRAN database as 3

Figure 3. Wing box model (upper skin not shown). The structural analysis model consisted of 309 elements. Different element types and materials were used for the various structural components. The characteristics of the finite elements and materials composing the wing box are listed in Tables 1 and 2. Component Element type Shell Shell Shell Shell Beam Beam Material Number of elements 40 40 44 30 55 100

Upper skin Lower skin Rib webs Spar webs Rib posts Spar beams, Stringers

Aluminum Aluminum Aluminum Aluminum Steel Steel

Table 1. FE properties of the wing box model. Material Aluminum Steel E (Msi) 10.5 30.0 0.3 0.3 y,t (Ksi) 43 75 y,c (Ksi) 39 60 y,s (Ksi) 23 -

Table 2. Material properties. The five columns of Table 2 correspond to the Young's modulus, the Poisson's ratio and to the

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geometric points. Then, very accurate curves and surfaces were generated based on the points created. Finally, all points were removed from the database since they were no longer necessary. 2) Inner wing geometry generation A PCL function was developed to automatically generate the inner geometry of the wing based on the outer geometry created in step 1). It was assumed that the wing box geometry changed if the wing shape changed. For modeling purposes it was assumed that all wing box structural components were in contact with the outer shape. For the case presented in this paper, the three spars were assumed to be located at 15%, 40%, and 65% of the wing chord length from the leading edge. However, the location of the spars and ribs could be changeable and treated as variables in a shape optimization context.

3) Finite element model generation If the geometry of the wing changed, the finite element nodal points had to be relocated. A function written in PCL automatically created the mesh and assigned materials, element properties, and boundary conditions to the various parts of the model. The number of element properties was chosen according to the desired optimization complexity. The optimization process acted mainly on element properties and several elements could therefore be grouped and given same properties to limit the number of design variables. 4) Pressure loading import and application The application of the aerodynamic loads on the structural model was the most important step of the coupling procedure. The developed technique aimed at reducing the number of steps that were performed outside the pre-processing capabilities of PATRAN. The load distribution was stored in a tabular form. For simplification, the same data points were used for the geometry and pressure data transfer. Accordingly, only one file needs to be transferred from aerodynamics to structures. The aerodynamic loads were kept as a distributed pressure load, as calculated by CFD, and were applied directly to the geometry mode. To do this, load fields were created using a developed PCL function that extracted the appropriate information from the datatransfer file exported by the aerodynamics discipline. These fields were then applied to the corresponding geometric surfaces as normal loads. This procedure had two advantages: i) The load distribution was automatically distributed among the nodal points as nodal forces, independent of the mesh. ii) The nodal forces were automatically oriented normal to the outer wing surface.

1) Outer wing geometry import

2) Inner wing geometry generation

3) FE model generation

4) Pressure loading import and application

5) Analysis setting

5) Analysis setting The last step was to initialize the analysis parameters and create the analysis input file. This step was performed automatically using a developed PCL function. A finite element solver, NASTRAN5 in the current case, was used to solve the updated structural problem.

Figure 4. Automatic model generation.

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STRUCTURAL OPTIMIZATION PROCESS 1) Initialization The structural optimization process encompassed several operations and used several software products. In the present case, PATRAN was used for pre- and post-processing purposes, NASTRAN was used for analysis purposes, and MATLAB Optimization Toolbox6 was used for optimization. Further, a program written in MATLAB7 was used as the framework controlling the various operations and exchange of data. A flow-chart of the process is presented in Figure 5. As seen in this figure, the structural optimization could be divided into three types of operations: 1) initialization, 2) analysis, and 3) optimization. Everything was done automatically and could be executed as a background process. The three types of operations are discussed below. A new analysis deck (NASTRAN input file) was generated after each aerodynamic analysis. Because the examples in this paper involved structural sizing only, this step needed to be performed only once for each structural optimization. As mentioned before, it was in this initialization stage that the information from aerodynamics was transferred to structures. The analysis deck creation is fully automatic. A methodology using PCL functions was developed to allow MATLAB to control the initialization phase, which could be performed in the background without a graphical interface. With this methodology, any part of the PATRAN database could be changed (shape, mesh, loads, materials, etc.) if needed. 2) Analysis

Geometry

Initial or previous design FE database generation (Figure 4)

