Vibration based damage detection in composite structures using computational intelligence tools

Obinna K. Ihesiulor

A thesis submitted in fulllment of the requirements for the degree of Master of Engineering (Research)

School of Engineering and Information Technology University College University of New South Wales Australian Defence Force Academy

31 August, 2012.

Originality Statement I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the projects design and conception or in style, presentation and linguistic expression is acknowledged.

Signed:

Copyright and DAI Statement I hereby grant the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microlms to use the 350 word abstract of my thesis in Dissertation Abstracts International (this is applicable to doctoral theses only). I have either used no substantial portions of copyright material in my thesis or I have obtained permission to use copyright material; where permission has not been granted I have applied/will apply for a partial restriction of the digital copy of my thesis or dissertation. Signed: Date:

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Abstract
This research work deals with aspects concerned with delamination detection in composite structures as revealed by an approach based on vibration measurements. Variations in vibration characteristics generated in composite laminates indicate the existence of delaminations. This is because degradation due to delamination causes reduction in exural stiness and strength of a material and as a result, vibration parameters like natural frequency responses are changed. Hence it is possible to monitor the variation in natural frequencies to identify the presence of delamination, and assess its size and location for online structural health monitoring (SHM). The approach in this thesis typically depends on undertaking the analysis of structural models implemented by nite element analysis (FEA). FE models also known as the simulator are used to compute the natural frequencies for delaminated and intact specimens of composite laminates. The FE models are validated using the analytical model. However, these FE models are computationally expensive and surrogate (approximation) models are introduced to curtail the computational expense. The simulator is employed to solve the inverse problem using dierent algorithms based on computational intelligence concepts. An articial neural network model (ANN) is developed to solve the inverse problem for delamination detection directly and to provide surrogate models integrated with optimization algorithms (the gradient based local search and Real-coded Genetic Algorithm (RGA)). This approach is termed as surrogate assisted optimization (SAO) and it is seen that the engagement of surrogate models in lieu of the FE models in the optimization loop greatly enhances the accuracy of delamination detection results within an aordable computational cost. It also provides control when handling dierent variables. Meanwhile, to aid the building of eective surrogate models using substantial number of training datasets, K-means clustering algorithm is harnessed and this eectively reduces the large training datasets usually required for ANN network training. Response surface methods (RSM) are also developed to directly solve the inverse problem. The principal advantage of the RSM is its ability to give physical mathematical models that are used to identify the size and location of delamination given any input changes in natural frequencies. A delamination detection strategy that uses K-means clustering algorithm for database selection and ANN, RSM and optimization algorithms integrated with surrogate models based on ANN have been successfully developed. It is demonstrated that these algorithms show immense potentialities for use in delamination damage detection scenarios when applied to composite beams and plates. The algorithms successfully performed delamination detection given limited amount of training datasets. Prediction errors of the algorithms were quantied and they were shown to be robust in the presence of articial errors and noise and

ii even when applied to experimental and simulation data. Results clearly indicate remarkable accurate delamination damage detection capability of the algorithms. The algorithms in their inverse formulations are capable of predicting accurately delamination parameters. These algorithms should hence be employed for application in the domain of SHM where their small computational requirements could be exploited for online damage detection.

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Acknowledgments
My gratitude goes to all the individuals who made this thesis possible. First and foremost, I would like to thank my supervisor, Dr. Krishna Shankar for the invaluable opportunity given to me to be his student. I thank him immensely for believing in me and allowing me undertake research under his supervision. With a busy schedule and many other research students to supervise, he has always been very responsive to my requests and always had great ideas for me and there has never been an occasion when I have knocked on his door and he did not give me time. My thanks also go to my joint-supervisor, Dr. Tapabrata Ray. Without his extraordinary optimization ideas and computational expertise, this thesis would have been a distant dream. I express my gratitude to Zhifang Zhang for her codes used for ANSYS simulations. My colleagues at Multi-Disciplinary Optimization (MDO) group who have enriched my research life in many ways, deserve a special mention. Thanks to Asaf for his great help in providing A the basic understanding of L TEX for writing this thesis. I would also like to acknowledge the help and support from the sta members. Vera Berra, Craig Edward and Liz Careys administrative help, suggestion and kindness is highly appreciated, as is the support services from the Library and ICTS department. Special thanks and love to Mum and Dad, my siblings and other family members for their endless inspiration, love and support throughout my studies. Their support through these years are greatly appreciated. I oer my regards to colleagues and friends @ UNSW@ADFA; Adura, Ahmed, Essam, Moysen, Chigozie, Chengjun, Ram, Sayem, Vishal, and many others who supported and helped me throughout my Masters program. My interactions with them and others too numerous to mention have been very fruitful. My housemates at my place of residence have also played a positive role in my nishing well. I would also like to express my gratitude to my family friends in Canberra who have made my social and academic life balanced. Not left out, I acknowledge the nancial support from UNSW@ADFA that made this venture worthwhile. It is impossible to remember all, and I apologize to those I have inadvertently left out. Finally, thank you all and thank God for making my journey thus far an unforgettable one!!

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Publications arising from the thesis work

Journal papers 1. Obinna K. Ihesiulor, Krishna Shankar, Zhifang Zhang, and Tapabrata Ray. Delamination detection with noise polluted measurements of natural frequencies using computational intelligence concepts. Computers and Structures (under review). 2. Obinna K. Ihesiulor, Krishna Shankar, Zhifang Zhang, and Tapabrata Ray. Validation of algorithms for delamination detection in composite structures using experimental data. Composite Structures (under review). 3. Obinna K. Ihesiulor, Krishna Shankar, Zhifang Zhang, and Tapabrata Ray. Eciencies of algorithms for vibration-based delamination detection: A comparative study. Optimization and Engineering (under review). Conference Papers 1. Obinna K. Ihesiulor, Krishna Shankar, Zhifang Zhang, and Tapabrata Ray. Eectiveness of articial neural networks and surrogate-assisted optimization techniques in delamination detection for structural health monitoring. In Proceedings of the 23rd IASTED International Conference on Modelling and Simulation (MS 2012), July, 2012, Ban, Canada. 2. Obinna K. Ihesiulor, Krishna Shankar, Zhifang Zhang, and Tapabrata Ray. Delamination detection using methods of computational intelligence. In Proceedings of the sixth Global Conference on Power Control and Optimization (PCO 2012), Las Vegas, USA, August, 2012. 3. Zhifang Zhang, Obinna K. Ihesiulor, Krishna Shankar, and Tapabrata Ray. Comparison of Inverse Algorithms for Delamination Detection in Composite Laminates. In Proceedings of the ASME 2012 Conference on Smart Materials, Adaptive Structures and Intelligent Systems (SMASIS 2012), September, 2012, Stone Mountain, Georgia, USA. (Accepted and to be published).

Contents
Abstract Acknowledgments Publications arising from the thesis work List of Figures List of Tables Nomenclature 1 Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural health monitoring (SHM) . . . . . . . . . . . . . . . . . . . . . . Approach to thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limitations of current approaches . . . . . . . . . . . . . . . . . . . . . . . . Objectives of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Research scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Organization of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i iii iv x xiv xix 1 1 3 5 8 9 10 11 12 14 14

2 Literature review 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CONTENTS 2.2 2.3 2.4 Overview of SHM methods for damage detection . . . . . . . . . . . . . . . . Potential SHM methods except vibration based methods . . . . . . . . . . . Methods based on vibration monitoring . . . . . . . . . . . . . . . . . . . . . 2.4.1 2.4.2 Studies on the eects of delaminations on natural frequencies . . . . . Solution to the inverse problem . . . . . . . . . . . . . . . . . . . . . 2.4.2.1 Intelligent inverse algorithms for vibration based damage detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct optimization based methods . . . . . . . . . . . . . . . Surrogate Assisted Optimization (SAO) based methods . . . . Articial-neural network based methods . . . . . . . . . . . . . 2.4.2.2 2.5

vi 16 18 19 20 24

24 24 26 27

RSM as inverse method for vibration based damage detection 29 30

Summary of research gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Mathematical formulation and modeling of composite laminates for beams and plates 3.1 3.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical formulation of the optimization problem . . . . . . . . . . . . 3.2.1 3.3 Solution methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 32 33 35 37 37 42 45 46 49 49 49 50

Modeling of the laminated composite beam . . . . . . . . . . . . . . . . . . . 3.3.1 3.3.2 3.3.3 Finite Element (FE) Analysis of the composite beam . . . . . . . . . Theoretical modeling of the laminated composite beam . . . . . . . . Validation of the FE Analysis for the composite beam . . . . . . . . .

3.4

FE modeling of composite plates . . . . . . . . . . . . . . . . . . . . . . . .

4 Algorithms for solution of the inverse problem 4.1 4.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization search algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Gradient based Local Search (GBLS) . . . . . . . . . . . . . . . . . .

CONTENTS 4.2.2 4.3 Global optimizer based on evolutionary algorithm (EA) . . . . . . . .

vii 51 55 56 57 57 59 60 61 62 64 65 65 66

Database creation for ANN training using K-Means clustering algorithm . . 4.3.1 Preparation of database used for ANN training . . . . . . . . . . . .

4.4

Articial Neural Network (ANN) as an inverse solver and surrogate creator . 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.4.6 4.4.7 4.4.8 4.4.9 Introduction to Articial Neural Network (ANN) . . . . . . . . . . . Pros of ANN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cons of ANN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modications to the basic ANN . . . . . . . . . . . . . . . . . . . . . The steps in designing ANN model . . . . . . . . . . . . . . . . . . . Generalization enhancement . . . . . . . . . . . . . . . . . . . . . . . Trial training algorithms and learning functions . . . . . . . . . . . . Performance study between single and ensemble neural nets . . . . . ANN conguration to analyze the best network architectures . . . . . 4.4.9.1 Training performance of trial neural networks for the forward problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.9.2 Training Performance of trial neural networks for the inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.10 Study of the eect of transfer functions on network performance . . . 4.4.11 Data preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.12 Summary of selected best neural network architectures . . . . . . . .

67

69 70 70 71 71 72 74 74 75 75 76

4.5

Surrogate approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Validation of the surrogate model . . . . . . . . . . . . . . . . . . . .

4.6

Response Surface Methodology (RSM) . . . . . . . . . . . . . . . . . . . . . 4.6.1 4.6.2 4.6.3 4.6.4 Introduction to RSM . . . . . . . . . . . . . . . . . . . . . . . . . . . Advantages of RSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of a rst-order model response surface . . . . . . . . . . . . Analysis of a second-order model response surface . . . . . . . . . . .

CONTENTS 4.6.5 Validation and adequacy check of the developed models . . . . . . . .

viii 78 81 81

5 Results and discussion 5.1 5.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-Variable problemPrediction of delamination location and size at known interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 5.2.2 5.2.3 5.2.4 Application of ANN for delamination prediction . . . . . . . . . . . . Application of RSM for delamination prediction . . . . . . . . . . . . Optimization without surrogates . . . . . . . . . . . . . . . . . . . .

81 82 83 84 85 86

Surrogate Assisted Optimization (SAO) . . . . . . . . . . . . . . . . . 5.2.4.1 5.2.4.2 Optimization using RGA with surrogates . . . . . . . . . . . Optimization using gradient based local search (GBLS) method with surrogates . . . . . . . . . . . . . . . . . . . . . . . . .

88

5.3

3-Variable problemPrediction of delamination location and size at unknown interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 91 91 91 93 93 95

5.4

Validation with published experimental results . . . . . . . . . . . . . . . . . 5.4.1 5.4.2 Experimental validation with Okafor et al.s results . . . . . . . . . . Experimental validation with Su et al.s published results . . . . . . .

5.5

Eect of articial errors and noise . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 5.5.2 Eect of articial errors . . . . . . . . . . . . . . . . . . . . . . . . . Eect of articial noise . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6

5-Variable problemDelamination detection in composite plates . . . . . . . 100 5.6.1 Prediction of delamination parameters for the composite plate laminates using RGA with surrogates . . . . . . . . . . . . . . . . . . . . 100

6 Comparison of delamination prediction eciencies of dierent algorithms 102 6.1 6.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Algorithm 1 - ANN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

CONTENTS 6.3 6.4 6.5 6.6 6.7 6.8

ix

Algorithm 2 - RSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Algorithm 3 - RGAW oS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Algorithm 4 - RGAW S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Algorithm 5 - GBLSW oS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Algorithm 6 - GBLSW S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Summary of comparative results . . . . . . . . . . . . . . . . . . . . . . . . . 109 114

7 Conclusions and recommendations 7.1 7.2

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . . 118 120

References A Trial performance of dierent ANN network architectures, transfer functions, training functions and scaled values

127

A.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 B Validation of the surrogate models 141

B.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

List of Figures
1.1 2.1 3.1 3.2 Approach to thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline for literature review. . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of solution methodology. . . . . . . . . . . . . . . . . . . . . . . . Sample lay plot display for [0/90/0/90]s sequence of ber orientation in the laminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 3.4 Locations of dierent interfaces of the [0/90/0/90]s composite laminates . . . 3D plots of frequency shifts (dF) as functions of delamination location (X) and size (a) for Mode 1 to Mode 4. . . . . . . . . . . . . . . . . . . . . . . . 3.5 3D plots of frequency shifts (dF) as functions of delamination location (X) and size (a) for Mode 5 to Mode 8. . . . . . . . . . . . . . . . . . . . . . . . 3.6 Representation of the integral and delaminated sections of a composite beam having midplane delamination. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 4.1 4.2 4.3 4.4 4.5 4.6 Delamination position in x-y plane for the plate problem. . . . . . . . . . . . Flowchart of RGA optimization framework. . . . . . . . . . . . . . . . . . . Design space for training and testing datasets. . . . . . . . . . . . . . . . . . Design space for 40 and 28 training datasets respectively. . . . . . . . . . . . Design space for 20 and 8 training datasets respectively. . . . . . . . . . . . . A schematic framework of one-hidden layer architecture . . . . . . . . . . . . Pros and Cons of ANN and RSM . . . . . . . . . . . . . . . . . . . . . . . . 45 48 54 57 57 57 58 61 42 41 39 40 8 16 36

LIST OF FIGURES 4.7 Comparison of performance of dierent network architectures on RMSE of training data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Comparison of performance of dierent network architectures on RMSE of testing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Comparison between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 1 training and testing datasets respectively . . . . . . . . . . . . . . . . . . . . . .

xi

68

69

73

4.10 Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 1 training and testing datasets respectively. . . . . . . . . . . . . . . . . . . . 5.1 5.2 Convergence plot RGA without and with surrogates. . . . . . . . . . . . . . Variation of frequencies with location and size of delamination damage with and without noise for modes 1 to 4. . . . . . . . . . . . . . . . . . . . . . . . 6.1 Performance comparison of dierent methods based on error of prediction in delamination location (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.2 Performance comparison of dierent methods based on error of prediction in delamination size (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 B.1 Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 2 training and testing datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 B.2 Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 2 training and testing datasets. . . . . . . . . . . . . . . . . . . . . . . . . . 142 B.3 Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 3 training and testing datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 97 74 85

LIST OF FIGURES B.4 Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode

xii

3 training and testing datasets. . . . . . . . . . . . . . . . . . . . . . . . . . 143 B.5 Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 4 training and testing datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 B.6 Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 4 training and testing datasets. . . . . . . . . . . . . . . . . . . . . . . . . . 144 B.7 Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 5 training and testing datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 B.8 Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 5 training and testing datasets. . . . . . . . . . . . . . . . . . . . . . . . . . 145 B.9 Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 5 training and testing datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 B.10 Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 5 training and testing datasets. . . . . . . . . . . . . . . . . . . . . . . . . . 146 B.11 Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 6 training and testing datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 B.12 Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 6 training and testing datasets. . . . . . . . . . . . . . . . . . . . . . . . . . 147

LIST OF FIGURES B.13 Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 7 train-

xiii

ing and testing datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 B.14 Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 7 training and testing datasets. . . . . . . . . . . . . . . . . . . . . . . . . . 148 B.15 Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 8 training and testing datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 B.16 Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 8 training and testing datasets. . . . . . . . . . . . . . . . . . . . . . . . . . 149

List of Tables
3.1 3.2 3.3 Material properties of the composite beam laminates [1]. . . . . . . . . . . . Natural frequencies for delamination along dierent interfaces (in Hertz). . . Quantitative comparison between the analytical and numerical natural frequencies (in Hertz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 4.1 4.2 4.3 Material properties of the composite plate laminates. . . . . . . . . . . . . . RGA Parameters setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trial training algorithms and functions . . . . . . . . . . . . . . . . . . . . . Adopted dierent network architectures with trainbr algorithm and Tan and Pur transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 4.5 Comparison of dFi between FE simulations and ANN approximations . . . . Summary of generated 4th order polynomial equation for delamination location (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 5.1 77 71 73 45 48 55 65 37 41

Summary of generated 4th order polynomial equation for delamination size (a) 78 % Errors of 8 test cases with inverse ANN modeling using 400 and 40 training datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2

% Errors of 8 test cases with inverse ANN modeling using 28 and 20 training datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 84

5.3 5.4

% Errors of 8 test cases using RSM with 400 data points. . . . . . . . . . . . % Errors of 8 test cases using RGA with 400 and 40 function evaluations via FE models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

xiv

LIST OF TABLES 5.5 % Errors of 8 test cases using RGA via surrogate models created with 400 and 40 training datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 % Errors of 8 test cases using RGA via surrogate models created with 28 and 20 training datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 % Errors of 8 test cases using RGA via surrogate models created with 8 training datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 % Errors of 8 test cases using GBLS via surrogate models created with 400 and 40 training datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 % Errors of 8 test cases using GBLS via surrogate models created with 28 and 20 training datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 % Errors of 8 test cases using GBLS via surrogate models created with 8 training datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 RGA prediction via surrogates for 5 test cases. . . . . . . . . . . . . . . . . . 5.12 Gradient based local search prediction via surrogates for 5 test cases . . . . . 5.13 Comparison of results with Okafor et al. [1]s experimental results. . . . . . . 5.14 Comparison of performance of results with Su et al. [2]s published experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Prediction results using RGA via surrogates with addition of articial errors. 5.16 Prediction results using GBLS optimizer via surrogates with addition of articial errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.17 Prediction results using ANN with addition of articial errors. . . . . . . . .

xv

87

87

87

88

88

89 90 90 92

93 94

95 95

5.18 Eect of errors accrued from noise addition on a delamination signature [1,54,24]. 98 5.19 Prediction results using RGA via surrogates with addition of articial random noise in the undamaged and damaged frequencies. . . . . . . . . . . . . . . . 5.20 Prediction results using GBLS optimizer via surrogates with addition of articial random noise in the undamaged and damaged frequencies. . . . . . . . 99 98

LIST OF TABLES 5.21 Prediction results using ANN with addition of articial random noise in the undamaged and damaged frequencies. . . . . . . . . . . . . . . . . . . . . . .

xvi

99

5.22 % Errors of test cases using RGAW S for the 5-variable problem . . . . . . . 101 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 List of proposed algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Selected 10-test cases to ascertain the method with best performance . . . . 104 % Errors of 10 test cases using ANN modeling . . . . . . . . . . . . . . . . . 104 % Errors of 10 test cases using RSM modeling . . . . . . . . . . . . . . . . . 105 % Errors of 10 test cases using RGAW oS . . . . . . . . . . . . . . . . . . . . 106 % Errors of 10 test cases using RGAW S via surrogate models over 10 runs . 107 % Errors of 10 test cases using GBLSW oS over 100 start points . . . . . . . . 108 % Errors of 10 test cases using GBLSW S over 10 start points. . . . . . . . . 109 Average completion time (ACT) and lowest minimum objective function values (Minimum Objfun) for dierent algorithms. . . . . . . . . . . . . . . . . 111 6.10 Summary of comparative prediction % error results ([(E-X), (E-a)]) for dierent proposed algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 A.1 RMSE analysis for the forward problem using dierent training algorithms with 400 and 41 datasets for network training and testing respectively . . . . 128 A.2 Trial of ensemble nets for the forward problem . . . . . . . . . . . . . . . . . 129 A.3 RMSE analysis for the inverse problem using dierent training algorithms with 400 and 41 datasets for network training and testing respectively . . . . 130 A.4 Trial of ensemble nets for the inverse problem . . . . . . . . . . . . . . . . . 130 A.5 RMSE analysis with 400/441 datasets (2-80-1 network architecture with 300 epochs gives the best network with an average time of 133 secs for completion 131 A.6 R2 values using 400/441 datasets with dierent network architecture . . . . . 132

LIST OF TABLES A.7 RMSE analysis with 40 data points for training (2-8-8-1 network architecture with 300 epochs gives the best network and it takes an average time of 55 secs

xvii

for completion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.8 R2 values using 40 datasets with dierent networks (2-8-8-1 network architecture with 300 epochs gives the best network) . . . . . . . . . . . . . . . . . . 133 A.9 RMSE analysis with 28 data points (2-8-8-1 network architecture with 1000 epochs gives best network and it takes an average time of 62 secs for completion)133 A.10 R2 values with 28 datasets using dierent networks . . . . . . . . . . . . . . 134 A.11 RMSE analysis with 20 data points (2-80-1 network architecture with 1000 epochs gives the best network and it takes an average time of 168 secs for completion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 A.12 R2 values of using 20 datasets with dierent networks . . . . . . . . . . . . . 134 A.13 RMSE analysis with 8 data points (2-80-1 network architecture with 1000 epochs gives the best network and takes an average time of 168 secs for completion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 A.14 R2 values for 8 datasets with dierent networks . . . . . . . . . . . . . . . . 135 A.15 RMSE analysis with 400/441 data points (8-20-20-1 network architecture with 300 epochs gives the best network and takes an average time of 77 secs for completion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 A.16 RMSE analysis with 40 data points (8-15-15-1 network architecture with 300 epochs gives the best network and takes an average time of 25 secs for completion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 A.17 RMSE analysis with 28 data points (8-8-8-1 network architecture with 300 epochs gives the best network and takes an average time of 14 secs for completion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

LIST OF TABLES A.18 RMSE analysis with 20 data points (8-2-2-1 network architecture with 300 epochs gives the best network and takes an average time of 11 secs for com-

xviii

pletion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 A.19 Network performance on scaled (-1 to +1) 400/441 dataset under dierent transfer functions combinations with RMSE . . . . . . . . . . . . . . . . . . 138 A.20 Network performance on scaled (-1 to +1) 400/441 dataset under dierent transfer functions combinations with R2 . . . . . . . . . . . . . . . . . . . . 139

A.21 RMSE analysis for the forward problem using dierent training algorithms with 218/22 datasets for network training and testing . . . . . . . . . . . . . 140

xix

Nomenclature
List of Abbreviations Acronyms ANN ADD BPNN DACE DOE EA FAA FEA FEM FFNN FRPs GA GBLS GBLSW oS GBLSW S MA MSE MLP NDE NN N SGA II R2 RBF RGA RGAW oS RGAW S RMSE RSM SAO SHM SBX QP SQP Trainbr Description Articial Neural Network Absolute Average Deviation Back-Propagation Neural Network Design and Analysis of Computer Experiments Design of Experiment Evolutionary Algorithm Federal Aviation Administration Finite Element Analysis Finite Element Method Feed-Forward Neural Network Fibre Reinforced Polymers Genetic Algorithm Gradient Based Local Search GBLS without Surrogates GBLS with Surrogates Memetic Algorithm Mean Square Error Multi-Layer Perceptron Non-Destructive Evaluation Neural Network Non-dominated Sorting Genetic Algorithm-II Coeecient of Determination Radial Basis Function Real-coded Genetic Algorithm RGA without Surrogates RGA with Surrogates Root Mean Square Error Response Surface Method Surrogate Assisted Optimization Structural Health Monitoring Simulated Binary Crossover Quadratic Programming Sequential Quadratic Programming Bayesian Regularization

xx
List of Symbols Symbols a A D11 dFi dFpi dFdi dFui dFpdi dFpui E Ei E1 E2 G12 h ith j th k L N ObjS Obj 12 W X Xa , Yb X ,Y [Z, X, a] [Z, X, Y, Xa , Yb ] Z

Description Delamination size Cross sectional area Eective exural stiness % change in natural frequency of an ith mode for beams % change in natural frequency of an ith mode for plates % change in natural frequency of an ith mode for delaminated beam % change in natural frequency of an ith mode for intact beam % change in natural frequency of an ith mode for delaminated plate % change in natural frequency of an ith mode for intact plate Error Error in individual modes of natural frequencies Longitudinal modulus Transverse modulus In-plane shear modulus Thickness of beam Thickness of plate i=1 to 8, Each mode number of natural frequencies i=1 to N, Each individual member in N runs Each individual variable in k Number of input variables for RSM modelling Length of beam Length of plate Initial population Total number of simulation runs Objective function with surrogates Objective function without surrogates Poissons Ratio Density Width of beam Width of plate Delamination location of a composite beam laminate Delamination location of a composite plate Length and width of delaminations for a composite plate In-plane x and y coordinates of the left lower corner of the delamination for the composite plate Delamination signature of a composite beam Delamination signature of a composite plate Interface position of a composite beam Interface position of a composite plate

Chapter

Introduction
1.1 Background
Composite materials are typically composed of structured combinations of discrete and continuous constituents. The stronger and stier discrete phase (the reinforcements) impart unique properties to the weaker and softer continuous constituent (the matrix material). The matrix surrounds and supports the reinforcement materials by maintaining their relative positions via interfacial bonding. The combined constituent materials work together to produce material properties that are dierent from that of each individual elements. Composite materials have been widely and extensively used over the years in the aerospace industry and other engineering applications where weight and cost savings are of great signicance. This is as a result of their excellent advantages such as: high strength and stiness to weight ratios, ease of manufacturing, light weight, ability to withstand fatigue and corrosion in contrast to their metallic counterparts and ability to tailor properties (both mechanical and thermal) [1]. Laminated bre reinforced polymers (FRPs) are one of the most favored kinds of composite congurations due to the ease to tailor their properties in preferred orientations. However, such composite structures in service can suer from a number of failure mechanisms, such as ply/bre breakage, matrix cracking and delaminations. These failures are 1

1.1. BACKGROUND

caused due to static overload, impact, fatigue, design/manufacturing errors, overheating, and lightening strikes [3]. Essentially, delamination is the greatest weakness of composite laminates. Delamination or interlaminar damage is the separation of the laminate plies as a result of low strength. This could lead to loss of structural integrity [4]. Delaminations can easily spread throughout the whole laminate of a composite structure upon repeated loading causing costly and/or disastrous failures when undetected. Online vibration based monitoring using shifts in natural frequencies can provide early warning of occurrence of delamination. While non destructive testing (NDT) techniques such as ultrasonic inspection, thermography, optical holography and mechanical impedance have been utilized for delamination assessment in composite laminates [5; 6; 7; 8], they cannot be used for real time and online damage detection [9]. Moreover, most of these techniques are mainly applicable for local inspection of limited areas of a structure and hence are labor intensive, time consuming, and cost ineective when considering large structures [10]. This leaves room for vibration based techniques as a structural health monitoring (SHM) tool to identify and assess delamination damage of an in-service structure globally. Vibration health monitoring and integrity assessment of structures are necessary to ensure rapid and immediate assessment of the health condition of a structure in terms of reliability and safety, as well as optimizing the economic impacts of structural failure before degradation hampers the target objectives of in-service structures. SHM exploiting vibration measurements are global methods based on the principle that degradation due to damage in a structure changes vibration parameters like; natural frequencies, mode shapes and damping characteristics. Therefore, it is feasible to analyze any of the measured vibration parameters to characterize and identify the presence of damage using inverse modeling techniques. Harnessing natural frequencies as a vibration parameter has proved successful for delamination damage identication because they can be determined easily [11; 12]. An ap-

1.2. STRUCTURAL HEALTH MONITORING (SHM)

proach for online SHM exploits the measured shifts in natural frequencies due to degradations to detect the nature and extent of damage. Utilizing natural frequencies for damage detection derives its benets from the fact that changes in natural frequencies are the easiest to measure accurately in real time when compared to mode shapes or damping parameters [13]. However, some notable limitations of natural frequencies for damage detection are their insensitivity to low severity structural damages and requirement of large data size [14]. Mode shapes are also commonly utilized as indicators for structural damage detection. Changes in mode shape measurements before and after damage have been applied indirectly as measures of modal strain energy changes [15]. Especially favorable in terms of mode shape based damage identication is the damage index (DI) method, which is dependent on changes in modal strain energy. In spite of the eectiveness of the DI method, some challenges such as the inability to identify light damage and the need to take measurements over a large number of points across the whole structure are still being faced [16].

1.2

Structural health monitoring (SHM)

In engineering applications, structural health monitoring (SHM) is the key to securing

condence in the utilization of FRPs. Before now, structures were monitored by carrying measuring devices to the site each time a set of readings are required. Currently, using vibration measurements for SHM can help in processing extensive amount of data o-site. In this work, delamination detection in composite laminates are evaluated in the context of SHM. SHM is a reliable system which has the ability to detect and interpret adverse changes in a structure as a result of damage. The key motivation driver for SHM is that knowing the integrity of in-service structures on a continuous real-time basis is vital. As a result, SHM helps to eliminate catastrophic failure through early detection of problems, optimal use of structures, prevention of regular shut downs of in-service structures [10]. It also helps to minimize downtown and aids the inspection of areas dicult to assess. SHM also replaces

1.2. STRUCTURAL HEALTH MONITORING (SHM)

periodic maintenance with long term maintenance schedules. This reduces maintenance labor and minimizes human involvement, and thus improving safety and reliability. Analogy In the eld of medicine, a doctor regularly monitors patients blood pressure to determine the health state of the patient by observing deviations in blood pressures using sophisticated equipments. Similarly, engineers monitor the integrity of a structure by measuring the changes in the responses of the structure which can lead to structural failure. Taking immediate response in both scenarios can avoid catastrophic implications [17]. An eective SHM system involves the use of expertise in many disciplines giving rise to solution of multidisciplinary problem using nite element analysis, optimization methods, structures and materials, computers, communication and electronics, real-time controllers, and intelligent processing etc. The aim of the technology is not simply to detect structural failure, but also provide an early indication of damage. The early warning provided by an SHM system can then be used to dene remedial strategies before the structural damage leads to failure. Benets of SHM Monitoring and evaluating the integrity, in-situ behavior and health condition of a structure accurately and eciently while it is in service optimizes resources for repair/replacement, reduces down time while increasing productivity and thus ensuring safety of lives and property. Classications of SHM SHM can be classied into two key subsystems: Periodic monitoring and Continuous monitoring (Passive and Active) [17]. Periodic monitoring involves tests to determine changes in structures as a result of damage. Passive monitoring infers that measured data are obtained from eld whereas in active monitoring the structure is actuated with predened signals. The real benets of SHM system is particularly in giving a wealth of information that are not easy to get from eld data. Hence, a typical SHM system is grouped into the following: A sensory system for acquisition of data A communication system

1.3. APPROACH TO THESIS Intelligent processing system Storage system for storing of processed data A modeling and diagnostics system for damage detection A retrieval system

The content of this research focuses on SHM as a diagnostic tool for delamination damage detection because it is the most important subset of SHM system. The functional diagnostic tool of SHM is integration of various sensing devices and ancillary systems [17]. In this aspect of SHM, the abstract data numbers are converted into quantities that relate directly to the responses of a structure. In this research, the natural frequency measurements will be converted into quantities of delamination parameters via inverse algorithms.

