3.Finaldissertation_yhliu - Efficient Methods for Structural Analysis of Wings

7/23/2019 3.Finaldissertation_yhliu - Efficient Methods for Structural Analysis of Wings

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EFFICIENT METHODS FOR STRUCTURAL ANALYSIS OF BUILT-UP WINGS

by

Youhua Liu

Dissertation Submitted to the Faculty of Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

In

Aerospace Engineering

Approved:

Rakesh K. Kapania, Chairman

Romesh C. Batra Zafer Grdal

Eric R. Johnson Efstratios Nikolaidis

April 2000

Blacksburg, Virginia

Keywords: Built-Up Wing, Structural Analysis, Continuum Model, Equivalent Plate Model,

Mindlin-Plate Theory, Ritz-Method, Neural Network, Sensitivity

Copyright 2000, Youhua Liu


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Efficient Methods for Structural Analysis of Built-Up Wings

by

Youhua Liu

Committee Chairman: Rakesh K. Kapania

Aerospace and Ocean Engineering

(ABSTRACT)

The aerospace industry is increasingly coming to the conclusion that physics-based high-

fidelity models need to be used as early as possible in the design of its products. At the preliminary

design stage of wing structures, though highly desirable for its high accuracy, a detailed finite

element analysis(FEA) is often not feasible due to the prohibitive preparation time for the FE

model data and high computation cost caused by large degrees of freedom. In view of this situation,

often equivalent beam models are used for the purpose of obtaining global solutions. However, for

wings with low aspect ratio, the use of equivalent beam models is questionable, and using an

equivalent plate model would be more promising.

An efficient method, Equivalent Plate Analysis or simply EPA, using an equivalent plate

model, is developed in the present work for studying the static and free-vibration problems of built-

up wing structures composed of skins, spars, and ribs. The model includes the transverse shear

effects by treating the built-up wing as a plate following the Reissner-Mindlin theory (FSDT). The

Ritz method is used with the Legendre polynomials being employed as the trial functions.

Formulations are such that there is no limitation on the wing thickness distribution. This method is

evaluated, by comparing the results with those obtained using MSC/NASTRAN, for a set of

examples including both static and dynamic problems.


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The Equivalent Plate Analysis (EPA) as explained above is also used as a basis for generating

other efficient methods for the early design stage of wing structures, such that they can be

incorporated with optimization tools into the process of searching for an optimal design. In the

search for an optimal design, it is essential to assess the structural responses quickly at any design

space point. For such purpose, the FEA or even the above EPA, which establishes the stiffness and

mass matrices by integrating contributions spar by spar, rib by rib, are not efficient enough.

One approach is to use the Artificial Neural Network (ANN), or simply called Neural Network

(NN) as a means of simulating the structural responses of wings. Upon an investigation of

applications of NN in structural engineering, methods of using NN for the present purpose are

explored in two directions, i.e. the direct application and the indirect application. The direct method

uses FEA or EPA generated results directly as the output. In the indirect method, the wing inner-

structure is combined with the skins to form an "equivalent" material. The constitutive matrix,

which relates the stress vector to the strain vector, and the density of the equivalent material are

obtained by enforcing mass and stiffness matrix equities with regard to the EPA in a least-square

sense. Neural networks for these material properties are trained in terms of the design variables of

the wing structure. It is shown that this EPA with indirect application of Neural Networks, or

simply called an Equivalent Skin Analysis (ESA) of the wing structure, is more efficient than the

EPA and still fairly good results can be obtained.

Another approach is to use the sensitivity techniques. Sensitivity techniques are frequently used

in structural design practices for searching the optimal solutions near a baseline design. In the

present work, the modal response of general trapezoidal wing structures is approximated using

shape sensitivities up to the second order, and the use of second order sensitivities proved to be

yielding much better results than the case where only first order sensitivities are used. Also

different approaches of computing the derivatives are investigated. In a design space with a lot of

design points, when sensitivities at each design point are obtained, it is shown that the global

variation in the design space can be readily given based on these sensitivities.


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Acknowledgments

This work would not have been accomplished without the support and guidance of my advisor

and committee chairman, Dr. Rakesh K. Kapania. Dr. Kapania's professional attitude influenced me

a lot, and his prompt responses to my questions and submitted work, encouragement during all

phases of my work, and his understanding are greatly appreciated. I am grateful to Dr. Romesh C.

Batra, Dr. Zafer Grdal, Dr. Eric R. Johnson, and Dr. Efstratios Nikolaidis for serving in my

committee. I would like to thank the financial support of NASA Langley Research Center on this

research through Grant NAG-1-1884 with Dr. Jerry Housner and Dr. John Wang as the Technical

Monitors. I am also thankful to other students for the helps I have received, especially Dr. Daniel

Hammerand, Dr. Luohui Long, and Mr. Erwin Sulaeman.

Finally, I would say this work could not have got started, let alone been finished, without the

unconditional support, trust and love of my wife, Ting, and my daughter, Lisa. I owe them a lot.


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Contents

List of Tables x

List of Figures xi

Nomenclature xvi

1. Introduction 1

1.1 The Trend of Early Analysis in Product Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Neural Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 History and Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Applications in Structural Engineering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Continuum Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Plate Theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Sensitivity Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8

1.6 Scope of the Present Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2. Neural Networks and Its Applications 11

2.1 Two Important Types of NN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

2.1.1 Feed-Forward Multi-Layer Neural Network. . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Radial Basis Function Neural Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Features of ANN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Algorithms in the MATLAB Neural Network Toolbox. . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Ways of Application of Neural Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

2.4.1 Direct Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19


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2.4.2 Indirect Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3. Continuum Model Approaches 21

3.1 Methods of Obtaining Continuum Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 An Example of NN Modeling of Continuum Models. . . . . . . . . . . . . . . . . . . . . . . . . .23

3.2.1 Neural Network with 2 Input Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24



4. An Approach for the Solution of Mindlin Plates 32

4.1 Assumptions and Formulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32

4.2 Strain Energy and Stiffness Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Kinetic Energy and Mass Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

5. Equivalent Plate Analysis of Built-Up Wing Structures 42

5.1 Numerical Integration of Stiffness and Mass Matrices. . . . . . . . . . . . . . . . . . . . . . . . .42

5.1.1 Skins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43

5.1.2 Spars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

5.1.3 Ribs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

5.2 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

5.3 Formulation for Vibration Problem of Wing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50

5.4 Convergence Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.5 Static Problem Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.6 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.6.1 Free Vibration Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.6.1.1 A Trapezoidal Plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55

5.6.1.2 A Trapezoidal Shell with a Camber. . . . . . . . . . . . . . . . . . . . . . . . .57

5.6.1.3 A Solid Wing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59

5.6.1.4 A Built-up Wing Composed of Skins, Spars and Ribs. . . . . . . . . . 61

5.6.1.5 A Box Wing used as a test case in Livne. . . . . . . . . . . . . . . . . . . . .64

5.6.2 Displacement under Static Loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65


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5.6.2.1 Tip Point Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.6.2.2 A Force Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.6.2.3 Tip Torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65

5.6.2.4 The Box Wing in Livne. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

5.6.3 Skin Stress Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

5.6.4 On Efficiency of EPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73

6. Modal Response Using Sensitivity Technique

and Direct Application of Neural Networks 75

6.1 Shape Sensitivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76

6.2 An Issue in Equivalent Plate Analysis (EPA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77

6.3 Approaches to Sensitivity Evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4 Application of Sensitivity Technique (ST) in Multi-variable Optimization. . . . . . . . 80