Pressure distribution

The second type of operations corresponded to the structural and flutter analysis of the design. In the present stage, stresses, displacements, flutter speed and the weight were extracted from the analysis. First, a static finite element analysis, performed by NASTRAN, evaluated the stress and displacement distributions. Second, a sequence of PATRAN operations written in PCL calculated the weight of the structure. This was done in the same way as the database was created in the initialization phase. Finally, the flutter speed was determined using NASTRAN and MATLAB. For the current case, the structural input of the flutter analysis was the wing box model used for static optimization, without the pressure loading. The aerodynamic loading in the flutter analysis was computed using Doublet-Lattice theory, as implemented in NASTRAN, and a surface spline was used to transfer this loading to the wing box. All of the nodes of a flat plane lying inside the box were used for the air­structure interaction. Ten modes were calculated to ensure that all useful modes for flutter were obtained. The PK method was used for flutter calculations since it provided an easily definable flutter condition. During each flutter analysis, attention was paid to local and horizontal modes, which were not relevant to flutter. Local modes were avoided by choosing appropriate element sizes. Horizontal modes were discarded from the NASTRTAN output by examining damping curves.

1 3 FE input file 2 Weight calculation Weight (Objective) Structural analysis Stress & displacements (Constraint) Flutter analysis Flutter speed (Constraint)

Yes Weight Converged? No Design update (Optimization))

Figure 5. Structural optimization process flow-chart.

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3) Optimization An optimization algorithm analyzed the output values obtained by NASTRAN and PATRAN and updated the wing box design for the next analysis step. This loop was performed until the optimal design was obtained. In the present case, the Sequentially Quadratic Programming (SQP) gradient-based algorithm of the MATLAB Optimization Toolbox was used. The optimization statement is defined in the next section. The design modifications were done in the analysis deck only. This avoided the regeneration of the database in each iteration. However, if shape design variables, such as wing taper ratio or swept angle, were included in the structural optimization formulation, full database updates were necessary. STRUCTURAL OPTIMIZATION STATEMENT The objective of each structural optimization was to obtain the lightest feasible design, for a specific aerodynamic load case, taking into consideration stress and displacement requirements. A description of the objective, design variables, and constraints used in the wing box problem is given in the following section. 1) Objective The objective of the structural optimization was to minimize the "half-total weight" WT of the airplane. This value corresponded to the weight that was supported by one wing and was calculated as WT = W0 + WW where W0 corresponds to half of the basic airplane weight and WW to the wing weight. These parameters were assumed to take the following values: W0 = 100000 lb and WW = WWB * 1.3 where WWB is the wing box weight calculated by PATRAN. The factor 1.3 corresponds to the structural overhead8. 2) Design variables

In addition to a set of fixed parameters, the wing box design was fully represented by a vector of design variables bounded by maximum and minimum allowable values. The ith component of this vector, and the corresponding lower and upper bounds, are written as xi, ximax, and ximin, respectively. The optimization process was therefore achieved through a search in a design space spanned by these design variables. For simplification, only seven variables were used to perform the structural optimization of the wing box at this stage. These variables and their bounds are listed in Table 3. More variables will be included in the optimization model when the coupled design process becomes more mature. 3) Constraints In addition to the bounds on the variables, other requirements had to be met to ensure design feasibility. Constraints on the stresses, displacements, and flutter speed were considered in the present example. Component 1- Upper skin 2- Lower skin 3- Rib webs 4- Spar webs 5- Posts 6- Spar beams 7- Spar beams Variable thickness thickness thickness thickness area width height Min 0.04 in 0.04 in 0.04 in 0.04 in 0.10 in2 0.10 in 0.10 in Max 0.80 in 0.80 in 1.00 in 1.00 in 0.20 in2 1.00 in 1.00 in

Table 3. Design variable definitions. The loading for this problem was multiplied by a load factor equal to 3, to consider the peak loads encountered during various flight maneuvers or caused by turbulent air. The chosen load factor is typical for cargo and passengers aircraft9. It was assumed that the allowable stress, which the structural components could sustain, was equal to 1.5 times the yield stress of their constitutive materials. Furthermore, the constraints were formulated differently for each type of components. For the skins, the major and minor principal stresses located at the corner of each element were used. Also, both surfaces of each element were considered. Therefore, 16 stress values per elements were compared to the maximum and minimum allowable stresses, resulting in 1280 values, or 2560 constraints.