1.3

Approach to thesis
Damage detection is essentially the solution to the inverse problem. However, it may be

necessary to solve the forward problem to generate data. Approaches dependent on variation of natural frequencies are thus categorized into inverse and forward problems as shown in Figure 1.1. The inverse problem involves deducing delamination parameters (inter-laminar position, size and location of delamination) from account of given shifts in natural frequencies by directly using Articial Neural Networks (ANNs), Response Surface Method (RSM) and other inverse methods. The forward problem is to determine the natural frequency changes of a given structure based on delamination parameters. These approaches however require a large pool of datasets which have to be harnessed from experiments or simulations. Therefore, in practice, extensive data characterizing the delamination damage as well as an inverse solver must be included in any delamination detection method. This means that in order to characterize the delamination damage parameters, computationally expensive numerical models are required to sift through all possible delamination signatures/congurations in

1.3. APPROACH TO THESIS

order to select the best estimates of delamination parameters. This sifting is an intrinsically iterative process that requires thousands of computationally expensive simulations which motivates the concept of surrogate modelling and surrogate assisted optimization (SAO). SAO techniques using the local and global optimizers are employed to determine the delamination parameters. This enhances the search performance and quality of results in a rapidly shorter time while curtailing the high computational cost. ANN is a system that contains many simple and highly interconnected neurons which process information based on an architecture inspired by the structure of the cerebral cortex of the brain [18]. It is a response surface approximation method that is based on the concept of Articial Intelligence (AI). ANN oers capabilities such as self-adaptiveness, generalization, abstraction and suitability for real-time applications. These features make ANNs powerful tools for vibration based damage identication. In recent years, the concept of ANN has been extensively applied in delamination detection in composite structures successfully [19]. In the current work, ANN is to provide surrogate models to computationally expensive FE models employed for the database generation. This is because the use of high delity computational analysis tools (FE methods) in the optimization process increases the optimization time to an inhibitively great extent, mainly due to the time taken for the nite element models to compute the vibration response. In this case, surrogate models come as great substitutes for the high delity models in the optimization process because of their advantage in reducing the optimization time. Thus, once a neural network is eectively trained, they are capable of being used for future interpolation and approximation. RSM approximates the output of a given system as a function of some input design variables by solving a system of non linear equations. This method is eectively employed as an inexpensive low order approximation model for delamination prediction. The selection of the sampling points for building inverse models is vital because the prediction capabilities of an approximation function is highly inuenced by the sampling points in the given design space [20]. The K-means clustering technique is used so as to

1.3. APPROACH TO THESIS

ensure that the sampling points are evenly distributed over the design space. This method gives a systematic and ecient means of analyzing the complete design space. It explores the high-dimensional design space and screens the most clustered design points corresponding to the set of design variables. Gradient based optimization schemes are local optimizers because they get stuck at the rst optimum obtained during the search process. When applied to continuous and nonmultimodal problems, this algorithm gives the best performance than any other optimization scheme. However, it is highly unsuitable when searching for global optimum in a multi-modal optimization problem [21]. This algorithm is eectively employed for delamination prediction because of its rapid and relative small number of function calls in comparison with its global search counterpart (RGA). Global optimizers like the Real-coded Genetic Algorithm (RGA) are most promising in both discrete and continuous problems due to their robust and random nature of search but with a signicantly high computing cost [22]. The high computing cost involved in deducing the function objectives can be greatly reduced using surrogate models, hence taking advantage of their global optimization behavior. RGA is adopted in this thesis as an ideal method to eectively manipulate the optimization task for detecting delaminations. A prediction strategy for detecting delaminations in composite beams and plates is proposed. The approach relies on the use of K-means clustering for identication of training data with adequate spread, surrogate models for approximation, ANN, RSM and gradientbased and global search strategies. The aim of using surrogates is to replace expensive numerical simulations while enhancing computation eciency. On the other hand, application of ANN in delamination prediction usually suers from the requirement of large data sets for training. Existing methods have employed large bank of simulation data for network training [1]. However, it is dicult to conduct a large number of runs of computations or experiments for delamination prediction. K-means clustering is eectively used to reduce the size of data set required for ANN training. The proposed approach is applied to 2, 3,

1.4. LIMITATIONS OF CURRENT APPROACHES

and 5-variable problems. To test the validity of the algorithms in detecting delamination parameters, results were compared with the predictions obtained using experimental results. Results of numerical and experimental investigations validated the viability of the proposed methodology. Uncertainty quantication on the performance and robustness of the proposed methodology is also studied. This strategy adds articial error and noise to natural frequency measurements. Performance analysis on test cases with noise free and noise polluted natural frequency data is used to demonstrate the eciency of the proposed approach.

Figure 1.1: Approach to thesis.

1.4

Limitations of current approaches

The following are some of the notable research limitations in vibration based structural

health monitoring of composites laminates; Almost all the work in literature deal with one or at most two variables i.e predicting

1.5. OBJECTIVES OF RESEARCH either delamination size or location at mid-planes in composite beams.

Most of the damage detection methods that have been reviewed attempt to identify delamination by solving an inverse problem without solving the forward problem via surrogate models, which often requires the construction of numerical models. This dependency on rsthand numerical models, which are computationally expensive makes direct inverse solution approach unpromising for SHM. While neural network based approaches have been found successful due to their adequate generalizing capabilities, they require a large number of training cases. Most of the work have employed large datasets for network training. In this work, the training data is eectively reduced to a minimum size using K-means clustering concept. Most of the works validate their results against numerical data without considering experimental uncertainties. To the best of authors knowledge, there are no reports of any SHM approach that investigates the eects of noise and error on its behavior.

1.5

Objectives of research
The main aim of this research work is to develop an ecient approach for delamination

damage detection. The specic objectives therefore include: To develop a SHM tool for delamination detection in composite laminates using vibration measurements To conduct numerical simulations by nite element modeling for composite beams and plates with and without delaminations To validate the more expensive numerical simulations with theoretical/analytical models for delaminated and undelaminated composite beams

1.6. RESEARCH SCOPE

10

To engage eective surrogate models in lieu of computationally expensive simulation driven design via FE models while enhancing optimization search performance To determine minimum datasets for network training using K-means clustering algorithm. This is done by investigating various types of damage training data scenarios to obtain the optimized dataset required for delamination prediction To investigate the approximation capability of the response surface model (RSM) in delamination detection and study the input variables that is most signicant or inuential to the RSM models To investigate dierent surrogate models and study their limitations and capabilities To investigate whether the solution of the forward problem via optimization algorithms are more ecient for delamination prediction i.e. to investigate whether the use of surrogate models in the optimization loop improve the accuracy of delamination detection at a low computation expense To investigate and compare the eciencies of dierent inverse algorithms in vibrationbased delamination damage identication To study the performance and robustness of dierent algorithms against noise in the numerical data (analysis of uncertainties in the prediction of failure in composite structures, and estimation of the amount of associated uncertainty).

1.6

Research scope
In this thesis, algorithms for delamination damage detection in laminated compos-

ite beams and plates are studied. Integration of surrogate models with optimizers is of peculiar interest in this study. The motivation is derived from the concept that constructing a surrogate model to approximate any expensive function can substantially reduce the

1.7. SUMMARY OF CONTRIBUTIONS

11

computational cost during the course of optimization and improve the optimization search performance. This is of signicant importance, as in many real-world problems function evaluations are computationally expensive. ANN, RBF, RSM, Kriging are some examples of surrogate models [23]. The choice of surrogate models depends on the problem under consideration. In this research, ANN is adopted to provide surrogate models to computationally expensive FE models and are found to be very eective. This approach justies the need for surrogate models in reducing the optimization cycle time and exploring the complete design space with minimal computational cost. Therefore, this thesis is within the following scope: The investigation of existing vibration based damage detection methods The study of ANN, RSM and optimization based damage identication methods suitable for delamination detection The validation of proposed delamination damage identication algorithms Investigation of the capabilities and limitations of employing dierent algorithms in vibration-based damage identication in composite beams and plates.

1.7

Summary of contributions
The main contribution of this research is the investigation and validation of inverse

algorithms for delamination damage detection in composite beams and plates using changes in natural frequencies. The contributions are summarized thus: Development of a working approach for delamination detection in composite laminates using variations in natural frequencies. Application of the developed approach to error and noise polluted measurements of natural frequencies to show robustness.

1.8. ORGANIZATION OF THESIS

12

The developed approach is validated on numerical and published experimental results for composite laminates.

1.8

Organization of thesis
The work done in this research is detailed in the following order: Chapter 1 outlines an introduction to the background of the research work, basic

concepts of SHM, ANN, RSM and optimization algorithms, objectives of the research study and motivation, statement of the problem, thesis scope and signicant contributions. In Chapter 2, a literature survey on vibration based SHM inclined to analytical, numerical and experimental studies of natural frequencies of delaminated and undelaminated composite structures is presented. A detailed account of work on inverse algorithms like ANNs, RSMs and optimization algorithms used for delmaination detection is presented. The research gaps that exist in the eld of vibration based SHM for delamination detection are also highlighted. The formulation of the optimization problem and nite element analysis (FEA) tool used in this work is described in chapter 3. The analytical/theoretical formulation for composite laminates is also presented in this chapter. Chapter 3 ends with validation of the numerical results by FE modeling with the theoretical results obtained. Chapter 4 introduces the computational intelligence methods used in this work such as ANN and numerical optimization techniques (RGA and GBLS by SQP). The concept of RSM is also described. Analysis and development of the background for the various proposed algorithms are detailed in this chapter. The selection of optimum datasets using the concept of K-means clustering algorithm is also well presented. In this chapter, the analysis of performance of dierent network architectures, transfer functions and training algorithms are studied. The approach of developing and integrating surrogate models in the optimization loop is also described. The models are integrated into an iterative optimization

1.8. ORGANIZATION OF THESIS procedure that is denoted as surrogate assisted optimization (SAO).

13

Chapter 5 presents results of the proposed current algorithms applied to delamination detection in composite beams with 2 and 3-variables. Also Chapter 5 presents validation of the developed algorithms with experimental data and the results of articial errors and noise on the developed algorithms. Chapter 5 ends by extending the application of the developed algorithms to more complicated composite plates (the 5-variable problem). Chapter 6 compares the prediction eciencies of dierent proposed inverse algorithms for delamination detection. The last chapter 7, summarizes the ndings and conclusions of the thesis and nally presents some of the challenges that need to be addressed, and recommendations for future work.

Chapter

Literature review
2.1 Overview
Composite materials oer a number of potential advantages in the aerospace eld, particularly in aircraft structures because of their low weight and critical safety requirements [4]. For instance, the advanced materials research program of Federal Aviation Administration (FAA), discovered that for every pound of weight saved on a commercial aircraft, there is a 100-300 US-Dollars cost saving during the service life of that aircraft [24]. This encourages many components of transportation airframes (whole empennages, main portions of wing and fuselage structures) to be designed from composites. Signicant weight savings is therefore the basic objective that has facilitated the increased utilization of composites, specically in aircrafts, over metallic structures. The study of damage detection methods in structures is essential because the ability of structures to fail due to damage in form of delaminations, matrix cracking and bre breakage causes undesirable risk in terms of safety, economics, management and overall performance of structural components. Delamination occuring as a result of separation of the laminate materials causes excessive degradation of structures thereby imposing great threats to safety and economic conditions of composite materials. This results in massive weakening of the mechanical properties of structures due to the absence of structural integrity [4; 25]. The growh of delaminations at the interface 14

2.1. OVERVIEW

15

of two layers in composites are due to impact, fatigue and imperfections as a result of manufacturing. They are the endpoint of interlaminar shear stresses, causing reduction in the compressive residual strength of the composite laminate [26]. Failure assessments provided from airworthiness directives or ADs (compulsory inspection documents which are analyzed in the occurrence of a problem in the aviation industry) reported that on November 12, 2001, a disastrous breakdown of a composite rudder and vertical n occurred on an Airbus 300, killing 265 on board passengers. Similarly, in March 2005, the rudder of another Airbus aircraft (A-310) collapsed. These disasters were caused as a result of intense and huge delamination patterns of the composite parts that were unpredictable leading to eventual loss of its excessive strength and stiness [27]. Structures like aircrafts should possess the ability to resist the wear and tear impacts subjected to them during their service life. Wear and tear subsurface kind of damage is dened as barely visible impact damage (BVID). This has the potential to multiply and deteriorate structures until they eventually fail with time [28]. Delaminations can be associated with this kind of damage. Delamination is also the most prevalent cause of failure in compression and the need for its detection. Among other non-destructive inspection methods, vibration based method can be used to eectively and eciently determine the extent of delamination damage via frequency shifts monitoring. The problem of delamination in composites is therefore a broad area covering eects on static loading, fatigue, damage tolerance, vibrations, etc [26]. Only the eect on modal frequencies is of interest in this study, i.e the forward problem which is the eect of delamination on natural frequencies, and the inverse problem (assessment of delamination parameters from shifts in frequencies). This literature is done bearing this in mind. However, the forward problem is not directly relevant to SHM, except in that it is required to generate data for the solution of the inverse problem (Figure 1.1). The inverse problem is of most relevance to this study. This chapter therefore reveals previous studies with respect to vibration based damage

2.2. OVERVIEW OF SHM METHODS FOR DAMAGE DETECTION

16

detection algorithms. The discoveries from previous research gave rise to the motivation and foundation on which this research study is carried out. It commences with a preamble of highlights associated with immense need for structural health monitoring. To enable a good survey of the related literature, Figure 2.1 is adopted as an outline for review and also classication of four levels of damage identication is introduced. In this chapter, numerical, analytical and experimental methods that are used to characterize structural damage by examining changes in measured natural frequencies are presented. This follows with relevant literature of ANNs, RSM and optimization techniques as inverse algorithms used in the eld of vibration-based damage identication. Finally, the research gaps that are addressed in this thesis are summarized.

Figure 2.1: Outline for literature review.

2.2

Overview of SHM methods for damage detection

Degradation in a structure results to changes in the natural frequencies, mode shapes,

damping and time series. Since the measurement of natural frequencies is easier to determine

2.2. OVERVIEW OF SHM METHODS FOR DAMAGE DETECTION

17

than that of changes in damping and mode shapes of a structure, damage can be detected from dynamic analysis using natural frequencies obtained analytically, experimentally as well as numerically. Therefore, several researchers have studied and reviewed damage detection techniques based on measuring changes in the structural dynamic characteristics. The principal concept of damage identication can be dened as approaches that indicate the changes in structural responses due to degradation. Damage can be dened as the introduction of changes into a system that adversely aects its current or future performance [29]. Structural damages can be classied as global and local. Traditional Non-destructive Inspection (NDI) techniques such as C-scan, radiography, eddy current and liquid penetrant are local methods with high sensitivity to damage detection while ensuring structural integrity. However, these conventional NDI techniques cannot be implemented easily online for SHM. They also require accessibility of the vicinity of damage prior to identication but in practical cases the location of damage under inspection is inaccessible [30]. These limitations have signicantly motivated the introduction of SHM systems in recent years. SHM systems basically involve incorporating a Non-Destructive Evaluation (NDE) technique into a structure to remotely monitor for damages continuously with less human intervention [30]. Due to the shortcomings of NDI methods, engineers have resorted to SHM tools that can be engaged online, require less human intervention and monitored remotely in real time via the use of intelligent algorithms. Potential SHM methods include; Lamb waves, Fibre optic sensors, acoustic emissions, impedance and vibration based methods. This research concentrates on vibration-based global damage identication method for SHM. However, an overview of other SHM methods is also described. To aid in this literature survey, the system of classication for damage identication methods, as proposed by Rytter [31], which denes four levels of damage identication are presented as follows: Level 1: Determination that damage is present in the structure Level 2: Determination of the geometric location of the damage

2.3. POTENTIAL SHM METHODS EXCEPT VIBRATION BASED METHODS 18 Level 3: Quantication of the severity of the damage Level 4: Prediction of the remaining service life of the structure Vibration-based damage identication methods which do not use any structural model but only use variation in changes in natural frequencies primarily provide Level 1 damage identication. When vibration based methods are coupled with structural models, Level 2 and Level 3 damage identication can be obtained. Level 4, prediction of failure life of a structure is generally associated with the elds of fracture mechanics, fatigue life analysis, or structural design assessment [29]. The scope of this research is to provide Level 1, Level 2 and Level 3 damage identication. This is the backbone for which this literature survey is based.

2.3

Potential SHM methods except vibration based methods

An area of non-destructive delamination monitoring is associated with smart structures

and technologies such as optical bres and lamb waves. The use of smart structures like Bragg gratings to measure frequencies and their use for detecting delaminations directly cannot be over looked [2]. Smart structures are structures which incorporate their own embedded sensors, actuators, identiers and controllers into part or whole SHM system [30]. Existence of delamination aects sensors or the signals they sense dierently. SHM systems are nowadays applied mainly to light-weight high-performance composite structures. Grouve et al. [32] measured experimentally the resonance frequencies as a function of delamination parameters (location and size) and laminate lay-up for laminated cantilever beams using embedded bre Bragg grating strain sensors. They showed that the experimental results were in agreement with analytical results for the undamaged and damaged test specimens. However, the major limitation of their technique is the low sensitivity of the

2.4. METHODS BASED ON VIBRATION MONITORING

19

bres at the surface for internal damages. Also, this technique cannot be used for existing structures because the bres have to be embedded during the manufacturing process. Nagesh Babu and Hanagud [33] and Hanagud et al. [34] proposed the use of piezoelectric sensors to detect delaminations in composite beams. They developed analytical models for automatic detection and control of delamination growth using piezoelectric sensors and actuators. Kim and Yiu [12] explored the use of simple anti-symmetric Lamb wave mode for delamination detection. Lamb waves as a non destructive method was applied to detect and locate through-the-width delamination in ber-reinforced plastic beams. This method comprises of a piezoelectric patch and an accelerometer both mounted near the support. Distortions of waveforms due to boundary reections are reduced by actuated frequencies. The developed methodology was applied to locate delaminations in some fabricated Kevlar/epoxy beam specimens. The errors associated with the predicted damage positions range from 4.5% to 8.5%.

2.4

Methods based on vibration monitoring

Damage detection methods of structural systems based on changes in their vibration

characteristics have been widely and extensively employed over the years in research laboratories. Generally, vibration based damage detection methods are classied into techniques based on measurement of; natural frequencies, damping, mode shapes, frequency and time responses. The key focus of this work is on measurement of natural frequecies and will be mainly reviewed. Doebling et al. [9] detailed an intensive review of vibration-based damage identication methods. The vibration-based damage detection methods based on changes in natural frequencies, curvature or strain modes, modal strain energy, dynamic exibility, and other signal processing methods such as Wavelet techniques, empirical mode decomposition and Hilbert spectrum methods are discussed in their work. In almost a similar manner,

2.4. METHODS BASED ON VIBRATION MONITORING

20

Doebling et al. [35] reported a summary of vibration-based damage identication methods. Sohn et al. [36] then gave an overview of the literature on vibration based health structural monitoring from 1996 upto 2001.

2.4.1

Studies on the eects of delaminations on natural frequencies

A great number of researchers have employed analytical, numerical and experimental techniques for identication of the presence of delamination damage from detected changes in natural frequencies. Reduction in natural frequencies indicates that damage has taken place providing damage identication only at Level 1. Previous studies [33; 37; 38; 39; 40; 41; 42] on the free vibration of delaminated beams have shown that by determining the variation in natural frequencies of beam specimens, it is possible to identify both the location and severity of damage. From their investigations, it can be deduced that natural frequencies can be greatly aected by the manifestation of interfacial delaminations. It is therefore an established fact that when damage occurs, the natural frequencies of a structure decrease. However, locating the damage from the changes in natural frequencies alone is dicult as modal frequencies are global properties of the structure and hence cannot provide spatial information about structural changes. This gives room to consider additional properties if necessary, such as displacement or curvature mode shapes, which provide spatial information about the damage. An intensive amount of research has been undertaken that utilizes changes of natural frequencies as damage indicators. The main reason for its use is that natural frequencies are easy to determine with a relatively high level of condence [11]. Also, natural frequencies have much less statistical variation from random error sources than other parameters, which make them a more robust means in the assessment of damage [29] Extensive and detailed review has been provided on free vibration analysis of delaminated composite laminates using numerical and analytical models by Della and Shu [43].

2.4. METHODS BASED ON VIBRATION MONITORING

21

Salawu [44] provided a review on using natural frequency data to identify structural damage. He noted that the advantages of using natural frequencies are because they are easy to implement and the cost of implementation involved is relatively low. Cawley [38] also published an excellent review of vibration methods used for nondestructive evaluation (NDE) of laminated structures. He reviewed a number of methods suitable for NDE using natural frequencies. Cawley and Adams [45] studied dynamic parameters such as natural frequencies and mode shapes for damage detection as the earliest workers. They proposed a method to detect and locate a variety of types of damages by noticing the change in natural frequency. Thereafter, Mujumdar and Suryanarayan [40] reported an analytical model based on the eect of delamination on the natural vibration characteristics of beam type structures with through-the-width delaminations parallel to the surface of the beam. They assumed that the delaminated layers are constrained and termed their model as Constrained Model. They validated their model with results from an extensive experimental investigation. Tracy and Pardoen [41] at the same time with Mujumdar and Suryanarayan investigated that the presence of delamination leads to decrease in natural frequencies using the same so called Constrained Model early proposed Mujumdar and Suryanarayan. They constrained the upper and lower intact of the delaminated segments to be equal for a simply supported composite beam. They validated their theoretical results with experimental vibration studies on the eects of delamination of natural frequencies on the composite beam specimens. Nagesh Babu and Hanagud [33] and Paolozzi and Peroni [42] used nite element models to study the problem of delaminations in composites. They performed nite element (FE) analysis to correlate the natural frequency shift with the extent of delamination damage. Their results conrmed the conclusions of Tracy and Pardoen regarding a reduction in natural frequencies as a result of delamination. Shen and Grady [39] studied the eect of inter-ply delaminations on natural frequencies

2.4. METHODS BASED ON VIBRATION MONITORING

22

and observed that frequency is very sensitive to the size and location of the delaminations. They considered the eect of coupling between longitudinal and bending vibration on free vibration of laminated composite beams with interply delaminations. Natural frequencies and mode shapes are obtained both analytically and experimentally. Coupling eect is shown to signicantly aect the calculated natural frequencies and mode shapes of the delaminated beam. They developed equations of motion for a delaminated beam using the Hu-Washizu variational principle and Timoshenko beam theory, and also reported their experimental results. Saravanos [46] developed an exact analytical procedure for the analysis of delamination eects on the natural frequencies, mode shapes and modal damping of composite beams. He presented an approach that represents a version of a layer-wise theory which allows tracing of a number of delaminations through the thickness of the laminate. Banerjee and Jayatunga [47] investigated the free vibration response of a composite beam using dynamic stiness matrix method and employed both geometric and material coupling between bending and torsional motions. Their analysis is based on Hamiltons principle resulting in the solution of the governing dierential equations of motion in free vibration using closed analytical form for harmonic oscillation and satisfying all the necessary boundary conditions. Thus, the dynamic stiness matrix with respect to frequency relating the amplitudes of loads to the vibrational responses is computed. Using Wittrick-Williams algorithm, the corresponding dynamic stiness matrix is used to compute the natural frequencies and mode shapes of a composite beam. The developed theory is applied for modal analysis of high aspect ratio composite wings. Luo and Hanagud [48] presented a novel model for composite beams with throughthe-width delaminations. The governing equations of vibration incorporates shear eects, rotary inertia terms, and bending-extension coupling. Nonlinear interaction, due to piecewise linear spring models between the delaminated sublaminates, is also included. Based on this model, eigen-solutions for vibrations of perfect and delaminated beams are found analytically.

2.4. METHODS BASED ON VIBRATION MONITORING

23

Dynamic behavior results predicted by their model is consistent with experimental results. The vibration of delaminated composite structures has been investigated by Lee [37]. He applied a layer-wise theory to the buckling of a composite beam and considered multiple delaminations through the thickness of the beam using a nite element formulation. He discovered that a layerwise beam theory approach is adequate for free vibration analysis of a laminated beam with delaminations. A comparative analysis between the results from theory and numerical results was carried out considering the inuence of lamination angle, delamination size and location on the natural frequencies of a delaminated beam. However, the bending-extension coupling induced by the delaminations was neglected or partially ignored in almost all the studies reviewed so far. Moreover, most studies were restricted to beams with only single (one) delamination. Ju et al. [49] conducted numerical simulations for the vibration analysis of composite beams with multiple delaminations using the nite element method. Their model incorporates bending-extension coupling and transverse shear deformation. This provides excellent eciency when applied to the vibration analysis of composite beams with multiple delaminations under arbitrary boundary conditions. To the best of authors knowledge, only few studies on the vibration of delaminated composite plates have been reported. Chakraborty and Mukhopadhyay [50] conducted both experimental and numerical investigations of the free vibration of composite FRP plates. It was seen that their experimental data were in agreement with the numerical results. Similarly, Campanelli and Engblom [51] and Ju et al [52] investigated the vibration characteristics of delaminated composite paltes. Tenek et al. [53] also studied a three-dimensional nite element method to analyze the natural frequencies of delaminated composite plates as well as the delamination dynamics over a broad range of frequencies. Such a three-dimensional analysis is accurate but is very computationally intensive.

