UNIVERSIDADE ESTADUAL PAULISTA "JÚLIO DE MESQUITA FILHO" FACULDADE DE ENGENHARIA CAMPUS DE ILHA SOLTEIRA JESSÉ AUGUSTO DOS SANTOS PAIXÃO DAMAGE QUANTIFICATION IN LAMINATED COMPOSITES USING GAUSSIAN PROCESS REGRESSION MODEL ILHA SOLTEIRA 2020 JESSÉ AUGUSTO DOS SANTOS PAIXÃO DAMAGE QUANTIFICATION IN LAMINATED COMPOSITES USING GAUSSIAN PROCESS REGRESSION MODEL Dissertation presented to the Faculdade de Engenharia de Ilha Solteira - UNESP as a part of the requirements for obtaining the Master in Mechanical Engineering. Knowledge area: Solid Mechanics. Advisor: Prof. Dr. Samuel da Silva ILHA SOLTEIRA 2020 Paixão Damage Quantification in Laminated Composites using Gaussian Process Regression ModelIlha Solteira31/08/202086 Sim Dissertação (mestrado)Engenharia MecânicaMecânica dos SólidosNão . FICHA CATALOGRÁFICA Desenvolvido pelo Serviço Técnico de Biblioteca e Documentação Paixão, Jessé Augusto dos Santos. Damage quantification in laminated composites using gaussian process regression model / Jessé Augusto dos Santos Paixão. -- Ilha Solteira: [s.n.], 2020 86 f. : il. Dissertação (mestrado) - Universidade Estadual Paulista. Faculdade de Engenharia de Ilha Solteira. Área de conhecimento: Mecânica dos Sólidos, 2020 Orientador: Samuel da Silva Inclui bibliografia 1. Damage quantification. 2. Gaussian process. 3. Composite structures. 4. Structural health monitoring. P149d ������������������������������ ����������������������� ��������������������������������������������������� ������������������������������������������������������� ������������������������������������������������ ������������������������ ������������������������������������������������������������������������������������������� ����������������� �������������������������������������� ��������������������������� ����������������������������������������������������������������������������������� ��������������������������������������������������������������� ������������������������� �������������������������������������������������������������������������������������� �������������������������������������� �������������������������������������������������������������������������������������� ��������������������������� ��������������������������������������������������� ����������������������������������� To my parents José and Marisa and to my sisters Josiane and Mirian for their love and teachings throughout my life. To my wife Giovanna for her love, patience and unwavering support. ACKNOWLEDGMENTS First I would like to thank God for giving me the knowledge, ability, and opportunity to complete this research project. I’m extremely grateful for his blessings in my life. I wish to express my deepest gratitude to my advisor Prof. Samuel da Silva for the unwavering guidance, patience, valuable advises, encouragement and friendship during the development of this research project. I would also like to extend my deepest gratitude to Prof. Eloi Figueiredo for the invaluable contribution to my work and for the support in the period I spent at Universidade Lusófona de Humanidades e Tecnologias. I’m also extremely grateful to Prof. Gyuhae Park for the helpful suggestions and support during my internship at Chonnam National University. I would also like to thank Prof. David Garcia Cava for providing the dataset of the wind turbine blade used in this work. I would like to extend my sincere thanks to all my friends from the Group of Intelligent Materials and Systems (GMSINT). Many thanks to Carlos Santana ("Grilo") for assistance and friendship. Special thanks should go to my friends who shared the room with me: João Trentin, Luis Villani, Marcus Omori, Rafael Teloli, and Lucas Zanovelo. Thank you for always contributing to the good atmosphere in the office, the numerous pieces of advice during the development of my work, and the friendship. Thanks also to my friend Kayc Wayhs Lopes with whom I shared good talks about some of the challenges found in our projects and uncountable cups of coffee. Thanks also should go to Bruna Pavlack with whom I worked in the last months on the dataset of the wind turbine blade. I would like to acknowledge the financial support provided by São Paulo Research Foundation (FAPESP) to this master project (grant number 2018/15671-1) and the re- search internship at Chonnam National University (grant number 2019/11755-9). I also acknowledge the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, that financed part of this study. My success and the completion of this work would not have been possible without the support and nurturing of my family to whom I dedicate this work. My parents Marisa and José for all the love and teachings throughout my life. My sisters Mirian and Josiane for the support in all my dreams. Finally, a huge thank you to Giovanna, my wife, for her love, patience and invaluable daily support! RESUMO Após detectar um dano em uma estrutura de material compósito laminado por meio de uma abordagem orientada a dados, o usuário precisa decidir se há uma falha estrutural iminente ou se o sistema pode ser mantido em operação sob monitoramento da progressão do dano e seu impacto na segurança estrutural. Neste cenário é essencial quantificar a extensão do dano para se tomar uma decisão. Portanto, este trabalho propõe uma metodologia para quantificação de dano baseada na aplicação de modelo de Regressão de Processos Gaussianos (GPR) para capturar a tedência e incertezas associadas a progressão do índice de dano global de acordo com a extensão do dano. O modelo GPR treinado de maneira supervisionada é utilizado para quantificar o dano através da interpolação estocástica dos índices de dano. Uma metodologia principal é proposta para quantificação de dalaminação em placas de compósito laminado utilizando índices de dano baseados no sinal de propagação de ondas Lamb. Modelos autoregressivos são aplicados para extrair características sensíveis ao dano e o algoritmo de distância quadrada de Mahalanobis é usado para calcular os índices de dano, embora diferentes técnicas de extração dos índices de danos possam ser adaptadas a metodologia. Para demonstrar a versatilidade da metodologia, uma versão modificada é apresentada com índices de dano baseados na análise espectral singular de sinais de vibração. Três aplicações experimentais são apresentadas para demonstrar a eficácia dessa abordagem – na primeira com uma placa de carbono-epóxi laminado com dano simulado e mudanças de temperatura para mostrar as etapas gerais do procedimento; na segunda envolvendo um conjunto de placas de polímeros reforçados com fibra de carbono com delaminação real causada em um ensaio de fadiga, para as quais a metodologia principal é aplicada; e, finalmente, um exemplo industrial envolvendo uma pá de turbina eólica com dano simulando a descolagem progressiva na borda de fuga, na qual aplica-se a versão modificada da metodologia proposta usando índices de danos baseados em sinais de vibração. O modelo GPR demonstra ser capaz de capturar a tendência e acomodar as incertezas relacionadas aos índices de dano versus o tamanho do dano nos pontos simulados nos testes. Os resultados demonstraram uma previsão adequada da área de delaminação simulada e real nas duas primeiras aplicações experimentais e da extensão de descolagem na pá de turbina eólica. Palavras-chave: Quantificação de dano. Processos gaussianos. Estruturas de material compósito. Monitoramento de integridade estrutural. ABSTRACT After detecting initial damage in a composite structure through a data-driven approach, the user needs to decide if there is an imminent structural failure or if the system can be kept in operation under monitoring to track the damage progression and its impact on the structural safety condition. In this scenario, it is essential to quantify the extension and the damage’s size to decide about these points. Therefore, this work proposes a damage quantification based on the application of the Gaussian Process Regression (GPR) model to capture the trend and uncertainties associated with the damage index progression ac- cording to damage extension. The GPR model trained in a supervised approach is used to quantify the damage by the stochastic interpolation of the damage indices. The cen- tral methodology is proposed for delamination area quantification in laminated composite plates using damage indices based on Lamb wave signals. Autoregressive models are ap- plied to extract damage-sensitive features from Lamb waves signals, and the Mahalanobis squared distance is used to compute damage indices, although any damage features ex- traction technique could be used and adapted to the proposed methodology. A modified version of the central methodology is proposed to demonstrate the methodology’s versa- tility, using damage indices based on singular spectrum analysis of vibration signals, a well-established technique in the literature. Three sets of tests are used to demonstrate the effectiveness of this approach — one in carbon-epoxy laminate with simulated damage under temperature changes to show the general steps of the procedure; a second test in- volving a set of carbon fiber reinforced polymer coupons with actual delamination caused by repeated fatigue loads, for which the central methodology is applied; and finally an industrial example involving a wind turbine blade with damage caused by debonding in the trailing edge and using traditional vibration-based damage indices. Various damage progression levels are measured during the tests and monitored using the sensors bonded to these structural surfaces. The GPR proved to be capable of capturing the trend and accommodating the uncertainties related to the damage indices versus the damage size in the simulated spots in the tests. The results manifest a smooth and adequate prediction of the size area of the simulated and real delamination damage in the two first application cases and the debonding size in the last application case. Keywords: Damage quantification. Gaussian process. Composite structures. Structural health monitoring. LIST OF FIGURES 1 Problem Statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Schematic representation of the methodology. . . . . . . . . . . . . . . . . 27 3 Flowchart of the proposed methodology for damage quantification in com- posite structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Composite plate and schematic view of the experimental setup (measure- ments in mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5 CFRP composite coupon and schematic view of the experimental setup in the application case I (measurements in mm). . . . . . . . . . . . . . . . . 39 6 Output signals from path between PZT 4 and 9 considering baseline (H) and damage (D8) conditions for the CFRP coupons (case II). . . . . . . . . 40 7 Example of X-ray image of composite coupon L1S11 under 40000 loading cycles and selected region of damaged conditions to estimate delamination area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 8 Effects of damage (a) and temperature variation (b) on output signals from application case I: carbon-epoxy plate. . . . . . . . . . . . . . . . . . . . . 42 9 Effects of damaged conditions on the output signal in the coupon L1S19 from application case II: CFRP coupon. . . . . . . . . . . . . . . . . . . . 43 10 Baseline measured and predicted by AR(20) and autocorrelation of resid- uals for whiteness test validation within 95 % confidence bound ( ) for the case I: Carbon-epoxy plate. . . . . . . . . . . . . . . . . . . . . . . . . 44 11 Space of features for application case I. . . . . . . . . . . . . . . . . . . . . 45 12 Feature space considering the coupon L1S11 in two different paths (appli- cation case II). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 13 Damage index computed for all signals in application case I: carbon-epoxy plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 14 Damage index computed for all acquired signals from coupon L1S11 (ap- plication case II). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 15 Progressive DI versus simulated delamination area S, and mean ( ) of the kriging metamodel with confidence interval of 3σ ( ). . . . . . . . . . 