6.5 Application of Neural Networks (NN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.6 Examples and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82

6.6.1 Results on sensitivity evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.6.2 Application of Sensitivity Technique (ST) and Neural Networks (NN) . . . .89

7. Equivalent Skin Analysis Using Neural Networks 95

7.1 Equivalent Skin Analysis (ESA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.1.1 The Constitutive matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.1.2 Mass distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.2 Examples and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.2.1 Results at a design point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99

7.2.2 Three-variable case: design space I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.2.3 Four-variable case: design space II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115

7.2.4 Six-variable case: design space III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126

7.2.5 Design space IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.3 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148

8. Conclusions and Future Work 149


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8.1 Conclusions of the Present Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.2 Recommendations for Future Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151

References 153

Appendix A The Constitutive Matrix for Various Cases 160

A.1 Rotation along z -axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160

A.2 Rotation along y -axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A.3 Skin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .161

A.4 Spar and Rib Cap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

A.5 Spar and Rib Web. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163

Appendix B Formulation for Multi-Plane Problem Using EPA 164

B.1 Strain Energy and Stiffness Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165

B.2 Kinetic Energy and Mass Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Appendix C Airfoil Sections Generated with Karman-Trefftz Transformation 167

Vita 171


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List of Tables

Table 3.1 Comparison of Continuum Model Properties for a Lattice Repeating Cell. . . . . . . . . . 31

Table 3.2 Comparison of Continuum Model Properties for a Lattice Repeating Cell. . . . . . . . . . 31

Table 5.1 Natural frequencies (Hz) of the cantilevered swept-back box wing. . . . . . . . . . . . . . . . 64

Table 5.2 Displacement (in) of the cantilevered swept-back box wing. . . . . . . . . . . . . . . . . . . . . 69

Table 5.3 Comparison of FEA and EPA in terms ofDOFand Number of Elements. . . . . . . . . . .74

Table 7.1 Differences between the natural frequencies by EPA and ESA. . . . . . . . . . . . . . . . . . 101

Table 7.2 Natural frequencies (rad/sec) of the wing in Fig. 7.20. . . . . . . . . . . . . . . . . . . . . . . . .148

Table 7.3 Natural frequencies (rad/sec) of the wing in Fig. 7.29. . . . . . . . . . . . . . . . . . . . . . . . .148


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List of Figures

Fig. 2.1 A feed-forward multi-layer neural network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

Fig. 2.2 Details of a neuron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

Fig. 2.3 Transfer functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Fig. 2.4 Radial basis function neural network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

Fig. 3.1 Geometry of repeating cells of a single-bay double laced lattice structure. . . . . . . . . . . . 22

Fig. 3.2 Evaluating continuum model properties for a repeating cell. . . . . . . . . . . . . . . . . . . . . . .24

Fig. 3.3 Training data for )( , cc LAfGA = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Fig. 3.4 Distributions of training and testing points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Fig. 3.5 Feed-forward NN simulation for )(, cc

LAfGA = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Fig. 3.6 Feed-forward NN simulation errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Fig. 3.7 Radial-basis function NN simulation for )( , cc LAfGA = . . . . . . . . . . . . . . . . . . . . . . . . . .28

Fig. 3.8 Radial-basis function NN simulation errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28

Fig. 3.9 Training history of a 3-10-1 feed-forward NN by trainbp. . . . . . . . . . . . . . . . . . . . . . . . .30

Fig. 3.10 Training history of a 3-10-1 feed-forward NN by trainbpa. . . . . . . . . . . . . . . . . . . . . . .30

Fig. 3.11 Training history of a 3-10-1 feed-forward NN by trainlm. . . . . . . . . . . . . . . . . . . . . . . .31Fig. 4.1 The coordinate system and its transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Fig. 4.2 The Legendre polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

Fig. 4.3 The Chebyshev polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Fig. 5.1 Wing skin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44


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Fig. 5.2 Sketches for wing spar and rib. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

Fig. 5.3 The first 10 natural frequencies of wing I as functions of boundary-condition-

simulating spring value, when 6 terms of Legendre polynomials are used. . . . . . . . . . . . . .48

Fig. 5.4 The first 10 natural frequencies of wing I as functions of boundary-condition-simulating spring value, when 8 terms of Legendre polynomials are used. . . . . . . . . . . . . .49

Fig. 5.5 Natural frequencies of wing I with regard to number of trial function terms. . . . . . . . . . 52

Fig. 5.6 Natural frequencies of wing II with regard to number of trial function terms. . . . . . . . . .53

Fig. 5.7 Mode Shapes and Natural Frequency f )/( srad for a Trapezoidal Plate. . . . . . . . . . . .56

Fig. 5.8 Mode Shapes and Natural Frequency f )/( srad for Wing-Shaped Shell

with a Camber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Fig. 5.9 Mode Shapes and Natural Frequency f )/( srad for the Solid Wing. . . . . . . . . . . . . . .60

Fig. 5.10 Wing cross-sections at rib positions and spar positions. . . . . . . . . . . . . . . . . . . . . . . . . .62

Fig. 5.11 Mode Shapes and Natural Frequency f )/( srad for a Built-up Wing

Composed of Skins, Spars and Ribs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Fig. 5.12 A box wing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Fig. 5.13 Comparison of Displacements for Load Case of Tip Point Force. . . . . . . . . . . . . . . . . .66

Fig. 5.14 Comparison of Displacements for Load Case of a Force Distribution. . . . . . . . . . . . . . 67Fig. 5.15 Comparison of Displacements for Load Case of Tip Torque. . . . . . . . . . . . . . . . . . . . . .68

Fig. 5.16 Comparison of Von Mises Stress on the Upper and Lower Skins

of a Wing under a Point Force at the Wing Tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70

Fig. 5.17 Distribution of Von Mises Stress on the Upper Skin


Fig. 5.18 Distribution of Von Mises Stress on the Lower Skin


Fig. 6.1 Plan configuration of a trapezoidal wing: .,),( 221 baAsbasA ==+= . . . . . . . . . .76

Fig. 6.2 Natural frequencies using equivalent plate analysis with mode tracking. . . . . . . . . . . . .84

Fig. 6.3 Effect of the finite difference step size on the sensitivities

of various natural frequencies to taper ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85


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Fig. 6.4 The 2ndnatural frequency w.r.t. wing plan area

using 1stand 2ndorder sensitivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Fig. 6.5 The 3rdnatural frequency w.r.t. wing sweep angle

using 1stand 2ndorder sensitivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Fig. 6.6 Comparison of the natural frequencies of the first 6 modes for wing structures

randomly chosen inside the box of design space, as obtained by the NN and ST

w.r.t. those obtained using a full-fledged EPA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Fig. 6.7 Comparison of the natural frequencies of the first 4 modes for wing structures

along a path inside the box of design space (n 1 =0.945, n 2 =8.200, n 3 =3.203) using

only the 1storder sensitivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Fig. 6.8 Comparison of the natural frequencies of the first 4 modes for wing structuresalong a path inside the box of design space (n 1 =0.945, n 2 =8.200, n 3 =3.203) using

sensitivities up to the 2ndorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Fig. 7.1 An example of mass density distribution generated using Eq. (7.8) . . . . . . . . . . . . . . . .101

Fig. 7.2 The stiffness matrix given by EPA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102

Fig. 7.3 The stiffness matrix given by ESA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103

Fig. 7.4 Difference between stiffness matrices given by EPA and ESA. . . . . . . . . . . . . . . . . . . .104