2.4. METHODS BASED ON VIBRATION MONITORING

24

2.4.2

Solution to the inverse problem

Solution to the inverse problem can be resolved directly using inverse algorithms such as ANN, RSM and optimization techniques. So many researchers have employed FEA for simulation of natural frequencies used for the solution of the inverse problem for delamination detection [1; 3; 4; 15; 54; 55; 56; 57]. 2.4.2.1 Intelligent inverse algorithms for vibration based damage detection

Vibration based damage detection is an inverse problem where causes are essentially deduced from their eects. Often a unique solution does not exist for an inverse problem, especially when the relationships are very complex and only limited data are avalaible. The review of works employing ANN and optimization of structural parameters using gradient based local search and genetic algorithms (GAs) will be described in this section. Many inverse algorithms based on computational intelligence concepts such as ANN and optimization of structural parameters using gradient based local search and genetic algorithms (GAs) have found their applications in delamination identication mostly in composite laminated beams. This is because measurement of variation in natural frequencies alone provides Level 1 type of damage and hence to provide Levels 2 and 3 kinds of damage, the need to adopt inverse algorithms arises. Vibration based SHM methods utilizing neural networks and optimization techniques as inverse algorithms appear in previous works [1; 2; 4; 9; 19; 54; 55; 57; 58; 59; 60]. Zou et al. also gave a thorough review of online damage identication and health monitoring techniques for delaminated composite structures [61]. Direct optimization based methods The usual approaches of vibration-based methods applied to detect structural damage incorporate optimization algorithms to minimize the errors between the measured vibration data and the numerical data. Optimization methods can be applied to identify structural damage by minimizing the objective function, which directly compares the changes in the measurements before and after damage between simu-

2.4. METHODS BASED ON VIBRATION MONITORING

25

lated frequencies and the actual ones. Optimization of structural parameters using gradient based local search and genetic algorithms (GAs) have been utilized in delamination identication in laminated composite beams. These techniques however require large pool of datasets which have to be harnessed from experiments or simulations. Nag et al. [54] identied only mid-plane delaminations in composite beams by using combination of spectral estimation of damaged spectral nite element through modelling and a genetic alorithm. They consisdered Graphite/Epoxy laminate with a unidirectional 0 ply stacking sequence. Their results show that by the introduction of damaged spectral element through modelling, ecient simulation and prediction of delaminations in composite beams can be obtained for online/in-service structural health monitoring. They used simulated data from spectral nite element model to compare with the baseline/reference response data from experiments for the tness function. In all the case studies, they considered the baseline congurations to have delamination near the mid-span of the beam. They accurately predicted delaminations. Harrison and Butler [58] employed two numerical optimizers (gradient-based local search and genetic algorithm techniques) to locate delaminations in composite beams. In most of their validation cases, it was seen that the interface position of the delamination was not identied perfectly and the delamination size was under-predicted. However, their adopted GA located better solutions than the gradient-based method as evident by a lower value in objective function at the expense of high computational cost. Krawczuk et al. [62] harnessed numerical models using genetic algorithm and ANN to detect delaminations in composite beams. They formulated an objective function which is based on changes in natural frequencies and the Damage Location Assurance Criterion (DLAC) using only the rst 4 natural frequencies. They generated 2277 datasets for network training. Their analysis are compared with those from a neural network in which the rst 4 natural frequencies are used as inputs, and the onset location of delamination, delamination end location, and the number of the delaminated layers are used as outputs. GA results

2.4. METHODS BASED ON VIBRATION MONITORING

26

are shown to be better than the NN results because of the very small size of the training population representing delamination layer numbers. The genetic algorithm converges after a reasonable number of generations and could be improved by incorporating more processes that are observed in nature such as elitism. The neural networks performance in detecting the delaminated layer location across the thickness was poor notably because of the use of small training data. Keilers and Chang [63] proposed a delamination identication method that minimizes the dierence between measured and predicted frequency responses using built-in sensors and actuators for composite structures. The approach comprises a damage selector, an analytical model in lieu of FE model, and a comparator, for predicting the location and size of a single mid-plane delamination in composite beams. Using the measured amplitude responses, the identication method predicts midplane delaminations greater than about 10 percent of the total beam length but failed to predict smaller delaminations. Research studies on the application of vibration-based assessments for health monitoring of composite structures for 3-variable problem are more recent. Most recently, Su et al. [2] solved a 3-variable problem (interlaminar position, delamination size and location) by exploring GA and ANN to evaluate delaminations in clamped composite beams. Their GA method required 2000 function evaluations and large population size. Using experimental data, their model reported a maximum error of 26.5%. Surrogate Assisted Optimization (SAO) based methods As described earlier in Section 1.3, the solution to the forward problem basically involves initial database generation and optimization of the delamination parameters using the generated database through inverse formulation. Authors [15; 57] adopted the forward approach to solve the inverse problem by building approximation models with initial database generation. Chen et al. [57] developed an approach that combines genetic algorithm and neural network technique with a large set of generated data for delamination detection in composite

2.4. METHODS BASED ON VIBRATION MONITORING

27

laminates using numerical analysis. They used ANN to train a back propagation neural network using 1624 datasets with a network achitecture of 3-70-70-10 and validated their trained neural network using 1104 datasets. The three inputs to the network are the thickness location, lengthwise location and delamination size and the outputs to the network are the rst ten natural frequencies. The trained ANN model provides an encapsulation/approximation of the numerical model. They customized GA using ANN approximations of the numerical model to detect delaminations in laminated composite beams and plates over 9600 function calls. Their approach failed to predict accurately small delamination sizes and near surface delaminations. Furthermore, they did not however validate their methodology with experimental results. In a very more similar approach, Deenadayalu et al. [15] adopted a muilti-layer feedforward network to train a neural network using simulation data (294 datasets). The trained ANN model provides an encapsulation/approximation of the numerical model. They formulated an objective function based on a damage index distribution which quanties the dierence in modal strains between the damaged and undamaged laminates. With the combination of ANN as function approximators and modied GA, they detected through-the width delaminations in composite laminates with 80000 function evaluations. Though they failed to validate their approach with experimental data. Articial-neural network based methods Vibration based damage identication is a pattern recognition problem where changes in the vibrational properties of a structure are associated with certain characteristics of damage [10]. The capability of ANN to map complex non-linear relationships and their robust nature in the presence of noise make them promising tools in vibration based damage identication. Since natural frequencies are affected to dierent extents depending on the location of the damage, analyses become too complicated to be handled. Therefore, ANNs can be used in the post-processing of vibrationbased data to extract the patterns and to solve this inverse problem. Bishop [64] reviewed

2.4. METHODS BASED ON VIBRATION MONITORING several applications of neural networks in vibration based damage detection.

28

Researchers have used ANN approximation capability successfully as surrogate models for computationally expensive FE models of composite laminates for delamination detection [15; 57]. Section 4.4 gives more detailed analysis of ANNs. Okafor et al. [1] and Islam and Craig [55] applied articial neural networks depending upon a pool of simulated frequencies for training purposes to predict the sizes and/or locations of delaminations in composite beams based on the rst few natural frequencies. Okafor et al. [1] demonstrated strategies based on ANN technique with 850 simulated datasets using natural frequencies as network input to predict only delamination size at the mid-plane. Their results, using experimental data show a prediction error between 0.25% and 19% for the delamination size. Islam and Craig [55] combined neural network and embedded piezoceramic sensors to detect the presence of damage in composite structures. Their approach is based on modal analysis using response measurements from embedded piezoelectric sensors for delamination detection. They predicted delamination with maximum error of 27% and 10% in location and size respectively. Watkins et al. [60] employed 1066 datasets generated from analytical model for feedforward back propagation neural network training, delamination size and location at midplane were predicted with an average error of 5.9% and 4.7% respectively. Zheng et al. [4] employed a nite element analysis to obtain simulated modal frequencies and applied a new neural network learning procedure, called genetic fuzzy hybrid learning algorithm (GFHLA) with 283 training datasets to predict delamination size and location at mid-planes with an average error of 4.7% and 4.3%, respectively. All their validation cases were based on simulation results. Valoor and Chandrashekhara [59] investigated the feasibility of neural networks in determining only mid-plane delaminations in laminated composite beams. They essentially

2.4. METHODS BASED ON VIBRATION MONITORING

29

modied the analytical model developed by Okafor et al. [1] by taking into account the poisson eect and the transverse shear deformation to obtain the natural frequencies. They trained a back-propagation neural network to predict the delamination size and location from the natural frequencies of the beam with a maximum average error of 15%. They found that errors were excessive for delaminations located near the beam end as a result of insucient number of training data. Chakraborty [3] applied a methodology based on articial back propagation neural network for predicting the presence of incorporated delaminations in bre reinforced plastic composite plates with respect to their size, shape and location using natural frequencies. The articial BPNN which serves as the emulating technique with three layers (input, hidden, and output) was used to train 165 datasets generated via nite element models. The rst 10 modes of natural frequencies were used as input to the network and their corresponding delamination size, shape and location were given as outputs from the network. However, he only reported results with simulation data and his results were ecient in delamination prediction. 2.4.2.2 RSM as inverse method for vibration based damage detection

RSM is a non-intelligent inverse method that have also been employed for delamination detection. Performance analysis of ANN and polynomial approximations have been done by Todoriki [56] based on their damage detection ability. The results obtained using ANN were compared to those derived using RSM (RSM basically uses regression surface tting to obtain approximated responses). Compared to ANN, polynomial based approximations were found more suitable as a diagnostic tool. Results showed that ANN predicted exact location and size of the delamination for all specimens used in the training but failed in the results of the new data not used in training as a result of the small dataset used in training. It was seen that ANN are not simple to manipulate as a tool of diagnosis for delamination detection. More runs could bring better ANN but the fact remains that ANN is based on

2.5. SUMMARY OF RESEARCH GAPS

30

the choice of network architecture, training algorithms used, etc to obtain improved results. The RSM does not have such limitations and resulted in better predictions. Furthermore, the poor predictions of ANN emanated as a result of the small dataset used in training. One other reason for the poor results relates to whether or not the training dataset spanned the whole range including that used in testing. Section 5.3 gives more detailed analysis of RSMs. Monitoring the delamination of composite laminates using changes in electrical resistance obtained from FEM analyses by means of ANN and RSM have been investigated by Todoroki [56]. The input data to the ANN and RSM is the electrical resistance ratio and the output information is the normalized delamination size and location. His developed ANN model gave poor results with testing data in contrast to the RSM. Inada et al. [65] developed an approximation relationship between natural frequencies and damage parameters (location and size) using response surface methodology for damage identication of carbon ber reinforced plastics beams and plates. Response surface in cubic polynomials was used to establish the relationship between the damage size and location to the measured frequencies. By conducting design of experiments for the response surface analysis the most suitable points for tting the surface successfully are determined using input data obtained from analytical models. The response surface uses regression analysis by the least squares method which needs training data sets of dierent damage scenarios.

2.5

Summary of research gaps

ANNs with their pattern recognition and classication capability have proved to be

an ecient tool for vibration based damage identication, and several research publications exist in this area. To verify the already proposed methods, researchers have applied dierent algorithms to dierent types of numerical and experimental structures, in rare cases, some techniques

2.5. SUMMARY OF RESEARCH GAPS

31

have been validated using experimental results. Most developed algorithms have been validated using numerical benchmarks. Although above reviewed previous works have demonstrated the feasibility of articial neural networks and optimization algorithms using mainly natural frequency shifts for delamination damage detection, some signicant hurdles still exist. While neural network based approaches have been found successful due to their adequate generalizing capabilities, they require a large number of training cases as justied by the works of Todoriki [56]. Finally, most of the works validate their results against numerical data without considering experimental uncertainties. To the best of authors knowledge, there are no reports of any SHM approach that investigates the eects of noise and error on its behavior. Hence, considerable eorts are needed for developing adequate identication methods which could identify more accurately delamination interface, size and location even with natural frequencies corrupted with error and noise and with optimized limited number of datasets.

Chapter

Mathematical formulation and modeling of composite laminates for beams and plates
3.1 Overview
This chapter introduces the concept of an optimization task by the formulation of an objective function. Because the objective function needs to be evaluated from analytical or numerical models, the nite element model and analytical model of a composite beam with and without delaminations are also presented. The validation of the numerical results with the analytical model of Tracy and Pardoen [41]s formulations is given thereafter in this chapter. Finally, the nite element modeling for the composite plate is presented.

32

3.2. MATHEMATICAL FORMULATION OF THE OPTIMIZATION PROBLEM 33

3.2

Mathematical formulation of the optimization problem

The goal of an optimization task is to ascertain a set of values that result in a maximum

or minimum of a function called the objective function. Objective function is a mathematical expression describing a relationship of the optimization parameters. Optimization is an activity, which could be single or multi-objective that aims at nding the best (i.e. optimal) solution to a problem. Single objective optimization is scalar valued with single unique solution whereas when the objective is vector valued, the optimization process is referred to as multi-objective [22]. In this delamination detection optimization problem, the objective function is dened as a single objective function and two key components are eectively required to solve the optimization problem: 1. The simulator (which essentially computes/simulates the natural frequencies) 2. The objective function (comparator) The optimization objective is to compare and minimize the errors between measured and predicted natural frequencies. The percentage change in frequency caused by delaminations for ith mode (dFi ) where i = 1, ..., n, is dened as: Fui Fdi ) 100 Fui

dFi = (

(3.1)

The objective function (damage comparator) is the norm of the dierence between the measured/actual (dFM i ) natural frequencies for which the delamination parameters are to be determined and numerically predicted natural frequencies(dFi ). All the computational analysis for the delaminated natural frequencies (in Hertz) is dened with respect to the beam in perfect condition and the deviation is expressed as a percentage between the change in natural frequencies.

3.2. MATHEMATICAL FORMULATION OF THE OPTIMIZATION PROBLEM 34 Where, Fui and Fdi are the numerically predicted natural frequencies of the undamaged and damaged composite beam respectively. dFM i is the measured percentage change in natural frequencies for which the delamination parameters are to be ascertained. The error, Ei , between the measured and predicted shifts in frequencies due to the delamination for the ith mode is given by; dFi dFM i 2 ) dFM i

Ei = (

(3.2)

The use of high delity simulation tools to compute the natural frequencies of composite structures for the objective function always comes with an unavoidable large computing time. For this reason, surrogate models is a key element to reduce the optimization cycle time by providing alternative function evaluations for the objective function. The objective function with surrogate, ObjS is therefore given by total sum of the errors;
n

ObjS =
i=1

Ei = E1 + E2 + E3 + ... + En

(3.3)

Similarly, the objective function without surrogate, Obj is also given by;
Obj = (dF1 dFM 1 )2 + ... + (dFn dFM n )2 (3.4)

Where n is the maximum ith mode. In order to evaluate the objective function, the natural frequencies of the undelaminated and delaminated composite beams are simulated via nite element analysis (FEA). The objective function is evaluated using frequency shifts due to delamination predicted using FEA and the measured frequency shifts. Very few simulations were used to build the database using K-means clustering and then surrogate assisted optimization.

3.2. MATHEMATICAL FORMULATION OF THE OPTIMIZATION PROBLEM 35

3.2.1

Solution methodology

A delamination detection method is proposed which consists of the simulator and the response comparator. When delamination occurs, the simulator is run repeatedly to compute the frequency shifts due to dierent possible delamination combinations. During the simulation runs, delamination interfaces, locations and sizes are compared randomly by the comparator, which is based on an optimization algorithm approach that updates agreement between the measured and predicted responses. In the response comparator, the agreement is quantied by using the objective function. When the objective function is minimized, the responses are found to be consistent and the delamination parameters are said to be estimated. The method has been validated using responses measured experimentally in the laboratory and has successfully identied arbitrary beam delaminations. An overview of the owchart of the approach is shown in Figure 3.1. A detailed description of the proposed methods to detect delaminations can be summarized in a procedure as follows: Optimization without surrogates: Step 1, develop an FE model by ANSYS that computes natural frequencies before and after damage. This is also known as the simulator. Step 2, engage the simulator directly by the optimizer to evaluate the objective function to be minimized to determine the delamination parameters. The objective function also known as the comparator essentially minimizes the sum of errors between the simulated natural frequency changes before and after damage and the actual ones to determine the delamination parameters for any number of variables (Nd ). Surrogate assisted optimization: Step 1, develop an FE model by ANSYS that computes natural frequencies before and after damage. Step 2, use the simulator to generate a database in the case of optimization with surrogates. Step 3, reduces the size of the generated database which is usually large by a K-means clustering method. This ensures that a small number of well clustered datasets within the entire design space is used for delamination prediction. Step 4, use the K-means clustered datasets to create

3.2. MATHEMATICAL FORMULATION OF THE OPTIMIZATION PROBLEM 36 a surrogate model. Step 5, engage the surrogate model directly by the optimizer to evaluate the objective function to be minimized to determine the delamination parameters for any number of variables (Nd ). Direct solution via ANN and RSM: Step 1, develop an FE model by ANSYS that computes a database of natural frequencies before and after damage. Step 2, do a Kmeans clustering if the size of the database is to be reduced. Step 3, use the database to train ANN and RSM models that give the delamination prediction for any number of variables (Nd ).

Figure 3.1: Schematic of solution methodology.

3.3. MODELING OF THE LAMINATED COMPOSITE BEAM

37

3.3

Modeling of the laminated composite beam

A laminated composite cantilever beam made of eight plies [0/90/90/0]s is considered.

Glass ber (E-Glass) is used as reinforcement in the form of unidirectional bers with epoxy resin as matrix for the composite beam. The laminates are reinforced unidirectionally. The material properties are given in Table 3.1 for the FE analysis and theoretical model. The composite beam has length, L = 267mm, width, W = 25.4mm, and each ply is 0.2222mm thick giving an overall thickness of h = 1.778mm. Considering through-width delaminations, delaminations are simulated extending through the width of the beam, with locations and sizes in the ranges of 0 < X 70 and 0 < a 58 respectively, satisfying the requirement that delamination must not extend outside the beam. For any delamination pattern ([Z, X, a]), Z deontes the interface; the normalized delamination location is expressed as X = XActual /L, where XActual is the distance from the middle of the delamination to the xed end of the beam and L is the total beam length. Similarly, the normalized delamination size is given as a = aActual /L, where aActual is the length of the delamination along the axis of the beam. Delaminations are simulated at dierent interfaces, Z = 1 to 4; 1 being mid-plane and 4 the outermost interface. Table 3.1: Material properties of the composite beam laminates [1]. Material Properties E1 42.34GPa E2 11.72GPa G12 3.0025Gpa Poissons ratio (12 ) 0.27 Density () 1901.5kg/m3

3.3.1

Finite Element (FE) Analysis of the composite beam

Manudha et al. [66] investigated the eects of delamination size and location in composite beams using nite element analysis model under quasi-static loading and an analytical

3.3. MODELING OF THE LAMINATED COMPOSITE BEAM

38

Euler-beam model. Ramanamurthy and Chanddrasekaran [67] used nite element modeling to develop a damage detection method in a composite cantilever beam with an edge crack. Ishak et al. [68] employed the application of Strip Element Method (SEM) and adaptive Multilayer Perceptron Networks (MLP) for inverse identication of interfacial delaminations in carbon/epoxy laminated composite beams. In this research work, FEA is used to solve the forward problem and generate data of frequency shifts for known delamination parameters. Numerical analysis is carried out using the commercial nite element program ANSYS 12.1 to build the FE models for both the undelaminated and delaminated glass bre reinforced composite beams in order to investigate their vibration behaviour. Analysis is carried out on a 3-D-8-Node layered solid element (SOLID185) with three degrees of freedom at each node. The three degrees of freedom used at each node in the nite element analysis are translations along the X and Y axes and the rotation along the Z axis. In actual structure, damage will often aect the stiness matrix but not the mass matrix of the system. In the theoretical development that follows, damage is assumed to cause a loss of stiness in damaged elements of the system. For the intact beam, a constant stiness is assumed for all elements. While for the damaged beam, reduction of stiness for damaged elements is applied. The shell section is adopted to dene the layer information as shown in Figure 3.2. The bottom layer is categorized as layer 1 and the other respective layers are stacked accordingly from bottom to top in the positive Z (normal) direction of the element coordinate system. To build the model, a mesh analysis was initially done to determine how ne the mesh should be in order to get convergence of the numerical results. The FE mesh employed was 10 x 1500, 10 elements across the entire thickness and 1500 elements along the length. The rst simulation was performed for the undelaminated beam model. The natural frequencies obtained from the undelaminated model is consistent with that from theory. In the FE model for the delaminated beam, nodes in the delaminated part are allowed to overlap but are left

3.3. MODELING OF THE LAMINATED COMPOSITE BEAM

39

Figure 3.2: Sample lay plot display for [0/90/0/90]s sequence of ber orientation in the laminate separate [69]. The geometry of the composite beam with an arbitrarily located throughwidth delamination is shown in Figure 3.6. For the purpose of analysis, the beam laminate is subdivided into two sublaminates, one below and one above the plane of the delamination. The undelaminated regions, indicated by sections 1 and 4, are modeled using layered eight noded shear deformable plate elements with each node have three degrees of freedom (ANSYS 12.1, Linear Layered Structural Solid elementSOLID185). In order to represent the delaminated region, the delaminated region is divided into the upper (section 2) and lower (section 3) plies, and is made from two separate segments joined at their ends to the integral segments. At the interface between plies, two sets of nodes, one belonging to the upper ply, the other to the lower ply, are placed. These nodes occupy the same geometric position in space, but are allowed to vibrate independently. Assuming that delamination continues to propagate in same plane. Ultimately, the sublaminates are modeled by nite elements with nodes oset to the top or bottom of the laminates which lied along the delamination. In the undelaminated segment, the nodes corresponding to the upper or lower sublaminates are connected by xed elements at the delamination plane. This provides continuity of displacements and rotation [70]. In the delaminated segment, the nodes of upper and lower portions

3.3. MODELING OF THE LAMINATED COMPOSITE BEAM

40

are connected by contact elements. This is based on the assumption that there is no friction against sliding of the sublaminates if they are in contact. To determine the natural frequencies of the delaminated beam, ANSYS Workbench batch mode simulation is used to setup the eigenvalue modal analysis. Further, using the Block Lanczos method, the natural frequencies of the rst eight bending modes are extracted discarding the torsion and in-plane bending modes. The natural frequencies of the undamaged and damaged glass bre/epoxy composite beam at any interface as shown in Figure 3.3 are computed.

Figure 3.3: Locations of dierent interfaces of the [0/90/0/90]s composite laminates The natural frequencies obtained from the delaminated beam model is compared to theoretical results for validation purposes. Table 3.2 shows the computation of natural frequencies of the delaminated and undelaminated composite laminates along dierent interfaces (interface 1 to interface 4). It is seen that the eects of delamination location and size on the delaminated and undelaminated natural frequencies results to decrease in natural frequencies for all the interfaces. The 3D plots of frequency shifts (dF) of rst eight modes as functions of delamination parameters for interface 1 is shown in Figure 3.4(a) through Figure 3.5(d). It could be seen that for the lower modes say Mode 1 and Mode 2, the change in frequency increases

3.3. MODELING OF THE LAMINATED COMPOSITE BEAM Table 3.2: Natural frequencies for delamination along dierent interfaces (in Hertz). S/N [Z, X, a] Mode 1 1 Undelaminated 16.039 2 [1, 18, 33] 15.666 3 [2, 18, 33] 15.695 4 [3, 18, 33] 15.747 5 [4, 18, 33] 15.839 Mode 2 100.458 82.666 83.755 85.787 89.991 Mode 3 281.042 233.711 225.727 257.246 254.698 Mode 4 550.068 475.201 518.556 509.658 516.924 Mode 5 907.913 740.05 847.82 802.211 831.369

41

monotonously with increase in size and location, therefore the non-linearity is not complex in these lower modes. However, in the higher modes, the plots are much more curved which means the function is much more complex and highly non-linear in the higher modes say Mode 3 to Mode 8.

(a) Mode 1.

(b) Mode 2.

(c) Mode 3.

(d) Mode 4.

Figure 3.4: 3D plots of frequency shifts (dF) as functions of delamination location (X) and size (a) for Mode 1 to Mode 4.

3.3. MODELING OF THE LAMINATED COMPOSITE BEAM

42

(a) Mode 5.

(b) Mode 6.

(c) Mode 7.

(d) Mode 8.

Figure 3.5: 3D plots of frequency shifts (dF) as functions of delamination location (X) and size (a) for Mode 5 to Mode 8.

3.3.2

Theoretical modeling of the laminated composite beam

The theoretical formulation of vibration characteristics models of beams with and without delaminations were based on the pioneering works in this eld. Ramkumar et al. [71] was the rst of these pioneers. They developed a simplied model to compute the free vibration frequencies of a cantilever laminated beam with a single through-the-width delamination at the interlaminar position. Their basic concept is to deduce mathematically the actual vibration properties of four dierent Timoshenko beams combined together by considering the delaminated and undelaminated portions of the beams. Their mathematical model of the eigenvalue problem fullled all the necessary boundary and continuity conditions between the adjoining beams. They also carried out experimental studies, which show that the analytical computations of the natural frequencies were uniformly lower than the experimental

3.3. MODELING OF THE LAMINATED COMPOSITE BEAM

43

ones. As a follow up, Wang et al [72] employed the analytical model of Ramkumar et al. using classical Bernoulli-Euler beam theory to obtain consistent results by incorporation of the coupling between exural and axial vibrations of the delaminated sub-laminates of the model. Then later on, Mujumdar and Suryanarayan [40] applied a pressure distribution between two respective delaminated sections to impose a constraint between the two beam sections in order to obtain similar exural deformation. This kind of proposed model was termed Constrained Model in contrast with the so called Free Model proposed by Wang et al. Their analytical model demonstrated for isotropic materials were found to have the ability to determine natural frequencies of a delaminated beam at any interface. Della and Shu [73] extended Mujumdar and Suryanarayans model to composite beams, by using the eective bending stiness terms of composite laminates. In what could be termed as a similar approach, Tracy and Pardoen [41] developed constrained model to predict natural frequencies of a simply-supported composite beam considering only mid-plane delaminations. In this analytical study, the model rst adopted by Mujumdar et al is customized for composite laminates, to determine the changes in natural frequencies due to delaminations located at dierent interfaces. The detailed theoretical background can be found in [41] but the fundamentals are described here. The classical or Euler-Bernoulli beam theory is employed to quantify the eects of position, size and location of delamination on the vibration response of a composite beam such as natural frequencies. This is developed based on the principle that straight lines perpendicular to the mid-plane before bending remain straight and perpendicular to the midplane after bending. As a result of this assumption, transverse shear strain/deformation is neglected [74]. The analytical model is based on a constrained vibration model with identical transverse displacements of the upper and lower sublaminates. The model gives room for independent extensional and bending stiness, but it does not contain bending/extensional coupling stiness terms. The natural frequencies, Fi (Hertz) of a symmetric and balanced cantilever composite

3.3. MODELING OF THE LAMINATED COMPOSITE BEAM beam without delaminations is evaluated using Equation 3.5, [2];
2 ki ] 2L2

44

Fi = [ Where;

D11 A

(3.5)

L = the beam Length; A = the cross sectional area; D11 = the eective exural stiness of the whole structure; and i is the ith mode of natural frequencies; = mass density. The parameter, ki , is the dimensionless frequency parameters (eigenvalues) which fullls the condition of Equation 3.6, [75];

cos(kn L)cosh(kn L) + 1 = 0

(3.6)

The beam shown in Figure 3.6 with a delamination at midplane is made up of four sections (i = 1 to 4); the free undelaminated end, the clamped undelaminated end, the sublaminate above the delamination, and the sublaminate below the delamination. The parameters for the delaminated beam in Figure 3.6 are dened as; the undelamianted segments having lengths L1 and L3 and the delaminated section has length L2 and beam thickness, h2 = h3 = h/2. Each beam section is expressed systematically by an extensional stiness Ai , a bending stiness Di , and a mass density per unit length mi . The controlling dierential equations for any section i are as follows; d4 wi m i 2 wi = 0 dx4

Di

(3.7)

Similarly; dPi d2 ui = Ai 2 = 0 dx dx (3.8)

where; is the natural frequency (radians/sec), wi is the transverse displacement, ui is the extensional displacement, and P is the extensional force along the beam at the x coordinate.

3.3. MODELING OF THE LAMINATED COMPOSITE BEAM Table 3.3: Quantitative comparison between the analytical and numerical natural frequencies (in Hertz).

45

[Z, X, a] Mode number Theory (Fui ) Theory (Fdi ) FE Model (Fui ) FE Model (Fdi ) % dierence (Fui ) % dierence (Fdi ) 1 16.06 15.97 16.04 15.94 0.159 0.14 2 100.67 100.54 100.46 100.33 0.211 0.2 3 281.9 252.38 281.04 251.49 0.349 0.3 4 549.4 522.12 550.07 519.81 0.443 0.12 [1, 54, 24] 5 913.09 780.59 907.91 776.82 0.482 0.57 6 1364.04 1192.95 1353.75 1184.85 0.679 0.75 7 1905.15 1636.83 1886.64 1624.22 0.771 0.97 8 2536.44 2191.25 2505.46 2170.49 0.947 1.22 1 16.06 16.032 16.039 16.007 0.154 0.14 2 100.67 100.333 100.458 100.120 0.213 0.2 3 281.9 275.044 281.042 274.152 0.324 0.3 4 549.4 504.202 550.068 501.887 0.459 0.119 [1, 62, 18] 5 913.09 894.735 907.913 889.365 0.600 0.57 6 1364.04 1199.253 1353.748 1191.622 0.636 0.75 7 1905.15 1725.247 1886.636 1709.563 0.909 0.97 8 2536.44 2291.992 2505.461 2271.168 0.909 1.22

Solving the deection functions in Equations 3.7 and 3.8 give 20 unknown coecients [76]. A Matlab program is coded to solve the simultaneous equations and generate the modal frequencies for various delamination positions, sizes and locations. The rst eight exural modal frequencies for dierent delamination signatures ([Z, X, a]) are computed.

Figure 3.6: Representation of the integral and delaminated sections of a composite beam having midplane delamination.

3.3.3

Validation of the FE Analysis for the composite beam

A comparison of the analytical and ANSYS numerical results for undelaminated (Fui ) and delaminated (Fui ) beams is shown in Table 3.3. In Table 3.3, it is observed that natural frequencies decrease as delaminations occur and the % errors increase for the higher modes. The objective of this exercise is to verify the results of the numerical model. The numer-

3.4. FE MODELING OF COMPOSITE PLATES

46

ical results are consistent with the analytical results with errors less than 1%. Numerical modeling is preferred to analytical modeling particularly in complex systems because it is realatively easy to incoporate dierent boundary conditions, loading congurations etc.

3.4

FE modeling of composite plates

Plates are two-dimensional (2D) bodies having one (thickness) dimension much smaller

than the other two. Delamination detection problem in composite plates is more complicated than the previous studies so far on composite beam laminates. The complexity of plate problems is due to the presence of 5-variables needed to describe a delamination signature in contrast to the 3-variables for composite beams. The 5-variables consist of two continuous variables (X, Y) representing the in-plane coordinates of the lower left corner of the delamination, two continuous variables (Xa and Yb ) denoting respectively the length and width dimensions of the delamination in the directions of x-y coordinates and one integer interface variable, (Z), specifying the interface location of the delamination. Hence, the delamination signature for the plate problem can be described as [Z,X,Y,Xa ,Yb ]. The coordinates of the centre of the delamination in x-y plane are indicated in Figure 3.7. Investigations to compute the natural frequencies of laminated composite plates in various modes of vibration by using the nite element method have been carried out previously [3; 53; 77; 78; 79]. In this work, a nite element analysis has been performed using ANSYS 12.1 for analyzing an 8-layer conguration [0,45,-45,90,90,-45,45,0] graphite/epoxy rectangular plate (length, L = 150mm, width, W = 100mm and thickness, t = 2mm, with all four edges simply supported) having an embedded delaminations at each of the interfaces (1 to 4) of the sub-laminates. The material properties of the plate are given in Table 3.4. The nite element model used for studying the dynamic behavior of the laminated plate is an eight node isoparametric layered element (SOLID185 in ANSYS) with orthotropic material

3.4. FE MODELING OF COMPOSITE PLATES

47

properties. This was employed for the analysis. For the analysis, in the delaminated region, corresponding nodes of the top and bottom sub-laminates are connected by contact elements (CONTAC173 in ANSYS) to simulate the contact force between the sub-laminates and to prevent element penetration. For each node, there are three degrees of freedom, i.e. translations along the global coordinate axes of x, y, and z. The element thickness is assigned to be equal to that of the corresponding individual lamina. The local element coordinate system (x1, x2, x3) is arranged with the rst axis being coincident with the ber direction. All physical parameters throughout an element are assumed to be the same. The current formulation incorporates the eects of transverse shear deformation as well as the bending-extension coupling (appearing in the element stiness matrix) caused by the presence of delaminations exiting in the upper and lower segments. Each sub-laminate was modeled individually. In the delaminated segments, all the sub-laminates arising from the delaminations are meshed separately by solid elements. Using the Lagranges principle, the equation of motion for free vibration of the composite plate is reduced to an eigenvalue problem. Therefore, the modal parameters such as natural frequencies can be obtained using the subspace iteration method of modal analysis. First eight natural frequencies have been extracted. A convergence study was also carried out due to the complexity of the numerical simulations, hence it was computationally convenient to determine the eect of element size on the accuracy of the nite element results. Dierent generated mesh sizes were considered and no remarkable changes in results were obtained when using the total number of elements in the X and Y locations respectively as 60 and 40 in comparison with results obtained with 120 and 80 elements in X and Y locations respectively. Hence, the 40x60 mesh size was in the subsequent computations to achieve less computation time.