48 16 Validation of the estimated percentage of delamination area with measured percentage of delamination area for test conditions in application cases I and II.The estimated percentage of delamination area (x) for all damage indices in each test condition and the mean of estimated percentage of delamination area ( ) for each test condition. . . . . . . . . . . . . . . . . 49 17 Flowchart of the proposed modified methodology for damage quantification based on GPR model using vibration-based damage indices. . . . . . . . . 53 18 Flowchart of vibration-based methodology for damage assesment in wind turbine blades proposed by Garcia and Tcherniak (2019) . . . . . . . . . . 54 19 Experimental setup of the SSP 34 m wind turbine blade. . . . . . . . . . . 59 20 Reconstructed reference state and measured spectrum frequency under healthy condition obtained from signals measured by accelerometer 1 in trailling edge and actuator location at A1. . . . . . . . . . . . . . . . . . . 61 21 Damage index by damage size for accelerometers in the trailing edge for actuator positions (a) A1, (b) A2, (c) A3 and (d) A4. The damage index in the healthy( ) and damaged conditions( ) The dashed line ( ) corre- sponds to the threshold defined by a risk of false alarm probability equal set to α = 0.01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 22 Stochastic interpolation of global damage index for each actuator position by damage size. The GPR model was trained using five conditions ( ) and tested with two conditions( ). The bold line ( ) corresponds to the mean and the gray colored region ( ) to the 95 % of confidence interval predicted by the GPR model. . . . . . . . . . . . . . . . . . . . . . . . . . 64 23 Validation of estimated damage size using the GPR model by the actual damage size for each actuator position and test condition. The estimated damage size for all damage indices (x) and the mean of estimated damage size ( ) for each test condition. . . . . . . . . . . . . . . . . . . . . . . . . 65 24 BIC computed from signals under baseline condition for application case I. 77 25 Algorithm of model order selection based on BIC. . . . . . . . . . . . . . . 78 26 Front panel of Labview application . . . . . . . . . . . . . . . . . . . . . . 81 27 Block diagram of Labview application. . . . . . . . . . . . . . . . . . . . . 82 LIST OF TABLES 1 Structural conditions tested for the carbon-epoxy laminate (case I). . . . . 39 2 Structural conditions tested for CFRP coupons (case II). . . . . . . . . . . 41 3 Results of the classifier performance for the two application cases. . . . . . 47 4 Mean of the estimated percentage of delamination and mean absolute error of prediction for test conditions in application cases I and II. . . . . . . . . 48 5 Damage size and number of signals measured on each experimental test condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6 Mean of estimated damage size and mean absolute error of prediction for test conditions (D20 and D60). . . . . . . . . . . . . . . . . . . . . . . . . 66 7 Optimized model order computed based on BIC for each path in the ap- plication case I. The value in bold highlighted represents the model order selected for AR model identification in this application case. . . . . . . . . 79 8 Optimized model order computed based on BIC for each path in the ap- plication case I. The value in bold highlighted represents the model order selected for AR model identification in this application case. . . . . . . . . 79 LIST OF ACRONYMS PZT - Piezoeletric transducer SPR - Statistical Pattern Recognition CFRP - Carbon Fiber Reinforced Polymer DI - Damage Index MSD - Mahalanobis Squared Distance GPR - Gaussian Process Regression NDE - Nondestructive Evaluation SHM - Structural Health Monitoring AR - Autoregressive ARX - AutoRegressive with eXogenous input AIC - Akaike information criterion BIC - Bayesian information criterion ASTM - American Society for Testing and Materials RMSE - Root Mean Squared Error LIST OF SYMBOLS ai - i-th coefficient of the autoregressive model A - polynomial containing the lag operator and cofficients of the autoregressive model Ab - polynomial containing the lag operator and coefficients of the autoregressive model identified in the baseline condition A - matrix of principal components CX - covariance matrix of the embedded matrix Di - local damage index corresponding to the i-th path D - dataset composed of damage index and damage severity vectors DI - global damage index EX - matrix of eigenvectors f(·) - Nonlinear regression function f - regression function relating damage index and damage severity f∗ - predicted output for a new input I - identity matrix K - covariance matrix associated with the GPR model k(·, ·) - kernel function associated with the GPR model L̄ - number of time sampling points of acceleration signal collected L - length of acceleration signal into frequency domain M - number of realizations of the signal vector M - Mahanobis squared distance p(·|·) - conditional probability function R - matrix of reconstructed components S - vector of damage severity TB - baseline feature matrix Ti - feature vector xi - i-th element of the vector of damage indices xj - j-th element of the vector of damage indices X1 - damage sensitive feature based on the residual error X2 - damage sensitive feature based on the coefficients of the autoregressive model X - vector of damage indices Xm - signal vector into frequency domain of m-th realization Xm - matrix containing signal vectors corresponding to the healthy condition X̌ - full embedded matrix of the reference state y - time series data Z - Test matrix in the Mahalanobis squared distance w - hyperparameter of kernel function W - number of legged copies used in the embedding operation GREEK LETTERS ΛX - diagonal matrix of eigenvalues θ - hyperparameter vector of kernel function µB - mean vector of the baseline feature matrix µ - mean vector of the training data in the baseline condition associated with Mahalanobis squared distance ε - residual error of autoregressive model εb - residual error of autoregressive model identified in the baseline condition εS - zero mean gaussian noise of the regression model σ2(·) - variance operator Σ - covariance matrix of the training data in the baseline condition associated with Mahalanobis squared distance σ2 S - variance of the Gaussian noise associated with the regression model σf - variance of the kernel function ΣB - covariance of baseline feature matrix CONTENTS 1 Introduction 18 1.1 Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Problem Statement and Method of Approach . . . . . . . . . . . . . . . . . . 22 1.3.1 Main problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.2 Secondary problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.4 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2 Damage quantification methdology based on GPR model using Lamb wave-based damage indices 26 2.1 Overview of the SHM methodology . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Damage features extraction using AR models . . . . . . . . . . . . . . . . . . 28 2.2.1 Model Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.2 Damage features extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Damage detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4 Damage quantification using GPR model . . . . . . . . . . . . . . . . . . . . 32 2.5 Methodology description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Experimental application of damage quantification methodology based on the GPR model in CFRP coupons 37 3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.1 Case I: CFRP plate with simulated damage . . . . . . . . . . . . . . . . . . . 37 3.1.2 Case II: CFRP coupons subject to tension-tension fatigue . . . . . . . . . . . 39 3.2 Application of the proposed methodology . . . . . . . . . . . . . . . . . . . . 41 3.2.1 Damage features extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.2 Damage detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.3 Damage quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4 Modified methodology for damage quantification in wind turbine blades based on the GPRmodel using vibration-based damage indices 51 4.1 Damage quantification using GPR model in wind turbine blades . . . . . . . 52 4.1.1 Damage detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1.2 Damage quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3.1 Damage detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3.2 Damage quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5 Final Remarks 67 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Suggestions for Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 REFERENCES 70 Appendix A Details on the identification of Autoregressive model 76 A.1 AR model order estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A.2 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Appendix B Labview application 81 Appendix C Publications 83 18 1 INTRODUCTION 1.1 Context and Motivation The use of composite materials has significantly expanded in the past decades. It has been driven by aerospace, automotive, agriculture, civil, and renewable energy industries due to its better properties than metallic materials, such as high strength and stiffness combined with a low-density and excellent corrosion resistance (GHRIB et al., 2017). However, composite materials present a complex internal structure, usually anisotropic, and multiple types of damage that make the damage identification and prediction of a failure challenge. This leads to overdesigned and less cost-efficient structures, in order to avoid catastrophic failures (SAXENA et al., 2011a). Traditional nondestructive evaluation (NDE) techniques are currently utilized for damage identification. However, they are often labor-intensive and may require disassembly of the structure, which increases the maintenance costs and system downtime, and it is usually limited to inspections when the structural degradation is suspected (YUAN, 2016). In this context, structural health monitoring (SHM) techniques have been the focus of intensive research and development in recent years as an alternative way for automatic and continual monitoring of composite structures even in service, reducing maintenance costs, downtime, and improving their safety and reliability. SHM is defined by Larrosa, Lonkar and Chang (2014) as the diagnostic process to extract meaningful health information from a structure, based on the sensing data from distributed, permanently installed sensors at certain structural regions. In the classifica- tion proposed by Rytter (1993), SHM techniques present four functional levels: Level 1 - damage detection; Level 2 - damage localization; Level 3 - damage quantification, and Level 4 - estimation of the remaining useful life. In the literature, the first three levels are also categorized as diagnosis, during the last level as prognosis. Typically, the diagno- sis, which is the focus of this work, includes an intelligent algorithm of signal processing from measurements in the structure to identify structural health’s current state. Several SHM techniques for application in composite materials have been reported in the litera- ture based on many different technologies such as ultrasonic inspection, imaging methods, acoustic emission, fiber optics sensors, piezoelectric transducers (PZT), and laser vibrom- etry (YUAN, 2016). Special SHM techniques based on active-sensing guided waves have 19 attracted substantial attention in the past two decades, especially for composite structures due to the high sensitivity to incipient damage. However, due to their high sensitivity to operating/environmental conditions, active-sensing SHM methods still face significant challenges, which will be addressed in this work. Composite materials present multiple types of damage, such as matrix-cracking, de- lamination, debonding, etc., which have a different impact on the performance of the structure (SAXENA et al., 2011a). Nevertheless, the