Fig. 7.5 The mass matrix given by EPA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Fig. 7.6 The mass matrix given by ESA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Fig. 7.7 Difference between mass matrices given by EPA and ESA. . . . . . . . . . . . . . . . . . . . . . 107

Fig. 7.8 49 randomly chosen wing plan forms in design space I. . . . . . . . . . . . . . . . . . . . . . . . . .110

Fig. 7.9 Comparison of the first 10 frequencies by EPA and ESA. . . . . . . . . . . . . . . . . . . . . . . .111

Fig. 7.10 The relative errors in Fig. 7.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112

Fig. 7.11 25 wing plan forms systematically varying through design space I. . . . . . . . . . . . . . . .113


Fig. 7.13 25 randomly chosen wing plan forms in design space II. . . . . . . . . . . . . . . . . . . . . . . .117

Fig. 7.14 Comparison of the first 10 frequencies by EPA and ESA. . . . . . . . . . . . . . . . . . . . . . .118

Fig. 7.15 The relative errors in Fig. 7.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119


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xiv

Fig. 7.16 16 wing plan forms systematically varying through design space II. . . . . . . . . . . . . . .120


Fig. 7.18 An arbitrarily chosen wing plan form in design space II. . . . . . . . . . . . . . . . . . . . . . . .122

Fig. 7.19 Comparison of displacements by EPA and ESA for 1 lb tip force . . . . . . . . . . . . . . . .123

Fig. 7.20 Comparison of the Von Mises stress at wing root by EPA and ESA

under 1 lb tip force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124

Fig. 7.21 Comparison of the Von Mises stress along central spar by EPA and ESA


Fig. 7.22 25 randomly chosen wing plan forms in design space III. . . . . . . . . . . . . . . . . . . . . . . 128



Fig. 7.25 16 wing plan forms systematically varying through design space III. . . . . . . . . . . . . . 131


Fig. 7.27 An arbitrarily chosen wing plan form in design space III. . . . . . . . . . . . . . . . . . . . . . . 133

Fig. 7.28 Comparison of displacements by EPA and ESA at 1 lb tip force . . . . . . . . . . . . . . . . .134





Fig. 7.31 16 randomly chosen wing designs in design space IV. . . . . . . . . . . . . . . . . . . . . . . . . .139



Fig. 7.34 16 wing designs systematically varying through design space IV. . . . . . . . . . . . . . . . .142


Fig. 7.36 An arbitrarily chosen wing design in design space IV. . . . . . . . . . . . . . . . . . . . . . . . . .144

Fig. 7.37 Comparison of displacements by EPA and ESA at 1 lb tip force . . . . . . . . . . . . . . . . .145




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xv



Fig. B.1 Sketch for a wing composed of main-body and wing-let. . . . . . . . . . . . . . . . . . . . . . . . 164

Fig. C.1 The Karman-Trefftz transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168Fig. C.2 Airfoils shapes obtained using Karman-Trefftz transformation. . . . . . . . . . . . . . . . . . . 170


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xvi

Nomenclature

a

A

gdcAAA ,,

ANN

b

{B}

c

rcc ,0

1

c

[C]

j

ib

b1,b2,b3

[D]

}{d

DOF

pqD

E

EA

EI

chord-length at wing tip

wing plan area

area of longitudinal bars, diagonal bars, and battens of a repeating cell

Artificial Neural Network

chord-length at wing root

Ritz base function vector defined in Eq. (4.12)

chord-length

chord-length at root

chord-length at tip

matrix defined in Eq. (4.20)

bias (threshold) of the i -th neural in the j -th layer

bias (threshold) vectors

constitutive matrix

displacement vector

number of Degrees Of Freedom

p -th row, q -th column term of constitutive matrix

Youngs Modulus

axial rigidity

bending rigidity


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xvii

EPA

ESA

)(f

FEA

FEM

FF

FSDT

GA

[G]

[H]

2,1h

i, j

I,J,K,L,M,N,

P,Q,R,S

initff

[J]

22211211 ,,, JJJJ

J

k

[K]

]~

[K

L

gdc LLL ,,

'logsig'

2,1l

[M]

Equivalent Plate Analysis

Equivalent Skin Analysis

transfer function

Finite Element Analysis

Finite Element Method

feed-forward

First-order Shear Deformation Theory

shear rigidity



spar, rib cap height

integers

integers

MATLABNN Toolbox feed-forward network initialization program

Jacobian matrix

terms of the inverse of Jacobian matrix

determinant of Jacobian matrix

integer

stiffness matrix based on {q}

stiffness matrix simulated by continuum model

Lagrangian, defined in Eq. (5.14)

length of longitudinal bars, diagonal bars, and battens of a repeating cell

Sigmoid transfer function

spar, rib cap width

mass matrix based on }{q


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xviii

]~

[M

MAC

m, n

N

NN

zN

n1, n2

),(4~1 N

ribn

sparn

p

{P}

)(xPi

p, q

'purelin'

zyx PPP ,,

}{q

j

ir

RBF

s

simuff

simurb

solverb

ST

t

mass matrix simulated by continuum model

modal assurance criterion

integers

dimension of [K] and [M], 25k=Neural Network

number of integration zones in z -direction

number of neurons in the 1st and 2nd hidden layer

transformation functions

number of ribs

number of spars

input training data matrix

generalized load vector defined in Eq. (5.22)

Legendre polynomials

integers

linear transfer function

force components

generalized displacement vector defined in Eq. (4.11)

input of the i -th neural in the j -th layer

Radial Basis Function

length of semi-span of wing

MATLABNN Toolbox FF network simulation program

MATLABNN Toolbox RBF network simulation program

MATLABNN Toolbox RBF network training program

sensitivity techniques

output training data matrix; time


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xix

0t

2,1t

T

[T]

'tansig'

trainbp

trainbpa

trainlm

)(xTi

{x}

x,y,z

j

ix

4~1x

4~1y

[ZZ]

U

u,v,w

000 ,, wvu

V

}{v

1jkiw

M

ij

K

ij ww ,

w1,w2,w3

)(piw

skin thickness

spar, rib thickness

kinetic energy


hyperbolic tangent sigmoid transfer function

MATLABNN Toolbox FF network training program with back-propagation

MATLABNN Toolbox FF network training program with back-propagation andadaptive learning

MATLABNN Toolbox FF network training program with Levenberg-Marquardt

Algorithm

Chebyshev polynomials

eigenvector

Cartesian coordinates

output of the i -th neural in the j -th layer

x-coodinates at quadrilateral wing plan corners

y-coodinates at quadrilateral wing plan corners


strain energy

displacements inx,y,zdirections

displacements inx,y,zdirections at plane 0=z

integration domain for a structure

velocity vector

weight between node kof the )1( j -th layer and node i of the j -th layer

weight coefficients defined in Eq. (7.9)

weight matrices of the 1st, 2nd, and 3rd layer

weight coefficient defined in Eq. (6.15)


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xx

w.r.t.

,,,zyx

yx ,

}{

}{

,,, zyx

zxyzxy ,,

}{ i

}{j

x

y

)( r

}{

,

)(s

with regard to

wing aspect ratio

linear spring coefficients

strain vector

vector defined in Eq. (4.18)

strain tensors

bending angle inxdirection

the i -th eigenvector for baseline design

the j -th eigenvector for perturbed design

rotation about the y direction

rotation about the x direction

rib position function

eigenvalue

wing sweep angle at leading-edge

Poisson's ratio

shear angle inxdirection

mass density; shape variable

stress vector

the taper ratio

frequency, rad/sec

transformed plane variables

spar position function


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1

Chapter 1

Introduction

1.1 The Trend of Early Analysis in Product Design

To reduce product development cycle is essential to a nowadays manufacturing enterprise not

only on economic savings in the process itself, but also to a broad business advantage in getting

product innovations to customers faster, and thereby increasing the company's market share1.