3.4. FE MODELING OF COMPOSITE PLATES

48

Figure 3.7: Delamination position in x-y plane for the plate problem.

Table 3.4: Material properties of the composite plate laminates.

Material Properties E1 E2 G12 Poissons ratio (12 ) Poissons ratio (23 ) Density () 138GPa 8.96GPa 7.1Gpa 0.3 0.019 1901.5kg/m3

The percentage change in the rst eight natural frequencies (dFpi ) of the simplysupported composite plate with delaminations as a function of ve variables can then be dened with respect to the that of a perfect plate as;

dFpi = dFupi dFdpi /dFupi 100

(3.9)

Where dFdpi and dFupi are the ith natural frequencies of a delaminated composite plate and corresponding undelaminated plate, respectively.

Chapter

Algorithms for solution of the inverse problem

4.1 Overview
This chapter details dierent algorithms implemented for delamination detection. ANNs, RSM and optimization techniques (GBLS and RGA) are the algorithms described with their respective pros and cons. The basis for adoption of each algorithm is explained. Also worth mentioning is the development of surrogate models integrated in the optimization loop to enhance the search performance of the optimization algorithms. Finally, the concept of K-means clustering for selection of ANN training dataset is discussed.

4.2

Optimization search algorithms

Global search methods are search schemes that are not prone to trapping from local

optima. They require huge amount of computational eort. Local search methods on the other hand restrict candidate potential solutions to a conned design space starting at an initial guess [80]. Thus, a combination of these is an attractive choice that has led to the development of the memetic algorithms (MAs). 49

4.2. OPTIMIZATION SEARCH ALGORITHMS

50

4.2.1

Gradient based Local Search (GBLS)

Gradient-based local search (GBLS) method is a kind of optimization algorithm that basically employs gradient information of the objective function for determining the direction of the subsequent search points from a given start point. This technique uses a function which seeks the minimization of a scalar (single) objective function of multiple design variables within a region specied by linear constraints and bounds using the SQP algorithm. The most famous gradient based algorithm is the Sequential Quadratic Programming (SQP). The SQP method is the most successful specically for problems of non-multimodal behavior. The gradient based local search used in this work is based on sequential quadratic programming (SQP). SQP methods portray the state of the art in nonlinear programming methods. An overview of SQP is found in Schittkowski [21]. The general method, however, is described here. In the SQP method, the function solves a quadratic programming subproblem at each iteration by obtaining and updating iteratively at every step an estimate or approximation of the Hessian of the Lagrangian function using a quasi-Newton updating method based on Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula [21]. SQP uses iteration from an initial guess until it reaches a feasible local optimum. During the process of every iteration, a quadratic non-linear programming is used to solve the objective at that point. If the problem is unconstrained, then the method reduces to Newtons method for nding a point where the gradient of the objective disappears. The solutions of the quadratic programming are used to initiate a search space towards a better solution until the optimum set is found. However, the gradient based local search can be trapped in local optima and hence a reinitialization is used to avoid stagnation. Reinitialization is a process adopted to prevent the trapping of the technique at the local optimum to some extent by starting the optimization process from dierent start points. The nal optimum may not be the global optimum but however this process eectively increases the optimization cycle time [80]. This technique

4.2. OPTIMIZATION SEARCH ALGORITHMS have been implemented in Multistart from the Matlab Optimization Toolbox.

51

4.2.2

Global optimizer based on evolutionary algorithm (EA)

To ensure global optimum results [81] and increase accuracy of results while reducing the search time via surrogates, biologically-inspired evolutionary algorithm (EA) are used. Genetic Algorithms (GAs), Particle Swarm Optimization (PSO) and Dierential Evolution (DE) are various kinds of EAs that encode (both numerical and non-numerical) design parameters to shue multiple candidates via parallel and interactive search. During the search process, there is rstly a selection performed based on survival of the ttest. To generate the next generation of possible candidate solutions, some parameter values are exchanged between two candidates (crossover) and new values introduced (mutation). They cannot be easily trapped in local minima or maxima as a result of crossover and mutation operations, which makes them an ideal method to eectively handle multi-modal optimization problems. Evolutionary Algorithms (EAs) are representatives of the class of stochastic (random) and robust optimization algorithms that do not require gradient information during the course of an optimization process but rather uses an objective function value. This makes them more computationally expensive than the GBLS which requires fewer number of iterations. However, they can handle a wide variety of problem characteristics such as discrete and continuous decision/design variables and multi-modality in contrast with the GBLS. On the other hand, surrogates are used to contain their computationally expensive nature hence exploring the advantage of their global optimization behavior. The choice of an ecient optimization algorithm is based on the problem under study. For the optimization problem in which the variables are combination of discrete and continuous, it is dicult to use conventional optimization algorithms such as the gradient based method to obtain the global optimum, since they rely on the use of continuous variables. In an attempt to solve a problem with mixed variables (i.e. discrete and continuous), a population based stochastic algorithm and in particular Real-coded Genetic Algorithm (RGA)

4.2. OPTIMIZATION SEARCH ALGORITHMS

52

with Simulated Binary Crossover (SBX) and Polynomial Mutation [82] is found as an ideal choice to eectively manipulate the optimization task for detecting delaminations in composite laminates because they are most promising in both discrete and continuous multimodal problems. Non-dominated Sorting Genetic Algorithm (NSGA-II) is capable of solving both single and multiobjective optimization problems. While for multiobjective problems, the ranks of feasible solutions are based on nondominated sorting, in a single objective case such ranks are based on the sorted values of the objective function. Hence NSGA-II in a single objective scenario is essentially a real coded genetic algorithm with simulated binary crossover and polynomial mutation. While an evolutionary algorithm has been used in the current study, other forms of evolutionary algorithms such as particle swarms, dierential evolution etc can also be used. Dierential Evolution (DE) is peculiar to solving single objective optimization problems with continuous variables. Several DE variants exist such as DE/rand/1/bin strategy and Self-adaptive DE (SaDE). In RGA with SBX and polynomial mutation, ultimately sharing function approach and non-dominated sorting implementations are incorporated to solve multimodal and discrete problems. It is noticed that the real-coded GAs outperform the binary coded GAs in solving a number of optimization problems [83]. Unique benet of the SBX operator is that it can constrain children solutions to any arbitrary proximity to the parent solutions, hence not involving any separate mating restriction scheme for enhanced performance. The parameters of RGA and their values are listed in Table 4.1 while its ow is depicted in Figure 4.1. The detailed description of RGA processes include: 1. Generation of Initial Population (G) : Generation of the rst parent population of size, N. This is randomly generated within the predened feasible region (the upper and lower bound of the design space). 2. Non-Dominated Sort: Individual population are evaluated and and sorted based on non-domination. A solution (s1 ) dominates (is preferred to) another solution (s2 ), if and only if s1 is better than s2 in the objective function. For every generation,

4.2. OPTIMIZATION SEARCH ALGORITHMS

53

fast non-dominated sorting is applied to identify non-dominated solutions to construct the non-dominated front. This produces a set of candidate solutions that are non dominated by any individual in the population. These solutions are then discarded from the population temporarily until the next best non dominated set are identied. This process goes on and on until all solutions are classied and assigned ranks equal to their non-domination level assuming tness minimization. 3. Crowding Distance: This is basically the niche safeguarding in the the design objective space. The crowding distance assigned equals the front density in the neighborhood (distance of each candidate solution from its nearest neighbors). 4. Selection by ranking: The initial population is sorted in ascending order according to tness functions. Potentially better solutions are ranked higher than the worst solutions. Individuals are selected by the use of a binary tournament selection with the crowded comparison operator. 5. Genetic Operators: A mating pool is formed by bonding of the parent and child populations. Simulated Binary Crossover operator and polynomial mutation is used to create an ospring of new population. Mutation occurs by random walk around individuals. The best population of parents and osprings with higher tness is selected to reproduce the next generation. 6. Recombination and selection: Combination of the traits of ospring population and and parent population is done to reproduce extended population of the next generation (2N) [84]. This is done to ensure elitism and to keep diversity in generating subsequent successive population [85]. The replacement criteria keeps the best among parents and osprings based on the fast non-dominated sorting and maintains the best diversied individuals to provide larger search space. New population size with the initial population is lled with individuals from the sorting fronts starting from the best. Crowding distance method is recalled to maintain diversity if a front incompletely lls the next

4.2. OPTIMIZATION SEARCH ALGORITHMS

54

generation. This ensures that convergence in one direction does not take place. This process repeats itself to generate subsequent generations until a stopping a criterion is reached.

Figure 4.1: Flowchart of RGA optimization framework.

4.3. DATABASE CREATION FOR ANN TRAINING USING K-MEANS CLUSTERING ALGORITHM 55 Table 4.1: RGA Parameters setting.
Parameter description Maximum population size Crossover probability Mutation probability Maximum number of exact function evaluations Evolutionary operators (Binary tournament selection) Evolutionary operators (Polynomial mutation) Evolutionary operators (Simulated binary crossover (SBX)) Evolutionary operators (Etilism, Non-domination rank and Crowded distance) Number of independent runs (Stochastic and could be run more than once) Parameter value N = 200 P c = 0 .9 P m = 0 .1 4000 s=2 m = 10 c = 10 80 10

4.3

Database creation for ANN training using K-Means clustering algorithm

ANN model and its results are highly dependent on the training data set provided. It

is hence necessary to maintain the diversity of the training set to obtain a good prediction model by ensuring that the training data is not clustered around one part of the design domain. The most vital strategy in the selection of the training datasets is to nd the ideal set that is a true representation of all the possible samples in the total design space. This is done using K-means clustering. K-Means clustering is used for grouping large data sets into smaller sets called clusters. The number of k-groups or objects needed is specied. Each object is represented by some feature vector in n-dimensional space, n being the number of all characteristics used to describe the objects to cluster. The algorithm then randomly chooses k-points in that vector space, these points serve as the initial centers of the clusters and all objects are each assigned to center they are closest to [86]. Basically, K-means clustering algorithm nds a subset of k-groups known as centroids that minimizes the mean squared distance from each data point to its nearest center in an entire dimensional space of n-data points [20]. The algorithm is implemented in the following steps: Step 1: Specify k-points into the space represented by the objects that are being clustered. These points represent initial group centroids.

4.3. DATABASE CREATION FOR ANN TRAINING USING K-MEANS CLUSTERING ALGORITHM 56 Step 2: Each object is allocated to the group with the closest centroid. Step 3: When all objects have been assigned, positions of the k-centroids are recomputed. Steps 2 and 3 are iterated until the centroids no longer move. This introduces a separation of the objects into groups from which the metric to be minimized can be computed.

4.3.1

Preparation of database used for ANN training

Building a database with large number of samples increases the number of high delity simulation runs. To limit the computational expense, the following scenarios in terms of database created from the FE models are studied for the ANN prediction and function approximations: Scenario 1 - 400/441 datasets (400 datasets for training and 41 datasets for testing as shown in Figure 4.2) Scenario 2 - Use K-means clustering to choose 40 out of 400 training datasets (Figure 4.3(a)) Scenario 3 - Use K-means clustering to choose 28 out of 400 training datasets (Figure 4.3(b)) Scenario 4 - Use K-means clustering to choose 20 out of 400 training datasets (Figure 4.4(a)) Scenario 5 - Use K-means clustering to choose 8 out of 400 training datasets (Figure 4.4(b))

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 57

(a) Design space for 400 training datasets. (b) Design space for 41 testing datasets.

Figure 4.2: Design space for training and testing datasets.

(a) Design space for 40 K-means clustered (b) Design space for 28 K-means clustered training datasets. training datasets.

Figure 4.3: Design space for 40 and 28 training datasets respectively.

(a) Design space for 20 K-means clustered (b) Design space for 8 K-means clustered training datasets. training datasets.

Figure 4.4: Design space for 20 and 8 training datasets respectively.

4.4

Articial Neural Network (ANN) as an inverse solver and surrogate creator

4.4.1

Introduction to Articial Neural Network (ANN)

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 58 structural engineering applications in the area of failure prediction, delamination identication, crack detection, etc [87; 88]. ANN is a very powerful interpolator that can be used to map functions and derive a relationship between a set of input parameters and their output responses. There are dierent types of ANN architectures, namely, multilayer perceptron (MLP), radial basis function (RBF), etc. Multi layer perceptron (MLP) type [89] is adopted in this study because it eectively provides a complex nonlinear mapping between the input and output variables. MLPs are feed-forward nets with one or more hidden layers between the input and output neurons as shown in Figure 4.5. ANN based on MLP is trained using back propagation neural network (BPNN) algorithm, a gradient based method that has emerged successful in the training of multi-layered neural nets using supervised learning. In supervised learning, the network learns using input and output data and provides an approximation of the functional mapping between the two.

Figure 4.5: A schematic framework of one-hidden layer architecture

A typical BPNN is based on the fact that a feed-forward neural net (FFNN) with at least one hidden layer can approximate any continuous nonlinear function arbitrarily accurate if the number of hidden neurons are sucient. MLP is a FFNN which consist essentially an input layer with several neurons (depending on the number of inputs say (dF i to dF n )), a layer of output neurons and one or more layers of hidden neurons that wholly perform

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 59 the application expected objective. The primary building blocks of MLP are the articial neurons or processors. The neurons in each layer are fully interconnected to preceding and subsequent layers. Every one of them is connected by adjustable associated weights to enable the network to map complex associations between the input and output data. The activation function (a) for each neuron is dened as the summation of all the inputs multiplied by their connection weights and biases (wt1 , wt2 ,...wtn ) given as;

a = dF1 wt1 + dF2 wt2 + ..... + dFn wtn

(4.1)

The activation function (a) is transmitted through a link to other neurons via a feedforward network design and then fed to a transfer function which could be linear or non-linear to generate the output. The structure of the network is a function of the interaction between these neurons. Functions such as sigmoidal, radial (Gaussian), are used to build the neuron activity. A tan-sigmoidal transfer function is used in the hidden layer and to avoid limiting the output to a small range, a linear transfer function is employed in the output layer. A Bayesian regularization (trainbr ) back propagation (BP) learning algorithm is utilized to speed up the convergence of the MLP model. Bayesian regulation is a robust iterative training algorithm that learns patterns based on input and output data and it essentially provides stronger and ecient generalization ability by regularization. In this study, ANN training process is stopped when a maximum number of 1000 epochs is reached. The error is measured based on the the root mean-square-error (RMSE) between the predicted values and the output for all elements in the training and testing set. The networks are initialized to return neural network nets with weight and bias values updated according to the network initialization function.

4.4.2

Pros of ANN

The most vital advantages of ANN (see Figure 4.6) include;

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 60 Its capability to model and map complex non-linear systems (Non-linearity) by deriving a relationship between a set of input & output responses (Input-output mapping) The ability to learn which allows the network to adapt to changes in the surrounding environment (Adaptivity) Once ANNs are properly trained, damage identication is relatively fast and mathematical models do not need to be constructed There are no limitations on the type of vibration parameters to be used as inputs for ANNs. The input and output parameters can be selected with much exibility without increasing the complexity of network training

4.4.3

Cons of ANN

A major demerit of ANNs is that the resulting weights and nets of the trained network are dicult to interpret. This is due to their inability to obtain adequate solution of complex problems with physical mathematical methods in contrast to RSM as shown in Figure 4.6. Others include; It is dicult to nd an appropriate network architecture. It usually suers from the problems of under-tting (inaccurate approximation of the training data) resulting from too small networks, and over-tting (inadequate generalization) due to too large networks. It requires that several networks of dierent architectures are trained, and their performance compared on a separate set of test data to estimate their generalization properties. Training data of the database should be large enough to have a close relationship with the associated parameters (i.e. sucient training data for complex ANNs are necessary (avalaibility of a large database)

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 61

Figure 4.6: Pros and Cons of ANN and RSM

4.4.4

Modications to the basic ANN

Extensive work has been done on the basic ANN to improve its generalization ability. Following are some contributions made to the ANN model to improve its generalization capability, accuracy of approximation, output variable handling and training time; Single and ensemble nets: The capabilities of articial neural networks to handle multidimensional outputs are well known. However, it is preferred to use individual ANN models to predict each output because it is not feasible to use ensemble (multiple) nets to predict all the output variables. Training single nets enhances the prediction accuracy because only one output is predicted. This improves the handling capability of the optimizers when coupled with surrogates during the optimization process. Tables A.1, A.3, A.2 and A.4 typically demonstrate that single nets outperforms ensemble nets based on RMSE analysis. The concept of K-means clustering algorithm introduced enables one to eectively reduce the size of data set required for ANN training and hence determine minimum amount of datasets just sucient for ANN training.

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 62 The adoption of the most ecient training algorithm showed great eectiveness in the ANN results. Tables A.1, A.3, A.2 and A.4 show the performance of dierent training algorithms and it was shown that trainbr outperforms all other algorithms with RMSE values very close to zero.

4.4.5

The steps in designing ANN model

The ANN training process is not an easy task and it involves nding an appropriate ANN model for a given problem. Hence, the necessary requirements for a successful ANN development include: sucient database, careful selection of parameters, network architectures (number of hidden layers, number of hidden and output nodes), transfer functions and eective training and learning algorithms. These choices completely depend on the approximation function. The process of building ANN model can hence be outlined and summarized as follows: Selection of number of input and output variables of the neural network and database creation (the data set obtained from FEM analysis after a K-means clustering is divided into a training set and a testing) Determination of network architecture by trial and error method (i.e. number of hidden layers and number of hidden nodesinformation processing occurs at many simple elements called neurons) required to generalize the design space Selection of the training algorithm and functions and measure of its eectiveness based on RMSE performance Invoking of a back propagation algorithm to train multilayer feed-forward networks with dierentiable transfer functions to perform function approximation. Training by back propagation is described as the process by which derivatives of network error with respect to network weights and biases are computed. This process is subdivided into:

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 63 Feed forward of the input training data (input vector of set of design variables from the input layer is fed forwardly by propagation to the hidden layer and subsequently to the output layer) in the process termed as the forward pass of the back propagation algorithm. The computation and back propagation of the associated weight and errors. The predicted values of the output layer are compared with the target values. The error between the predicted and target values is calculated and propagated back toward hidden layer in the process known as backward pass of the back propagation algorithm. The error is used to update weight matrices between input-hidden layers and hidden-output layers. The mean square error (MSE) of the network is computed by calculating the amount between the predicted and target values. Consequently, this error is minimized by a predetermined training algorithm using optimization algorithms based on gradient based back propagation process which repeatedly changes the performance values depending on network connection weights. The trainbr function is used for this purpose. This function trains the network by randomly initializing and updating the weights and bias values according to Bayesian regularization algorithm. Terminating the network training using 1000 maximum number of epochs to reduce the eect of random weights on training the network. This method stops the training when the the maximum number of training cycles given is reached. Computing the average performance of each of the network architecture with RMSE as the performance criterion and select the best network performance with the minimum RMSE. The best model is applied to test data to investigate the performance on an unseen set of data

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 64

4.4.6

Generalization enhancement

Multi-layer feed-forward neural networks can provide a functional relationship between input and output given sucient number of neurons in the hidden layers, though excess neurons in the hidden layers leads to over-tting (i.e. the process when the error on the training set is very small but cannot be used to make future predictions on test data). The ANN adopted is provided by Matlab Neural Network Toolbox. The ANN model is based on Bayesian regulation backpropagation (trainbr ) network training function that updates the weight and bias values according to Levenberg-Marquardt optimization. It also minimizes a combination of squared errors and weights, and then determines the correct combination so as to produce a network that generalizes well. The process is called Bayesian regularization. Hence, a customized network that uses trainbr with new is created. This is done by setting net.trainFcn to trainbr. The most eective methods for improving network generalization include: Generalization by Bayesian regularization which produces a network that performs well with the training data and exhibits smoother behavior when presented with new data. Properly generalized trained multilayer networks tend to give reasonable answers when presented with inputs that they have never seen. Increasing the number of hidden neurons and layers. Larger number of neurons in the hidden layer can give the network more exibility because the network has more parameters it can optimize. However, the number of layers should be increased gradually because large hidden layers lead to under-characterization of the network since the network must optimize more parameters than there are data vectors to constrain these parameters. Introduction of additional training data since providing extra data for the network is more likely to produce a network that generalizes well to new data.

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 65 Table 4.2: Trial training algorithms and functions S/N Function Algorithm 1 trainbr Bayesian Regularization 2 trainlm Levenberg-Marquardt 3 trainscg Scaled Conjugate Gradient 4 trainrp Resilient Backpropagation 5 trainbfg BFGS Quasi-Newton

4.4.7

Trial training algorithms and learning functions

Training and learning functions are mathematical procedures used to automatically adjust the networks weights and biases. The training function dictates a global algorithm that aects all the weights and biases of a given network. Neural Network Matlab Toolbox supports a variety of training algorithms, including several gradient descent methods, conjugate gradient methods, the Levenberg-Marquardt algorithm (LM), and the resilient backpropagation algorithm (Rprop) as given in Table 4.2. A trial of most of training algorithms is done to select the best training algorithm that suits the problem under consideration. It is observed from Tables A.1 & A.3 that trainbr gives the best networks with least RMSE in most cases and will be used in all analysis. Though trainlm gives a close performance with trainbr at a shorter time but trainbr was preferred over it because one is only interested in a network that gives the best performance irrespective of the training time. Tables A.1 & A.3 show that the fastest training function is typically trainlm even when the networks are large, and it is the default training function for feed-forward net.

4.4.8

Performance study between single and ensemble neural nets

Tables A.1, A.3, A.2 and A.4 illustrate the comparative results with the ANNs designed with single and multiple nets respectively. It is noted that individual nets are observed to perform better than ensemble nets in terms of lower RMSE values and hence are adopted in all analysis.

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 66

4.4.9

ANN conguration to analyze the best network architectures

Choosing the number of hidden layers and neurons in the hidden layer is also a demanding task and it is the principal limitation of ANN. Since the ANN conguration has a great inuence on the predictive output, various arrangements have been considered. It is essential to designate a formula to describe the ANN conguration as (I -n1 -n2 -O). For example, 2-20-1 means a one hidden layer ANN with two input and one output parameters, with the hidden layers containing 20 elements (neurons); 3-10-10-2 denotes a three input and two output ANN, with 10 neurons in two hidden layers. The MATLAB neural network toolbox has been used as the basis in which the networks can be congured in a very wide variety of architectures, and the training algorithms can be also chosen with ease. Evaluation of the network performance is measured based on RMSE analysis and coecient of determination (R2 ). The power of prediction can be quantied by the root mean square error (RMSE) of the predicted output from test data. The smaller the RMSE of the test dataset is, the higher is the predictive capability of the network. The RMSE (i.e. the root mean square of the dierences between the actual and predicted values which should be very close to zero) is expressed as;

RM SE =

mean((RA RP )2 )

(4.2)

Where, RA and RP are the actual and predicted values from the network respectively. Similarly, R2 (i.e. a measure of how well the variation in the output is explained by the targets, if this number is equal to 1 there is a perfect correlation between targets and outputs) is given by; A )(RP R P ) (RA R A )2 ( (RP R P ))2 (RA R

R2 =

(4.3)

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 67 A and R P are dened as the mean of the actual and predicted values from the network. R 4.4.9.1 Training performance of trial neural networks for the forward problem

Several neural network architectures were tried since there is no laid down rule for choosing the optimal number of hidden layers and processing elements for each layer and it is basically problem dependent. However, incorporation of more processing elements enhances the generalization capability of the network for a larger number of training data points. The motivation was to determine the best network with the least RMSE and good coecient of determination (R2 ) irrespective of the time. In other words, the length of computation time was not considered as a factor for choosing the best network. However, larger network architecture takes longer running time in all analysis; and smaller networks take a shorter time. It is shown that when enough training data sets have been used, the root mean square of the output error converges to zero. The neural network optimum network architecture in most cases consisted of one input layer, 1 or 2 hidden layers, and one output layer with the circles representing processing elements or neurons in Figure 4.5. For the inverse problem, the eight inputs were the dierences between the damaged beam and those of the undamaged beam for the rst eight natural frequencies and the outputs were the interface, delamination size and location and vice versa for the forward problem. The hidden layers contained maximum of 80 neurons. Firstly, consider the solution of a forward problem trained with ANN using dierent network architectures and with dierent dataset scenarios. Tables A.5 and A.6 demonstrate the trial network architectures with their training and testing RMSE values for the individual eight percentage frequency changes (individual nets) taking as output to the network with two inputs (location and size of delamination). It is shown that 2-80-1 is the best network with the least RMSE. From Table A.5 and Table A.6, it is seen that high network architectures outperform (yield better results) than smaller networks as evident in low RMSE values and high R2 values. Similarly, Figure 4.7 and Figure 4.8 illustrate performance of dierent

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 68 network architectures and it is evident that high network architectures give better results than the low network architecture for both training and testing datasets.

Figure 4.7: Comparison of performance of dierent network architectures on RMSE of training data Secondly, consider dataset scenario 2 with 40 data points used for training of network. Tables A.7 and A.8 list the trial network architectures with their RMSE and R2 values for the eight percentage frequency change taking as output to the network. It is shown that 2-8-8-1 is the best network with the least RMSE and highest R2 values. Thirdly, consider dataset scenario 3 with 28 data points used for training of network. Tables A.9 and A.10 list the trial network architectures with their RMSE and R2 values for the eight percentage frequency change taking as output to the network. It is shown that 2-8-8-1 is the best network with the least RMSE and highest R2 values. Fourthly, consider dataset scenario 4 with 20 data points used for network training. Tables A.11 and A.12 list the trial network architectures with their RMSE and R2 values

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 69

Figure 4.8: Comparison of performance of dierent network architectures on RMSE of testing data for the eight percentage frequency change taking as output to the network. It is shown that 2-80-1 is the best network with the least RMSE and highest R2 values. Finally for the forward problem, considering dataset scenario 5 with 8 data points used for network training. Tables A.13 and A.14 demonstrate the trial network architectures with their RMSE and R2 values for the eight percentage frequency change taking as output to the network. It is shown that 2-80-1 is the best network with the least RMSE and highest R2 values. 4.4.9.2 Training Performance of trial neural networks for the inverse problem

For the inverse problem, where the eight natural frequencies are taken as input to the network and delamination location and size as network outputs. High network architectures are also found to give better results in terms of low RMSE and high values of R2 than small

4.4. ARTIFICIAL NEURAL NETWORK (ANN) AS AN INVERSE SOLVER AND SURROGATE CREATOR 70 network architectures as shown from Table A.16 through Table A.18 for dierent network architectures using all the dataset scenarios under consideration.

4.4.10

Study of the eect of transfer functions on network performance

To determine suitable transfer functions, one or more of the following transfer functions were tried to adopt the most suitable one for the problem study. Transfer functions adopted include: Logsig (Log-sigmoid transfer function) Purelin (Linear transfer function) Tansig (Hyperbolic tangent sigmoid transfer) Tables A.19 and A.20 demonstrate that network architectures with tansig and purelin transfer functions yield better results as evident in low RMSE values and high R2 values than other possible combinations tried. Where in Tables A.19 and A.20; Tan, Log and Pur are dened as the transfer functions Tansig, Logsig and Purelin respectively.

4.4.11

Data preprocessing

Preprocessing of the network inputs and targets improves the eciency of neural network training. It basically reduces the dimensions of the input vectors to increase network performance. This is essentially the normalization or scaling of inputs and targets so that they fall in the range [-1,1]. A normalization transformation ranging from values of -1 to 1 is tested as shown in Tables A.19 and A.20 to show if there is signicant improvement in the network performance. However, it was seen from Tables A.19 and A.20 that scaling has no great signicant eect on the network performance for the proposed problem under consideration.

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71

4.4.12

Summary of selected best neural network architectures

In conguring the MLP neural network, the number of hidden elements to be used was decided from the results of the previous analysis. Several dierent congurations of neural networks with one and two hidden layers with large number of neurons have been chosen to optimize network performance since the performance of networks with small number hidden neurons are not acceptable. Table 4.3 illustrates the best ANNs designed and used in all our analysis with a variety of input and output pairs. Where I -n1 -n2 -O respectively signies the input to the neural network (Z, X, a), number of neurons in the rst and second layer, and output (dFi ) to the network for the forward problem (i.e. surrogate based modeling) and vice versa for the inverse problem. It is noted that individual nets are observed to perform better than ensemble nets and hence are adopted in all analysis as shown in Tables A.1 to A.4. NA stands for not applicable with 8 datasets. Table 4.3: Adopted dierent network architectures with trainbr algorithm and Tan and Pur transfer functions
Datasets Architecture (I -n1 -n2 -O) 400 8-20-20-X , 8-20-20-a 40 8-15-15-X , 8-15-15-a 28 8-8-8-X , 8-8-8-a Solution to the inverse problem 20 8-2-2-X , 8-2-2-a 8 NA 639 4-20-20-X , 4-20-20-a 400 2-80-dFi 40 2-8-8-dFi 28 2-8-8-dFi Solution to the forward problem 20 2-80-dFi 8 2-80-dFi 145 3-22-22-dFi 181 3-15-15-dFi Application of ANN

4.5

Surrogate approach
Sul et al. [23] have adopted surrogate modeling to predict fatigue life in short bre

composites at elevated temperatures. In the proposed surrogate approach, ANN is used

4.5. SURROGATE APPROACH

72

to build surrogate models for approximation of the computationally expensive FE models by solving the forward problem. This approach is extensively solving the forward problem to approximate the output of the natural frequencies simulated using the FE models from given corresponding delamination parameters. Later, these surrogates are used in the optimization loop instead of direct optimization via FE models as shown in Figure 3.1. The surrogate approach is extensively employed as an inexpensive approximation of the true function evaluations instead of the computationally expensive FEM simulations. Surrogates are especially interesting for expensive objective functions, since the necessary computational eort to build the surrogate is smaller than the eort of the objective function evaluation. In this research, RSM was rst tried and failed to consistently provide a true approximation model due to high multi-modality of the delamination detection problem in higher modes say Mode 4 to Mode 8. Since ANN was found to perform very well for the solution of the problem under consideration, it was selected for the unique purpose of building surrogate models to expensive FE models. Other surrogate models were not tried.