major concern is delamination dam- age, because it degrades the strength significantly and is usually the ultimate cause of the structural failure (LARROSA; LONKAR; CHANG, 2014). Many authors demonstrated the application of active-sensing with Lamb waves in composite structures for delamina- tion diagnosis. Typically, an array of PZT attached in the structure is used to propagate Lamb waves transmitted by the examined material. The damage is identified by analyzing the received waves from multiple PZT sensors. Numerous methods have proved feasibility, especially for the levels of damage detection and localization (YUAN, 2016; TIAN; YU; LECKEY, 2015; FARRAR et al., 2007; MITRA; GOPALAKRISHNAN, 2016). However, the damage quantification level is still little explored, which will focus on this work. 1.2 Literature Review Lamb waves are a type of guided wave which propagates between two parallel surfaces of plate-like structures (IHN; CHANG, 2004). These waves are useful for SHM purposes due to its high sensitivity to small internal defects and capacity to inspect vast areas even in materials with high attenuation ratio such as composites (TIAN; YU; LECKEY, 2015). Some papers have been handled to explain the effects of interaction between damage and Lamb waves using numerical or data-driven identification models (TOYAMA; NODA; OKABE, 2003; RAMADAS et al., 2009). Toyama, Noda and Okabe (2003) showed that transverse cracking and delamination decrease the stiffness of the structure and change the propagation velocity of the symmetric mode S0 in cross-ply laminates. Ramadas et al. (2009) examined the interaction of the anti-symmetric mode A0 with symmetric delamination, and the occurrence of mode conversion from A0 to S0 when the wave crosses the damaged region. They recommended that the time-of-flight between the through- transmitted mode-converted could be used to estimate the size of delamination. However, Lamb waves present numerous modes of propagation, and it is also influenced by a variety of interference sources, including high-frequency ambient noise, low-frequency structural vibration, temperature fluctuation, operational variability, inhomogeneity, and anisotropy 20 of materials (SU; YE, 2009). Thus, the development of a robust SHM method depends on the extraction of a damage feature with reduced sensitivity to interference sources or a compensation method, which is a more sophisticated solution due to the Lamb wave propagation complexity. In this context, numerous damage features are proposed in the literature involving the propagation velocity changes, amplitude attenuation, wave scattering, reflection, and others in the time or frequency domain (FIGUEIREDO et al., 2012). However, the application of time-series predictive models, such as Autoregressive (AR) models, have been few explored for Lamb wave-based techniques, despite to be widely implemented in vibration-based SHM procedures since the last decade (SILVA; JUNIOR; JUNIOR, 2008; FIGUEIREDO et al., 2012; NARDI et al., 2016). Figueiredo et al. (2012) proved the capability of AutoRegressive models with eXogenous inputs (ARX) as a sensitive feature for damage detection in a composite plate. Silva (2018) illustrated some practical aspects of the implementation of ARX models using Lamb wave for system identification of a composite plate assuming temperature effects. Moreover, it was discussed the potential regarding the purpose of SHM. Unfortunately, the scientific community has not extensively addressed damage quan- tification due to the limited understanding of the interaction mechanisms between guided waves and damage in composite structures (GHRIB et al., 2018; TIAN; YU; LECKEY, 2015). Damage quantification is crucial to enhance the safety and the service life of com- posite structures, motivating the scientific community to develop some damage quantifi- cation methods. These strategies are not generic and are restricted to the typical damage found in composite structures: the delamination (LARROSA; LONKAR; CHANG, 2014). Tian, Yu and Leckey (2015) introduced a methodology for delamination quantification in laminated composites using wavenumber analysis. The results show that this procedure can be used not only for assessing the delamination area but also for the ply depth of the damage. Si and Wang (2016) proposed a damage identification technique based on a rapid multi-damage index algorithm. The damage quantification is achieved by apply- ing a metric based on the energy density released extracted from the Hilbert spectral analysis of the Lamb wave signals, which presents a linear correlation with the delamina- tion extent provided by polynomial regression. Paixao, Silva and Figueiredo (2020) used AutoRegressive (AR) models upon Lamb wave signals measured in a composite plate to compute damage-sensitive features for computation of indices using Mahalanobis squared distance. The quantification was achieved by training a set curve utilizing cubic spline functions to predict the delamination area. Silva et al. (2019) performed a related idea by 21 manipulating the possibility of extrapolation of these trend curves to prognostic future states when the delamination grows in the same spot and with a similar effect. The re- sults demonstrated that the features using AR models are accurately correlated with the structural state and a smooth trend that enables using cubic splines functions. A practical obstacle to overcome is the natural variation of the measured signals caused by inherent uncertainties, e.g., imposed by operational and environmental conditions, sensor positioning, boundary conditions etc. (YADAV et al., 2017). Some authors have exhaustively investigated those variations in the last few years. Yadav et al. (2017) used a multi-path unit-cell approach to propose a robust SHM approach for damage quantifica- tion using a curve of damage indices. A calibration procedure established by a hyperbolic tangent function was conducted in the training step, assuming severity changes within a similar aluminum plate set. Another set of plates was employed to observe the unknown damage severity and test the learned model. The simulated damage was a circular-cut introduced progressively by electrical discharge machining. Saxena et al. (2011a) sug- gested a similar procedure to damage quantification in composite structures. The tests were performed using the same dataset from the NASA prognostics data repository. More advanced probabilistic methods are recommended to treat data from composite materi- als. Corbetta et al. (2018) employed particle filters to realize the damage quantification and prognosis using the same dataset in the Carbon Fiber Reinforced Polymer (CFRP) coupons from NASA. Another not sufficient explored stochastic procedure to perform the same task in composite structures is the Gaussian Process Regression (GPR)(SAXENA et al., 2011a). The GPR is a probabilistic nonlinear regression model and a helpful method for prediction with essential information about the uncertainty (FUENTES et al., 2014). Recent works in the literature demonstrate its potential to mitigate the effects of op- erational variability on the damage detection level as reported by Valencia, Chatzi and Tcherniak (2020) and in the damage quantification and prognosis levels as reported by Chen, Yuan and Wang (2020) and Yuan, Wang and Chen (2019). However, it should be noted that most of the damage quantification techniques are deterministic, which do not provide confidence intervals for damage size estimation and require a large amount of data from a population of identical structure to account for changing conditions (AMER; KOPSAFTOPOULOS, 2019). A stochastic approach to estimate the damage size would provide a more accurate and robust prediction with a reduced data footprint in this context. Recently, Corbetta et al. (2018) employed the stochastic method of particle filters, but focusing on the damage prognosis using the same dataset in the Carbon Fiber Reinforced Polymer (CFRP) coupons from NASA. As sug- 22 gested by Saxena et al. (2011a), another not sufficiently explored stochastic procedure to perform the same task in composite structures is the Gaussian Process Regression, which will be employed in this work. The GPR is a probabilistic nonlinear regression, a help- ful method for prediction with essential information about the uncertainty (FUENTES et al., 2014). The GPR model’s application is based on the assumption that there is a direct correlation between the defined damage index and damage severity, which is rea- sonable assumption verified for numerous damage indices and different application cases (YADAV et al., 2020; GHRIB et al., 2018; TIAN; YU; LECKEY, 2015; AMER; KOP- SAFTOPOULOS, 2019). The GPR model can be applied to capture the defined damage index’s trend and uncertainties related to the damage severity progression in a supervised learning approach under defined conditions. After training the GPR model, it can be used to monitor the damage severity progression inside the defined conditions using a stochastic interpolation using the damage index as an input of the model. 1.3 Problem Statement and Method of Approach 1.3.1 Main problem statement In the main problem statement of this work, it is proposed the application of a new SHM methodology for damage quantification level in laminated composite plates under the simplified case of delamination propagation in a hotspot region, e.g., the potential location of delamination initiation is known a priori based on the historical data or other preliminary methods. Figure 1 provides the schematic idea of the problem statement.The main focus of the SHM system is to detect and estimate the delamination size. Despite the damage initiation of a hotspot region being a particular case, it is quite common in structures submitted to fatigue loading and a problem of great interest to the aerospace industry, as stated by Yadav et al. (2017). The utilization of a GPR model for delamination area quantification in composite structures is proposed in this work, which is the central contribution of this research project. The best authors’ knowledge is the first time to be applied in the damage quantification level for SHM procedures in composite structures. A secondary contribution is applying AR models for feature extraction from Lamb wave signals should also be noted, which, although not new, has been little explored in the literature despite the advantage of simplicity and successful, related applications in vibration-based methods of SHM. AR model identification of Lamb wave signals is used for feature extraction, and the damage indices (DI) are computed based on the Mahalanobis squared distance (MSD) set for a 23 Figure 1: Problem Statement. Source: Prepared by the author. reference condition; then, the GPR model is trained to associate the damage indices with the delamination area. Two applications are herein proposed to demonstrate the reliability of the methodol- ogy. First, a composite plate with progressive simulated damaged conditions is tested, in which the GPR model is trained and used in the same structure to predict intermediaries’ delamination areas. Second, four similar CFRP composite coupons submitted to real pro- gressive delamination states generated in a fatigue loading are tested. The GPR model is trained with three of them and tested to predict the last one’s delamination area. The hypothesis here is that the learned model can be assumed for the quantification delam- ination area in identical structures for hotspot monitoring. This setup is the same used in Larrosa, Lonkar and Chang (2014) and Corbetta et al. (2018) in order to compare the performance. 