One of the most valuable CAE (Computer Aided Engineering) tools is finite-element analysis

(FEA), which assists in analyzing structures to detect areas that might undergo excessive stress,

deformation, vibration, or other potential problems. Yet, instead of assisting in reducing time to

market, the traditional, full-blown FEA actually became a bottleneck and was often done only

toward the end of product design.

The experience of manufacturers in many industries has shown that 85~90% of the total time

and cost of product development is defined in the early stages of product development, when only

5~10% of project time and cost have been expended 2,1 . This is because in the early concept stages,

fundamental decisions are made regarding basic geometry, materials, system configuration, and

manufacturing processes.

The process, however, can be re-oriented so that analysis is performed much earlier to shorten

the product development cycle. This moves CAE/analysis forward into conceptual design, where

changes are much easier and more economical to make in correcting poor designs earlier. The


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CHAPTER 1 INTRODUCTION 2

major benefits of up-front analysis includes giving designers the ability to perform "what-if?"

simulations that enable them to evaluate alternative approaches and explore options early in the

design cycle to arrive at a superior design. This methodology employs CAE to help avoid "fires" in

the early design stage, rather than uses CAE to put out "fires" in the later design stage as the

traditional practice does 1 .

Therefore, instead of being the last thing to do, CAE is now one of the first things for a

designer to do to make sure that the best design possible is to be obtained 3 .To facilitate this

methodology of early analysis in product design, there have emerged the following two issues

concerning the development of CAE.

The first issue is the lack of integration between CAD and analysis programs. The need totranslate, clean up, and further process design data for use in analysis has limited the effectiveness

of both CAD and CAE software. Over the past few years, software vendors have been moving to

tightly couple CAD and CAE software programs by tying them into suites using a shared database

and a single user interface. Sharing database means that engineers no longer have to translate

design data to formats that the analysis program can recognize, and vice versa. It also allows

updates in one system to be reflected immediately in the other. CAD and CAE sharing the same

user interface makes it easier for a user to switch from one program to the other.

The second issue is the inappropriateness of FEA as the tool of CAE in many cases. Usually

FEA can only be integrated in the early design stage of structurally simple products or components

of a structurally complex product. For instance, at the preliminary design stage of built-up wing

structures, though highly desirable for its high accuracy, a detailed finite element analysis(FEA) is

often not feasible because: (i) the preparation time for the FEM model data may be prohibitive,

especially when there is little carry-over from design to design; (ii) for complex structures

composed of large number of components, a detailed FEA involves huge number of degrees of

freedom, and needs large amount of CPU time and computation capacity, which makes the cost too

high. For such cases, unconventional methods that are more efficient than FEA are needed.


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People have employed continuum models, assuming the complex structures to behave

similarly, for analysis at the early stage of the design process of a complex product. This includes

using beam, plate or shell models to simulate complex structures. In the present work,

methodologies are developed in employing the first-order shear deformation theory (the Mindlin

plate) to simulate the structural responses of built-up wing structures, incorporating neural

networks and other tools to further enhance analysis efficiency. It is hoped that the methodologies

developed in the present work can be used in the early design stages of aerospace wings and other

plate-like complex structures, therefore a superior design can be obtained in a development process

of shorter cycle and less expenses.

1.2 Neural Networks

1.2.1 History and Concepts

The working mechanism in brains of biological creatures has long been an area of intense

study. It was found around the first decade of the 20-th century that neurons(nerve cells) are the

structural constituents of the brain. The neurons interact with each other through synapses, and are

connected by axons(transmitting lines) and dentrites(receiving branches). It is estimated that there

are on the order of 10 billion neurons in the human cortex, and about 60 trillion synapses 4 .

Although neurons are 5~6 orders of magnitude slower than silicon logic gates, the organization of

them is such that the brain has the capability of performing certain tasks (for example, pattern

recognition, and motor control etc.) much faster than the fastest digital computer nowadays.

Besides, the energetic efficiency of the brain is about 10 orders of magnitude lower than the best

computer today. So it can be said, in the sense that a computer is an information-processing system,

the brain is a highly complex, nonlinear, and efficient parallel computer.

Artificial Neural Networks (ANN), or simply Neural Networks (NN) are computational

systems inspired by the biological brain in their structure, data processing and restoring method,

and learning ability. More specifically, a neural network is defined as a massively parallel


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distributed processor that has a natural propensity for storing experiential knowledge and making it

available for future use by resembling the brain in two aspects: (a) Knowledge is acquired by the

network through a learning process; (b) Inter-neuron connection strengths known as synaptic

weights (or simply weights) are used to store the knowledge 4 .

With a history traced to the early 1940s, and two periods of major increases in research

activities in the early 1960s and after the mid-1980s, ANNs have now evolved to be a mature

branch in the computational science and engineering with a large number of publications, a lot of

quite different methods and algorithms and many commercial software and some hardware. They

have found numerous applications in science and engineering, from biological and medical

sciences, to information technologies such as artificial intelligence, pattern recognition, signal

processing and control, and to engineering areas as civil and structural engineering.

1.2.2 Applications in Structural Engineering

In the field of structural engineering, there have been a lot of attempts and researches making

use of NN to improve efficiency or to capture relations in complex analysis or design problems.

The following are a few examples. Abdalla and Stavroulakis 5 applied NN to represent

experimental data to model the behavior of semi-rigid steel structure connections, which are related

to some highly nonlinear effects such as local plastification etc. Several cases of neural network

application in structural engineering can be found in Vanluchene and Sun 6 . All the problems

treated in Ref. 6 had been reproduced in Gunaratnam and Gero 7 with a conclusion that

representational change of a problem based on dimensional analysis and domain knowledge can

improve the performance of the networks. There is a summary of applications of NN in structural

engineering in Ref. 8. In Liu, Kapania and VanLandingham

9

, methodologies of applying Neural

Networks and Genetic Algorithms to simulate and synthesize substructures were explored in the

solution of 1-D and 2-D beam problems.


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1.3 Continuum Models

As has been indicated in 1.1, it is estimated that about 90% of the cost of an aerospace product

is committed during the first 10% of the design cycle2

. As a result, the aerospace industry is

increasingly coming to the conclusion that physics-based high fidelity models (Finite Element

Analysis for structures, Computational Fluid Dynamics for aerodynamic loads etc.) need to be used

earlier at the conceptual design stage, not only at a subsequent preliminary design stage. But an

obstacle to using the high fidelity models at the conceptual level is the high CPU time that are

typically needed for these models, despite the enormous progress that has been made in both the

computer hardware and software.

In view of this situation, often equivalent continuum models are used to simulate complex

structures for the purpose of obtaining global solutions in the early design stages. This idea is

reasonable as long as the complex structure behaves physically in a close manner to the continuum

model used and only global quantities of the response are of concern. During the late seventies and

early eighties, there was a significant interest in obtaining continuum models to represent discreet

built-up complex lattice, wing, and laminated wing structures. These models use very few

parameters to express the original structure geometry and layout. The initial model generation and

set-up is fast as compared to a full finite element model. Assembly of stiffness and mass matrices

and solution times for static deformation and stresses or natural modes are significantly less than

those needed in a finite element analysis. All these make continuum models very attractive for

preliminary design and optimization studies.