4.5.1

Validation of the surrogate model

A validity check of the surrogate models is necessary to prevent misleading of search by the optimizers due to poor approximations. To validate the performance of the trained NN, a perfect match of true function evaluation by ANSYS and the approximated functions by ANN is shown in Table 4.4 for delamination signatures, [1, 54, 24] and [1, 62, 18] for the rst eight percentage changes in frequencies. The results are shown to have negligible error not more than 0.17%. Also, perfect ts plot and regression (R2 ) plots for the 400 training and 41 test datasets between the simulated % change in frequencies and the predicted output from the neural network training for Mode 1 are shown in Figures 4.9 to 4.10. The predicted points are encircled by the true points for the perfect ts. For all other modes (Modes 2 to 8) are shown in Appendix B.1.

4.5. SURROGATE APPROACH

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Table 4.4: Comparison of dFi between FE simulations and ANN approximations

[Z, X, a] Mode dF1 dF2 dF3 [1, 54, 24] dF4 dF5 dF6 dF7 dF8 dF1 dF2 dF3 [1, 62, 18] dF4 dF5 dF6 dF7 dF8 FE 0.8013 0.3182 10.6770 5.6542 14.5527 12.5625 13.9558 13.3759 0.4049 0.5266 2.6310 8.9072 2.1741 12.0654 9.4308 9.3618 ANN 0.8026 0.3179 10.6769 5.6545 14.5530 12.5636 13.9579 13.3778 0.4044 0.5266 2.6310 8.9068 2.1740 12.0639 9.4367 9.3599 % Error 0.1711 0.0831 0.0006 0.0050 0.0020 0.0083 0.0155 0.0134 0.1312 0.0075 0.0013 0.0039 0.0017 0.0126 0.0627 0.0199

(a) Actual vs predicted plot for (b) Actual vs predicted plot for Mode 1 training dataset Mode 1 testing dataset

Figure 4.9: Comparison between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 1 training and testing datasets respectively

4.6. RESPONSE SURFACE METHODOLOGY (RSM)

74

(a) Regression (R2 ) Plot for Mode 1 train- (b) Regression (R2 ) Plot for Mode 1 testing dataset. ing dataset.

Figure 4.10: Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 1 training and testing datasets respectively.

4.6
4.6.1

Response Surface Methodology (RSM)

Introduction to RSM

Response Surface Methodology (RSM) can be dened as development of mathematical and statistical techniques applied in the the modeling and analysis of engineering problems in which the output of interest is governed by some input variables and the key objective is to optimize this output response [90]. RSM is essentially a statistical method that employs quantitative data from appropriate simulations or experiments to determine and solve simultaneously multivariant equations. RSM also known as polynomial tness function modeling adopts regression curve tting to obtain mathematical approximations of responses of a given system as functions of some input design variables [56]. This method is widely employed as an inexpensive low order approximation model instead of the more time consuming but accurate calculations using FEM simulations. Response surfaces can easily be tted to data by least squares approach. The order of the polynomial is important; quadratic or cubic polynomials are mainly used, with quadratic polynomials best suited for continuous, unimodal problems. RSM models

4.6. RESPONSE SURFACE METHODOLOGY (RSM)

75

are widely used in polynomial approximation schemes due to their exibility and ease of use. In the polynomial approximation method, the response surface model is a polynomial of nth degree whose coecients are determined from a linear system of equations. The linear system is solved using least square minimization of the error between the predicted and the actual values.

4.6.2

Advantages of RSM

In the current study, the response surface methodology is adopted as another inverse problem solver because of the following benets it enjoys over ANN (see Figure 4.6): 1. It does not require too much computational eort and resources to generate its mathematical models (ease of calculations and use) 2. Its solution to the inverse problems can be approximately solved without the constraint of modeling and 3. its approximation model can be easily validated through statistical means.

4.6.3

Analysis of a rst-order model response surface

In response surface methodology, the factors that are considered as most important are used to build a polynomial model in which the independent variable is response from experiments or numerical simulations. A rst-order multiple regression model with N simulation runs carried out on k input variables and an output response R can be expressed as follows;

Ri = C0 + C1 dFi1 + C2 dFi2 + ... + Ck dFiq + i (i = 1, 2, ..., N )

k

(4.4) (4.5)

Ri = C0 +
j =1

Cj dFij + i (j = 1, 2, ..., k )

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76

The response Ri is a function of the input variables; dF1 , dF2 ,...,dFn , plus the error. Cj is the regression coecient. dFij represents the ith sample and j th independent variable. Equation 4.5 can be expressed in matrix form as;

R = dF C +

(4.6)

R1 1 dF11 dF12 dF1k C0 1 R2 1 dF21 dF22 dF2k C1 2 = + . . . . . . . . . . . . . . . . . . . . . Ck 1 dFN 1 dFN 2 dFN k N RN

R dF C

of the coecient Invoking the least square error method, the estimated coecient C vector (C) can be given as;

= (dF T dF )1 dF T R C

(4.7)

4.6.4

Analysis of a second-order model response surface

The presence of high curvature in the response surface makes the rst order models unsuitable for complex problems. A second-order model becomes handy in approximating the true response surface. The second-order model accommodates all the terms in the rst-order
2 model, in addition to quadratic terms (C11 dF1 i ) and all cross product terms (C13 dF1i dF3j ).

The method of least squares can also be applied to estimate the coecients in the model. The equation based on a second-order polynomial is given by;
k k k 1 k

Ri = C0 +
j =1

Cj dFj +
j =1

Cjj dFj2 +
i=1 j =i+1

Cij dFi dFj +

(4.8)

Hence, to develop the relationship between the variations in the simulated natural

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77

frequencies and the corresponding size and location of delamination, response surfaces of fourth-degree polynomials were generally adequate in this study. The RSM procedure described above is employed to t a fourth-order polynomial equation using the simulation data for dataset scenario 1 (441/400) where 400 FE model simulations are performed and 41 datasets used for testing. From the Minitab output, the fourth-order polynomial equation for predicting location and size is given below respectively where the output responses are the delamination location and size and the input are the rst percentage changes in natural frequencies. The least square error method is adopted to obtain the unknown coecients of the polynomials. All the insignicant interaction terms are removed from the models using ANOVA table. Table 4.5: Summary of generated 4th order polynomial equation for delamination location (X)
4th order polynomial equation in terms of input % change in natural frequencies 78.1 + 28.1 *dF1 - 29.6* dF2 + 11.1* dF3 + 16.5* dF4 - 7.65* dF5 - 1.74*dF6 + 7.90* dF7 + 6.22* dF8 + 6.62*dF1 *dF2 + 12.2*dF1 *dF3 - 0.647*dF1 *dF4 - 0.289*dF1 *dF5 + 1.79*dF1 *dF6 - 1.70* dF1 *dF7 + 1.05dF1 *dF8 - 1.90*dF2 *dF3 + 0.642*dF2 *dF4 + 0.279 *dF2 *dF5 + 0.253*dF2 *dF6 + 0.591 *dF2 *dF7 - 0.0537*dF2 *dF8 - 0.0150*dF3 *dF5 Location (X) + 0.546*dF3 *dF6 + 0.284*dF3 *dF7 - 0.170*dF3 *dF8 - 0.234* dF4 *dF5 - 0.0908*dF4 *dF6
2 2 2 2 - 0.150*dF4 *dF7 - 0.0582 *dF4 *dF8 - 50.5*dF1 - 0.015*dF2 -1.21* dF3 + 0.666*dF4 3 2 2 2 2 3 + 0.314*dF5 + 0.274*dF6 - 0.015*dF7 - 0.087*dF8 + 2.14 *dF1 + 0.0324 *dF2 3 3 3 3 3 3 + 0.0286*dF3 - 0.00935*dF4 - 0.0156*dF5 - 0.0186*dF6 + 0.00317*dF7 + 0.0029*dF8 4 4 4 4 4 4 - 0.0570*dF1 - 0.00114*dF2 - 0.000677*dF3 + 0.000030*dF4 + 0.000114*dF5 + 0.000264*dF6 4 4 - 0.000055*dF8 - 0.000075*dF7

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78

Table 4.6: Summary of generated 4th order polynomial equation for delamination size (a)
4th order polynomial equation in terms of input % change in natural frequencies 13.8 - 6.16 *dF1 + 1.23* dF2 + 0.667* dF3 + 0.214* dF4 + 0.0664* dF5 - 0.0667* dF6 - 0.276* dF7 + 0.541* dF8 + 0.247* dF1 *dF2 - 0.0635* dF1 *dF3 + 0.0177* dF1 *dF4 + 0.0152* dF1 *dF5 + 0.0530* dF1 *dF6 - 0.0268* dF1 *dF7 - 0.168* dF1 *dF8 + 0.00728* dF2 *dF3 + 0.0328* dF2 *dF4 - 0.0118* dF2 *dF5 - 0.00396* dF2 *dF6 + 0.00520* dF2 *dF7 + 0.0443 *dF2 *dF8 + 0.00523* dF3 *dF5 - 0.00468* dF3 *dF6 Size(a) + 0.00086* dF3 *dF7 + 0.0228* dF3 *dF8 - 0.00051* dF4 *dF5 - 0.00669* dF4 *dF6
2 2 2 + 0.00139* dF4 *dF7 - 0.00449* dF4 *dF8 + 0.733 *dF1 - 0.213 *dF2 - 0.00885* dF3 2 2 2 2 2 + 0.0203* dF4 + 0.0339* dF5 + 0.0289 *dF6 + 0.0547*dF7 - 0.0752* dF8 3 3 3 3 3 - 0.0746* dF1 + 0.00782* dF2 + 0.000083* dF3 - 0.00118* dF4 - 0.00184* dF5 3 3 3 4 - 0.00128* dF6 - 0.00289* dF7 + 0.00389* dF8 + 0.00197* dF1 4 4 4 4 + 0.000034* dF5 + 0.000020* dF4 + 0.000020* dF3 - 0.000149 *dF2 4 4 4 + 0.000021* dF6 + 0.000047* dF7 - 0.000061* dF8

4.6.5

Validation and adequacy check of the developed models

In order to ascertain the adequacy and goodness of the developed response surface approximation models, coecient of determination (R2 ) as already dened in Section 4.4.9 and absolute average deviation (AAD) are adopted. The eciency of the model in terms of its predictive power can be determined by both (R2 ) and AAD because only R2 cannot be eectively used to measure the performance of the developed models. The R2 is basically a measure of how well the variation in the output is explained by the targets, if this number is equal to 1 there is a perfect correlation between targets and outputs. However, a large value of R2 does not necessarily imply that the regression model is a good one [91]. Thus, it is possible for models that have large values of R2 to yield poor predictions of new observations or estimates of the mean response. Plotting actual results versus predicted results from the model gives a straight line passing the origin with the angle of 45 deg but in practical cases the model fails to give accurate results to new data. This limitation of (R2 ) is eliminated

4.6. RESPONSE SURFACE METHODOLOGY (RSM)

79

by using absolute average deviation (AAD) analysis, which is a direct method for describing the deviations in the actual and predicted outputs by the models [91]. Minitab was used to conduct all analysis. The AAD is calculated by the following equation;
N

ADD =
i=1

([abs(RA RP )/RA ]/N ) 100

(4.9)

2 ) is dened as the imThe expression for R2 is given in Equation 4.3. Also Adjusted (R provement of R2 when the the number of terms in a model is adjusted and always lower than R2 value. It is given as; 1 R2 N 1/N k 1

2 = 1 R

(4.10)

Where RA and RP are the actual and predicted responses, respectively, and N is the number of simulation runs. Evaluation of R2 and AAD values together was just adequate to check the accuracy of the developed models. R2 must be close to 1 and the AAD between the predicted and actual output must be as small as possible tending towards 0. Acceptable values of R2 and AAD values mean that the model equations denes the true behavior of the system and it can be used for interpolation in the simulation design space. From the equation shown in Table 4.5 for X, the R2 and adjusted R2 values for delamination location (X) are calculated as 99.3% and 99.2% respectively. The near perfect prediction (R2 value) of the model is an indicator that the model generated has been perfected to t the given data and thus highly signicant. The high R2 is a good indication of the predictive power of the developed model. Simialarly, from the equation shown in Table 4.6 for a, the R2 and adjusted R2 values for delamination size (a) is deduced to be 100% and 100% respectively. This shows that the developed model for delamination size has more predictive power in terms of accuracy of prediction results than that of the model for

4.6. RESPONSE SURFACE METHODOLOGY (RSM) delamination location.

80

Also, from the equations shown in Tables 4.5 and 4.6, the calculated AAD for the delamination location and size are respectively obtained as 2.78% and 0.37%. This indicates that the fourth-order polynomial model for delamination size (a) is highly signicant and adequate to represent the actual relationship between the response and the signicant input variables, with very small ADD value (0.37%) and a satisfactory coecient of determination (R2 1). For the response surface model for delamination location (X), it was found that the low AAD was obtained due to high non linear curvature in delamination location with respect to percentage change in frequencies.

Chapter

Results and discussion

5.1 Overview
This chapter presents the results and discussions of the earlier proposed methods. It starts with reported results of the proposed methods for delamination predictions at midplanes or at any known interface also referred to as the 2-variable problem. Thereafter, delamination prediction results of location and size at unknown interface is presented (3variable problem). This follows results of the experimental validations of the proposed solution methodology. Subsequently, the results of uncertainty quantication is reported. Finally, this chapter extends the application of the proposed solution methodology to the more complicated composite plates (the 5-variable problem).

5.2

2-Variable problemPrediction of delamination location and size at known interface

For the 2-variable problem (predicting the delamination size and location at a known

interface), considering only midplane delaminations, 441 FE models equally spaced at gaps of 2% were run in batch process for 2hrs assuming normalized delaminations located from

81

5.2. 2-VARIABLE PROBLEMPREDICTION OF DELAMINATION LOCATION AND SIZE AT KNOWN INTERFACE 82 30% to 70% (0.3:0.02:0.7) of the total beam length and having normalized sizes ranging from 18% to 58% at gaps of 2% (0.18:0.02:0.58). The 441 generated datasets were randomized using K-means clustering. 90.7% (400) of the 441 (shown in Figure 4.2(a)) simulations run in ANSYS is used for training a neural network model and the remaining 9.3% (41) of the dataset is used for testing of the network as shown in Figure 4.2(b). Eight random test cases from the reserved 41 test cases 4.2(b) with dierent delamination signatures at midplane (interface 1) for the dataset scenarios (400, 40, 28, 20, and 8) are used to study the performance of the proposed methodology.

5.2.1

Application of ANN for delamination prediction

A multilayer feed forward back propagation neural network with optimum network architectures is run for the dierent scenarios as shown in Table 4.3 by trial and error. Almost all the network structures give a root mean square error (RMSE) of 10e4 and 10e3 in training and test data respectively as illustrated in Table A.16 through Table A.18. The average time taken for the network training is 77secs. Table 5.1 reports the prediction results of ANN with 400 and 40 datasets for network training and Table 5.2 lists the prediction errors on 28 and 20 training datasets. %(E-X) and %(E-a) stand for the percentage error between actual and predicted values for location and size respectively. Delamination prediction via ANN takes an average time of 10 secs. It is shown that ANN prediction with 400 datasets yields excellent results with error less than 0.02%. However, as the training dataset decreases, the prediction error increases for all the test cases. With 40 training datasets, results are somewhat satisfactory with maximum error of 3.06% while ANN technique becomes almost inecient and unsatisfactory with 28 and 20 datasets as evident in prediction errors as high as 4.57% and 36.26% respectively for the 8 test cases. Therefore, with datasets as small as 20, ANN fails to give reasonable delamination prediction.

5.2. 2-VARIABLE PROBLEMPREDICTION OF DELAMINATION LOCATION AND SIZE AT KNOWN INTERFACE 83 Table 5.1: % Errors of 8 test cases with inverse ANN modeling using 400 and 40 training datasets.
S/N Actual [X, a] Predicted [X, a] (400) Predicted [X, a] (40) % 1 [54, 24] [53.99, 23.99] [52.44, 23.89] 2 [54, 20] [53.99, 19.99] [54.01, 20.02] 3 [62, 18] [61.99, 17.99] [69.39, 18.22] 4 [68, 24] [67.99, 23.99] [68.05, 23.88] 5 [52, 56] [51.99, 55.99] [51.81, 55.92] 6 [42, 56] [41.99, 55.99] [42.86, 56.21] 7 [68, 22] [67.99, 21.99] [67.58, 21.82] 8 [32, 44] [31.99, 43.99] [31.02, 43.61] (E-X) (400) % 0.014 0.009 0.02 0.0006 0.0099 0.0002 0.019 0.0035 (E-a) (400) % (E-X) (40) % (E-a) (40) 0.008 2.88 0.45 0.009 0.02 0.1 0.0006 2.25 1.22 0.0015 0.08 0.48 0.009 0.36 0.14 0.0002 2.05 0.38 0.009 0.62 0.83 0.002 3.06 0.89

Table 5.2: % Errors of 8 test cases with inverse ANN modeling using 28 and 20 training datasets.
S/N Actual [X, a] Predicted[X, a] (28) Predicted [X, a] (20) % (E-X) (28) % (E-a) (28) % 1 [54, 24] [53.36, 23.93] [45.57, 23.36] 1.19 0.29 2 [54, 20] [55.84, 20.05] [43.95, 20.99] 3.41 0.24 3 [62, 18] [61.18, 17.98] [39.52, 20.42] 1.33 0.08 4 [68, 24] [68.02, 23.92] [56.24, 24.28] 0.02 0.35 5 [52, 56] [50.12, 55.69] [63.06, 53.60] 3.62 0.54 6 [42, 56] [42.66, 55.98] [38.17, 54.11] 1.58 0.03 7 [68, 22] [69.41, 21.96] [56.90, 22.67] 2.08 0.16 8 [32, 44] [30.54, 44.13] [36.87, 45.90] 4.57 0.3 (E-X) (20) % (E-a) (20) 15.61 2.66 18.61 4.93 36.26 13.47 17.29 1.18 21.27 4.28 9.11 3.67 16.31 3.03 15.21 4.32

5.2.2

Application of RSM for delamination prediction

The developed response surface models in equations shown in Tables 4.5 and 4.6 were used for delamination prediction. Results (see Table 5.3) show that the RSM gives adequate approximations as an inverse tool using the variations in natural frequencies for delamination detection. From Table 5.3, it is seen that RSM can be used to successfully predict delamination location and size with maximum prediction errors of 10% and 1% in location and size respectively. The test data helps to establish that a model that closely matches the actual values is developed and it is still a mathematical t over the training data. Though it is seen that the results in predicting delamination size are more accurate than the delamination location because as shown in Figures 5.2, % change in frequency increase monotonously with increase in delamination size whereas for the delamination location versus % frequency

5.2. 2-VARIABLE PROBLEMPREDICTION OF DELAMINATION LOCATION AND SIZE AT KNOWN INTERFACE 84 change, the frequency changes increase with a very sharp curvature resulting to a high non linear complex function. Table 5.3: % Errors of 8 test cases using RSM with 400 data points. S/N Actual [X, a] Predicted [X, a] % (E-X) % (E-a) 1 [54, 24] [54.265, 24.104] 0.490 0.435 2 [54, 20] [53.058, 20.074] 1.745 0.371 3 [62, 18] [59.264, 18.151] 4.413 0.841 4 [68, 24] [69.070, 23.981] 1.573 0.081 5 [52, 56] [54.450, 56.325] 4.711 0.580 6 [42, 56] [46.034, 56.609] 9.605 1.087 7 [68, 22] [68.672, 21.992] 0.988 0.037 8 [32, 44] [33.978, 44.200] 6.183 0.455

5.2.3

Optimization without surrogates

Global search optimizer using RGA is adopted due to the discrete nature of the problem study in the interface variable. RGA is applied directly on ANSYS FE model of the objective function using a population of 40 members over 10 generations as shown in Figure 5.1(a). In other words, the RGA calls ANSYS to create data for each member of the population selected in each generation. The optimization takes a total CPU run time of 2hrs. This approach as shown in Figure 3.1 is limited to 400 function evaluations to be consistent with the number of function evaluations used to create the database for surrogate modeling. Table 5.4 shows the results of RGA without surrogates with 400 and 40 function evaluations respectively. In most test cases, prediction error of less than 1% is observed with a maximum error of 4.82% for the 400 function evaluations. However, prediction results are very poor with 40 function evaluations, hence the need for the adoption of optimization via surrogates.

5.2. 2-VARIABLE PROBLEMPREDICTION OF DELAMINATION LOCATION AND SIZE AT KNOWN INTERFACE 85

(a) Convergence plot of RGA without sur- (b) Convergence plot of RGA with surrorogates. gates.

Figure 5.1: Convergence plot RGA without and with surrogates. Table 5.4: % Errors of 8 test cases using RGA with 400 and 40 function evaluations via FE models.
S/N Actual [X, a] Predicted [X, a] (400) Predicted [X, a] (40) % (E-X) (400) % (E-a) (400) % 1 [54, 24] [53.96, 23.70] [52.76, 23.64] 0.08 1.25 2 [54, 20] [53.97, 19.04] [30.19, 19.69] 0.05 4.82 3 [62, 18] [61.74, 17.55] [37.76, 16.87] 0.42 2.48 4 [68, 24] [67.98, 23.83] [24.83, 23.54] 0.03 0.72 5 [52, 56] [51.88, 55.04] [37.95, 52.72] 0.24 1.71 6 [42, 56] [41.92, 55.26] [30.19, 23.99] 0.19 1.32 7 [68, 22] [67.59, 21.92] [32.05, 20.82] 0.59 0.36 8 [32, 44] [31.79, 42.92] [30.23, 31.85] 0.66 2.45 (E-X) (40) % (E-a) (40) 2.29 1.52 44.09 1.56 39.09 6.27 52.24 1.92 9.65 5.85 28.12 57.15 52.87 5.38 5.53 27.61

5.2.4

Surrogate Assisted Optimization (SAO)

Online optimization problems involve tasks that are solved within minutes whereas oine optimization tasks are processes that take hours to arrive at optimum solutions [92]. A methodology that converts oine optimization to online optimization with the use of surrogates is proposed. This is because the computation of natural frequencies of laminated composite beams demands a huge volume of calculations and consumes a lot of time, direct optimization of expensive objective functions become inecient. Hence, surrogates are built and integrated with the optimizers to solve the optimization problem. The use of surrogates therefore saves considerable time relatively i.e reduces the solution of inverse algorithms from hours to minutes. Many dierent surrogate modeling techniques like Kriging, RSM, ANN,

5.2. 2-VARIABLE PROBLEMPREDICTION OF DELAMINATION LOCATION AND SIZE AT KNOWN INTERFACE 86 etc have been developed and applied to optimization problems. ANN is considered as an eective surrogate model for the problem under consideration. In the surrogate approach, an objective function model is constructed using K-means clustered training data sets. An optimization algorithm then searches for the optimum using the surrogate model. To achieve the solution to the optimization problem via surrogates, an articial backpropagation neural network is developed that takes the percentage change in natural frequencies as outputs from the neural network and the interface, delamination size and location as inputs to the network. During the ANN training process, the surrogate model is rst trained using training sets and then a test set is given to measure its eectiveness for the unknown data. The nal approximation model and corresponding weights are saved and used for the optimization process. For 400, 40, 28, 20, and 8 training datasets used for building of the surrogate models, the network architectures assuming midplane delaminations are also given in Table 4.3. With these network architectures, low RMSEs are achieved. The average time of network training is 133secs. 5.2.4.1 Optimization using RGA with surrogates

40 member populations are allowed to evolve for 10 generations. In this method as shown in Figure 3.1, RGA is applied on the surrogate ANN model generated with 400, 40, 28, 20, and 8 datasets respectively. Unlimited number of function evaluations via surrogate can be allowed during the course of search. The optimization process takes an average time of 190secs. The convergence plot is shown in Figure 5.1(b) and it is evident that RGA with surrogates converges faster than RGA without surrogates even at the fourth generation when compared to Figure 5.1(a). Tables 5.5, 5.6 and 5.7 show results of optimization with surrogates using 400, 40, 28, 20 and 8 training data sets respectively. It can be seen that the results are good in all cases. Using datasets 400 and 40 for building of the surrogates, a maximum error of 1.82% is observed and the error increases to a maximum of 15.78% as the number of datasets

5.2. 2-VARIABLE PROBLEMPREDICTION OF DELAMINATION LOCATION AND SIZE AT KNOWN INTERFACE 87 is reduced to 8. Hence, optimization with surrogates yields better accurate results than optimization without surrogate even with very small number of datasets. Table 5.5: % Errors of 8 test cases using RGA via surrogate models created with 400 and 40 training datasets.
S/N Actual [X, a] Predicted [X, a] (400) Predicted [X, a] (40) % (E-X) (400) % (E-a) (400) % 1 [54, 24] [54.01, 23.98] [53.95, 23.94] 0.01 0.1 2 [54, 20] [54.03, 20.00] [54.13, 20.09] 0.05 0.003 3 [62, 18] [61.99, 18.00] [61.69, 18.00] 0.008 0.04 4 [68, 24] [67.98, 23.99] [67.96, 24.14] 0.03 0.03 5 [52, 56] [52.11, 56.16] [51.49, 55.52] 0.22 0.28 6 [42, 56] [41.98, 56.00] [42.05, 55.99] 0.05 0.013 7 [68, 22] [68.05, 22.01] [67.98, 22.39] 0.07 0.062 8 [32, 44] [32.09, 43.99] [31.84, 43.87] 0.28 0.02 (E-X) (40) % 0.097 0.235 0.5 0.06 0.98 0.12 0.03 0.497 (E-a) (40) 0.264 0.43 0.0007 0.57 0.86 0.009 1.82 0.28

Table 5.6: % Errors of 8 test cases using RGA via surrogate models created with 28 and 20 training datasets.
S/N Actual [X, a] Predicted [X, a] (28) Predicted [X, a] (20) % (E-X) (28) % 1 [54, 24] [53.83, 23.99] [55.12, 24.53] 0.31 2 [54, 20] [54.02, 20.25] [55.40, 21.70] 0.04 3 [62, 18] [62.03, 18.00] [61.15, 18.09] 0.04 4 [68, 24] [67.87, 24.03] [66.58, 24.07] 0.185 5 [52, 56] [52.49, 56.46] [51.67, 55.49] 0.94 6 [42, 56] [41.56, 55.61] [42.01, 56.07] 1.05 7 [68, 22] [67.81, 22.06] [36.83, 18.12] 0.28 8 [32, 44] [32.39, 44.36] [31.99, 43.91] 1.22 (E-a) (28) % 0.048 1.24 0.02 0.123 0.813 0.703 0.282 0.826 (E-X) (20) % (E-a) (20) 2.08 2.2 2.59 8.51 1.38 0.47 2.092 0.27 0.64 0.89 0.027 0.13 3.452 2.67 0.004 0.2

Table 5.7: % Errors of 8 test cases using RGA via surrogate models created with 8 training datasets. S/N Actual [X, a] Predicted [X, a] % (E-X) % (E-a) 1 [54, 24] [52.49, 23.71] 2.8 1.21 2 [54, 20] [52.54, 17.77] 2.7 11.14 3 [62, 18] [60.77, 17.17] 1.99 4.61 4 [68, 24] [67.73, 23.97] 0.39 1.46 5 [52, 56] [43.79, 55.93] 15.78 0.13 6 [42, 56] [37.70, 55.32] 10.23 1.21 7 [68, 22] [65.84, 21.61] 3.18 1.77 8 [32, 44] [30.00, 42.08] 6.25 4.37

5.2. 2-VARIABLE PROBLEMPREDICTION OF DELAMINATION LOCATION AND SIZE AT KNOWN INTERFACE 88 5.2.4.2 Optimization using gradient based local search (GBLS) method with surrogates A local search algorithm can also be used as the underlying search strategy. Optimization takes an average time of 16 seconds with 700 maximum number of iterations evaluated. Tables 5.8, 5.9 and 5.10 show the prediction results with the local search solver via surrogate models created with 400, 40, 28, 20, and 8 datasets respectively. Maximum prediction error of 0.69% is observed using 400, 40 and 28 datasets for surrogate creation. With 20 and 8 dataset scenarios, the maximum prediction error is found to be 16.13%. Thus, results are also satisfactory and accurate with the gradient based local search performing bettter results than RGA via surroagte prediction results shown in Table 5.7. Table 5.8: % Errors of 8 test cases using GBLS via surrogate models created with 400 and 40 training datasets.
S/N Actual [X, a] Predicted [X, a] (400) Predicted [X, a] (40) % (E-X) (400) % (E-a) (400) % (E-X) (40) % (E-a) (40) 1 [54, 24] [53.99, 24.00] [53.97, 24.01] 0.006 0.03 0.05 0.045 2 [54, 20] [54.00, 19.99] [54.12, 20.08] 0.0002 0.02 0.2 0.38 3 [62, 18] [62.00, 17.99] [61.83, 17.88] 0.008 0.009 0.3 0.69 4 [68, 24] [68.00, 24.00] [67.94, 24.06] 0.003 0.01 0.09 0.25 5 [52, 56] [52.00, 55.99] [51.85, 55.88] 0.001 0.005 0.3 0.22 6 [42, 56] [42.00, 56.00] [41.99, 55.97] 0.001 0.004 0.01 0.055 7 [68, 22] [68.00, 21.99] [67.89, 22.05] 0.07 0.0006 0.15 0.2 8 [32, 44] [31.99, 44.00] [32.03, 43.98] 0.002 0.005 0.082 0.04

Table 5.9: % Errors of 8 test cases using GBLS via surrogate models created with 28 and 20 training datasets.
S/N Actual [X, a] Predicted [X, a] (28) Predicted [X, a] (20) % (E-X) (28) % (E-a) (28) % (E-X) (20) % (E-a) (20) 1 [54, 24] [53.75, 24.02] [55.27, 24.42] 0.47 0.095 2.34 1.74 2 [54, 20] [54.26, 20.05] [56.03, 20.83] 0.48 0.25 3.75 4.15 3 [62, 18] [62.10, 17.93] [61.10, 18.06] 0.16 0.4 1.44 0.34 4 [68, 24] [67.89, 24.05] [66.48, 23.86] 0.155 0.2 2.23 0.57 5 [52, 56] [52.32, 56.28] [51.62, 55.47] 0.62 0.5 0.72 0.95 6 [42, 56] [41.92, 55.88] [42.03, 56.12] 0.18 0.2 0.08 0.22 7 [68, 22] [67.85, 22.07] [65.61, 21.41] 0.22 0.335 3.52 2.69 8 [32, 44] [32.22, 44.17] [31.99, 43.90] 0.69 0.38 0.02 0.22

5.3. 3-VARIABLE PROBLEMPREDICTION OF DELAMINATION LOCATION AND SIZE AT UNKNOWN INTERFACE 89 Table 5.10: % Errors of 8 test cases using GBLS via surrogate models created with 8 training datasets. S/N Actual [X, a] Predicted [X, a] % (E-X) % (E-a) 1 [54, 24] [52.19, 23.82] 3.34 0.75 2 [54, 20] [52.62, 17.98] 2.55 10.09 3 [62, 18] [58.01, 17.39] 6.43 3.35 4 [68, 24] [67.66, 23.62] 0.503 1.58 5 [52, 56] [46.97, 55.95] 16.13 0.09 6 [42, 56] [39.32, 55.51] 6.39 0.88 7 [68, 22] [65.86, 21.61] 3.15 1.78 8 [32, 44] [28.31, 41.58] 11.53 5.5 Results tabulated so far indicate that using surrogate assisted optimization with training datasets as small as 20, delamination prediction is satisfactory though with higher number of training datasets upto 400 results are more accurate. Furthermore, it can be seen that prediction errors are maintained using 40, 28 and 20 training data sets. This is evident with the dierences in errors hardly noticeable. However, as the training datasets reduce to 8, the prediction results become unsatisfactory. This leads to the conclusion that with a minimum of 20 datasets, delamination damage can be predicted eectively.