1.3.2 Secondary problem statement The GPR model’s application is based on the assumption that there is a trend between the damage index and damage size, which is verified for different damage indices reported in the literature (YADAV et al., 2020; GHRIB et al., 2018; TIAN; YU; LECKEY, 2015; AMER; KOPSAFTOPOULOS, 2019). Then, it could be extended for different scenar- ios and techniques of damage feature extraction. In this context, the application of a modified version of the methodology used in the main problem statement addresses the damage quantification problem using vibration-based damage indices in an industrial ex- 24 ample involving a wind turbine blade. Given a wind turbine blade instrumented with accelerometers to capture the impact response signals caused by an actuator, the SHM system’s objective is to detect and estimate the debonding size in the adhesive joint of the blade. The experimental application is proposed using the same dataset used by Garcia and Tcherniak (2019) that explored the damage detection level using vibration-based damage indices. Then, the damage indices proposed by Garcia and Tcherniak (2019) are combined with the application of the GPR model to address the damage quantification level. This practical application demonstrates the versatility of the proposed methodology based on the GPR model’s application to be applied with different damage features extraction techniques and types of damage. 1.4 Objective This research project aims to propose and validate an SHM methodology experi- mentally for damage quantification in composite structures based on the GPR model’s application. 1.5 Main Contributions The main contributions of the present work are the following: • Development of a damage quantification methodology based on the GPR model application using Lamb-wave-based indices for hotspot monitoring of delamination in laminated composite plates(PAIXAO; SILVA; FIGUEIREDO, 2020). • Application of damage features extraction method based on the AR model identifi- cation of Lamb-wave signals (PAIXÃO; SILVA, 2019); • Proposition of a modified damage quantification methodology based on the GPR model application using vibration-based damage indices for SHM in wind turbine blades. 1.6 Outline The work is organized as follows: 25 • Chapter 1: presents the motivation of this work, a literature review to contextual- ize the proposed methodology, the problem statement of this project, and the main objective of this research project. • Chapter 2: describes the proposed methodology together with the theoretical background of the damage feature extraction process using the AR model and the application of the two machine learning algorithms proposed for damage detection and quantification. • Chapter 3: presents a practical application of the proposed methodology in two application cases. In the first case, it is considered a unique CFRP plate with simulated delamination. In the second case, four CFRP coupons are tested with real delamination caused by fatigue loading. The results from the application of the methodology are discussed. • Chapter 4: presents a modified methodology for damage quantification based on the application of the GPR model using vibration-based damage indices and an experimental application in a wind turbine blade. • Chapter 5: the main conclusions are presented, and the next steps of this research project are outlined. • Appendix A - Details on the identification of Autoregressive models: provides a complement of Chapter 2 by presenting the techniques applied for model order selection and validation. • Appendix B - Labview application: shows the application developed in Labview to control the experimental setup in the application case I. • Appendix C - Publications: presents the publications in the subject of this research project. 26 2 DAMAGE QUANTIFICATION METHDOLOGY BASED ON GPR MODEL USING LAMB WAVE-BASED DAMAGE INDICES This chapter describes the methodology proposed for delamination area monitoring of the hotspot region in laminated composite plates. Firstly, Section 2.1 presents an overview of the SHM methodology proposed. Section 2.2 provides the theoretical background for applying the AR model to extract damage features from Lamb wave signals. In Section 2.3, the machine learning algorithms for statistical modeling and feature discrimination are presented and discussed. Then, in Section 2.4, a detailed description of the proposed methodology for the application in hotspot monitoring of the delamination area is carefully outlined. Finally, Section 2.5 presents the conclusions of this chapter. 2.1 Overview of the SHM methodology The methodology proposed in this work can be cast in the context of the statistical pattern recognition paradigm (SPR) for SHM systems proposed by Farrar and Worden (2012), which is defined through a four-part process: • Operational evaluation: involves the definition of the damage in the structure being investigated, the analysis and setting of limitations for operational and environmen- tal conditions under which the system will be monitored. • Data acquisition: comprises selecting the excitation method, the sensor types, num- ber and locations, and the hardware for data acquisition. • Feature extraction: to identify the data features that allow us to distinguish between healthy and damaged states of the structure. • Statistical modeling for feature discrimination involves applying statistical tech- niques for discrimination between features from healthy and damaged conditions. A more detailed discussion of the SPR paradigm can be found in Farrar and Worden (2012). The methodology proposed in this work focuses on parts 3 and 4 of the process, which have arguably received the most attention in the literature. An overview of the 27 proposed methodology, according to the SPR paradigm, is presented in Fig. 2. The operational evaluation (part 1) has already been addressed on the problem statement in Chapter 1, where the type of damage is defined as delamination, and the conditions of monitoring are discussed. Also, most of the data acquisition (part 2) has been tackled once the methodology is based on the active-sensing of Lamb waves using PZTs, and the damage location is supposed to be known, which consequently defines the excitation method, type of sensors, and locations. The feature extraction process (part 3) is based on the application of the AR model for Lamb wave propagation signals identification. Damage sensitive features are extracted based on a comparison between statistical measures from coefficients and residual errors of AR models identified in a reference condition with those of a different condition, possibly damaged. In the statistical modeling (part 4), two machine learning algorithms are implemented: MSD for outlier identification in the delamination detection level; and the GPR model to estimate the delamination size. Figure 2: Schematic representation of the methodology. AR Model Feature Extraction Statistical Modelling GPR Model MSD for Outlier Detection Source: Prepared by the author. The application of this methodology for hotspot monitoring of the delamination area in laminated composite plates is based on the concept of statistical learning from training data. Initially, the machine learning algorithms for delamination detection and quan- tification have to be trained in a supervised learning approach using experimental data from known structural conditions by considering healthy and progressive damaged con- ditions. The MSD algorithm is trained to detect delamination by the identification of outliers values of the damage index. The GPR model is trained to capture the trend and uncertainties of the damage indices related to the delamination growth, which can be used to estimate the delamination area based on the damage indices using the stochastic interpolation provided the model. After that, the methodology can be applied to mon- itor the delamination area in laminated composite plates under the training conditions. The methodology’s ultimate goal is to provide the capability to transfer the data-based knowledge of damage quantification from a single or a population of the representative laminated composite plate to other plates, or even scalable structural components. 28 2.2 Damage features extraction using AR models The utilization of AR models in SHM has been extensively explored for vibration- based methods, especially for civil structures as bridges and buildings (FIGUEIREDO et al., 2010). However, it has not been substantially utilized for Lamb wave-based propaga- tion methods. The first application is reported by (LYNCH, 2005), which uses an ARX model to detect damage in an aluminum plate. Figueiredo et al. (2012) apply a similar for damage detection in composite plates and point out the advantages of using AR models, such as the reduced computational resources compared with frequency-based signal pro- cessing techniques. This makes the application of AR models very appealing for real-time health monitoring systems. In some recent papers, Silva (2018) addressed the use of time series methods for modeling narrowband signals of Lamb wave propagation and Cano and Silva (2020), demonstrated its effectiveness and suitability for damage detection. The use of time series models to extract damage features from the signals in SHM applications is based on the capability of these models to capture the internal structure of time series, such as statistical correlations or systematic trends of varying time scales, which can be associated with the structural health state of the monitored system when compared with a reference state (PARK et al., 2010). Therefore, regardless of the type of signal, if the presence of damage causes essential changes in the signal’s internal structure, time series methods can be useful to detect it. 2.2.1 Model Identification Given the time series y(1), ..., y(N), a model that expresses y(n) as linear combination of past p-th samples y(n− 1), ..., y(n− p) and residual error ε(n) is called an AR model (KITAGAWA, 2010), y(n) = p∑ i=1 aiy(n− i) + ε(n) (1) where p is the model order, ai, ..., ap are the AR coefficients and ε(n) is the residual error at the sample value n after fitting the AR model. The residual error is a portion of the signal that cannot be explained by the linear combination of the explanatory variables of the AR model, and it is assumed to be an independent random variable. If the AR model is adequately identified, the residual error follows a normal distribution with mean 0 and 29 variance σ2 ε (KITAGAWA, 2010). For the representation of Equation 1 in more compact notation is convenient to define the operator lag q−k the way that y(n)q−k = y(n − k), for which the following equation is obtained, A(q)y(n) = ε(n) (2) where A(q) = 1 + a1q −1 + · · · + apq −p is the polynomial containing the lag operator and the coefficients of the model. The AR model identification requires a priori the selection of the model order, which is initially unknown. Various techniques to address this problem have been proposed in the literature, and, among these, the Akaike information criterion (AIC) and Bayesian information criterion (BIC) have attracted much attention, especially in the literature focused on SHM. Although BIC is closely related to the AIC, the penalty term of BIC is a function of the data size, and so it is more severe than AIC, and it often presents a faster convergence (SILVA, 2017; FIGUREIDO, 2010). Therefore, BIC is used in this work to select the appropriate model order. More details about the model order selection procedure are presented in Appendix A. Identification of AR models is a well-established research topic, and many identifica- tion methods have been developed in the literature, such as Yule-Walker, least-squares, and partial autocorrelation. The adequate performances of the least-squares method allied with its simplicity, low computational resources consumption, and robustness compared with other methods make it an appealing option for AR model identification (ZHENG, 2003). Therefore, the AR models are identified in this work using the least-squares method. A detailed review of these methods can be found in Kitagawa (2010). 