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For the posts, modeled as rods, the axial stress of each element was compared to the allowable tension and compression stresses, resulting in 55 values, or 110 constraints. For the spar and rib webs, modeled as shells, the maximum shear stress in each element was compared to the maximum allowable shear value, resulting in 74 values, or 148 constraints. For the spar beams and stringers, modeled as bars, the maximum and minimum stresses at both ends were used, resulting in 400 values, or 800 constraints. Another constraint specified that the maximum deflection encountered in the model must be less than 5% of the wingspan. This resulted in 110 additional constraints. Finally, the last constraint stipulated that the flutter speed must be higher than the flight speed. There are a total of 3618 stress constraints, 110 displacement constraints, and one flutter speed constraint in the optimization model. Each constraint, written as g, was normalized with respect to its allowable values as g = value ­ allowable value . allowable value With this formulation, all the constraints were scaled to the same order of magnitude. 4) Statement The optimization problem can be stated as minimize WT(x) subject to gj g k f 0 ; j = 1, .., 3618 0 ; k = 1, .., 110 g 0 ximin xi ximax ; i = 1, .., 7

Case 2: structural and aerodynamic optimization with aeroelastic analysis. Furthermore, two different ways of treating the air pressure were applied in Case 2. The first one (Case 2-1) did not constrain the pressure distribution. The second one (Case 2-2) used modifications in the mutation and crossover processes of the GA to screen the unreasonable pressure seeds1. The processes for cases 1, 2-1 and 2-2 were completed in about 1, 93 and 30 hours, respectively. The initial and final wing box designs for these cases are listed in Table 4. The variable numbers are defined in Table 3. x 1234567WW: Initial 0.40 in 0.40 in 0.50 in 0.50 in 0.10 in2 0.50 in 0.50 in 12315 lb Final 1 0.33 in 0.33 in 0.04 in 0.10 in 0.10 in2 0.10 in 0.10 in 5678 lb Final 2-1 0.05 in 0.05 in 0.04 in 0.04 in 0.10 in2 0.10 in 0.10 in 1149 lb Final 2-2 0.36 in 0.36 in 0.04 in 0.11 in 0.10 in2 0.10 in 0.10 in 6253 lb

Table 4. Initial and final designs. The wing weight was reduced by a factor of 53% for Case 1, 91% for Case 2-1 and 51% for Case 2-2. The complete multidisciplinary process required four loops for Case 1, ten loops for Case 2-1 and three loops for Case 2-2, each containing an aerodynamics analysis/optimization and a structural optimization. The weight and drag coefficient iteration histories are illustrated in Figure 6 and Figure 7, respectively.

20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 Case 1 - Wing weight (lb) Case 2-1 - Wing weight (lb) Case 2-2 - Wing weight (lb)

where the definitions of the objective WT, stress, displacement and flutter speed constraints gj, gk and gf, and design variables x are given in the previous paragraphs. RESULTS The multidisciplinary design of the wing was performed on a SGI Origin 2000 computer. The two cases described above were implemented. They are: Case 1: structural optimization with aerodynamic and aeroelastic analysis; 7

0 0 1 2 3 4 5 6 7 8 9 10

Iteration number

Figure 6. Weight change history.

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0.014 0.0135 0.013 0.0125 Case 1 - Drag coefficient Case 2-1 - Drag coefficient Case 2-2 - Drag coefficient

sufficient to distribute the mass and stiffness in a manner that challenged the flutter constraint.

Case 1 & case 2 original Case 2-1 final Case 2-2 final

20 10

y (in)

0.012 0.0115 0.011 0.0105 0.01 0 1 2 3 4 5 6 7 8 9 10

0 100 200 300

-10 0 -20

x (in)

Figure 8. Initial and final wing cross-sections.

Iteration number

Figure 7. Drag coefficient change history. It can be seen that the three cases started the MDO process in the same way. First, the initial wing weight WW = 12315 lb was found. This weight was then used to solve the aerodynamics problem, resulting in a drag coefficient of 0.013758. Finally, the first structural optimization reduced the weight of the wing box subject to the initial aerodynamics loads to 6231 lb. The following aerodynamics step was different for all cases. For Case 1, the aerodynamic loads were sufficiently changed after the first two loops to permit noticeable wing box weight reduction. This result outlined the importance of multidisciplinary analyses, as opposed to independent structural optimization. Furthermore, it is seen that the MDO process resulted in drag reduction.