5.3

3-Variable problemPrediction of delamination location and size at unknown interface

For the 3-variable problem, three variables; interface (Z), delamination size (a) and

location (X) are to be predicted. However, due to the discrete nature of the interface variable, ANN and gradient based local search are not employed for this kind of problem and only the global search method by RGA is applied. For the 8-ply laminate considered, four thickness/interlaminar positions or interfaces are obtainable. 40 datasets for each interface were generated giving a total of 160 datasets. 91% of the 160 datasets (145 data points) are used to create a surrogate model with an ANN architecture of 3-22-22-4 and the rest

5.3. 3-VARIABLE PROBLEMPREDICTION OF DELAMINATION LOCATION AND SIZE AT UNKNOWN INTERFACE 90 Table 5.11: RGA prediction via surrogates for 5 test cases. S/N Actual [Z, X, a] Predicted [Z, X, a] % (E-X) % (E-a) 1 [1, 70, 26] [1, 69.59, 26.06] 0.6 0.23 2 [2, 60, 26] [2, 60.02, 25.98] 0.03 0.08 3 [3, 64, 48] [3, 64.54, 48.49] 0.84 1.02 4 [4, 58, 18] [4, 58.02, 18.00] 0.03 0.02 5 [4, 54, 24] [4, 53.97, 24.02] 0.07 0.08 for testing the network performance. Only the rst four natural frequencies were employed for delamination prediction. Surrogate assisted optimization is applied to 5 test cases from the reserved test data for the purpose of validation. Optimization without surrogates (with 145 function evaluations) yield poor results for the 3-variable problem and hence its results are not reported. With optimization using RGA via surroagtes, 2000 function evaluations were done. Table 5.11 shows predictions with RGA via surrogate model with a maximum prediction error of 1.02%. The interfaces are predicted correctly in each case and the results for location and size show excellent prediction accuracy. Table 5.12 also shows predictions with the gradient based local search solver by reinitialization via surrogate model. In GBLS the interface is taken as a continuous variable, but when the predictions are rounded to the nearest integer, they are 100% accurate. The maximum error is 0.6%, for the X-location in case 1. Results are also satisfactory and accurate with the gradient based local search performing bettter than the surrogate assisted optimization with RGA. However, since GBLS is not suited for solving discrete problems, the results of the GBLS were discarded even though the results are satisfactory. Table 5.12: Gradient based local search prediction via surrogates for 5 test cases S/N Actual [Z, X, a] Predicted [Z, X, a] % (E-X) % (E-a) 1 [1, 70, 26] [1, 69.59, 26.06] 0.6 0.23 2 [2, 60, 26] [1.9994, 60.02, 26.02] 0.03 0.08 3 [3, 64, 48] [3, 64.00, 48.00] 0 0.004 4 [4, 58, 18] [4, 58.00, 18.00] 0 0 5 [4, 54, 24] [3.9995, 53.96, 23.98] 0.6 0.08

5.4. VALIDATION WITH PUBLISHED EXPERIMENTAL RESULTS

91

5.4
5.4.1

Validation with published experimental results

Experimental validation with Okafor et al.s results

Okafor et al. [1] used 850 datasets for NN training with rst four modes of natural frequencies to predict only the size of delamination. They studied midplane delaminations of a cantilever composite beam using experimental data. With same dimensions and material properties, a comparative analysis is shown in Table 5.13 based on these experimental results reported in [1]. A surrogate model using 90.7% of 160 datasets clustered using K-means algorithm were built. The data points span across interface 1 through interface 4, each with 40 data points. The architecture of the ANN surrogate model was 3-22-22-4. Optimization is applied on the surrogate model. The gradient based local search by reinitialization was adopted to avoid the local search from getting stuck at the local optimum due to the discrete nature of the interface. 2000 function evaluations were computed using RGA to get good prediction results using experimental data. For the prediction using neural network method, a network architecture of 4-20-20-2 is used. 639 datasets created with the combinations (44:0.5:48) and (18:0.5:53) for X-location and a-size respectively is utilized for the network training to ensure a more robust generalization of the network. The developed techniques are shown to be more ecient in terms of prediction accuracy. In Table 5.13, AN No = Okafor et al. ANN predictions and AN Nbr = ANN based on trainbr learning algorithm. NA means not applicable for predicting interface and location using AN No and also not applicable for predicting interface variable using AN Nbr .

5.4.2

Experimental validation with Su et al.s published results

Su et al. [2] developed a theoretical model of a clamped composite beam xed at both ends containing delamination and validated their analysis using experimental data. They considered 10-ply Graphite/Epoxy (GF/EP) composite laminate. They engaged only the

5.4. VALIDATION WITH PUBLISHED EXPERIMENTAL RESULTS Table 5.13: Comparison of results with Okafor et al. [1]s experimental results.
S/N Variables Actual(cm) Predicted [AN No ] (cm) Interface(Z) 1 NA 1 XActual 11.75 NA aActual 5.08 5.33 Interface(Z) 1 NA 2 XActual 11.75 NA aActual 10.16 10.57 Interface(Z) 1 NA 3 XActual 11.75 NA aActual 15.24 18.14 Predicted [AN Nbr ] (cm) NA 11.06 4.32 NA 11.43 6.94 NA 11.58 14.81 Predicted [RGA] (cm) 1 14.34 4.84 1 14.59 6.57 1 11.39 14.56 %Error [AN No ] NA NA 10.25 NA NA 4 NA NA 19 %Error [AN Nbr ] NA 5.87 14.96 NA 2.72 31.7 NA 1.45 2.82

92

%Error [RGA] 0 22.04 4.72 0 24.17 35.24 0 3.06 4.6

rst three modal eigenvalues for their analysis. Their GA method required 2000 function evaluations to get the optimal set. The maximum percentage error between the measured experimental and the theoretical frequencies for their analysis is calculated to be 15%. Using the same dimensions, boundry conditions and material properties of their composite beam, natural frequencies with corresponding delamination parameters are computed using the methodologies adopted in this thesis. The obtained results have been compared with their experimental results in Table 5.14. To apply RGA and the gradient based local search optimizers to their experimental data, a surrogate model with 200 data points clustered using K-means algorithm was built. The data points span across the interlaminar position in the beam thickness (z ), 0.1 through 0.5, i.e. z = 0.1, 0.2, ..., 0.5, each with 40 data points. But z = 0.5 (1/plyn) (interf ace 1), hence z = 0.1, 0.2, ..., 0.5 corresponds to Z = 5, 4, ...1, where plyn is the ply number given as 10. For example, when the interface, Z = 1; z = 0.5. Notably, only 181 data sets were used for the neural network training with an architecture of 3-15-15-3. Consequently, RGA is applied on the surrogate model over 2000 function evaluations. The results of the proposed SAO technique are competetive with the predictions of the SAO method with RGA better in Cases 1 and 2, but worse in Cases 3 and 4. However, ANN could not be used for prediction because of the discrete nature of

5.5. EFFECT OF ARTIFICIAL ERRORS AND NOISE

93

the interface variable. In Table 5.14, GAsu = Su et al. GA predictions and AN Nsu = Su et al. ANN predictions. Table 5.14: Comparison of performance of results with Su et al. [2]s published experimental results
S/N Variables Actual Predicted [GAsu ] 0.576 0.037 0.251 0.224 0.382 0.233 0.633 0.378 0.279 0.503 0.278 0.399 Predicted [AN Nsu ] 0.508 0.054 0.238 0.219 0.336 0.237 0.464 0.387 0.261 0.464 0.287 0.437 Predicted [RGA] 0.5 0.049 0.255 0.2 0.38 0.24 0.5 0.309 0.382 0.5 0.266 0.3512 %Error [GAsu ] 15.2 25.8 0.5 12 1.9 6.9 26.5 11.4 11.4 0.7 7.5 0.3 %Error [AN Nsu ] 1.6 8.1 4.8 9.5 10.4 5.2 7.2 3.2 4.4 7.2 4.3 9.3 %Error [RGA] 0 1.4 2 0 1.3 4 0 17.6 52.8 0 11.7 12.2

Interface(z) 0.5 Location(X) 0.05 Size(a) 0.25 Interface(z) 0.2 Location(X) 0.375 Size(a) 0.25 Interface(z) 0.5 Location(X) 0.375 Size(a) 0.25 Interface(z) 0.5 Location(X) 0.3 Size(a) 0.4

5.5

Eect of articial errors and noise

The objective of this exercise is to ascertain the amount of errors and noise in the

measured frequencies, the developed techniques can tolerate. The ANN model used for prediction is developed with 400 training datasets and the test cases are chosen from data not used during the network training. The surrogate models used for the optimization process is also developed with the same 400 training datasets.

5.5.1

Eect of articial errors

The error that was added articially is the dierence between computed percentage changes in natural frequencies (dFi ) by FE model and the actual percentage changes in natural frequencies that were input as test cases into the inverse algorithm solvers to determine

5.5. EFFECT OF ARTIFICIAL ERRORS AND NOISE

94

the delamination signatures as dened in Equation 5.2. The same percentage of errors were added across all frequency modes. dFM i dFi dFActual dFSimulated = dFSimulated dFi

%E =

(5.1)

Given any actual frequencies (dFM i ) for which the delamination signature is to be ascertained, a certain amount of articial error (E) is added using Equation 5.2 to obtain dFM ie .

dFM ie = dFi [1 + E/100]

(5.2)

dFM ie is the actual percentage changes in natural frequencies for the ith mode (say Mode 1) with errors; E = error % ranging from 10,...,20. Tables 5.15 to 5.17 show the results with articial errors for 5 delamination signatures at midplane from reserved test data. It is shown that with errors as high as 10%, prediction results are satisfactory with maximum errors of 5.56%, 5.54% and 6.46% using RGA via surrogates, gradient based local (GBLS) optimizer via surrogates and ANN respectively. With GBLS giving the best predictions with least errors. Table 5.15: Prediction results using RGA via surrogates with addition of articial errors. S/N [X, a] Error level 0 % Error 10 % Error 15 % %(E-X) %(E-a) %(E-X) %(E-a) %(E-X) 0.01 0.1 0.04 4.17 0.12 0.13 0.78 3.33 4.35 3.33 0.05 0.003 0.24 5.0 0.11 0.008 0.04 0.27 5.56 0.02 0.27 0.06 0.76 4.17 0.31

1 2 3 4 5

[54, [30, [54, [62, [60,

24] 46] 20] 18] 22]

Error %(E-a) 8.33 6.52 10.01 5.56 4.55

20 % %(E-X) 0.44 3.33 0.87 0.99 0.19

Error %(E-a) 8.33 8.7 10.01 5.56 9.09

5.5. EFFECT OF ARTIFICIAL ERRORS AND NOISE Table 5.16: Prediction results using GBLS optimizer via surrogates with addition of articial errors. S/N [X, a] Error level 0 % Error 10 % Error 15 % %(E-X) %(E-a) %(E-X) %(E-a) %(E-X) 0.003 0.03 0.44 5.54 0.65 0.004 0.006 3.21 5.01 4.16 0.0002 0.02 0.24 5.44 0.39 0.008 0.009 0.09 3.95 0.13 0.0006 0.01 0.37 4.12 0.52

95

1 2 3 4 5

[54, [30, [54, [62, [60,

24] 46] 20] 18] 22]

Error %(E-a) 8.08 7.4 8.11 5.8 6.03

20 % %(E-X) 0.83 4.04 0.56 0.16 0.65

Error %(E-a) 10.48 9.69 10.73 7.53 7.86

Table 5.17: Prediction results using ANN with addition of articial errors. S/N [X, a] 0% %(E-X) 0.01 0.0008 0.01 0.01 0.009 Error %(E-a) 0.002 0.002 0.0002 0.0006 0.002 Error 10 % Error %(E-X) %(E-a) 4.7 6.46 3.71 6.19 7.6 5.684 8.75 5.44 8.3 6.37 level 15 % %(E-X) 5.68 7.08 9.82 12.47 10.89

1 2 3 4 5

[54, [30, [54, [62, [60,

24] 46] 20] 18] 22]

Error %(E-a) 9.74 10.84 8.52 8.15 9.63

20 % %(E-X) 6.21 11.84 11.13 15.72 12.42

Error %(E-a) 13.06 17.23 11.37 10.86 12.93

5.5.2

Eect of articial noise

In experimental setups, noise in the function evaluation may be as a result of electrical uctuations, changing environmental conditions or limited measuring precision. The computed natural frequencies of the damaged and undamaged composite beams were corrupted with random variations which are referred to as noise. Articial random noise is added to simulated natural frequencies of both the undamaged and damaged composite beam data, in order to simulate the experimental uncertainties as shown in Figure 5.2. Uncertainty from physical experiments assumes the form of measurement errors, electrical uctuations, etc. To induce noise in our simulated natural frequency data, a random number generator, randn is used to create random numbers with a normal distribution having zero mean and a variance and standard deviation of one [93].

5.5. EFFECT OF ARTIFICIAL ERRORS AND NOISE

96

The natural frequencies of the undamaged composite beams (dFM uni ) with a certain amount of noise (N) is given as;

dFM uni = dFui [1 + N randn(1, D)/100]

(5.3)

The natural frequencies of the damaged composite beams (dFM dni ) with a certain amount of noise (N) is also given as;

dFM dni = dFdi [1 + N randn(1, D)/100]

(5.4)

Where; N = noise % ranging from 0,...,5; and D = number of datasets. Using Equations 5.3 and 5.4, the eect of noise is added on the natural frequencies of the undamaged and damaged composite beams. Actual percentage changes in natural frequencies with noise for the ith mode (dFM ni ) is given in Equation 5.5. dFM uni dFM dni 100 dFM uni

dFM ni =

(5.5)

The variation of percentage changes in natural frequencies of the rst four modes of the delaminated beam relative to those of the undamaged composite beam in the presence and absence of noise in terms of the delamination size and location are illustrated in Figure 5.2. The eects of the errors that accrue from the added noise on the percentage changes in natural frequencies for a delamination signature [1, 54, 24] simulated numerically are shown in Table 5.18. It is shown that when the natural frequency data are contaminated with noise, they are prone to error. From the error results, it is noticeable that the inuence of noise is greatly felt in the lower modes than the higher modes as also evident in the roughness of the mode shape plots of the lower modes shown in Figure 5.2.

5.5. EFFECT OF ARTIFICIAL ERRORS AND NOISE

97

(a) Mode 1 Frequency variation with (b) Mode 1 Frequency variation with no noise. 1% noise.

(c) Mode 2 Frequency variation with (d) Mode 2 Frequency variation with no noise. 1% noise.

(e) Mode 3 Frequency variation with (f) Mode 3 Frequency variation with no noise. 1% noise.

(g) Mode 4 Frequency variation with (h) Mode 4 Frequency variation with no noise. 1% noise.

Figure 5.2: Variation of frequencies with location and size of delamination damage with and without noise for modes 1 to 4.

5.5. EFFECT OF ARTIFICIAL ERRORS AND NOISE Table 5.18: Eect of errors accrued from noise addition on a delamination signature [1,54,24].
Noise Mode dFi (Noise free) dFM ni (Noisy) number (i) 1 0.803 0.803 2 0.32 0.32 3 10.68 10.68 4 5.66 5.66 0% 5 14.55 14.55 6 12.566 12.566 7 13.96 13.96 8 13.38 13.38 1 0.803 0.93 2 0.32 0.45 3 10.68 10.79 4 5.66 5.78 1% 5 14.55 14.7 6 12.566 12.68 7 13.96 14.07 8 13.38 13.49 1 0.803 1.67 2 0.32 1.19 3 10.68 11.46 4 5.66 6.48 2% 5 14.55 15.3 6 12.566 13.33 7 13.96 14.71 8 13.38 14.14 % Error

98

0 0 0 0 0 0 0 0 15.75 39.71 1.07 2.13 0.75 0.89 0.79 0.83 51.88 74.97 7.3 14.56 5.12 6.07 5.38 5.65

Table 5.19: Prediction results using RGA via surrogates with addition of articial random noise in the undamaged and damaged frequencies.
S/N [X, a] 0 % Noise %(E-X) %(E-a) 1 [54, 24] 0.01 2 [30, 46] 0.13 3 [54, 20] 0.05 4 [62, 18] 0.008 5 [60, 22] 0.27 0.1 0.78 0.003 0.04 0.06 Noise level 1 % Noise %(E-X) %(E-a) 0.383 0.03 1.28 0.43 1.53 4.74 0.03 0.89 0.78 5.23 2 % Noise %(E-X) %(E-a) 14.93 0.01 11.13 14.07 0.03 9.67 3.33 25.01 0.66 3.15 3 % Noise %(E-X) %(E-a) 20.41 1.2 16.5 37.6 35.7 46.5 3.38 19.95 31.46 10.5 16.9 0.99 15.3 1.17 33.8 5 % Noise %(E-X) %(E-a) 21.25 44.2 4.8 0.9 14.5

5.5. EFFECT OF ARTIFICIAL ERRORS AND NOISE

99

Table 5.20: Prediction results using GBLS optimizer via surrogates with addition of articial random noise in the undamaged and damaged frequencies.
S/N [X, a] 0 % Noise %(E-X) %(E-a) 1 [54, 24] 0.003 2 [30, 46] 0.004 3 [54, 20] 0.0002 4 [62, 18] 0.008 5 [60, 22] 0.0006 0.03 0.006 0.02 0.009 0.01 0.20 0.2 1.02 0.28 1.47 Noise level 1 % Noise %(E-X) %(E-a) 3.78 0.52 18.03 10.94 5.01 2 % Noise %(E-X) %(E-a) 14.88 1.39 12.27 13.93 0.16 9.9 3.41 17.34 11.9 6.03 3 % Noise %(E-X) %(E-a) 20.51 1.39 0.43 0.82 2.85 45.5 3.51 17.19 27.59 12.83 5 % Noise %(E-X) %(E-a) 16.88 11.03 0.09 16.98 2.78 21.72 46.22 6.64 60.0 16.11

Table 5.21: Prediction results using ANN with addition of articial random noise in the undamaged and damaged frequencies.
S/N [X, a] 0 % Noise %(E-X) %(E-a) 1 [54, 24] 0.01 2 [30, 46] 0.0007 3 [54, 20] 0.01 4 [62, 18] 0.01 5 [60, 22] 0.009 0.002 0.002 0.0002 0.0006 0.002 2.61 0.208 5.01 11.33 5.14 Noise level 1 % Noise %(E-X) %(E-a) 0.82 0.38 2.07 7.43 1.95 2 % Noise %(E-X) %(E-a) 15.62 1.462 119.2 11.46 1.97 5.84 2.46 13.28 4.42 6.35 3 % Noise %(E-X) %(E-a) 47.18 1.262 44.32 62.36 58.9 58.0 2.48 1.458 32.94 11.94 31.9 177.7 10.77 77.95 28.62 5 % Noise %(E-X) %(E-a) 17.15 33.21 3.29 13.5 27.34

Tables 5.19 to 5.21 show the prediction results with varying levels of noise added using RGA via surrogates, gradient based local search (GBLS) optimizer via surrogates and ANN respectively. It is shown that the accuracy of prediction results decreases as the noise % level increases. Prediction accuracy is maintained until noise % level rises to 2%. Hence from the obtained results, it is shown that the proposed methods is robust enough to tolerate errors and noise up to 10% and 2% respectively. It is observed that when compared with the results of addition of articial errors, that the results of articial noise addition is better even with 2% noise than 10% error. This is because of the same amount of error that was

5.6. 5-VARIABLE PROBLEMDELAMINATION DETECTION IN COMPOSITE PLATES 100 added accros all the modes for the error addition scenario.

5.6

5-Variable problemDelamination detection in composite plates

5.6.1

Prediction of delamination parameters for the composite plate laminates using RGA with surrogates

The nite element modeling for the composite plate is given in Section 3.4. For solution of the plate problem, 240 datasets were created for about 2hrs with rst eight modes of natural frequencies. Surrogate model was built using 212 of 240 datasets and the rest was used for testing. The data points using K-means clustering span across interface 1 through interface 4, each with 60 data points. The best architecture of the ANN surrogate model was 5-22-22-dFpi using trainbr as shown in Table A.21. It is seen that even when compared to other networks the RMSE values for the test data is not very good but even worse with smaller network architectures. This could be attributed due to the increase in the number of input variables to the network from 2 in the beam problem to 5. Optimization is applied on the surrogate model. 2000 function evaluations were computed using RGA for 576 secs. Table 5.22 shows the prediction results with RGA via surrogates and the results are somewhat satisfactory for some cases. However, Case 7 ([6, 23.3, 40, 30, 16.7]*) and Case 8 ([5, 48.3, 67.5, 33.3, 30]*) gives better results with the least prediction error as evident in the lower minimum objective function (M inObjS ) value as well. This is because Cases 7 and 8 are datasets from the training dataset. It can be concluded that poor results obtained could be because of over-tting problem of the surrogate model where the RMSE of the training cases are very low but very high for the testing data. Better results could be obtained by generating a more large database upto 1000 datasets, which could take more than 8hrs for each interface. This can be seen as computationally demanding.

5.6. 5-VARIABLE PROBLEMDELAMINATION DETECTION IN COMPOSITE PLATES 101 Table 5.22: % Errors of test cases using RGAW S for the 5-variable problem
S/N Actual [Z, X , Y , Xa , Yb ] Predicted [Z, X , Y , Xa , Yb ] M inObjS % Error [X , Y , Xa , Yb ] 1 2 [1, 35, 40, 30, 23.3] [2, 18.3, 60, 13.3, 26.7] [1, 34.92, 39.51, 29.87, 23.85] 0.0018 [2, 17.64 60.07 13.77 25.60] 0.0109 0.0140 0.0756 0.0054 [0.22, 1.24, 0.45, 2.38] [3.62, 0.11, 3.56, 4.12] [0.11, 7.82, 9.34, 16.11] [0.58, 1.53, 17.00, 22.69] [6.34, 3.79, 2.01, 3.93] [0.46, 0.10, 2.66, 1.71] [0.09, 0.02, 0.19, 0.11]

3 [4, 21.7, 67.5, 26.7, 33.3] [4, 21.72 72.78 29.19 27.94] 4 [2, 48.3, 52.5, 13.3, 23.3] [2, 48.02 51.69 15.56 18.01] 5 6 7 8 [4, 20, 70 33.3, 13.3] [1, 18.3, 42.5, 26.7, 15] [3, 23.3, 40, 30, 16.7]* [4, 18.07, 57.91 19.46 15.52] [1, 19.46 40.89 27.24 15.59]

0.1428 [9.66, 17.27, 41.57, 16.70]

[3 23.41, 39.96, 29.20, 16.98] 0.0036

[2, 48.3, 67.5, 33.3, 30]* [2, 48.26, 67.51, 33.24, 30.03] 0.0000

Chapter

Comparison of delamination prediction eciencies of dierent algorithms

6.1 Overview
The approach or algorithm a person might take or use can have a great eect on the accuracy of his/her results. In solving the problem of delamination detection in composite laminates, dierent algorithms were developed as seen in the previous sections and all the algorithms yielded comparable results. Hence, it is worthwhile to do a comparative analysis of these algorithms to ascertain the most eective and ecient in terms of accuracy of prediction. Eciency of algorithms with respect to the minimization optimization algorithms could be measured as the minimum time needed to lower the error below a certain specied value associated with the value of the objective function after a given number of runs. Whereas the eciency of ANN and RSM algorithms could be dened as the measure of performance in terms of RMSE and AAD respectively. Accordingly, one is only interested in the algorithm that gives the least prediction error regardless of the time but however the evaluation time should not be too large before expected results are evaluated for eective use in online SHM. With these factors under consideration, a comparative analysis is made in this chapter between four algorithms (ANN, RSM, GBLS and RGA) under six dierent 102

6.1. OVERVIEW Table 6.1: List of proposed algorithms Algorithm number Algorithm name Algorithm 1 Articial Neural network (ANN) Algorithm 2 Response Surface Method (RSM) Algorithm 3 GBLS with surrogates (GBLSW oS ) Algorithm 4 GBLS without surrogates (GBLSW S ) Algorithm 5 RGA with surroagtes (RGAW S ) Algorithm 6 RGA without surrogates (RGAW oS )

103

approaches (ANN, RSM, GBLSW S , RGAW S , GBLSW oS , RGAW oS ) as shown in Table 6.1. Where, GBLSW S and GBLSW oS are GBLS methods with and without surrogates respectively. Similarly, RGAW S and RGAW oS are RGA methods with and without surrogates respectively. So far it is been shown that most algorithms are fast and contain several capabilities such as: Ease of use and implementation and allowing a user to specify an input data for which the delamination signature is to be ascertained easily (RSM). Generation of random results upto a signicant number of runs and selection of the best result in terms of the minimum objective function value (RGA) For this study, in terms of dataset generation, dataset scenario 1 earlier described containing 400 training datasets and 41 test datasets were considered for this comparative analysis and 2hrs is required to generate the database. The training data used for this comparison was simulation data benchmarked at 400 datasets for all algorithms and 10 damage cases as shown in Table 6.2 out of the 41 reserved tests were selected for the performance comparative study. This comparison was based on the two variable problem i.e. predicting delamination location and size at mid-planes.