2.2.2 Damage features extraction The AR model application in SHM is based on the assumption that structural damage changes the structure’s physical properties, which is reflected in the time-series response measured. In general, AR models are used as a damage-sensitive feature extractor based on two approaches: (1) using the residual errors; and (2) using the AR coefficients. In the first approach, the model is identified from the structure’s signal in the baseline condition and used to predict the response obtained from a potentially damaged condition. As the damage will introduce changes in the structure response, the identified model will no 30 longer accurately predict the response, which causes changes in the residual errors that can be used as a damage-sensitive feature. In the second approach, an AR model is identified for the baseline and damaged conditions. The AR coefficients are then used directly as a damage-sensitive feature to distinguish between the conditions (FIGUEIREDO et al., 2010). Suppose a measured signal from the Lamb wave propagation in a composite plate considering the baseline condition. The signal is identified by an AR model, which is described as follows : Ab(q)yb(k) = εb(k) (3) where Ab(q) = 1 + a−1 b1q + · · · + abpq −p is the new AR regressor polynomial and εb is the residual error in the baseline condition. Note that if a signal from baseline condition is adequately identified and no significant changes are present in the signal, the parameters should be constant, and the residual errors should be white noise process. If a newly acquired signal under a different and unknown condition is identified using an AR model using the same selected order p, the identified model can be described by Eq. 2. Then, the new AR coefficients and residual errors change because the output signal is not the same in the new condition, and it carries new information. Various statistical features can be used to capture the changes in the distribution of coefficients and residual errors. In this work, using the two damage- sensitive features (i.e., coefficients and residuals) is recommended by using a statistical feature based on the variance, which has already been employed by Sohn and Farrar (2001). First, using the residual errors as follows: X1 = σ2(ε) σ2(εb) (4) where ε and εb are the vectors that contain all samples of residual errors from the AR models identified under an unknown and the baseline conditions, respectively. Second, using the AR coefficients in the form of: X2 = σ2(A) σ2(Ab) (5) 31 where A and Ab are the vectors containing the coefficients for the AR models identified under an unknown and the baseline conditions. Once the features extraction process is defined based on the AR model identification, the structural health assessment can be addressed by the statistical modeling for feature discrimination. As the SHM algorithm’s objective is the hotspot monitoring, the problem of the structural health assessment can be categorized into two levels: (1) damage detec- tion; and (2) damage quantification. Note that for the hotspot monitoring problem, the damage localization is supposed to be known. Consequently, the damage localization is not necessary. 2.3 Damage detection The damage detection level is addressed using the Mahalanobis Squared Distance (MSD), a distance measure for multivariate statistics outlier detection (WORDEN; MAN- SON; FIELLER, 2000). As two damage features are considered in the proposed approach, the MSD cast into a bi-variate space of features. The machine-learning algorithm based on the MSD is applied in the bi-variate feature space to compute the distance from the centroid of a well-known cluster of points to any other point. This cluster of points is defined by a training matrix X ∈ Rm×2 that is formed by the set of m samples of features (X1 X2) extracted from the conditions under the healthy state. As the algorithm is trained, one can compute the distance in the feature space to all l samples, defined by the test matrix Z ∈ Rl×2 : M = (Z− µ) Σ−1(Z− µ)T (6) where µ is the mean vector and Σ, is the covariance matrix of the training matrix from the baseline condition X. Given that the instrumented structure with the PZTs presents l actuators and s sensors, this PZT network will provide n = s× l output signals from the n PZT actuator- sensor paths. The MSD is computed for each path independently, yielding aM vector n element. DI = 1 n n∑ i=1 Mi (7) 32 where MSDi is the computed distance in the space of features assuming the i-th trans- ducer path. Once the DI is computed, the structural health classification can be implemented utilizing the outlier detection strategy by setting a threshold for the most significant DI value, considering all signals corresponding to the baseline condition. 2.4 Damage quantification using GPR model In the quantification level of the methodology, it is proposed the application of a supervised machine learning algorithm employing GPR, also named by kriging. This non- parametric Bayesian approach has the benefit of working on small datasets and the ability to provide uncertainty measurements on the predictions. The goal here is to establish a direct ratio among the damage index and the surface delamination area in the structure using a GPR model. This model can learn the ratio and the uncertainties in the training step associated with environmental and/or operational variability, fabrication differences, defects, area measurements, etc., in order to predict the delamination area using a damage index extracted from a structure in an unknown condition. Another beneficial point to use GPR is calculating the variance that enables us to provide the confidence interval of the regression associating the damage index with the delamination area. The surface area of delamination S(i) ∈ R can be written as an output of a nonlinear regression: S(i) = f(DI(i)) + ε (i) S (8) where f(·) is a nonlinear function, DI(i) ∈ R is an input vector, and ε(i) S is a zero mean Gaussian noise: ε (i) S ∼ N ( ε (i) S |0, σ2 S ) (9) where σ2 S is a Gaussian noise variance with zero mean. For N tests, the training data set 33 is: D = ( DI(i),S(i) )N i=1 ≡ (X ,S) (10) where X ∈ RN×1 is the input vector (damage indices) and S ∈ RN×1 is the output vector (delamination areas). Here the S(i) is assumed to be the percentage of delamination area about the total area of the structure. The function f(·) is assumed a priori as a Gaussian multivariate (RASMUSSEN, 2004): f = f(X ) ∼ N (f |0,K) (11) where K ∈ RN×N is the covariance matrix with Kij = k(xi, xj), and k(·, ·) is the kernel function, also named by covariance function. The kernel function can assume different varieties depending on the treated problem. A general kernel function is the exponential kernel: k(xi, xj) = σ2 f exp [ −1 2w 2 (xi − xj)2 ] (12) where θ = [σf , w 2] is the hyperparameter vector that command the model’s covariance. The hyperparameter is found by solving a maximization of a likelihood p(S|f) = N ( S|f , σ2 fI ) (13) where I ∈ RN×N is the identity matrix. When a new input x∗ is measured, using a bayesian inference, a new output f∗ is identified by: p(f∗|S,X , x∗) = N ( f∗|µ∗, σ2 ∗ ) (14) 34 µ∗ = k∗N ( K + σ2 SI )−1 S (15) σ2 ∗ = k∗∗ − k∗N ( K + σ2 SI )−1 kN∗ (16) where k∗N = [k(x∗, x1), · · · , k(x∗, xN)] kN∗ = kT N∗ k∗∗ = k(x∗, x∗) The predictive distribution of S∗ is similar to f∗, but σ2 S has the variance added. The hyperparameters are determined by using the likelihood of log-marginal using training data (MATTOS et al., 2016): log p(S|X ,θ) = −1 2 log ∣∣∣K + σ2 SI ∣∣∣− 1 2ST ( K + σ2 SI )−1 S − N 2 log (2π) (17) A detailed discussion about the optimization methods of this function is given by Rasmussen (2004). A broad sensitivity is observed depending on the training data; a kernel function is chosen, regression order, and possible overparameterization. In this work, the UqLab software1 was employed to optimize the kriging model (LATANIOTIS; MARELLI; SUDRET, 2015). The optimization of the hyperparameters is performed just to train the GPR model. After that, it can be used to estimate the delamination area just using as input the damage index. 2.5 Methodology description The proposed damaged quantification methodology is illustrated in Figure 3 and can be split into two steps: 1) training and 2) test. In the training step, the supervised learning of the GPR model is performed using labeled training data from the structure under the 1https://www.uqlab.com/ https://www.uqlab.com/ 35 baseline condition and known progressive damaged conditions. The set of training data consists of damage indices extracted from the output signals and the delamination area measured in the structure. The damage index is based on the MSD of two features (X1 and X2) extracted from AR models identified for the output signals. In the test step, a similar or the same structure in an unknown condition is interrogated about the presence of damage by performing an outlier detection test. When an outlier is detected, the delamination area is estimated from the damage index using the trained GPR model. In this work, each condition is defined by three pieces of information: a structural state that can be healthy or damaged; damage severity, which is the delamination area; and temperature changes in some cases. The condition referred to as the baseline is characterized by the structure in the healthy state, i.e., no delamination or ambient (or one specified) temperature. Figure 3: Flowchart of the proposed methodology for damage quantification in composite structures. AR Model Identification Model Order Selection Damage Index DI ≥ 𝛽 ? AR Model Identification Damage Index Learned Model I ) Training II) Test GP Regression Training y baseline n a ε A(q) 𝛽 DI n a Delamination Area DI Delamination Area Baseline Condition Unknown Condition? Progressive Damaged Conditions DI Source: Prepared by the author. 2.6 Conclusions In this chapter, the methodology for hotspot monitoring of the delamination area in composite structures was presented. The procedure of damage-sensitive feature extraction based on AR models’ application to identify Lamb wave signals were detailed. Then, 36 two machine learning algorithms for damage detection and quantification were presented. The application of the outlier detection algorithm based on the MSD was proposed in the damage detection level. In the damage quantification, it was formulated the application of the algorithm for stochastic regression model GPR to map the delamination area as a function of the damage index extracted. The methodology proposed can be used to monitor the delamination growth in hotspot regions of composite structures, where the potential damage location is known from his- torical data or other preliminary methods of analysis. This is a practical problem in the monitoring of structures submitted to fatigue loading. The methodology is based on the supervised learning of the machine learning algorithms, which requires a training step in controlled conditions of the delamination growth that can be expensive. How- ever, it should be noted that the stochastic regression model’s application can reduce the number of samples in the training dataset and provide a more robust prediction of the de- lamination area compared with ordinary deterministic methods of regression. The GPR model application for delamination size prediction allows us to adequately capture the uncertainties associated with the training step and improve the accuracy of the damage quantification in the structure’s monitoring. 37 3 EXPERIMENTAL APPLICATION OF DAMAGE QUANTIFICATION METHODOLOGY BASED ON THE GPR MODEL IN CFRP COUPONS This chapter presents the practical application and discussions of the proposed SHM method. Firstly, Section 3.1 describes the experimental setup of the two cases proposed. Section 3.2 shows the results obtained in the application of the proposed methodology for hotspot delamination monitoring. Finally, Section 3.3 presents the conclusions obtained in this chapter. 