The optimal wing cross-section, as determined by the aerodynamics processes, is illustrated in Figure 8. The Case 1 fixed geometry, which is also the initial geometry for case 2-1 and 2-2, is also shown. Finally, the resulting deformed wing box with associated von Mises stress distribution for the optimal wing box design of Case 1 is presented in Figure 9.

MSC.Patran 2000 r2 26-Mar-02 13:21:51 1.12+05 1.05+05 9.75+04 9.00+04 8.25+04 7.50+04 6.75+04 6.00+04 1.12+05 0. 5.25+04 4.50+04 3.75+04 3.00+04 2.25+04 1.50+04 7.50+03 Y -1.46-02 default_Fringe : Max 1.12+05 @Nd 59 Min 0. @Nd 1 default_Deformation : Max 1.86+01 @Nd 175

Case 2, including aerodynamic wing shape optimization, generated quite different convergent histories. In Case 2-1 both weight and drag were reduced significantly more than in Case 1. However, the final air pressure distribution was not reasonable since it was not controlled in the optimization process. In Case 2-2, both weight and drag converged rapidly and the air pressure distribution was improved. However, the pressure constraint was too restrictive since the weight and drag improvements were not as good as in Case 1. More detailed explanations can be seen in the paper1. For the three cases, the flutter speed was found to be significantly higher than the flight speed. One reason is that the leading edge and the trailing edge were not considered in the wing box model. Another reason is that wing box design flexibility was not

Z

X

Figure 9. Wing box deformed shape and stress contours (Case 1). CONCLUDING REMARKS A three-discipline coupled design process was established and successfully used in a simplified wing design. Several advanced computational techniques were developed to ease the data transfer from aerodynamics to structures and to automate the operations. Furthermore, the developed coupling techniques were effective and easily adaptable. By maximizing the number of PATRAN internal operations, these techniques minimized the risk of errors. Similarly, the automatic model generation was a powerful tool allowing sizing and shape optimization

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with little initial time investment in a representative model development. The presented test cases showed that the multidisciplinary coupling process worked. Furthermore, the analysis and optimization models were simple enough to develop the desired methodology without involving high computational load. In the future, the flexibility and complexity of models in each discipline will be increased to analyze realistic wing models. Furthermore, a stronger coupling will be developed by including the effects of structural deformations in the aerodynamic models. Finally, several decomposition MDO formulations will be implemented on the wing design process. ACKNOWLEDGEMENT 7. This work was carried out under the IAR projects Development of MDO strategies for Aerospace system and Computational Structures Technology development. The financial assistance received from the IAR New Initiative Funding program and the Department of National Defense of Canada is gratefully acknowledged. REFERENCES 1. Chen, S., Zhang, F., Renaud, G., Shi, G. and Yang, X., "A Preliminary Study of Wing

2.

3.

4. 5.

6.

8.

9.

Aerodynamic, Structural and Aeroelastic Design and Optimization", The 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Atlanta, GA, Sept. 4 ­6, 2002. Kafyeke, F., "An Analysis Method for Transonic Flow about Three Dimensional Configurations", Technical Report, Canadair Ltd., Montreal, Canada, 1986. Sobieszczanski-Sobieski, J. and Haftka, R. T., Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments. AIAA Paper 96-0711, January 1996. MSC/PATRAN User's Manual, Version 5, vols. 1-4, The MacNeal-Schwendler Corporation, 1996. MSC/NASTRAN Quick Reference Guide, Version 69, The MacNeal-Schwendler Corporation, 1996. Optimization Toolbox for use with MATLAB, User's Guide, The MathWorks Inc., 1999. MATLAB the Language of Technical Computing, Using MATLAB Version 6, The MathWorks Inc., 2000. Garcelon, J. H., Balabanov, V. and Sobieski, J., Multidisciplinary Optimization of a Transport Aircraft Wing using VisualDOC. 40th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, St. Louis, MO, April 12-15 1999. Michael Chun-Yung Niu, Airframe Structural Design, Conmilit Press ltd., 1988, p. 21-23.

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