Despite its great potential, however, the continuum approach has gained a limited popularity in

the aerospace designers community. This might be due to the fact that, all the developments have

been made by keeping specific examples (e.g. periodic lattices or specific wings) in mind. Also,

with some exceptions, most of these approaches were rather complex. The key obstacle, though,

appears to be the fact that if the designer makes a change in the actual built-up structures, the

continuum model has to be determined from scratch.


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The complex nature of the various methods and the large number of problems encountered in

determining the equivalent models are not surprising given the fact that determining these models

for a given complex structure (a large space structure or a wing) belongs to a class of problems

called inverse problems. These problems are inherently ill-posed and it is fruitless to attempt to

determine unique continuum models. The present work deals with investigating the possibility that

a more rational and efficient approach of determining the continuum models can be achieved by

using artificial neural networks.

The following are examples of work on using beam or plate models to simulate repetitive

lattice structures: Noor, Anderson, and Greene 10 ; Nayfeh, and Hefzy11 ; Sun, Kim, and

Bogdanoff

12

; Noor

13

; Lee

16~14

. Specifics of these methods will be discussed inChapter 3

. In the area of analyzing aerospace wing structures, a number of studies have been conducted on

using equivalent beam models to represent simple box-wings composed of laminated or anisotropic

materials, which include Kapania and Castel 17 , Song and Librescu 18 , and Lee 19 . They have given

some fine results for the specific problems. However, for wings with low aspect ratio, the use of

equivalent beam models is questionable, and using an equivalent plate model would be more

promising.

1.4 Plate Theories

There exists a considerable body of work on the static or dynamic behaviors of all kinds of

plates. A thorough description of literature on the study of plates was given by Lovejoy and

Kapania 21,20 , where more than 300 references has been listed about all plates. The plates studied

include thin, thick, laminated or composite, whose geometry can be rectangular, skew, or

trapezoidal, and the lamina can be of similar or dissimilar material and isotropic, orthotropic, or

anisotropic in nature

One way of classifying existing methods for the solution of plates is according to the

deformation theory used, namely: the Classical Plate Theory (CPT), the First-order Shear


26/190


Deformation Theory (FSDT), or the Higher-order Shear Deformation Theory (HSDT) etc. The

CPT is based on the Kirchhoff-Love hypothesis, that is, a straight line normal to the plate middle

surface remains straight and normal during the deformation process. This group of theories work

well for truly thin isotropic plates, but for thick isotropic plates and for thin laminated plates they

tend to overestimate the stiffness of the plate since the effects of through-the-thickness shear

deformation are ignored 23,22 . The FSDT is based on the Reissner-Mindlin model 25,24 , where the

constraint that a normal to the mid-surface remains normal to the mid-surface after deformation is

relaxed, and a uniform transverse shear strain is allowed. The FSDT is the most widely used theory

for thick and anisotropic laminated plates owing to its simplicity and its low requirement for

computation capacity. For more accurate results or more realistic local distributions of the

transverse strain and stress, one should use the HSDT 26 , or the CFSDT (Consistent First-order

Shear Deformation Theory) proposed by Knight and Qi 27 .

Methods of solving the CPT, FSDT or HSDT mainly include finite element, Galerkin, and

Rayleigh-Ritz methods 21,20 . In the context of using equivalent plate to represent the behaviors of

wing structures at the conceptual stage at least, it is obvious that while the computationally costly

finite element method is to be avoided, the Rayleigh-Ritz method is attractive.

There have been several studies using equivalent plate models to model wing structures.

Giles 29,28 developed a Ritz method based approach, which considers an aircraft wing as being

formed by a series of equivalent trapezoidal segments, and represents the true internal structure of

aircraft wings in the polynomial power form. In Giles 28 the CPT was used, but this shortcoming

was removed subsequently 29 . Tizzi 30 presented a method whose many aspects are similar to that

of Giles. In Tizzi's work several trapezoidal segments in different planes can be considered, but the

internal parts of wing structures (spars, ribs, etc.) were not considered. Livne 31 formulated the

FSDT to be used for modeling solid plates as well as typical wing box structures made of cover

skins and an array of spars and ribs based on simple-polynomial trial functions, which are known to

be prone to numerical ill-conditioning problems. Livne and Navarro then further developed the

method to deal with nonlinear problems of wing box structures 32 .


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1.5 Sensitivity Techniques

Sensitivity techniques are frequently used in structural design practices for searching the

optimal solutions near a baseline design35~33

. The design parameters for wing structure include

sizing-type variables (skin thickness, spar or rib sectional area etc.), shape variables (the plan

surface dimensions and ratios), and topological variables (total spar or rib number, wing topology

arrangements etc.). Sensitivities to the shape variables are extremely important because of the

nonlinear dependence of stiffness and mass terms on the shape design variables as compared to the

linear dependence on the sizing-type design variables.

Kapania and coworkers have addressed the first order shape sensitivities of the modal response,

divergence and flutter speed, and divergence dynamic pressure of laminated, box-wing or general

trapezoidal built-up wing composed of skins, spars and ribs using various approaches of

determining the response sensitivities 42~36 .

1.6 Scope of the Present Work

The aim of the present work is trying to develop efficient methods for the structural analysis of

built-up wings at the early design stage, such that with a fraction of the computational cost of a

detailed FEA, sufficiently accurate results for the global properties of the wing can be obtained. In

the present study, continuum models, neural networks and some other efficient simulation tools are

going to be used to make the objective possible.

As a preparation for application in later chapters, basic concepts and formulations about two

most commonly used neural networks, the Feed-Forward NN and Radial Basis Function NN, are

described in Chapter 2. Details of how to use some basic functions in the MATLABNN Toolbox

for training and testing networks are provided, together with two ways of application of neural

networks: the direct approach and the indirect approach.


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Chapter 3 is composed of an introduction of the continuum models used by several authors,

and an example of treating a lattice structure with repeating cells by continuum modeling applying

neural networks, compared with results obtained by other authors.

The present study is an extension of the previous works of Kapania and Singhvi 44,43 , Kapania

and Lovejoy 46,45,21,20 , and Cortial 47 , who all used the Rayleigh-Ritz method with the Chebyshev

polynomials as the trial functions, and applied the Lagranges equations to obtain the stiffness and

mass matrices. In Kapania and Singhvi 44,43 , the CPT was used to solve generally laminated

trapezoidal plates, while in Kapania and Lovejoy 46,45,21,20 , the FSDT was used. In all these studies,

only uniform plates were considered. In Cortial 47 , efforts were made to use the method of Kapania

and Lovejoy 46,45,21,20 to calculate natural frequencies of box-wing structures, but an assumption of

constant wing thickness makes it difficult to apply the method to general wing structures.

In the present work, it is assumed that the wing plan form is quadrilateral, and the wing

structure is composed of skins, spars and ribs. The wing is represented as an equivalent plate

model, and the Reissner-Mindlin displacement field model is used. The Rayleigh-Ritz method is

applied to solve the plate problem, with the Legendre polynomials being used as the trial functions.

After the stiffness matrix and mass matrix are determined by applying the Lagranges equations,

static analysis can be readily performed and the natural frequencies and mode shapes of the wing

can be obtained by solving an eigenvalue problem. Formulations are such that there is no limitation

on the wing thickness distribution as was the case in Cortial 47 . This basic part of work, a method to

solve the Mindlin plates, is contained in Chapter 4. Then the method, being called the Equivalent

Plate Analysis (EPA), is applied for solving built-up wing structures in Chapter 5. As examples of

verifying EPA, a wing-shaped plate, a wing-shaped plate with camber, a solid wing and a general

built-up wing are analyzed respectively, and the results are compared with those obtained from a

detailed FE analysis using MSC/NASTRAN.