3.1 Experimental Setup Two experimental application cases are used in this work to demonstrate the proposed approach. In the application case I, a carbon-epoxy composite plate with a unidirectional layup, is instrumented with four PZT transducers. Progressive damaged conditions are simulated by increasing the area covered by an industrial putty bonded on the surface’s plate (SILVA et al., 2019; PAIXAO; SILVA; FIGUEIREDO, 2020). In the application case II, four similar composite coupons with dogbone geometry are monitored during run-to- failure experiments under fatigue loading using two patches of six PZT transducers (LAR- ROSA; LONKAR; CHANG, 2014). The experimental dataset used in this second case is available on the Prognostics Center of Excellence at NASA data repository2(SAXENA et al., 2011b). 3.1.1 Case I: CFRP plate with simulated damage In this case, the CFRP plate is composed of 10 layers of carbon fibers unidirectionally aligned along a 0◦ direction and stacked in a matrix of epoxy resin. The dimensions of the plate are 500 x 500 x 2 mm3. This material is commonly used in the aircraft industry, and it has been manufactured by a company in the sector, which has the name and the procedure of manufacturing omitted in this work due to intellectual property restrictions. The composite plate is instrumented with four PZT transducers SMART Layer man- ufactured by Acellent Technologies Inc, each one measuring 6.35 mm in diameter and 0.25 2https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognostic-data-repository/ https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognostic-data-repository/ 38 Figure 4: Composite plate and schematic view of the experimental setup (measurements in mm). Source: Prepared by the author. mm in thickness. The structure is excited by a five cycle sinusoidal tone burst with 35 V of amplitude and center frequency of 250 kHz applied on PZT 1. Then, PZT 2, PZT 3, and PZT 4 are used as sensors to acquire the output signal by using a sampling frequency of 5 MHz and timespan of 200 µs. The actuator signal is generated by NI USB 6353 from National Instrument coupled with a power amplifier EL 1225 from Mide QuickPack, and the output signal is acquired using an oscilloscope DSO7034B from Keysight. The application developed Labview and presented in Appendix B is used to control the gen- eration/acquisition system. Figure 4 presents a schematic overview of the experimental setup used in the experiments. Damage is simulated by inserting an industrial adhesive putty on the surface’s plate. This additional localized mass simulates local changes in the plate’s damping, which has some similarities to delamination in composites structures, as reported by Lee et al. (2011). A common practice in the literature is to simulate damage reversibly, i.e., without damnifying the structure. This will be discussed later and compared with the real delamination damage in application case II. The experiments were conducted inside a temperature chamber SM-8 Thermotron at seven temperature levels from 0◦C to 60◦C with a variation of 10◦C in each level. The composite plate was placed in a free-free boundary condition in order to avoid operational variability. The conditions tested are presented in Table 3.1.1. Healthy conditions are referred to asH(t) and damaged asD(t) (s), where the superscript denotes the temperature of the test and 39 Table 1: Structural conditions tested for the carbon-epoxy laminate (case I). H(t) D (t) 1 D (t) 2 D (t) 3 D (t) 4 D (t) 5 D (t) 6 D (t) 7 D (t) 8 D (t) 9 D (t) 10 D (t) 11 S [%] 0 0.196 0.282 0.384 0.502 0.785 1.13 1.53 0.95 2.01 2.27 2.54 Note: (t) superscript indicates the temperature of test in the range : 0, 10, 20, 30, 40, 50, 60 ◦C. Source: Prepared by the author. subscript s the number of damaged state. Baseline condition is defined as the structure in a healthy state at 30◦C, represented by Hbaseline. For each condition, 100 output signals were collected sequentially. The progressive damage conditions were simulated by the increase in the area covered with the putty, which has a circular format and constant mass. 3.1.2 Case II: CFRP coupons subject to tension-tension fatigue Figure 5: CFRP composite coupon and schematic view of the experimental setup in the application case I (measurements in mm). 123456 121110987 Actuators Sensors (a) 152 353 R203 R2.5 12.5 50 50 25 Actuator Patch Sensor Patch (b) Source: Adapted from Saxena et al. (2011b). The dataset of the fatigue test experiments available on NASA data repository ex- hibits three types of CFRP laminates with different stacking sequences. Only one type of laminate is selected in this application case, which has the following stacking sequence: [02/904]s. Four samples, labeled as L1S11, L1S12, L1S18, and L1S19, were manufactured with Torayca T700G unidirectional carbon-prepreg material. The structure presents a dogbone geometry with a side-notch element to induce stress concentration (see Figure 5). The structure was instrumented with 12 PZTs SMART Layer from Acellent Technolo- gies Inc. A pitch-catch configuration was used to monitor the wave propagation through the samples, where six transducers were used as actuators and the other six as sensors. 40 Figure 6: Output signals from path between PZT 4 and 9 considering baseline (H) and damage (D8) conditions for the CFRP coupons (case II). 0 30 60 90 120 150 Time[µs] −80 −40 0 40 80 P Z T 4 − 9 [m V ] H D7 Source: Prepared by the author. The input signal was a five cycles sinusoidal tone burst with an a◦verage amplitude of 50 V. Seven frequencies were available on the dataset, from 150 kHz to 450 kHz, the frequency of 250 kHz was selected because as investigated by Larrosa, Lonkar and Chang (2014) the wave propagation presents the fundamental symmetric and anti-symmetric modes as distinguishable as possible according to dispersion curves. The signal acquisition and gen- eration were performed using Acellent’s ScanScentry hardware and SmartPatch interface software. Figure 6 illustrates an output signal in baseline condition versus one damage condition for the path between PZT 4 and 9. The fatigue test was performed using an MTS machine and following the American Society for Testing and Materials (ASTM) Standards D3039 and D3479. The coupon samples were subjected to cyclic loading at a frequency of 5 Hz and a load ratio of ap- proximately 0.14, with a maximum peak equal to 31 kN (LARROSA; LONKAR; CHANG, 2014; CORBETTA et al., 2017). Periodically, the test was paused, and the coupon was removed from the machine, to perform the acquisition of signals from Lamb wave propa- gation in the structure and to collect X-ray images from the specimen in order to visualize the internal damage growth. The open-source image processing package Fiji was used to estimate the delamination area based on X-ray images, as already performed in previ- ous works using this dataset (LARROSA; LONKAR; CHANG, 2014; CORBETTA et al., 2017). Figure 7 illustrates the selected area to estimate delamination in the X-ray image. There is only one signal per path in this application case, i.e., there are only 36 signals per condition. To introduce significant variability, 50 signals per path were generated from the experimentally acquired ones by adding white noise with 40 dB relative to the output signal in the path with maximum power under the baseline condition. The conditions tested are presented in Table 2. 41 Figure 7: Example of X-ray image of composite coupon L1S11 under 40000 loading cycles and selected region of damaged conditions to estimate delamination area. Source: Adapted from Saxena et al. (2011b). Table 2: Structural conditions tested for CFRP coupons (case II). Coupon H D1 D2 D3 D4 D5 D6 D7 D8 D9 L1S11 0 0.23 0.33 0.44 0.84 1.84 2.35 2.84 3.25 4.22 L1S12 0 0.35 0.81 1.80 2.37 2.85 3.06 3.17 4.20 - L1S18 0 1.32 1.87 1.82 2.49 2.95 3.08 3.09 3.57 - L1S19 0 0.63 2.16 3.36 4.27 4.91 5.22 7.08 - - Source: Prepared by the author. 3.2 Application of the proposed methodology The results and discussions from the application of the proposed SHM methodology in the two cases are presented together in this section in a complementary way by follow- ing the natural path in which the validation of the methodology was conducted in this research: first, case I under more controlled conditions with simulated damage and then, case II, with real damage. The proposed SHM methodology is divided into two steps: training and test. However, as most parts are executed in both steps, the results are presented simultaneously following the SPR paradigm parts presented in Chapter 2. A detailed analysis of the effects of damage and temperature on the signal of wave propagation is presented first to provide a clearer understanding of the sensitivity of the damage in the feature extraction process. The effect of the simulated damage in the carbon-epoxy plate produced on the wave propagation is illustrated in Figure 8(a), where it can be identified a signal amplitude reduction as the area covered by the mass increases, which also increases the damping locally and causes a higher attenuation of the wave (LEE 42 et al., 2011). It is essential to note that this effect is more notable on the wave paths crossing the damage (between PZT 1 and PZT 2) and is dependent on the wave mode propagation. Furthermore, the effects of temperature changes on guided wave propagation have been under investigation by the scientific community, where a general conclusion is that temperature variations provoke a shift of phase and an amplitude variation in the signal compared with the baseline condition (SALAMONE et al., 2009; SCALEA; SALAMONE, 2008). These effects are also demonstrated in Figure 8(b), where it can be observed that a temperature variation changes the amplitude and introduces a time delay in the response signal. Figure 9 presents the effect of real delamination damage in the CFRP coupons (case II). It is noted that variations in output signal under damaged conditions with progressive delamination area compared with baseline conditions. As damage severity progress, it is recognized a wave attenuation and a signal lag. As observed in Figures 8(a) and 9, the effect of wave attenuation caused by the damage simulated with the industrial putty is similar to that caused by the actual delamination. Although the goal is not to establish a direct relationship between the simulated damage and a specific delamination area, the similar effects observed justifies the application of the proposed technique to simulate the damage in order to demonstrate the methodology under more controllable conditions. Figure 8: Effects of damage (a) and temperature variation (b) on output signals from application case I: carbon-epoxy plate. 40 50 60 Time[µs] −90 −45 0 45 90 P Z T 1 − 2 [m V ] H30 D30 4 D30 11 (a) Damage effect 40 50 60 Time[µs] −90 −45 0 45 90 P Z T 1 − 2 [m V ] H30 H0 H60 (b) Temperature effect Source: Prepared by the author. 3.2.1 Damage features extraction First, the model order was selected based on BIC. The BIC for each path was evaluated according to the procedure proposed by Figueiredo et al. (2009). Appendix A detailed this issue. Since, the optimized model order did not present significant variations among all 43 Figure 9: Effects of damaged conditions on the output signal in the coupon L1S19 from application case II: CFRP coupon. 