6.2. ALGORITHM 1 - ANN Table 6.2: Selected 10-test cases to ascertain the method with best performance S/N Delamination Signature [X, a] 1 [50, 26] 2 [68, 24] 3 [50, 20] 4 [30, 36] 5 [34, 44] 6 [58, 40] 7 [48, 34] 8 [58, 30] 9 [48, 28] 10 [62, 36]

104

6.2

Algorithm 1 - ANN
The ANN model used in the benchmark tests, had 8 input nodes, 80 hidden nodes, and

2 output nodes (i.e network architecture of 8-80-2). The training data contained 400 input and output vector pairs. The training time for the ANN algorithm is measured to be about 133secs. Table 6.3 gives the prediction results of the selected 10 cases using ANN and it is seen that ANN gives for the 10 cases a total % errors of 0.07% and 0.009% in location and size respectively. This shows that ANN is very unique in performance giving a maximum error of 0.014% in all the cases under consideration. Table 6.3: % Errors of 10 test cases using ANN modeling S/N Actual [X, a] Predicted [X, a] % ([(E-X), (E-a)]) 1 [50, 26] [49.9946 26.0000] [0.0109, 0.0001] 2 [68, 24] [68.0095 23.9990] [0.0139 0.0042] 3 [50, 20] [50.0068 19.9999] [0.0136 0.0003] 4 [30, 36] [30.0009 36.0000] [0.0031 0.0001] 5 [34, 44] [33.9994 44.0000] [0.0019 0.0001] 6 [58, 40] [57.9997 40.0004] [0.0006 0.0011] 7 [48, 34] [48.0056 34.0001] [0.0116 0.0002] 8 [58, 30] [57.9919 30.0003] [0.0139 0.0011] 9 [48, 28] [47.9980 27.9999] [0.0043 0.0004] 10 [62, 36] [61.9995 36.0005] [0.0008 0.0013] Total [0.0745, 0.0088]

6.3. ALGORITHM 2 - RSM

105

6.3

Algorithm 2 - RSM
Invoking the RSM regression models developed in Equations 4.5 and 4.6 using 400

datasets for delamination location and size respectively, the prediction results of the 10 selected cases are shown in Table 6.4. It is seen that the total sum of errors for the 10 test cases are 11% and 2% for delamination location and size respectively. However the maximum error for individual cases is shown to be 3% and 0.5% in delamination location and size respectively. These results suggest that the performance of the RSM algorithm results is quite satisfactory. It is also worth mentioning that this approach requires less time for computation with less than 5secs. The only reasonable time involved in this approach is the time required to create the 400 data points used for the surface tting. Table 6.4: % Errors of 10 test cases using RSM modeling S/N Actual [X, a] Predicted [X, a] % ([(E-X), (E-a)]) 1 [50, 26] [50.0724, 25.9779] [0.1448, 0.0850] 2 [68, 24] [69.0697, 23.9806] [1.5731, 0.0808] 3 [50, 20] [50.4786, 19.9550] [0.9572, 0.2250] 4 [30, 36] [29.9083, 36.1363] [0.3057, 0.3786] 5 [34, 44] [35.1090, 44.2064] [3.2618, 0.4691] 6 [58, 40] [59.3110, 40.0181] [2.2603, 0.0452] 7 [48, 34] [48.1630, 34.0942] [0.3396, 0.2771] 8 [58, 30] [58.4583, 29.9824] [0.7902, 0.0587] 9 [48, 28] [47.9457, 27.9744] [0.1131, 0.0914] 10 [62, 36] [62.8340, 35.9788] [1.3452, 0.0589] Total [11.0909, 1.7698]

6.4

Algorithm 3 - RGAW oS
RGAW oS approach is basically the minimization of the objective function directly from

the FE models. This approach is very time consuming, in other words computationally expensive. To be consistent with the amount of datasets used for database creation only 400 function evaluations of the FE models are allowed for about 2hrs. Because of this signicant amount of time required to produce 400 function evaluations, only one test run

6.5. ALGORITHM 4 - RGAW S

106

was allowed. Over this limit of function evaluations, Table 6.5 shows that prediction results are highly unsatisfactory with the total sum of errors for the 10 test case given as 35% and 15% in location and size respectively. The individual maximum errors in location and size predictions are seen to be 27% and 2.6% respectively. Table 6.5: % Errors of 10 test cases using RGAW oS S/N Actual [X, a] M inimumObj 1 [50, 26] 0.09550 2 [68, 24] 0.00553 3 [50, 20] 0.00852 4 [30, 36] 1.56888 5 [34, 44] 0.69202 6 [58, 40] 0.73237 7 [48, 34] 0.00528 8 [58, 30] 2.82593 9 [48, 28] 0.00543 10 [62, 36] 0.00481 Total Predicted [X, a] % ([(E-X), (E-a)]) [49.6446, 25.4675] [0.7108 ,2.0482] [68.0706, 23.8523] [0.1039, 0.6156] [50.0634, 19.4827] [0.1268, 2.5863] [31.3309, 36.1582] [4.4363, 0.4396] [34.6365, 43.4115] [1.8721, 1.3376] [57.8042, 38.9887] [0.3376, 2.5282] [48.0291, 33.2507] [0.0606, 2.2037] [42.2571, 30.0895] [27.1428, 0.2983] [48.0293, 27.3201] [0.0610, 2.4284] [62.1006, 35.6956] [0.1622, 0.8456] [35.0140, 15.3315]

6.5

Algorithm 4 - RGAW S
RGAW S approach is essentially the minimization of the objective function via the

surrogate models instead of the FE models. This approach is very time saving and increases optimization performance and its results. At 400 function evaluations, Table 6.6 shows the prediction results of the best and mean predicted values over 10 runs for each test case. The results are highly satisfactory with the total sum of errors for the 10 test case given as 1% and 2% in location and size respectively. The individual maximum errors in location and size predictions are seen to be 0.9% and 1% respectively. The mean and best predicted results over the 10 runs are also tabulated. It takes 107secs by the surrogate model to get to the lowest minimum objective function value at 1.95E 08. The standard deviation (Std) over the 10 runs for each case is found to be reasonable enough conrming the correlation of results. The total time taken for this approach is calculated as the time taken for building the database for the surrogate models and the optimization time using surrogates which is

6.6. ALGORITHM 5 - GBLSW OS given as;

107

Total run time = time taken to build database for surrogate modeling + average optimization time for the 10 runs = 7135 + 107 = 7242 secs Table 6.6: % Errors of 10 test cases using RGAW S via surrogate models over 10 runs
S/N Actual [X, a] Best [X, a] Mean [X, a] Best M inObjS Best % ([(E-X), (E-a)]) Std 1 [50, 26] [49.9291, 25.8786] [49.6955, 25.5635] 1.03E-04 [0.1419, 0.4668] [0.5757, 0.6222] 2 [68, 24] [68.0011, 24.0027] [68.0426, 24.0560] 1.36E-06 [0.0016, 0.0111] [0.2205, 0.1768] 3 [50, 20] [50.1820, 20.0756] [49.9154, 19.8753] 4.41E-04 [0.3640, 0.3781] [0.3117, 0.2228] 4 [30, 36] [30.0322, 36.0383] [30.2427, 36.0153] 3.10E-06 [0.1073, 0.1065] [0.2449, 0.2519] 5 [34, 44] [33.9957, 44.0029] [34.0788, 43.8958] 1.95E-08 [0.0126, 0.0066] [0.4450, 0.3539] 6 [58, 40] [57.9630, 40.0507] [58.0399, 40.0705] 4.46E-06 [0.0637, 0.1267] [0.3116, 0.4229] 7 [48, 34] [47.8901, 34.0011] [48.1582, 34.1024] 7.75E-05 [0.2290, 0.0032] [0.5521, 0.4906] 8 [58, 30] [58.0443, 29.8654] [57.7149, 29.9079] 6.91E-05 [0.0764, 0.4485] [0.8881, 0.3545] 9 [48, 28] [47.9878, 28.0775] [47.7617, 27.6020] 4.85E-05 [0.0255, 0.2767] [0.6261, 0.9934] 10 [62, 36] [61.9884, 36.0114] [62.0230, 36.0974] 6.10E-07 [0.0187, 0.0318] [0.1003, 0.2432] Total [1.0406, 1.8559]

6.6

Algorithm 5 - GBLSW oS
GBLSW oS approach is basically the minimization of the objective function directly

from the FE models. This approach is very time consuming requiring about 3hrs to execute average 603 function calls over 100 dierent start points. The function calls cannot be controlled to 400 function evaluations used in other algorithms and it takes a bit more function calls to obtain its result shown in Table 6.7. Prediction results are unsatisfactory with maximum total sum of errors for the 10 test case given as 29% in both location and size. The individual maximum errors in location and size predictions are seen to be 7% and 6% respectively. This approach requires no initial database creation and takes an average total time of 10016secs.

6.7. ALGORITHM 6 - GBLSW S Table 6.7: % Errors of 10 test cases using GBLSW oS over 100 start points S/N Actual [X, a] Predicted [X, a] M inimumObj % ([(E-X), (E-a)]) 1 [50, 26] [50.7110, 27.0946] 2.7400 [1.4220, 4.2100] 2 [68, 24] [67.8292, 24.8455] 2.9079 [0.2512, 3.5229] 3 [50, 20] [52.4490, 18.7351] 3.0489 [4.8980, 6.3245] 4 [30, 36] [30.7860, 36.3687] 1.0421 [2.6200, 1.0242] 5 [34, 44] [34.7297, 45.7929] 4.1068 [2.1462, 4.0748] 6 [58, 40] [54.8648, 39.4625] 3.9815 [5.4055, 1.3438] 7 [48, 34] [51.1505, 34.1076] 1.5220 [6.5635, 0.3165] 8 [58, 30] [59.9530, 31.3064] 4.4239 [3.3672, 4.3547] 9 [48, 28] [48.2997, 29.0019] 2.4912 [0.6244, 3.5782] 10 [62, 36] [61.0460, 35.9119] 1.8632 [1.5387, 0.2447] Total [28.8367, 28.9942]

108

6.7

Algorithm 6 - GBLSW S
GBLSW S approach is essentially the minimization of the objective function via the

surrogate models instead of the FE models. This approach is very time saving and increases optimization performance and its results. At an average 524 function calls over 10 dierent start points for the 10 test cases, the prediction results using this method is shown in Table 6.8. Results are also highly satisfactory with the total sum of errors for the 10 test case being 0.04% and 0.12% in location and size respectively. The individual maximum errors in location and size predictions are seen to be 0.06% and 0.04% respectively. The excellent results of GBLSW S can be attributed to very low objective function, with the lowest at 3.63E 09. The total time taken for this approach is calculated as the time taken for building the database for the surrogate models and the optimization time using surrogates which is given as; Total time = time taken to build database for surrogate modeling + average optimization time for the 10 runs =7135 + 124 = 7259secs.

6.8. SUMMARY OF COMPARATIVE RESULTS Table 6.8: % Errors of 10 test cases using GBLSW S over 10 start points.

109

S/N Actual [X, a] Predicted [X, a] Number of calls M inimumObjS % ([(E-X), (E-a)]) 1 [50, 26] [49.9983, 26.0013] 657 4.40E-07 [0.0034, 0.0050] 2 [68, 24] [68.0023, 24.0063] 477 1.26E-06 [0.0034, 0.0262] 3 [50, 20] [50.0011, 19.9926] 292 1.08E-05 [0.0022, 0.0370] 4 [30, 36] [30.0017, 36.0010] 530 4.45E-09 [0.0057, 0.0028] 5 [34, 44] [33.9993, 44.0010] 545 3.63E-09 [0.0021, 0.0023] 6 [58, 40] [57.9984, 40.0036] 635 5.50E-08 [0.0028, 0.0090] 7 [48, 34] [48.0014, 34.0022] 661 5.22E-08 [0.0029, 0.0065] 8 [58, 30] [58.0036 ,29.9994] 552 7.76E-08 [0.0062, 0.0020] 9 [48, 28] [48.0024, 28.0058] 670 2.83E-07 [0.0050, 0.0207] 10 [62, 36] [61.9991, 36.0038] 523 4.48E-08 [0.0015 ,0.0106] Total [0.0350 ,0.1220]

6.8

Summary of comparative results

Tables 6.9 and 6.10 give a comprehensive comparative analysis for in terms of com-

pletion time and least minimum objective function value and prediction errors respectively for the dierent algorithms denoted as ANN, RSM, RGAW oS , RGAW S , GBLSW oS , and GBLSW S . The prediction % errors in delamination location and size for the 10 test cases are plotted for better comparison in Figures 6.1 and 6.2 respectively. From Tables 6.9 and 6.10 and Figures 6.1 and 6.2, the following deductions can be made: Giving a general conclusion on the algorithm with best performance in terms of negligible prediction error irrespective of the time, it is seen from in Table 6.10 that ANN outperforms all other methods with a maximum prediction error of 0.012% in predicting delamination location and size. This is followed by GBLSW S , with a maximum prediction error of 0.06% in terms of delamination location and size and then RGAW S with maximum prediction error of 0.5%. Fouth in the performance order is the highly exible RSM with a maximum prediction error of 3.3% and 0.4 in predicting delamination location and size respectively. The optimiztion methods without surrogates

6.8. SUMMARY OF COMPARATIVE RESULTS

110

(GBLSW oS and RGAW oS ) both perform badly with high amount of errors which justies the approach and objective of this study since optimization without surrogates is not only computationally demanding but yields poor results in terms of prediction accuracy because of the huge amount of computational eort required to explore the design space. ANN and GBLSW S give the best delamination predictions. Results show not more than 0.06% error using both methods for predicting both delamination location and size at a known interface. However, ANN results are achieved by solving only the inverse problem whereas the optimization methods requires the solution of the forward problems. When ANN and RSM are compared. It is evident that ANN results beats the RSM in terms of prediction results; however the ease and exibility of the RSM methods when compared to ANN and even other methods oer vital advantages over other methods. This is basically because unlike other methods, it gives mathematical expressions (models) that can be used to predict delamination location and size of a modeled structure at any given time. GBLSW S in comparison with RGAW S oers better results in terms of accuracy as evident in the lowest minimum objective function value for the former and later given to be 3.63E 09 and 1.95E 08, respectively as shown in Table 6.9. However, subsequent comparisons between the two approaches when the number of variables considered increases from 2 to 5 as that of a more complicated composite plate revealed that RGAW S performs consistently better than the GBLSW S due to their ability to accommodate both discrete and continuous variables which are limited in other methods. When the minimum objective function values are compared in Table 6.9 for the optimization techniques, it is shown that GBLSW S gives the lowest minimum value of

6.8. SUMMARY OF COMPARATIVE RESULTS

111

objective function followed by RGAW S . The objective functions for optimization without surrogates are very high leading to their ineciency in delamination predictions. The number of iterations or function calls for GBLS cannot be controlled unlike the RGA and hence it yields better results even with function evaluations less than 300 calls. This justies the need to use surrogate models. It can also be deduced from Table 6.9 that it takes about the same completion time for all the algorithms to successfully predict delaminations. Table 6.9: Average completion time (ACT) and lowest minimum objective function values (Minimum Objfun) for dierent algorithms. S/N Algorithm *ACT =DT+RT (secs) Minimum Objfun 1 ANN 7135+133=7268 N/A 2 RSM 7135+5=7140 N/A 3 GBLSW oS 10016 4.42 4 GBLSW S 7135+124=7259 3.63E-09 5 RGAW oS 7135 2.82 6 RGAW S 7135+107=7242 1.95E-08 *ACT = DT+RT, where DT=time to create database and RT=run time for algorithm Table 6.10: Summary of comparative prediction % error results ([(E-X), (E-a)]) for dierent proposed algorithms.
S/N ANN RSM GBLSW oS GBLSW S RGAW oS RGAW S 1 [0.0109, 0.0001] [0.1448 0.0850] [1.4220 4.2100] [0.0034 0.0050] [0.7108, 2.0482] [0.1419, 0.4668] 2 [0.0139, 0.0042] [1.5731, 0.0808] [0.2512 3.5229] [0.0034, 0.0262] [0.1039, 0.6156] [0.0016, 0.0111] 3 [0.0136, 0.0003] [0.9572, 0.2250] [4.8980 6.3245] [0.0022, 0.0370] [0.1268, 2.5863] [0.3640, 0.3781] 4 [0.0031, 0.0001] [0.3057, 0.3786] [2.6200, 1.0242] [0.0057, 0.0028] [4.4363, 0.4396] [0.1073, 0.1065] 5 [0.0019, 0.0001] [3.2618, 0.4691] [2.1462, 4.0748] [0.0021, 0.0023] [1.8721, 1.3376] [0.0126, 0.0066] 6 [0.0006, 0.0011] [2.2603, 0.0452] [5.4055, 1.3438] [0.0028, 0.0090] [0.3376, 2.5282] [0.0637, 0.1267] 7 [0.0116, 0.0002] [0.3396, 0.2771] [6.5635, 0.3165] [0.0029, 0.0065] [0.0606, 2.2037] [0.2290, 0.0032] 8 [0.0139, 0.0011] [0.7902, 0.0587] [3.3672, 4.3547] [0.0062, 0.0020] [27.1428, 0.2983] [0.0764, 0.4485] 9 [0.0043, 0.0004] [0.1131, 0.0914] [0.6244, 3.5782] [0.0050, 0.0207] [0.0610, 2.4284] [0.0255 ,0.2767] 10 [0.0008, 0.0013] [1.3452, 0.0589] [1.5387, 0.2447] [0.0015, 0.0106] [0.1622, 0.8456] [0.0187, 0.0318] Total [0.0745, 0.0088] [11.0909, 1.7698] [28.8367, 28.9942] [0.0350, 0.1220] [35.0140, 15.3315 ] [1.0406, 1.8559]

6.8. SUMMARY OF COMPARATIVE RESULTS

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Figure 6.1: Performance comparison of dierent methods based on error of prediction in delamination location (X)

6.8. SUMMARY OF COMPARATIVE RESULTS

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Figure 6.2: Performance comparison of dierent methods based on error of prediction in delamination size (a)

Chapter

Conclusions and recommendations

7.1 Conclusions
In this research, an approach for structural health monitoring using methods based on computational intelligence and response surface models have been presented. The approach relies on the use of changes in natural frequencies between undamaged and damaged structures to identify the damaged interface, location and its size. The performance and comparison of the schemes are studied with varying sizes of training data sets using ANN, RSM and dierent optimization approaches (surrogate and non-surrogate based) via two optimization strategies (GBLS and RGA). The performance of the proposed approach is benchmarked against numerical models and previously published experimental results on delamination detection in composite beams. The methods are robust and ecient as illustrated using two and three variable formulations that deal with interlaminar position, size and location of delaminations. Results show that most of the proposed methods are capable of solving the delamination detection problems with limited datasets often with prediction errors within 1%. The results clearly highlight that the ANN and RSM models are ecient in solving only the 2-variable problem. The GBLS also can be eectively used to solve the 2 and 3-variable problems whereas the global search method by RGA eectively solves both the 2, 3 and 5-variable problems. The performance of the surrogate assisted opimtization 114

7.1. CONCLUSIONS

115

techniques are better than the ones without the aid of surrogates. While a maximum of 2000 function evaluations are required by the surrogate-assisted optimization models to get to feasible solutions in not more than 3 minutes, a similar number of function evaluations could take 10 hours by direct optimization via FE models. The inverse modeling developed by ANN is ecient in generalizing the learning ability of delamination patterns for delamination prediction in the presence and absence of large database for network training. The eectiveness of the ANN model is not surprising as the network has been trained with points spread through out the entire design space using K-means clustering algorthm, hence its global ability to detect delaminations is encouraging. Also the developed RSM models have a very good predictive power for delamination prediction as evident in the low values of absolute average deviation (AAD). The solution strategies which uses K-means clustering for data selection and one of the inverse algorithms (ANN, RSM, GBLS and RGA) successfully predicts delamination interface, location and size in laminated composite beams with tolerance of errors and noise up to 10% and 2% respectively. The methods need a minimum of 20 training datasets and rst three modes of natural frequency changes for delamination prediction in composite beams. For the composite plates, with a substantial large training datasets for building surrugate models, better results could be achieved. The conclusions from this thesis can be summarized thus: Dierent inverse algorithms for delamination damage detection for SHM have been successfully developed and tested with numerical and experimental results. The notable excellent delamination prediction results obtained reiterates the robustness and accuracy of the algorithms as well as the approach. Results also underline the advantages of using K-means clustering for the choice of database while integrating surrogates in the optimization loop. An interesting nding in this research is that related with the superiority of incorporation of surrogate models in the optimization loop. Surrogates with adequate approximation accuracy can be used to replace computationally expensive analysis.

7.1. CONCLUSIONS

116

Management of such surrogates i.e. training regimes, selection of training data, validation schemes play an important role in any surrogate assisted optimization exercise. Hence, surrogates enhance the optimization search performance by exploring the entire design space at a relatively shorter time. The use of optimization techniques directly in the optimization loop for delamination detection in composite beams and plates require evaluations of large numbers of candidate solutions and thereby making the evaluations computationally expensive. The research reported in this thesis is focused on improving the eciency of delamination detection results by allowing minimum computational cost. The approach adopted was to use surrogate models in lieu of expensive simulations to evaluate the natural frequencies of the laminates. Further, an optimization strategy has been developed by integration with surrogate models. The optimization methodology was successfully applied to delamination prediction with the objective of detecting lack of structural integrity in composites while minimizing computational costs. As already explained, the principal utility of adopting K-means clustering is that it provides a smart way of rigorously choosing a few points in a design space to eciently represent all possible points. Training points are chosen using this algorithm to create surrogates and also establish the mapping between target response and input variables. Because the main focus of this research thesis was to reduce computational expense, one key advantage of the K-means clustering is that it was used to reduce the number of computational experiments necessary to explore the response design space. While RSM and ANN enjoy so many similar benets in terms of straight forward relationships between target response and input variables, the number of training datasets required to achieve remarkable results are very high for the ANN in contrast to the RSM. For the RSM models, due to high non-linearity and complexity in the higher frequency modes used for delamination detection, higher order polynomials are em-

7.1. CONCLUSIONS ployed.

117

An articial neural network is a superior and more accurate modeling technique as compared to RSM as it represents the nonlinearities in a much better way. However, a major disadvantage of ANN is that the resulting weights of the trained network are dicult to interpret unlike the RSM which its models are easy to compute and interpret. Another problem peculiar to ANN in contrast to the RSM, is the diculty of nding an appropriate network architecture. The RSM algorithm is straight forward in that it gives a physical mathematical expression for delamination prediction and its model can easily be validated by statistical means. Both the RSM and ANN require a large number of numerical experiments for obtaining trained ANN model or regressed response surfaces for predicting the delamination location and size from the measured data. By means of K-means clustering, the total number of numerical experiments required to build ANN and RSM model is reduced to a substantial size. This represents a signicant computational cost reduction associated with the developed approach. It is shown that with small training data sets, ANN and RSM fail to give accurate prediction results. But when solving a forward problem by creating surrogates with small datasets (say 40) and doing surrogate assisted optimization with GBLS and RGA, prediction results are very satisfactory. When one is interested in solving only the 2-variable problem, RSM and ANN algorithms deliver a good job. But however for the 3-variables, surrogate assisted optimization using GBLS and RGA are very ecient and nally for the 5-variable problem as in composite plates, RGA is the most preferred.

7.2. RECOMMENDATIONS FOR FUTURE WORK

118

7.2

Recommendations for future work

Though the results so far reported have been satisfactory, some areas still need to be

looked into. Validation of the algorithms using numerical results have been proved very successful. However, when validations are made from experimental results, the results are only fair enough. This could be because the available experimental results suered from too many discrepancies and errors in the actual measurements. Results could be greatly improved by using validation from a more accurate experimental studies. Delamination damage indicators in this thesis have focused on using changes in natural frequencies to determine the delamination parameters. Future algorithms should incorporate combination of changes in natural frequencies and changes in mode shapes for delamination detection (which gives spatial information of damage in a structure unlike the natural frequencies). Also, other parameters such as damping which have been proven as an eective damage indicators should be considered. It will be worthwhile to study the performance of Dierential Evolution (DE) strategies compared to a Real Coded Genetic Algorithm (RGA) adopted in this work. Since DEs are most suitable for single objective optimization schemes, their performance study in the proposed problem study under consideration is ultimately recommended. Some extensions of the current work may be aligned in the way of exploring other surrogate models such as RBFs, Kriging, etc. Based on the results of this thesis, it seems to be a promising area of research. Considering that both (RGA and GBLS) search algorithms reached good results in some instances. As another future work, a global-local search (memetic) algorithm as part of the future work is to be pursued.

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It is evident from the results, that extensive work still needs to be done for the composite plates. Regardless of the computational expense of the simulations for the plate problem, a large database should be generated to solve the forward problem for composite plates. Delamination prediction results for plates should also be validated with results from experiments.

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References
[1] A Chukwujekwu Okafor, K Chandrashekhara, and Y P Jiang. Delamination prediction in composite beams with built-in piezoelectric devices using modal analysis and neural network. Smart Materials and Structures, 5, no. 3:338347, 1996. [2] Zhongqing Su, Hang. Yin. Ling, Li. Min. Zhou, Kin. Tak. Lau, and Lin. Ye. Eciency of genetic algorithms and articial neural networks for evaluating delamination in composite structures using bre bragg grating sensors. Smart Materials and Structures, 14, no. 16:15411553, 2005. [3] D. Chakraborty. Articial neural network based delamination prediction in laminated composites. Materials & Design, 26, no 1:17, 2005. [4] Shi. J. Zheng, Zheng. Q. Li, and Hong. Tao. Wang. A genetic fuzzy radial basis function neural network for structural health monitoring of composite laminated beams. Expert Systems with Applications, 38:1183711842, 2011. [5] Buynak. C.F., Moran. T.J., and Martin. R.W. Delamination and crack imaging in graphite-epoxy composites. Materials Evaluation, 47:438441, 1989. [6] Manjunath Rao Pawar Dasarath Rao. Review of nondestructive evaluation techniques for FRP composite structural components. Masters thesis, West Virginia University, Morgantown, West Virginia, 2007. [7] W. R. Broughton, M.J. Lodeiro, G.D. Sims, B. Zeqiri, M. Hodnett, R. A Smith, and L. D. Jones. Standardized procedures for ultrasonic inspection of polymer matrix composites. In ICCM 12, Paris, France., 1999. [8] W J Cantwell and J Morton. The signicance of damage and defects and their detection in composite materials: A review. The Journal of Strain Analysis for Engineering Design, 27, no. 1:2942, 1992. [9] Scott W. Doebling, Charles R. Farrar, Michael B. Prime, and Daniel W. Shevitz. A review of damage identication methods that examine changes in dynamic properties. Shock and Vibration Digest, 30, no. 2:91105, 1998. [10] Ulrike Dackerman. Vibration-based damage identication methods for civil engineering structures using articial neural networks. PhD thesis, Faculty of Engineering and Information Technology, University of Technology, Sydney, 2010. [11] P. Cawley and R. D. Adams. The location of defects in structures from measurements of natural frequencies. The Journal of Strain Analysis for Engineering Design, 14, no. 2:4957, 1979. [12] H.L. Kim and W.M Yiu. Delamination detection in smart composite beams using lamb waves. Smart Mater. Struct., 13:544551, 2004.

REFERENCES

121

[13] Fang. X and Tang. J. Frequency response based damage detection using principal component analysis. In Proceedings of the IEEE International Conference on Information Acquisition, Hong Kong and Macau, China, pages 407412, 2005. [14] H.L. Chen, C.C. Spyrakos, and G. Venkatesh. Evaluating structural deterioration by dynamic response. Journal of Structural Engineering, 121, no. 8:11971204, 1995. [15] Chaitanya. A. Deenadayalu, Hsin. Piao Chen, and Aditi Chattopadhyay. Characterization and detection of delamination in smart composite structures. In 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, New York, 2004. [16] Barroso L.R and Rodriguez R. Damage detection utilizing the damage index method to a benchmark structure. Journal of Engineering Mechanics, 130, no. 2:142151, 2004. [17] Aftab Mufti. Guidelines for structural health monitoring. Technical report, ISIS Canada, Intelligent Sensing for Innovative Structures, A Canadian Network of Centres of Excellence, 227 Engineering Building, University of Manitoba, 2001. [18] L. H. Tsoukalas and R. E. Uhrig. Fuzzy and neural approaches in engineering. New York: Wiley Interscience, 1997. [19] A. O. Addina, S. M. Sapuanb, E. Mahdic, and M. Osmand. Prediction and detection of failures in laminated composite materials using neural networks - a review. Polymers and Polymer Composites, 14, no. 4:433441, 2006. [20] T. Kanungo, D. M. Mount, N. S. Netanyahu, C. D. Piatko, R. Silverman, and A. Y. Wu. An ecient k-means clustering algorithm: analysis and implementation. Pattern Analysis and Machine I, 24(7):881892, 2002. [21] K. Schittkowski. NLPQL: A fortran subroutine solving constrained nonlinear programming problems. Annals of Operations Research, 5, no. 4:485500, 1986. [22] T. Ray., T. Kang, and K.C. Seow. Multi-objective design optimization by an evolutionary algorithm. Engineering Optimization, 33, no. 4:399424, 2001. [23] J.H. Sul, G. Prusty, and T. Ray. Prediction of low cycle fatigue life of short bre composites at elevated temperatures using surrogate modelling. Composites PartB, 42:no. 6, 2011. [24] Joe Soderquist Peter Shyprykevich. Aircraft advanced materials research and development program plan. Technical report, U.S. Department of Transportation Federal Aviation Administration Technical Center Atlantic City International Airport, NJ 08405, 1994. [25] Roberto A. Osegueda. Damage evaluation of structures using frequencies and mode shapes extracted from laser interferometry. Technical report, U.S. Army Research Oce, 1992.

REFERENCES

122

[26] R. Olssonm, J. Iwarsson, G G. Melin, A. Sjogren, and J. Solti. Experiments and analysis of laminates with articial damage. Composites Science and Technology, 63, no 2:199 209, 2003. [27] Another case for adequate setbacks. Springwater Preservation Committee P.O. Box 128 Springwater, NY 14560. [28] R.P Taylor. Fibre composite aircraft - capability and safety. Technical report, Australain Transport Safety Bureau, 2008. [29] Charles R. Farrar and Scott W. Doebling. An overview of modal-based damage identication methods. Technical report, Engineering Analysis Group Los Alamos National Laboratory Los Alamos, NM, 1998. [30] S.S Kessler, M.S. Spearing, and C. Soutis. Damage detection in composite materials using lamb wave methods. Smart Materials and Structures, 11, no. 2:269278, 2002. [31] Rytter. A. Vibration based inspection of civil engineering strucures. PhD thesis, Aalborg University, Aalborg, Denmark, 1993. [32] W.J.B. Grouve, L. Warnet, A. de Boer, R. Akkerman, and J. Vlekken. Delamination detection with bre bragg gratings based on dynamic behaviour. Composites Science and Technology, 68:24182424, 2008. [33] Nagesh Babu G. L and Hanagud S. Delaminations in smart composite structures: A parametric study on vibrations. In 31st AIAA/ASME/ASCE/AHS/ASC SDM Conference, pages 24172426, 1990. [34] Hanagud. S., Nagesh Babu. G.L, and Won. C.C. Delaminations in smart composite structures. In Proceedings, The 1990 SEM Spring Conference on Experimental Mechanics, Bethel, CT, Society for Experimental Mechanics, Inc., pp. 776-781, 1990. [35] S.W. Doebling, C.R. Farrar, and M.B. Prime. A summary review of vibration based damage identication methods. Shock and Vibration Digest, 30:91105, 1998. [36] H. Sohn, C.R. Farrar, M. Hemez, D. Shunk, W. Stinemates, and R. Nadler. A review of structural health monitoring literature: 1996-2001. Technical report, Los Alamos National Laboratory, 2003. [37] J Lee. Free vibration analysis of delaminated composite beams. Computers Structures, 74 no.2:121129, 2000. [38] Cawley. P. Low frequency NDT techniques for the detection of disbonds and delaminations. British Journal of Nondestructive Testing, 32:454461., 1990. [39] M H H Shen and J.E Grady. Free vibrations of delaminated beams. 1992. [40] P.M. Mujumdar and S. Suryanarayan. Flexural vibrations of beams with delaminations. Journal of Sound and Vibration, 125 no. 3:441461, 1988.