The EPA as explained above can also be used as a basis for generating other efficient methods

in the design of wing structures, such that can be incorporated with optimization tools into the

process of searching for an optimal design. In the search for an optimal design, it is essential to


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CHAPTER 1 INTRODUCTION

10

assess the structural responses quickly at any design space point. For such purpose, the FEM or

even the above EPA, which establishes the stiffness and mass matrices by integrating contributions

spar by spar, rib by rib, are not efficient enough.

One approach is to use Neural Networks as a means of simulating the structural responses of

wings. This is the so called direct application of neural networks, as discussed in Chapter 2.

Another approach is to use the sensitivity techniques. Sensitivity techniques are frequently used in

structural design practices for searching the optimal solutions near a baseline design 34,33 . In the

present work, the modal response of general trapezoidal wing structures is approximated using

shape sensitivities up to the 2ndorder, and the use of second order sensitivities proved to be

yielding much better results than the case where only first order sensitivities are used. Also

different approaches of computing the derivatives are investigated. These two approaches of direct

simulation of modal wing responses are described in Chapter 6, along with an example showing

results giving by both approaches.

Finally, in Chapter 7, a method more efficient than the EPA with indirect application of neural

networks is developed. Instead of evaluating the matrices over all components of the wing

structure, evaluation is performed only over the skins, whose "equivalent" material constitutivematrix and mass density distribution are changed accordingly to incorporate the effects of spars and

ribs. The new skin material properties are simulated using Neural Networks in terms of the wing

design variables. As it is shown, while the Neural-Network-aided EPA, which can be called

Equivalent Skin Analysis (ESA), gives almost equally good results, it uses only a fraction of the

CPU time spent in the ordinary EPA in generating the matrices.

Major parts of the present work are published. They include Chapter 4and 5 in Ref. 48 and 49,

Chapter 6 in Ref. 50 and 51, and Chapter 7 in Ref. 52.


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11

Chapter 2

Basics of Neural Networks

In this chapter a brief description is given to the most extensively used neural network in civil and

structural engineering, Multi-Layer Feed-Forward NN, and another kind of NN, Radial Basis

Function NN, which is very efficient in some cases. Some conceptual features of NN are listed.

Several functions of MATLABNN Toolbox are introduced, which will be used as the major tools in

the present work. At the end of this chapter a brief discussion is made on approaches of application

of neural networks.

2.1 Two Important Types of NN

As simplified models of the biological brain, ANNs have lots of variations due to specific

requirements of their tasks by adopting different degree of network complexity, type of inter-

connection, choice of transfer function, and even differences in training method.

According to the types of network, there are Single Neuron network (1-input , 1-output, and no

hidden layer), Single-Layer NN or Percepton (no hidden layer), and Multi-Layer NN (1 or more

hidden layers). According to the types of inter-connection, there are Feed-Forward network (values

can only be sent from neurons of a layer to the next layer), Feed-Backward network (values can

only be sent in the different direction, i.e. from the present layer to the previous layer), and

Recurrent network (values can be sent in both directions).


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CHAPTER 2 BASICS OF NEURAL NETWORKS 12

In the following a brief description is given to two kinds of extensively used neural networks

and some of the pertinent concepts.

2.1.1 Feed-Forward Multi-Layer Neural Network

An example of feed-forward multi-layer neural network is shown in Fig. 2.1, where the

numbers of input and output are 3 and 2 respectively, and there are two hidden layers with 5

neurons in the first hidden layer, and 3 neurons in the second hidden layer. The details of a neuron

is illustrated in Fig. 2.2. As shown in Fig. 2.2, in the j -th layer, the i -th neuron has inputs from

the )1( j -th layer of value ),,1( 11

= j

j

k nkx ,and has the following output

)( j

i

j

i

rfx = (2.1)

where

j

i

n

k

j

k

j

ki

j

ibxwr

j

=

=

1

1

11 (2.2)

in which 1j

kiw is the weight between node kof the )1( j -th layer and node i of the j -th layer,

and jib is the bias (also called threshold). The above relation can also be written as

=

=1

0

11

jn

k

jk

jki

ji xwr (2.3)

where jij

bx =10 and 11 =joiw , or 1

1

0 =j

x and jij

oi bw =1 .


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inputlayer

hiddenlayers

outputlayer

Fig. 2.1 A feed-forward multi-layer neural network

f( ). x i

jrij

w k i j- 1

-1

.

.

.

.

.

.

b ij

x 1 j- 1

x 2 j- 1

k= nj-1

w 1 ij- 1

w 2 ij- 1

x kj- 1

Transferfunction

Summingjunc tio n

Output

Inputsignals

Synapticweights

Threshold

Fig. 2.2 Details of a neuron


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The transfer function(also called activation functionor threshold function) is usually specified

as the following Sigmoid function

re

rf+

=1

1)( . (2.4)

Other choices of the transfer function can be the hyperbolic tangent function

r

r

e

erf

+

=1

1)( , (2.5)

thepiece-wise linear function

+

=

.5.0,0

;5.05.0,5.0

;5.0,1

)(

r

rr

r

rf (2.6)

and, sometimes, the 'pure' linear function

.)( rrf = (2.7)

These transfer functions are displayed in Fig. 2.3.

r

f(r)

-5 0 5-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Linear

Hyperbolic tangent

Sigmoid

Piecewise-linear

Fig. 2.3 Transfer functions


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n1 n2 n3number ofneurons:

inputlayer

hiddenlayer

outputlayer

Fig. 2.4 Radial basis function neural network

2.1.2 Radial Basis Function Neural Network

Radial Basis Function (RBF) NN usually have one input layer, one hidden layer and one output

layer, as shown in Fig. 2.4.

For the RBF network in Fig. 2.4, we have the relations between the input 1ix (here 1,,1 ni = )

and the output 2kx (here 3,,1 nk = ) as follows:

2

1

2

,

22

k

n

j

jjkk brwx += =

(2.8)

=

=1

1

1

,

11 ),,(n

i

jijij bwxGr (2.9)

where 2w , 2b are the weights and bias respectively, and the Gaussian function is used as the

transfer function:

)}{}{exp(),,( 21121,1

,

11

jijijiji wxbbwxG = (2.10)

where 1w is the center vector of the input data, and 1b is the variance vector.


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2.2 Features of ANN

Some important features of NN are briefed as follows.

Many NN methods are universal approximators, in the sense that, given a dimension (number

of hidden layers and neurons of each layer) large enough, any continuous mapping can be

realized. Fortunately, the two NNs we are most interested in, the multi-layer feed-forward NN

and the radial basis function NN, are examples of such universal approximators 54,53 .

Steps of utilizing NN: specification of the structure(topology) training(learning)

simulation(recalling).

(1)Choosing structural and initial parameters (number of layers, number of neurons of each

layer, and initial values of weights and thresholds, and the kind of transfer function) is usually

from experiences of the user and sometimes can be provided by the algorithms. (2)The training

process uses given input and output data sets to determine the optimal combination of weights

and thresholds. It is the major and the most time-consuming part of NN modeling, and there are

lots of methods regarding different types of NN. (3)Simulation means using the trained NN to

predict output according to new inputs (This corresponds to the 'recall' function of the brain).