30 40 50 60 70 Time[µs] −80 −40 0 40 80 P Z T 4 − 9 [m V ] H D3 D5 D7 Source: Prepared by the author. paths, as can be observed in the results presented in Appendix A, it was chosen a different model order for all transducer paths in each case. It was selected in the application cases I and II, a model order of 18 and 16, respectively. Thus, each case has defined the model order used to identify an AR model for all acquired signals. The validation of each identified model was performed by comparing the model’s prediction value with the measured signals and the autocorrelation analysis from residuals by considering the baseline condition. This procedure of AR model validation is detailed in Appendix A. Figure 10(a) shows the comparison of measured signal and predicted by the AR model for the application case I under the baseline condition. The residuals were tested based on the autocorrelation function; as observed in Figure 10(b), the prediction errors are from a white noise process, which suggests that the model has been adequately identified. The result of the model validation process was similar in both cases and for all evaluated paths. Therefore, for simplicity, due to a large number of paths, just the analysis of the path presented previously will be shown in this work. After the AR model identification of the signals, the damage feature extraction us- ing the residuals and coefficients were performed. Figure 11 shows the space of damage features extracted from the signals of the three paths in the application case I. A dis- tribution pattern of the features related with the conditions can be noted, in the form of clusters, which is more pronounced for PZT 2, as a result of the damage localization. Compared with the cluster from the baseline condition, temperature variation shifts the cluster smoothly to the bottom right corner, increasing the variance of residuals related to the delay introduced in the signal and decreasing the variance of coefficients due to the smooth amplitude signal changes. On the other hand, damaged conditions shift the features to the left-side region of the baseline cluster; additionally, as the area of simulated 44 Figure 10: Baseline measured and predicted by AR(20) and autocorrelation of residuals for whiteness test validation within 95 % confidence bound ( ) for the case I: Carbon- epoxy plate. 0 40 80 120 160 200 Time[µs] −100 −50 0 50 100 P Z T 1 − 2 [m V ] Measured Predicted (a) Output signal (b) Autocorrelation Source: Prepared by the author. damage increases, it moves to the lower region, which means a significant reduction on the residual and coefficients variance, as the simulated damage causes a significant atten- uation of the amplitude signal in the path where the damage is placed. It is important to note that the model order was estimated for the baseline condition, and it was kept the same for each signal in all conditions; therefore, as the signal complexity increases when damage or temperature variation are introduced, the model can no longer fit the signal, and so the distribution of the residuals and coefficients change. For application case II, Figure 12 shows the feature space for coupon L1S11 and two straight paths, one close and another far from the damaged region. In general, it can be observed a similar pattern from application case I, where the clusters of points under damaged conditions presented a reduction on the two features. It is important to note that this reduction is more pronounced for paths closer to the damage, and it is dependent on the delamination area, as demonstrated in Figure 14 by the damage indices computed. In this application case, the four coupons which present 36 paths each one presented the pattern demonstrated using the two selected paths shown in Fig. 12. 3.2.2 Damage detection The damage feature pattern is now explored to classify the structure’s condition using damage indices based on the MSD, trained using 70 % of data from the structure in the healthy state, including those with temperature variation. The ellipse of the confidence level of 95 % from the covariance matrix is presented in Figure 11 as well as the mean value of the cluster. Figure 13 shows, for the application case I, the damage index computed 45 Figure 11: Space of features for application case I. 0.5 1.0 1.5 X1 0.5 1.0 X 2 H30 H0,10,20 H40,50,60 D30 1:11 (a) PZT 2 0.5 1.0 1.5 X1 0.5 1.0 X 2 H30 H0,10,20 H40,50,60 D30 1:11 (b) PZT 3 0.5 1.0 1.5 X1 0.5 1.0 X 2 H30 H0,10,20 H40,50,60 D30 1:11 (c) PZT 4 Source: Prepared by the author. Figure 12: Feature space considering the coupon L1S11 in two different paths (application case II). 0.7 0.8 0.9 1.0 1.1 X1 0.6 0.8 1.0 X 2 H D1:9 (a) PZT 5 - 8: Path through the damaged area 0.9 1.0 1.1 X1 0.9 1.0 X 2 H D1:9 (b) PZT 2 - 11: Path ahead the damaged area Source: Prepared by the author. sequentially for all tested conditions, which is the average MSD from all three paths. As can be observed, the DI is capable of distinguishing damaged conditions from those of average and temperature variation and still capture the effect of simulated increasing delamination area. Figure 14 presents the damage index by delamination area for the coupon L1S11 in the application case II. It should be noted that in both cases, the damage 46 index demonstrated to be well correlated with the delamination area, and it presents a smooth trend. Figure 13: Damage index computed for all signals in application case I: carbon-epoxy plate. 0 20 40 60 DI 0.0 0.5 1.0 1.5 2.0 2.5 S[ % ] H0:60 D1:11 Source: Prepared by the author. Figure 14: Damage index computed for all acquired signals from coupon L1S11 (applica- tion case II). 0 1000 2000 3000 DI 0 1 2 3 4 S[ % ] H D1:9 Source: Prepared by the author. The damage detection was performed based on the proposed outlier detection strategy, where the threshold is defined as the most significant DI value under the healthy state. Table 3 summarize the results of the classification for the two cases. The percentages of true detections refer to the cases were the damaged states were correctly classified, and the false alarms indicate the cases where the damage was detected in the structure with no damage. In the application case I has observed a low rate of false alarms due to the conditions of temperature changes. However, the true detection rate still high. The classifier’s performance in the application case II is maximum, with no false alarm and accurate detection. This is justified by the large variations in the damaged states’ damage index compared with the healthy state. Furthermore, in this case, it was not considered changes in environmental or operational conditions. 47 Table 3: Results of the classifier performance for the two application cases. Case I Case II False Alarm [%] 1.8 0 True Detection [%] 99.9 100 Source: Prepared by the author. The minimum delamination size detected was equivalent to a delamination area of 110 mm2, which corresponds to 0.23 % of the surface area of the coupon L1S11. Due to Lamb waves’ high sensitivity to small internal defects, it is expected to detect delamination area as small as that. 3.2.3 Damage quantification Finally, Figure 15(a) illustrates the GPR of a progressive DI versus the area covered by the adhesive for application case I. The training of the GPR model associated DI with the percentage of the covered area S with a confidence interval of 3σ revealed a trend and an adequate estimative for intermediary test conditions. It should be noted that in this case, the training and test step were performed in the same structure. However, the data from different conditions were considered in the two steps, as indicated in Fig. 15(a). On the other hand, for application case II, Figure 15(b) presents the stochastic inter- polation considering as training data observations in coupon L1S19, L1S18, and L1S12 to obtain the kriging metamodel, i.e., the GPR model. The performance of the metamodel was tested with the observation measured in the coupon specimens L1S11. It is worth to note that all test data is inside the uncertainty bounds assuming 3σ, even assuming some possible imprecision to measure the delamination area in both cases during the training step. It is also important to note that in both cases was used a polynomial cubic trend with correlation function was chosen based on an exponential and ellipsoidal family, with optimization of the maximum likelihood estimation performed by gradient method to set the kriging metamodel (LATANIOTIS; MARELLI; SUDRET, 2015). The validation of the GPR model prediction is presented in Figure 16 for both ap- plication cases. The estimated percentage of the delamination area using the damage indices from test conditions converged on the diagonal line. The Root Mean Squared Error (RMSE) computed for both cases, which were respectively 0.15 and 0.56, showed a better overall prediction performance in application case I compared with application II revealed by the lower RMSE value. The assessment of the prediction performance in each test condition is presented in Table 4, where it is showed the mean of the est values and the 48 Figure 15: Progressive DI versus simulated delamination area S, and mean ( ) of the kriging metamodel with confidence interval of 3σ ( ). DI 0 10 20 30 40 50 60 70 S [% ] -1 0 1 2 3 4 (a) Application case I - carbon-epoxy plate o training data and x test data DI 0 1000 2000 3000 4000 5000 S [% ] -2 0 2 4 6 8 10 (b) Application case II - CFRP Coupons: training data are× L1S19, � L1S18 and� L1S12, test data o L1S11. Source: Prepared by the author. Table 4: Mean of the estimated percentage of delamination and mean absolute error of prediction for test conditions in application cases I and II. Condition D3 D5 D8 Mean of estimated S 0.27 0.70 1.04 Mean absolute error 0.06 0.20 0.12 (a) Application case I. Condition D1 D2 D3 D4 D5 D6 D7 D8 D9 Mean of estimated S 0.21 0.34 0.78 1.72 2.61 2.79 3.72 4.66 4.43 Mean absolute error 0.02 0.02 0.34 0.87 0.77 0.43 0.88 1.41 0.20 (b) Application case II. Source: Prepared by the author. mean absolute error. As can be observed, the maximum values of the mean absolute error in application cases I and II were respectively 0.2 and 1.41. The higher mean absolute errors observed in application case II and consequently, the lower prediction performance of the GPR model is explained because, in this case, the model was trained and tested in different coupons, which involves more sources of uncertainties. The GRP model’s application to associate the DI with the damage size is based on establishing a functional relationship upon a supervised learning strategy. It has been explored for a particular case of hotspot monitoring of delamination. However, as long as a function describes the relationship between the DI and the damage size, the GPR model can be applied to learn and to predict the damage size in different scenarios. Then it could be extrapolated for applications in more complex structures and different types 49 Figure 16: Validation of the estimated percentage of delamination area with measured percentage of delamination area for test conditions in application cases I and II.The estimated percentage of delamination area (x) for all damage indices in each test condition and the mean of estimated percentage of delamination area ( ) for each test condition. (a) Application case I - test conditions D3, D5 and D8. (b) Application case II - all conditions from test coupon L1S11 Source: Prepared by the author. of damage. 3.3 Conclusions This chapter presented the application of the proposed SHM methodology for the problem of hotspot monitoring of delamination in composite structures. The damage feature extraction results based on AR models demonstrated an adequate sensitivity to the presence of the delamination and a direct correlation with the damaged area. Con- sequently, the simple classifier using the outlier detection strategy based on the damage index presented high performance. Many other damage features and detection methods are proposed in the literature. They could also be used in these levels; however, the application of time-series predictive models was chosen because few explored for wave propagation-based methods and their strong appeal for implementation on embedded microelectronics hardware, as already performed by Lynch (2005). The GPR model’s application to establish the mapping relationship between the damage index and the de- lamination area demonstrated to capture the trend and uncertainties adequately. The trained GPR model performed well to estimate the delamination area in both cases: first, by considering the training and test on the same structure specimen and simulated dam- age, where the uncertainties caused by temperature variation were included in the healthy state; and second, for the training and test in different specimens nominally identical and 50 real delamination induced by fatigue loading, which represents a more challenging case due to the higher uncertainty level and it is closer to the practical application of the method. Although the application of the proposed SHM methodology presented herein was performed entirely in offline mode, in a practical implementation situation, the test step should be implemented online, allowing the hotspot monitoring in real-time. The results obtained with the GRP model’s utilization reflect the central contribution of this work, applying a stochastic method to estimate the delamination area. It is essential to consider the Lamb wave signals’ inherent uncertainties for damage quantification levels in compos- ite structures. This can provide more accurate results than deterministic methods since the various sources of uncertainties discussed are inevitably in the practical application. 