REFERENCES

123

[41] J. J. Tracy and G. C. Pardoen. Eect of delamination on the natural frequency of composite laminates. Journal of composite materials, 23, Issue 12:12001215, 1989. [42] Paolozzi. A. and Peroni. I. Detection of debonding damage in a composite plate through natural frequency variations. Journal of Reinforced Plastics and Composites, 9:369389, 1990. [43] Christian N. Della and Dongwei Shu. Vibration of delaminated composite laminates: A review. Applied Mechanics Reviews, 60, no 1:120, 2007. [44] Salawu. O.S. Detection of structural damage through changes in frequency: a review. Engineering Structures, 19, no. 9:718723, 1997. [45] Peter Cawley and Robert D. Adams. A vibration technique for non-destructive testing of bre composite structures. Journal of Composite Materials, 13 no. 2:161175, 1979. [46] Saravanos. D.A. Mechanics for the eects of delaminations on the dynamic characteristics of composite laminates. In Dynamic Characteristics of Advanced Materials, Eds. P.K. Raiu and R.F. Gibson, ASME Winter Annual Meeting, NCA-Vol 16/AMD-Vol. 172, ASME, pp. 11-21, 1993. [47] J. R. Banerjee, H. Su, and C. Jayatunga. A dynamic stiness element for free vibration analysis of composite beams and its application to aircraft wings. Computers and Structures, 86 Issue 6, 2008. [48] H. Luo and S. Hanagud. Dynamics of delaminated beams. International Journal of Solids and Structures, 37, 10:15011519, 2000. [49] F. Ju, H. P. Lee, and K. H. Lee. Free-vibration analysis of composite beams with multiple delaminations. Composites Engineering,, 4, no.7:715730, 1994. [50] S. Chakraborty and M. Mukhopadhyay. Free vibrational responses of frp composite plates: Experimental and numerical studies. Journal of Reinforced Plastics and Composites, 19, no. 7:535551, 2000. [51] R.W. Campanelli and J.J. Engblom. The eect of delaminations in graphite/peek composite plates on modal characteristics. Composite Structures, 31:195202, 1995. [52] F. Ju, H.P. Lee, and K.H. Lee. Free vibration of composite plates with delaminations around cutouts. Composite Structures, 31:177183, 1995. [53] Henneke II. E. G. Tenek. L. H. and Gunzburger. M. D. Vibration of delaminated composite plates and some applications to non-destructive testing. Composite Structures, 23:253262, 1993. [54] A Nag, D Roy Mahapatra, and S Gopalakrishnan. Identication of delamination in composite beams using spectral estimation and a genetic algorithm. Smart Materials and Structures, 11, no. 6:899908, 2002.

REFERENCES

124

[55] Abu S Islam and Kevin C Craig. Damage detection in composite structures using piezoelectric materials (and neural net). Smart Materials and Structures, 3, no. 3:318 328, 1994. [56] Akira Todoroki. The eect of number of electrodes and diagnostic tool for monitoring the delamination of CFRP laminates by changes in electrical resistance. Composites Science and Technology, 61, no. 13:18711880, 2001. [57] H.P. Chen, Hieu Le, Jin. Hwan. Kim, and Aditi Chattopadhyay. Delamination detection problems using a combined genetic algorithm and neural network technique. In 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, New York, 2004. [58] C. Harrison and R. Butler. Locating delaminations in composite beams using gradient techniques and a genetic algorithm. AIAA Journal, 39, no. 7:13831389, 2001. [59] Manish T. Valoor and K. Chandrashekhara. A thick composite-beam model for delamination prediction by the use of neural networks. Composites Science and Technology, 60, no. 9:17731779, 2000. [60] Steve E Watkins, Gilbert. W Sanders, Farhad Akhavan, and K Chandrashekhara. Modal analysis using ber optic sensors and neural networks for prediction of composite beam delamination. Smart Materials and Structures, 11, no. 4:489495, 2002. [61] Zou. Y, Tong. L, and Steven. G.P. Vibration based model dependent damage (delamination) identication and health monitoring for composite structures - a review. Journal of Sound and Vibration, 230, no. 2:357378, 2000. [62] M Krawczuk, W. Ostachowicz, and G Kawiecki. Detection of delaminations in cantilevered beams using soft computing methods. In System Identication and Structural Health Monitoring, number 243-252, Madrid, Spain, 2000. [63] C.H. Keilers and F.K. Chang. Identifying delaminations in composite beams using builtin piezoelectrics: part 1. experiments and analysis. Journal of Intelligent Materials and Structures, 6:649663, 1995. [64] Bishop. C.M. Neural networks and their applications. Review of Scientic Instruments, 65, no. 6:18031832, 1994. [65] T. Inada, Y. Shimamura, A. Todoroki, H. Kobayashi, and H. Nakamura. Damage identication method for smart composite cantilever beam with piezoelectric materials. In Structural Health Monitoring 2000, pages 986994. Stanford University, Palo Alto, California, 1999. [66] Manudha T. Herath, Kaustav Bandyopadhyay, and Joshua D. Logan. Modelling of delamination damage in composite beams. In 6th Australasian Congress on Applied Mechanics, ACAM 6, 2010.

REFERENCES

125

[67] E.V.V. Ramanamurthy and K. Chandrasekaran. Vibration analysis on a composite beam to identify damage and damage severity using nite element method. International Journal of Engineering Science and Technology (IJEST), 2011. [68] S.I. Ishak, G.R. Liu, H.M. Shang, and S.P. Lim. Locating and sizing of delamination in composite laminates using computational and experimental methods. Composites Part B: Engineering, 32, no. 4:287298, 2001. [69] Z. Zhang, K. Shankar, M. Tahtali, and E. V. Morozov. Vibration modelling of composite laminates with delamination damage. In Proceedings of 20th International Congress on Acoustics, ICA, Sydney, Australia, 2010. [70] A nite element for modeling delaminations in composite beams. B. v. sankar. Computers and Structures, 38, no. 2:239246, 1991. [71] Ramkumar. R. and Kulkarni. S. Free vibration frequencies of a delaminated beam. In 34th Annual Technical Conference, Reinforced/Composites Institute, Society of Plastics Industry, 1979. [72] Wang. J. and Liu. Y. Vibration of split beams. Journal of Sound and Vibration, 84, no. 4:491502, 1982. [73] Della. C. N. and Shu. D. Vibration of beams with double delaminations. Journal of sound and vibration, 282:919935, 2005. [74] Galal.A. Hassan, Mohammed.A. Fahmy, and Ibrahim Goda Mohammed. Eects of ber orientation and laminate stacking sequence on out-of-plane and in-plane bending natural frequencies of laminated composite beams. [75] J.R. Vinson, R.L. Sierakowski, and C. W. Bert. The behavior of structures composed of composite materials. Journal of Applied Mechanics, 54, no. 1:249249, 1987. [76] J. J. Tracy. The eect of delamination on the response of advanced composite laminates. PhD thesis, University of California, Irvine, 1987. [77] Alnefaie. K. Finite element modeling of composite plates with internal delamination. Composite Structures, 90, no. 1:2127, 2009. [78] D.X. Lin, R.G. Ni, and R.D. Adams. Prediction and measurement of the vibrational damping parameters of carbon and glass bre-reinforced plastics plates. Composite Materials, 18:132152, 1984. [79] L.H. Yam, Z. Wei, L. Cheng, and W.O. Wong. Numerical analysis of multi-layer composite plates with internal delamination. Composite Structures, 82:627637, 2004. [80] P.J. Markus Fromherz Yi Shang, Yungchun Wan and Lara S. Crawford. Toward adaptive cooperation between global and local solvers for continuous constraint problems. In CP01 Workshop on cooperative Solvers in Constraints Programming, Pahos, Cyprus, 2001.

REFERENCES

126

[81] Efren Mezura-Montes, Mariana Edith Miranda-Varela, and Rubi del Carmen GomezRamon. Dierential evolution in constrained numerical optimization: An empirical study. Information Sciences, 180, no. 22:42234262, 2010. [82] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2):182197, 2002. [83] Kalyanmoy Deb and Amarendra Kumar. Real-coded genetic algorithms with simulated binary crossover: Studies on multimodal and multiobjective problems. Complex Systems, 9:431454, 1995. [84] Helon Vicente Hultmann Ayala and Leandro dos Santos Coelho. A multiobjective genetic algorithm applied to multivariable control optimization. In ABCM Symposium Series in Mechatronics, volume 3, pages 736745, 2008. [85] Kumara Sastry, D.D. Johnson, A.L. Thompson, D.E. Goldberg, T.J. Martinez, J. Leiding, and J. Owens. Multiobjective genetic algorithms for multiscaling excited -state direct dynamics in photochemistry. In GECCO 06 Proceedings of the 8th annual conference on Genetic and evolutionary computation ACM New York, NY, USA, 2006. [86] J. B. MacQueen. Some methods for classication and analysis of multivariate observations. In Proceedings of 5-th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, University of California, 1967. [87] M. Man and B.G. Prusty. Neural network modelling for composite damage behaviour using a continuum approach. Composite Structure, 93:383391, 2011. [88] El Sadig Mahdi and Hany El Kadi. Crushing behavior of laterally compressed composite elliptical tubes: Experiments and predictions using articial neural networks. Composite Structures, 83, no. 4:399412, 2007. [89] H. Delashmit Walter and T. Manry Michael. Recent developments in multilayer perceptron neural networks. In Proceedings of the 7th Annual Memphis Area Engineering and Science Conference, MAESC, 2005. [90] Montgomery. Douglas. C. Design and Analysis of Experiments: Response surface method and designs. New Jersey: John Wiley and Sons, Inc., 2005. [91] Deniz Bas and Ismail H. Boyaci. Modeling and optimization i: Usability of response surface methodology. Journal of Food Engineering, 78, no. 3:836845, 2005. [92] A. Caponio, G. L. Cascella, F. Neri, N. Salvatore, and M. Sumner. A fast adaptive memetic algorithm for online and oine control design of pmsm drives. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 37(1):2841, 2007. [93] R.A. Shenoi M. Sahin. Quantication and localisation of damage in beam-like structures by using articial neural networks with experimental validation. Engineering Structures, 25, no. 14:17851802, 2003.

Appendix

Trial performance of dierent ANN network architectures, transfer functions, training functions and scaled values
A.1 Overview

The understanding of the best network architecture of a problem for maximum performance is crucial towards extending the capabilities of ANN for delamination detection and surrogate modelling. To obtain the best network performance, several network architectures are tried for the forward and inverse problems. The performance criteria using RMSE and R2 values are described in detail in Sections 4.4.8 and 4.4.9. In those sections, I made a quantitative comparison between dierent possible network architectures with respect to individual and ensemble nets and adopted the networks with best performance. The best network architectures using dierent training dataset schemes were selected as shown in Table 4.3. The following tables below show dierent network architectures (high and low), dierent training functions and use of single and ensemble nets. Also the performance the networks on scaled values and dierent transfer functions are also presented.

127

Table A.1: RMSE analysis for the forward problem using dierent training algorithms with 400 and 41 datasets for network training and testing respectively
Training RMSE dF1 Network 2-10-10-1 (trainbr=123secs) 0.0018 0.0031 0.0035 0.0037 0.0062 0.0385 1.1875 0.0363 2-80-1 (trainbr=186secs) 0.0016 0.0002 0.0002 0.0006 0.0006 0.0013 0.0032 0.0036 0.0021 0.0032 0.0076 0.0084 0.0091 0.0632 1.2426 0.0767 0.0030 0.0011 0.0008 0.0038 0.0024 0.0045 0.0068 0.0077 dF2 dF3 dF4 dF5 dF6 dF7 dF8 dF1 dF2 dF3 Test RMSE dF4 dF5 dF6 dF7 dF8

A.1. OVERVIEW

2-10-10-1 (trainlm=63secs) 0.0017 0.0002 0.0005 0.0059 0.0190 0.0312 0.2498 0.4632 2-80-1 (trainlm=90secs) 0.0019 0.0001 0.0001 0.0007 0.0035 0.0026 0.0022 0.0043

0.0035 0.0003 0.0008 0.0092 0.0342 0.0514 0.3133 0.6301 0.0034 0.0006 0.0004 0.0013 0.0084 0.1061 0.0107 0.0197 0.1079 0.3170 0.1734 0.2777 0.2276 0.3737 0.3490 0.5856 0.5647 0.8886 0.8430 0.8840 1.1946 1.1738 1.0614 1.2071 0.0598 0.2722 0.2236 0.3376 0.7148 0.6813 0.6386 0.8183 0.5185 2.1048 1.7781 1.9701 2.0133 1.3977 1.4648 1.2342 0.1092 0.1053 0.1078 0.1922 0.2976 0.3480 0.3994 0.4438 0.0797 4.7235 5.1217 4.7299 4.2384 4.4016 0.4523 0.6085

2-10-10-1 (trainscg=116secs) 0.0615 0.2148 0.1696 0.1997 0.2460 0.2423 0.3299 0.4250 2-80-1 (trainscg=110secs) 0.5092 0.6941 0.6256 0.8059 0.9327 0.9684 0.8011 0.9753 2-10-10-1 (trainrp=95secs) 0.0422 0.1647 0.1635 0.2799 0.3750 0.4681 0.6761 0.7166 2-80-1 (trainrp=87secs) 0.4059 1.3169 1.3578 1.2508 1.2410 1.1000 1.0916 0.8973

2-10-10-1 (trainbfg=151secs) 0.0615 0.0906 0.1065 0.1289 0.1670 0.2212 0.3417 0.3540 2-80-1 (trainbfg=213secs) 0.0582 4.6443 5.3398 4.9446 4.2159 4.6519 0.3468 0.5016

128

Table A.2: Trial of ensemble nets for the forward problem

Training RMSE dF1 Network 2-10-10-8 (trainbr = 56secs) 0.2419 0.1618 0.1818 0.1704 0.1816 0.2349 0.3440 0.3500 2-80-8 (trainbr = 172secs) 0.0021 0.0025 0.0040 0.0057 0.0092 0.0095 0.0089 0.0139 2-10-10-8 (trainlm = 34secs) 0.2931 0.1728 0.1806 0.2054 0.3488 0.3606 0.4357 0.9119 2-80-8 (trainlm = 57secs) 0.0019 0.0040 0.0049 0.0049 0.0080 0.0094 0.0090 0.0125 0.3115 0.2826 0.2738 0.2018 0.2446 0.3438 0.5067 0.7172 0.0066 0.0094 0.0071 0.0148 0.0166 0.0147 0.0182 0.0333 0.3158 0.1753 0.2314 0.2837 0.4930 0.4669 0.5562 0.9917 dF2 dF3 dF4 dF5 dF6 dF7 dF8 dF1 dF2 dF3 Test RMSE dF4 dF5 dF6 dF7 dF8

A.1. OVERVIEW

0.0034 0.0050 0.0119 0.0101 0.0115 0.0199 0.0166 0.0174

129

A.1. OVERVIEW

130

Table A.3: RMSE analysis for the inverse problem using dierent training algorithms with 400 and 41 datasets for network training and testing respectively Training RMSE Location (X) Size (a) Network 8-20-20-1 (trainbr = 225secs) 8-10-10-1 (trainbr = 76secs) 8-20-20-1 (trainlm = 15secs) 8-10-10-1 (trainlm = 12secs) 0.0002 0.0026 0.0430 0.0429 0.0003 0.0019 0.0009 0.0128 0.0044 0.0077 0.0841 0.0821 0.0038 0.0080 0.0090 0.0354 Test RMSE Location (X) Size (a)

Table A.4: Trial of ensemble nets for the inverse problem Training RMSE Location (X) Size (a) Network 8-20-20-2 (trainbr = 170secs) 8-10-10-2 (trainbr = 48secs) 8-20-20-2 (trainlm = 19secs) 8-10-10-2 (trainlm = 14secs) 0.0019 0.0055 0.0051 0.0795 0.0008 0.0034 0.0041 0.0522 0.0059 0.0114 0.0179 0.1846 0.0030 0.0054 0.0093 0.0627 Test RMSE Location (X) Size (a)

Table A.5: RMSE analysis with 400/441 datasets (2-80-1 network architecture with 300 epochs gives the best network with an average time of 133 secs for completion Training RMSE dF1 Network 2-80-1 2-60-1 0.002 0.0002 0.0002 0.0006 0.0006 0.0013 0.003 0.004 0.002 0.0002 0.0004 0.0005 0.005 0.005 0.005 0.004 0.003 0.001 0.0008 0.004 0.002 0.005 0.007 0.008 0.003 0.0003 0.0005 0.0001 0.007 0.007 0.0105 0.011 0.002 0.003 0.008 0.008 0.009 0.06 1.24 0.08 0.002 0.02 0.04 0.10 2.5 0.32 1.23 0.89 0.5 A.1. OVERVIEW dF2 dF3 dF4 dF5 dF6 dF7 dF8 dF1 dF2 dF3 Test RMSE dF4 dF5 dF6 dF7 dF8

2-10-10-1 0.0018 0.003 0.004 0.004 0.006 0.04 1.19 0.04 2-10-1 0.002 0.014 0.04 0.08 1.996 0.28 1.19 0.8997 0.5

2-5-5-1 0.0018 0.002 0.007 0.06 0.127 0.107 1.19

0.002 0.002 0.009 0.08 0.17 0.25 1.24

2-30-30-1 0.002 0.0004 0.0009 0.0024 0.009 0.016 0.036 0.045

0.002 0.0004 0.001 0.005 0.014 0.03 0.072 0.08

131

Table A.6: R2 values using 400/441 datasets with dierent network architecture R2 Training dF1 dF2 Network 2-80-1 2-60-1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.9999 0.9999 dF3 dF4 dF5 dF6 dF7 dF8 dF1 dF2 dF3 R2 Test dF4 dF5 dF6 dF7 dF8

0.9999 0.9999 0.9999 0.9999 A.1. OVERVIEW 0.9999 0.9999 0.9779 0.9999

2-10-10-1 1 2-10-1

0.9999 0.9793 0.9999

1 0.9999 0.9999 0.9999 0.9566 0.9989 0.9792 0.9874

1 0.9999 0.9999 0.9999 0.9337 0.9988 0.9778 0.9887

132

A.1. OVERVIEW

133

Table A.7: RMSE analysis with 40 data points for training (2-8-8-1 network architecture with 300 epochs gives the best network and it takes an average time of 55 secs for completion) Network dF1 dF2 dF3 dF4 dF5 dF6 dF7 dF8 0.786

2-60-1 0.001064 0.038205 4.914536 0.036805 1.714118 1.896684 1.3243 2-8-8-1 0.001005 0.000803 0.012

0.017355 0.008592 1.896121 1.312762 0.782065

2-10-10-1 0.000994 0.000496 4.917948 0.002407 0.044896 1.893866 1.311767 0.781884 2-15-1 0.001031 0.004409 0.083403 0.041637 1.714118 1.891287 1.325211 0.785993 2-3-3-1 0.002949 0.02468 0.170554 0.265078 1.624476 1.910973 1.325815 0.783474

Table A.8: R2 values using 40 datasets with dierent networks (2-8-8-1 network architecture with 300 epochs gives the best network) Network dF1 2-8-8-1 1 dF2 1 dF3 dF4 dF5 dF6 dF7 dF8

0.9999 0.9999 0.9999 0.9511 0.9764 0.9908

2-3-3-1 0.9999 0.9999 0.9998 0.9994 0.9733 0.9503 0.9759 0.9908

Table A.9: RMSE analysis with 28 data points (2-8-8-1 network architecture with 1000 epochs gives best network and it takes an average time of 62 secs for completion) Network dF1 dF2 dF3 dF4 dF5 dF6 dF7 dF8

2-8-8-1 0.000597 0.000586 0.000462 0.046986 2.152408 1.919299 1.367984 0.694562 2-10-10-1 0.000597 0.00058 4.950606 0.000591 2.152339 1.917633 1.366599 0.694153 2-20-1 0.001481 0.004529 4.956622 1.24704 2.160104 1.922691 1.367506 0.6998 2-3-3-1 2.195086 5.622086 9.656511 7.772794 7.01001 6.157217 5.924651 5.313779

A.1. OVERVIEW

134

Table A.10: R2 values with 28 datasets using dierent networks Network dF1 dF2 dF3 2-8-8-1 1 1 1 dF4 dF5 dF6 dF7 dF8

0.9999 0.9508 0.9488 0.9721 0.9912 1 0.9507 0.9489 0.9721 0.9912

2-10-10-1 1

1 0.8569

Table A.11: RMSE analysis with 20 data points (2-80-1 network architecture with 1000 epochs gives the best network and it takes an average time of 168 secs for completion) Network dF1 dF2 dF3 dF4 dF5 dF6 dF7 dF8

2-80-1 0.000694 0.000639 0.024681 0.383107 1.653734 1.461385 1.113525 0.744244 2-8-8-1 0.000705 0.000659 4.835092 0.000668 1.091619 1.430818 1.046889 0.739667

Table A.12: R2 values of using 20 datasets with dierent networks Network dF1 dF2 dF3 2-80-1 2-8-8-1 1 1 dF4 dF5 dF6 dF7 dF8

1 0.9999 0.9986 0.9739 0.9619 0.9831 0.9913 1 0.8515 1 0.9887 0.9635 0.9851 0.9914

Table A.13: RMSE analysis with 8 data points (2-80-1 network architecture with 1000 epochs gives the best network and takes an average time of 168 secs for completion) Network dF1 dF2 dF3 dF4 dF5 dF6 dF7 dF8

2-80-1 0.000735 0.000991 4.264345 4.048564 1.655025 0.14345 1.006374 0.332408

A.1. OVERVIEW

135

Table A.14: R2 values for 8 datasets with dierent networks Network dF1 dF2 2-80-1 1 dF3 dF4 dF5 dF6 dF7 dF8

1 0.895356 0.85095 0.960239 0.99979 0.9817 0.997563

Table A.15: RMSE analysis with 400/441 data points (8-20-20-1 network architecture with 300 epochs gives the best network and takes an average time of 77 secs for completion) Training RMSE Location (X) Size (a) Network 8-20-20-1 8-60-1 8-10-10-1 8-8-8-1 8-12-12-1 8-80-1 8-30-30-1 8-25-25-1 0.000417 0.013488 0.007225 0.015738 0.005491 0.018507 0.000762 0.000354 0.001477 0.003493 0.003266 0.004685 0.003112 0.004062 0.004574 0.001895 0.004799 0.021696 0.013166 0.022569 0.013252 0.070018 0.006052 0.013112 0.006989 0.01959 0.007882 0.012461 0.010944 0.023322 0.015538 0.004017 Test RMSE Location (X) Size (a)

A.1. OVERVIEW

136

Table A.16: RMSE analysis with 40 data points (8-15-15-1 network architecture with 300 epochs gives the best network and takes an average time of 25 secs for completion) Network Location (X) Size (a) 8-15-15-1 8-12-12-1 8-25-25-1 8-8-8-1 0.000496 0.00049 0.000494 0.000459 0.000445 0.000487 0.001063 0.00263

Table A.17: RMSE analysis with 28 data points (8-8-8-1 network architecture with 300 epochs gives the best network and takes an average time of 14 secs for completion) Network Location (X) Size (a) 8-8-8-1 8-8-8-8-1 8-20-1 0.000552 12.43124 0.001776 0.000422 11.7047 0.053727

A.1. OVERVIEW

137

Table A.18: RMSE analysis with 20 data points (8-2-2-1 network architecture with 300 epochs gives the best network and takes an average time of 11 secs for completion) Network Location (X) Size (a) 8-2-2-1 8-8-8-1 8-3-3-3-1 8-5-5-1 8-15-1 8-6-6-1 0.040928 12.26377 0.000532 0.000483 12.26377 0.000609 0.007107 11.95826 0.199025 0.227743 0.263237 0.221155

Table A.19: Network performance on scaled (-1 to +1) 400/441 dataset under dierent transfer functions combinations with RMSE Training RMSE dF1 Network 2-60-1(Tan,Pur) 0.0003 0.0002 0.0002 0.0002 0.0002 0.0002 0.0004 0.0004 2-60-1(Log,Pur) 0.0003 0.0002 0.0002 0.0002 0.0002 0.0002 0.0003 0.0005 2-60-1(Tan,Log) 0.62 0.71 0.45 0.3 0.3 0.3 0.03 0.3 0.04 0.35 0.09 0.011 0.034 0.03 0.014 0.013 0.06 0.13 0.03 0.011 0.03 0.03 0.014 0.013 0.06 0.13 0.03 0.65 0.744 0.468 0.36 0.25 0.33 0.39 0.31 0.02 0.03 0.025 0.016 0.178 0.074 0.13 0.09 0.02 0.03 0.026 0.021 0.178 0.086 0.17 0.093 A.1. OVERVIEW dF2 dF3 dF4 dF5 dF6 dF7 dF8 Test RMSE dF1 dF2 dF3 dF4 dF5 dF6 dF7 dF8

2-8-8-1(Tan,Pur) 0.0004 0.002 0.0032 0.007 0.15

2-8-8-1(Log,Pur) 0.0004 0.0012 0.006 0.0097 0.146 0.0329 0.0967 0.0891

138

Table A.20: Network performance on scaled (-1 to +1) 400/441 dataset under dierent transfer functions combinations with R2 R2 Training dF1 dF2 dF3 dF4 dF5 dF6 dF7 Network 2-60-1(Tan,Pur) 2-60-1(Log,Pur) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.99 1 1 1 1 0.99 0.99 0.999 0.99 0.999 0.999 0.999 0.999 A.1. OVERVIEW dF8 dF1 dF2 dF3 R2 Test dF4 dF5 dF6 dF7 dF8

2-60-1(Tan,Log) 0.79 0.768 0.88 0.91 0.91 0.88 0.92 0.909 2-8-8-1(Tan,Pur) 1 0.999 0.999 0.999 0.956 0.998 0.996 0.98739 2-8-8-1(Log,Pur) 1 0.999 0.999 0.999 0.957 0.997 0.98 0.987

0.843 0.8693 0.87 0.91 0.897 0.88 0.943 0.914 1 1 0.99 0.999 0.999 0.934 0.996 0.994 0.989 0.9998 0.9999 0.999 0.94 0.99 0.98 0.99

139

Table A.21: RMSE analysis for the forward problem using dierent training algorithms with 218/22 datasets for network training and testing
Training RMSE dF1 Network 5-22-22-dFi (trainbr=306secs) 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 5-10-10-dFi (trainbr=137secs) 0.0025 0.0127 0.0085 0.0285 0.0347 0.0185 0.0342 0.0602 5-22-22-dFi (trainlm=127secs) 0.0030 0.0000 0.0000 0.0002 0.0000 0.0006 0.0001 0.0001 5-10-10-dFi (trainlm=44secs) 0.0532 0.1113 0.1192 0.2668 0.4115 0.2139 0.2452 0.3751 0.0540 0.1288 0.2113 0.4697 0.5275 0.9528 0.3243 0.6801 0.0557 0.2026 0.1867 0.9847 0.2972 0.3662 0.5211 1.0429 dF2 dF3 dF4 dF5 dF6 dF7 dF8 dF1 dF2 dF3 Test RMSE dF4 dF5 dF6 dF7 dF8

A.1. OVERVIEW

0.1728 0.6068 0.3541 0.4317 0.9666 0.9785 0.9943 1.3592 0.0954 0.3006 0.2814 0.5693 0.8624 0.6874 0.8772 1.4715

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Appendix

Validation of the surrogate models

B.1 Overview

In Section 4.5.1, detailed analysis of the validation of the developed surrogate models used in the optimization loop was given. The 45 deg plots and perfect t plots for Mode 1 % change in frequencies were given in that section. Here, the plots of Mode 2 to Mode 8 are given. It is worth mentioning that the whole of this thesis document was built in Adobe
A using L TEX. Figures were drawn with Dia and converted to EncapsulatedPostScript (eps)

les. Graphs were generated with Matlab and converted to EncapsulatedPostScript (eps) les.

(a) Actual vs predicted plot for (b) Actual vs predicted plot for Mode 2 training dataset Mode 2 testing dataset

Figure B.1: Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 2 training and testing datasets

141

B.1. OVERVIEW

142

(a) Regression (R2 ) Plot for Mode 2 train- (b) Regression (R2 ) Plot for Mode 2 testing dataset. ing dataset.

Figure B.2: Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 2 training and testing datasets.

(a) Actual vs predicted plot for (b) Actual vs predicted plot for Mode 3 training dataset Mode 3 testing dataset

Figure B.3: Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 3 training and testing datasets

B.1. OVERVIEW

143

(a) Regression (R2 ) Plot for Mode 3 train- (b) Regression (R2 ) Plot for Mode 3 testing dataset. ing dataset.

Figure B.4: Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 3 training and testing datasets.

(a) Actual vs predicted plot for (b) Actual vs predicted plot for Mode 4 training dataset Mode 4 testing dataset

Figure B.5: Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 4 training and testing datasets

B.1. OVERVIEW

144

(a) Regression (R2 ) Plot for Mode 4 train- (b) Regression (R2 ) Plot for Mode 4 testing dataset. ing dataset.

Figure B.6: Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 4 training and testing datasets.

(a) Actual vs predicted plot for (b) Actual vs predicted plot for Mode 5 training dataset Mode 5 testing dataset

Figure B.7: Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 5 training and testing datasets

B.1. OVERVIEW

145

(a) Regression (R2 ) Plot for Mode 5 train- (b) Regression (R2 ) Plot for Mode 5 testing dataset. ing dataset.

Figure B.8: Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 5 training and testing datasets.

(a) Actual vs predicted plot for (b) Actual vs predicted plot for Mode 5 training dataset Mode 5 testing dataset

Figure B.9: Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 5 training and testing datasets

B.1. OVERVIEW

146

(a) Regression (R2 ) Plot for Mode 5 train- (b) Regression (R2 ) Plot for Mode 5 testing dataset. ing dataset.

Figure B.10: Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 5 training and testing datasets.

(a) Actual vs predicted plot for (b) Actual vs predicted plot for Mode 6 training dataset Mode 6 testing dataset

Figure B.11: Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 6 training and testing datasets

B.1. OVERVIEW

147

(a) Regression (R2 ) Plot for Mode 6 train- (b) Regression (R2 ) Plot for Mode 6 testing dataset. ing dataset.

Figure B.12: Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 6 training and testing datasets.

(a) Actual vs predicted plot for (b) Actual vs predicted plot for Mode 7 training dataset Mode 7 testing dataset

Figure B.13: Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 7 training and testing datasets

B.1. OVERVIEW

148

(a) Regression (R2 ) Plot for Mode 7 train- (b) Regression (R2 ) Plot for Mode 7 testing dataset. ing dataset.

Figure B.14: Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 7 training and testing datasets.

(a) Actual vs predicted plot for (b) Actual vs predicted plot for Mode 8 training dataset Mode 8 testing dataset

Figure B.15: Perfect ts between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 8 training and testing datasets

B.1. OVERVIEW

149

(a) Regression (R2 ) Plot for Mode 8 train- (b) Regression (R2 ) Plot for Mode 8 testing dataset. ing dataset.

Figure B.16: Regression (R2 ) plot between simulated percentage change in natural frequencies (Actual) and the predicted output (Predicted) from ANN model for Mode 8 training and testing datasets.