The input and output relationship of NN is highly nonlinear. This is mainly introduced by thenonlinear transfer function. Some networks, e.g. the so-called "abductive" networks, use double

even triple powers besides linear terms in their layer to layer input-output relations 55 .

A NN is parallel in nature, so it can make computation fast. Neural networks are ideal for

implementation in parallel computers. Though NN is usually simulated in ordinary computers

in a sequential manner.

A NN provides general mechanisms for building models from data, or give a general means to

set up input-output mapping. The input and output can be continuous (numerical), or not

continuous (binary, or of patterns).

Training a NN is an optimization process based on minimizing some kind of difference

between the observed data and the predicted while varying the weights and thresholds. For


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numerical modeling, which is of our major concern for the present study, there is a great

similarity between NN training and some kind of least-square fitting or interpolation.

Simulation using NN gives better results in interpolation than in extrapolation, the same as any

other data fitting or mapping methods.

Where and when to use NN depend on the situation, and NN is not a panacea. The following

comment on NN application on structural engineering seemingly can be generalized in other

areas:

"The real usefulness of neural networks in structural engineering is not in reproducing existing

algorithmic approaches for predicting structural responses, as a computationally efficient

alternative, but in providing concise relationships that capture previous design and analysis

experiences that are useful for making design decisions"7.

Despite the above features and wide application in a lot of areas, there seems to be no evidence

for neural networks to claim superiority over some other mapping tools. For instance, in a recent

paper of Nikolaidis, Long, and Ling 73, it is claimed that the response surface polynomials with

stepwise regression and the neural network models appear to be almost equally accurate, but it took

considerably less time to develop the polynomials than the neural networks.

2.3 Algorithms in the MATLABNeural Network Toolbox

When using MATLABNN Toolbox, one should first choose the number of input and output

variables. This is accomplished by specifying the two matrices p and t, where p is a nm

matrix; m is the number of input variables, and n the number of sets of training data; and tis a

nl matrix; l is the number of output variables. The number of network layers, and numbers of

neurons of each layer are other factors that need to be specified.

MATLABgives algorithms for specifying initial values of weights and thresholds in order that

training can be started. For feed-forward NN, function initffis given for this purpose. The

following is an example of using the algorithm


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[w1,b1,w2,b2,w3,b3]=initff(p,n1,'logsig',n2,'logsig',t,'logsig');

where w1, w2, andw3are initial values for the weight matrices of the 1st (hidden), 2nd (hidden)

and 3rd (output) layer respectively, b1, b2, andb3are the bias (threshold) vectors, n1and n2the

number of neurons in the 1st and 2nd hidden layer respectively, and 'logsig'means that the Sigmoid

transfer function is used.

The present version of MATLABNN Toolbox can support only 2 hidden layers, but the number

of neurons is only limited by the available memory of the computer system being used. For the

transfer function, one can also use other choices, such as 'tansig'(hyperbolic tangent sigmoid),

'radbas'(radial basis) and 'purelin'(linear) etc.

Experiences of using initffindicated that it seems to be a random process since it is found that

the result of the execution of this algorithm each time is different. And other conditions kept the

same, two executions of this function usually give quite different converging histories of training

by the training algorithm 8 .

Shown in the following is the MATLABalgorithm for training feed-forward network with back-

propagation:

[w1,b1,w2,b2,w3,b3,ep,tr]=trainbp(w1,b1,'logsig',w2,b2,'logsig',w3,b3,'logsig',p,t,tp);

where most of the parameters which the user should take care of have been mentioned in the above

paragraphs. The only parameter that the user sometimes need to specify is the 41 vector tp,

where the first element indicates the number of iterations between updating displays, the second the

maximum number of iterations of training after which the algorithm would automatically terminate

the training process, the third the converging criterion (sum-squared error goal), and the last the

learning rate. The default value of tpis [25, 100, 0.02, 0.01].

Other algorithms for training include trainbpa(training feed-forward NN with back-

propagation and adaptive learning), solverb(designing and training radial basis network), and

trainlm(training feed-forward NN with Levenberg-Marquardt algorithm) etc.

trainbpaand trainlmhave very similar formats for using as that of trainbp. The radial basis

network designing and training algorithm has the following format


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[w1,b1,w2,b2,nr,err]=solverb(p,t,tp);

where the algorithm chooses centers for the Gaussiansand increases the neuron number of the

hidden layer automatically if the training cannot converge to the given error goal. So it is also a

designing algorithm.

After the NN is trained, one can predict output from input by using simulation algorithms in

terms of the obtained parameters w1, b1, w2, b2, etc. For feed-forward network one use

y=simuff(x,w1,b1,'logsig',w2,b2,'logsig',w3,b3,'logsig');

wherexis the input matrix, andythe predicted output matrix. Similarly, after a radial basis

network has been trained one uses

y=simurb(x,w1,b1,w2,b2);

to predict the output.

Once a NN is trained, we can use the formulations in 2.1or 2.2together with the obtained

parameters (weights etc.) to setup the network to do prediction anywhere and not necessarily within

the MATLABenvironment.

2.4 Ways of Application of Neural Networks

For the efficient simulation of the structural performances of complex wings, there can be two

directions to apply NN as specified in the following:

2.4.1 Direct Application

In this case, the input layer includes all the design variables of interest (for instance, the four

shape parameters of the wing plan form: the sweep angle, the aspect ratio, the taper ratio, and the

plan area). The output layer gives the desired structural responses, such as natural frequencies etc.

The EPA is being used as the training data generator, though if necessary, results obtained using

the FEA can also be used as the training data. Preparation of training data is very important, and the

training algorithm used also greatly impacts the process of training 8 . Caution must be taken in

specifying the network parameters and training criterion, such that the results of the trained


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network would not oscillate around the training data. Once the networks are trained, structural

responses at any design point can be recalled in a fraction of a second and this is really favorable in

a design situation 51 .

2.4.2 Indirect Application

Here it is desired to find a way of incorporating NN into the application of the equivalent plate

analysis (EPA) of complex wing structures, other than just making use of results generated by EPA

as the training data base. Note that in the EPA of a complex wing, the computational effort is

mainly spent on integrals for generating the contribution from the inner-structural components of

the wing, i.e. the spars and the ribs, in the stiffness and mass matrices. If an anisotropic material

can be found to replace the inner components, in terms of an equivalent skin, such that the new

composite wing has very close global properties as the original one, then the EPA can be

performed more efficiently. Solution of the adequate material properties of the anisotropic material

is the major obstacle here. The role of NN will be relating the material properties to all kinds of

wing design parameters, and it can be trained when there exists enough data base for training. This

way of applying NN has been claimed to be the best use of the Neural Networks in structural

engineering7

. This is the path that is to be followed in Chapter 7.


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21

Chapter 3

Continuum Model Approaches

3.1 Methods of Obtaining Continuum Models

A lot of methods have been used to develop continuum models to represent complex structures.

Many of these methods involve the determination of the appropriate relationships between the

geometric and material properties of the original structure and its continuum models. An important

observation is that the continuum model is not unique, and determining the continuum model for a

given complex structure is inherently ill-posed therefore diverse approaches can be used. This can

be clearly shown in the following example of determining continuum models for a lattice structure.

The single-bay double-laced lattice structure shown in Fig. 3.1 has been studied in Ref. 10, 12,

and 14 with different approaches to the continuum modeling. This lattice structure with repeating

cells can be modeled by a continuum beam if the beam's properties is properly provided.