51 4 MODIFIED METHODOLOGY FOR DAMAGE QUANTIFICATION IN WIND TURBINE BLADES BASED ON THE GPR MODEL USING VIBRATION-BASED DAMAGE INDICES The methodology for damage quantification proposed in Chapter 2 is based on the stochastic interpolation using the GPR model to capture the trend and uncertainties be- tween the damage index and damage size. Although the application presented in Chapter 3 has been focused on a specific case of delamination quantification in laminated com- posite plates using Lamb wave propagation signals, it could be extended for different scenarios and damage feature extraction techniques. The methodology is adjustable to other damage feature extraction procedures, which could be more sensitive to damage in different applications. The GPR model’s application is based on the assumption that there is a trend be- tween the damage index and damage size, which has been demonstrated in the application cases proposed in Chapter 3. This type of trend has been reported for different techniques of damage features extraction and other applications cases (YADAV et al., 2020; GHRIB et al., 2018; TIAN; YU; LECKEY, 2015; AMER; KOPSAFTOPOULOS, 2019). In re- cent work, Garcia and Tcherniak (2019) proposed a data-driven methodology for damage detection in large wind turbine blades using acceleration data. The methodology demon- strated to be useful for debonding detection in the trailing edge of blades. Also, the results demonstrated a monotonic increasing function of the damage index according to damage size progression. The damage index progression analysis suggests that it could be used for damage quantification, which is a problem of great interest to the clean energy industry because of the potential to improve the blades’ operation and maintenance. In this context, in this chapter, the methodology proposed by Garcia and Tcherniak (2019) is extended for the damage quantification level based on the application of GPR model in a similar way to the methodology presented in Chapter 2. It demonstrates the versatility of damage quantification methodology presented in Chapter 2 for damage quantification based on the GPR model application using a different technique for damage feature extrac- tion. Here, the damage features extraction method based on vibration signals proposed by Garcia and Tcherniak (2019) is used instead of the damage features extraction based on AR model identification of Lamb wave signals. 52 This chapter is organized as follows: firstly, a brief description of the methodology proposed by Garcia and Tcherniak (2019) and its extension for damage quantification based on the GPR model is given in Section 4.1. In Section 4.2, it is presented an overview of the experimental setup and procedures of data collection of the data set used for practical application in an SSP 34 m wind turbine blade. Then, the results and discussions obtained by the methodology’s practical application are presented in Section 4.3. Finally, Section 4.4 presents the conclusions of this chapter. 4.1 Damage quantification using GPR model in wind turbine blades The methodology presented in this section for damage quantification using the GPR model in wind turbine blades combines the methodology for damage detection proposed by Garcia and Tcherniak (2019) and the damage quantification methodology proposed in Chapter 2. As will be noticed, the methodology is a modified version of the methodology proposed in Chapter 2. This modified methodology proposed for damage quantification is illustrated in Figure 17 and can be split into two steps: 1) training and 2) test. Similar to the methodology presented in Chapter 2, the application of the GPR model still identical. In the training step, the GPR model’s supervised learning is performed using labeled training data from the structure under the baseline condition and known progressive damaged conditions. In the test step, a similar or the same structure in an unknown condition is interrogated about the presence of damage by performing a hypothesis test with a defined threshold value. When damage is detected, the damage index is estimated from the damage index using the trained GPR model. The differences in this methodology compared with that proposed in Chapter 2 is restricted to the type of signal collected from the structure, which in this case is the acceleration of the impulse response instead of the Lamb wave propagation, and the methodology for damage detection, which is the same proposed and validated by Garcia and Tcherniak (2019). 4.1.1 Damage detection The methodology proposed by Garcia and Tcherniak (2019) for damage detection in wind turbine blades is based on singular spectrum analysis. It is assumed a sim- ple non-parametric method for data compression and information extraction (GARCIA; TCHERNIAK, 2019). The methodology is organized into four steps: data collection, cre- ation of the reference state, feature extraction, and decision making. Figure 18 provides a scheme of the procedure, which will be summarized in the following subsections. 53 Figure 17: Flowchart of the proposed modified methodology for damage quantification based on GPR model using vibration-based damage indices. Features extraction Data Collection Damage Index DI ≥ 𝛽 ? Damage Index Lerned Model I ) Training II) Test GP Regression Training 𝛽 Damage Size DI Damage Size Reference Condition Unknown Condition Progressive Damaged Conditions DI Creation of the reference space Data Collection Features extraction Data Collection X R R X T X T No Yes Source: Prepared by the author. Data collection In the data collection step, the discrete acceleration signals with L̄ time sampling points collected from the structure are standardized to have zero mean and unit variance. Then, each signal is transformed to the frequency domain to obtain signal vectors Xm with length L = L̄/2 where m = 1, . . . ,M is the number of signal vector realizations measured as: Xm = (X1,m,X2,m, . . . ,XL,m)t (18) The signal vectors are stored in columns of the matrix X defined as: X = (X1,X2, . . . ,XM) (19) The matrix X presents dimension [L × M ], and it is constructed for each healthy structural condition of the wind turbine blade. The matrix X corresponding to the healthy 54 Figure 18: Flowchart of vibration-based methodology for damage assesment in wind tur- bine blades proposed by Garcia and Tcherniak (2019) Data collection Xm = (x1,m, x2,m, · · · , xN,m)t X = (X1,X2, · · · ,XM) Creation of the reference state X̌m = x1,m . . . xW,m x2,m . . . x(W+1),m ... . . . ... xN,m . . . 0     X̌ = (X̌1, X̌2, . . . , X̌M) CX = ˇX t ˇX N Eigenvalues Et XCXEX = ΛX A = X̌EX Reconstruction reference state Rk m,n = 1 Wn W∑ W=1 Ak n−W+1E k m,W Feature extraction Tj = N∑ n=1 xnRnj Ti = (T1,i, T2,i, . . . , Tp,i) Di = √ (Ti − µB)t ∑−1(Ti − µB) H0 : Di ≤ DT Undamaged H1 : Di > DT Damaged Decision making Source: Prepared by the author. condition is used to create a reference state. Creation of the reference state The reference state’s creation is performed using three operations: embedding the signal vector into a matrix with lagged copies, decomposition of the embedded matrix into principal components, and reconstruction of the reference state using a set of principal components. In the embedding operation, a full embedded matrix X̌ is created. Each element of this full matrix corresponds to a single embedded matrix X̌m created by W-lagged copies of each vector signal Xm. The dimension of each matrix X̌m is [L×W ], where W corresponds to the sliding window size. X̌ = ( X̌1, X̌2, . . . , X̌M ) (20) In order to compute the principal components of data it is necessary to perform the eigendecomposition of the covariance matrix CX defined as follows, which has dimension 55 [(MW )× (MW )]: CX = X̌ t X̌ N (21) The eigendecomposition of CX is written as: Et XCXEX = ΛX (22) where ΛX contains all eigenvalues stored in the diagonal matrix and EX contains all eigenvectors Ek with dimension {Ek : 1 < k ≤MW}. The principal component Ak associated with each eigenvector Ek is computed by projecting the matrix X̌ onto EX as described by: A = X̌EX (23) The reference state is formulated based on the reconstructed components obtained by the linear combination of a set of principal components. The reconstructed components are computed by convolving the principal components Ak with the associated Ek: Rk m,l = 1 Wn W∑ w=1 Ak l−w+1E k m,w (24) where Wn is a normalization factor described by Wn =  n 1 6 n 6 W − 1 W W 6 n 6 N (25) The reconstructed components are arranged in columns into the matrix R with di- mension [L × (MW )]. Each column corresponds to the reconstructed component of the signal associated with the respective principal component. Therefore, R can be used as the reference state of the structure to which the observation signal vectors are compared (GARCIA; TCHERNIAK, 2019). 56 Feature extraction In the feature extraction step, the similarity of each signal is compared with the reference state. It is performed by computing the feature vector defined in the following equation, which represents the inner product between each observed signal vector and the reference state’s reconstructed components R, where j = 1, ...,W : Tj = L∑ l=1 xlRlj (26) Inspection phase and decision making A baseline feature matrix is constructed with a selected number of feature vectors from signals corresponding to the structure in the healthy state. This baseline matrix is defined in the following equation and it has dimension [s × u], where u is the dimension selected from the FV {T : u ≤ W} and s is the number of signal vectors utilized to define the baseline matrix: TB =  T1,1 T1,1 . . . Tu,1 T1,2 T2,2 . . . Tu,2 ... ... . . . ... T1,s T2,s . . . Tu,s  (27) Once the baseline feature matrix is defined, the damage index is computed based on Mahalanobis distance between all the observed feature vectors and the baseline feature matrix: Di = √ (Ti − µB)tΣ−1 B (Ti − µB) (28) where µB is the mean row of the baseline feature matrix TB; ΣB is its corresponding covariance matrix, and Di is the damage index. It is important to note that the procedure described for damage index computing is based on the signals extracted from a single accelerometer. Therefore, it is considered a local damage index. The structural health classification is performed based on the detection of outliers, 57 which is necessary to define a local damage index threshold. As demonstrated by Garcia and Tcherniak (2019), the distribution of damage index in the reference state (obser- vations from the healthy wind turbine blade) can be fitted by a log-normal probability density function. Then, the threshold is selected by a particular risk level (α), which de- termines the false alarm probability equal to the log-normal density function (GARCIA; TCHERNIAK, 2019). 4.1.2 Damage quantification The damage detection methodology proposed by Garcia and Tcherniak (2019) is based on the local damage index Di. Then, to extend its application for damage quantification level using the GPRmodel in structure instrumented with multiple sensors, it is convenient to define a new global damage index, which combines the features extracted from all paths. Otherwise, it should be necessary to train one GPR model for each sensor. Then, in an analogous way of the methodology proposed in Chapter 2, it is proposed here the following global damage index: DI = 1 Ns Ns∑ i=1 Di (29) where Ns is the number of sensors used to monitor the structure. The GPR model’s application is proposed based on the methodology presented in Chapter 2 to extend the methodology proposed by Garcia and Tcherniak (2019) to the dama