Noor et al's method include the following steps 10 : (1)introducing assumptions regarding the

variation of the displacements and temperature in the plane of the cross section for the beamlike

lattice, (2)expressing the strains in the individual elements in terms of the strain components in the

assumed coordinate directions, (3)expanding each of the strain components in a Taylor series, and

(4)summing up the thermoelastic strain energy of the repeating elements which then gives the

thermoelastic and dynamic coefficients for the beam model in terms of material properties and

geometry of the original lattice structure.


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CHAPTER 3 CONTINUUM MODEL APPROACHES 22

Lg

Lc

longitudinal bar

diagonal bar

batten

In Sun et al 12 , the properties of the continuum model is obtained respectively by relating the

deformation of the repeating cell to different load settings under specified boundary conditions. For

example, the shear rigidity GAis obtained by performing a numerical shear test in which a unit

shear force is applied at one end of the repeating cell and the corresponding shear deformation is

calculated by using a finite element program. The mass and rotatory inertia are calculated with a

averaging procedure.

Lee put forward a method that he thought to be more straightforward 14 . He used an extended

Timoshenko beam to model the equivalent continuum beam. By expressing the total strain and

kinetic energy of the repeating cell in terms of the displacement vector at both ends of the

continuum model, and equating them to those obtained through the extended Timoshenko beam

Length of longitudinal bars: cL Length of battens: gL

Length of diagonal bars: 21

)( 22 gcd LLL += Areas: ,cA ,gA dA

Fig. 3.1 Geometry of repeating cells of a single-bay double laced lattice structure


42/190


theory, he got a group of relations. The number of these relations, 2N(1+2N), whereNis the degree

of freedom of the continuum model, is usually larger than that of the equivalent continuum beam

properties to be determined. Lee then introduced a procedure in which the stiffness and mass

matrices for both the lattice cell and the continuum model are reduced and so is the number of

relations. Yet how to reduce the number of relations to be equal to the number of unknowns seems

to depend on luck.

All the above three methods give close results for the continuum model properties, and the

continuum models also generate promising global results for the lattice structure.

3.2 An Example of NN Modeling of Continuum Models

Emphasizing the application of NN, we choose an approach similar to that in Ref. 12, that is, to

derive the properties of the beam by investigating the force-deformation relationships of the

repeating cell in certain boundary conditions. The approach is illustrated in Fig. 3.2, where the

beam's axial rigidityEA, bending rigidityEI, and shearing rigidity GAare calculated respectively

by using the results of finite element analysis of the repeating cell in different load conditions.

Concerning the finite element analysis of 3-D lattice structures one can consult Ref. 56.

There are five parameters of the repeating cell for the lattice structure in Fig. 3.1 that can be

varied, the longitudinal bar length cL , the batten length gL , and the longitudinal, batten and

diagonal bar area, cA , gA and dA . Generally, a function with more variables will be more complex

and it will be more difficult for a neural network to simulate its performance. A NN with more

input variables needs much more training data since in the training data each variable should vary

separately. As can be shown in the following, this kind of "coarse" training data pose an obstacle to

most of the training algorithms.

Three scenarios were investigated, with the number of input variables set to be 2, 3 and 4

respectively.


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1

2

3

4

5

6

x, u

y, v

1/31/3

1/3

z, w

1

2

3

4

5

6

x, u

y, v

1/31/3

1/3

z, w

1

2

3

4

5

6

x, u

y, v

1/21/2

1

z, w

3.2.1 Neural Network with 2 Input Variables

The input variables are cL and cA . The number of training data sets is 400=20 20. The

number of testing data, most of which are located at centers among the training data mesh, is also

400=20 20. Part of the results, as the training data, about GA, is shown in Fig. 3.3.Positions of the

training as well as the testing data points are shown in Fig. 3.4. Simulations on the testing data and

the relative errors of a 2-10-1 FF NN (feed-forward neural network with 2 inputs, one hidden layer

of 10 neurons, and 1 output) trained with Levenberg-Marquardt algorithm ( trainlm) are shown in

Axial rigidityEA:

641

521

441 vvv

L

v

LEA cc

++

=

Bending rigidityEI:

45

2

4

3

2

3

2

31

vv

LLLLEI

LEILM

cgcg

c

g

=

==

Shear rigidity GA:

641

521

441

641

521

441

11

www

L

GA

c

++=

++=

Fig. 3.2 Evaluating continuum model properties for a repeating cell


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Figs. 3.5 and 3.6. Results of a radial-basis-function (RBF) NN doing the same job are shown in

Figs. 3.7 and 3.8. In both cases the training error criteria were set to be 3104.0 .

From Figs 3.5 and 3.7 we can see that both the FF NN and RBF NN give a very good

simulation of the relation ),( cc LAfGA = , except at points outside the training data range of

variable cL ( cf. Fig. 3.4). At the "inside" points, or positions where interpolations are made, the

abstract values of the relative errors are well below the 1%. On the other hand, at points outside the

training data range of variable cL , the relative errors can be as high as 3~5%. This provides another

proof to the fact mentioned before, that interpolation using NN will give results more accurate than

extrapolation.


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1.5

2

2.5

3

GA

0

1

2Ac

(m2)

0

2

4

6

8

10

12

Lc(m)

X Y

Z

GA Training Data

x10-4

x106

Lc(m)

Ac(m

2)

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Testing points

Training points

Training & Testing Points

x10-4

Fig. 3.3 Training data for )( , cc LAfGA=

Fig. 3.4 Distributions of training and testing points


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2

3

GA(FF)

0

0.5

1

1.5

2Ac

(m2)

0

2

4

6

8

10

12

Lc(m)

X Y

Z

FF Simulation on Testing Data

x10-4

x106

-4

-2

0

2

{GA

-GA(FF)}/GA

0.5

1

1.5

2

Ac(m

2)

2

4

6

8

10

Lc(m)

0.49

-0.070

.49

-2.85

-1.18-

0.62 -0.

07-0.

07

-0.07

-0.07 -0.

07

X Y

Z

FF Simulation Errors

x10-4

x10-2

Fig. 3.5 Feed-forward NN simulation for )( , cc LAfGA=

Fig. 3.6 Feed-forward NN simulation errors


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2

3

GA(RBF)

0

1

2Ac

(m2)

0

2

4

6

8

10

12

Lc(m)

X Y

Z

RBF Simulation on Testing Data

x10-4

x106

0

2

4

{GA-G

A(RBF)}/GA

0

1

2Ac

(m2)

0

2

4

6

8

10

12

Lc(m)

0.90

0.90

-0.15

1.96

0.380.9

0

1.96

-0.15

1.43

1.43

1.962

.48

0.90

1.43

-0.15

-0.15

-0.15

-0.15

X Y

Z

RBF Simulation Errors

x10-2

x10-4

Fig. 3.7 Radial-basis function NN simulation for )( , cc LAfGA=

Fig. 3.8 Radial-basis function NN simulation errors


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3.2.2 Neural Network with 3 Input Variables

The input variables were chosen as cL , cA and dA . The number of training data sets is

343=7 7 7. For this case, the effectiveness of different training algorithms can be seen clearly in

Figs. 3.9~3.11. When ordinary back-propagation training algorithm, i.e. trainbpis used, it is very

hard to train the NN to the error level of 110 , as shown in Fig. 3.9. When the adaptive learning

technique is included, an improvement can be made, but it is still hard to reach the 210 error level,

as can be seen in Fig. 3.10. Now if the alg

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3.Finaldissertation_yhliu - Efficient Methods for Structural Analysis of Wings