Applied Science, Faculty of
Graduate Studies, College of (Okanagan)
DSpace
UBCO
Wang, Teng
Doctor of Philosophy - PhD
University of British Columbia
Fatigue cracks pose a critical threat to the structural integrity of mechanical and structural components. In damage tolerance design philosophy, it is allowed that an in-service component has growing fatigue cracks, as long as the cracks can be reliably detected during scheduled inspections, and will not cause the failure of the component in a catastrophic manner. In this context, the accurate prediction of fatigue crack growth is crucial to ensure the reliability of in-service components while reducing maintenance costs. This thesis research aims to develop a set of probabilistic models for fatigue crack growth prediction considering a variety of uncertainty factors. First, to model the material uncertainty of individual structure, a probabilistic model for the fatigue crack growth prediction is proposed based on a closed-form solution. In this model, the stress intensity factor is related to the crack length and load condition through the finite element model. The material parameters and model error are regarded as random variables. Based on conjugate Bayesian analysis, a closed-form solution is derived to update the posterior distribution of the material parameters according to the crack growth observation. Given the posterior distribution, a modified Paris–Erdogan equation is adopted to predict the crack growth in a probabilistic view. The effectiveness of the model is validated based on the fatigue experimental data of a set of aluminum alloy specimens. Second, to consider multi-source prior knowledge, a probabilistic model for the fatigue crack growth prediction with hybrid prior is proposed. This model comprises two inference steps. In the first inference step, a set of candidate priors are input. Then, the Monte Carlo integration is adopted in the calculation of the posterior belief of each candidate prior. In the second inference step, the particle filter is extended to conduct the Bayesian inference with a hybrid prior. Numerical studies on the spur gear tooth show integrating multiple priors can increase the robustness of the fatigue crack growth prediction. Last but not least, to consider crack location randomness on a pipe component, a probabilistic model is proposed for the fatigue crack prediction based on the localization of the right and left crack tips. A real-world application of pipeline fatigue crack prediction is specified. In this model, the guided-wave sensor network is adopted for the crack tip localization where the probabilistic of detection and measurement accuracy are calculated first. Then, the finite element model is established for stress intensity factor calculation. To conduct a Monte Carlo analysis, a data-driven model is built as the surrogate of the finite element model. Finally, the crack growth is predicted based on the state space model, with the findings from the guided-wave sensor network. The research outcomes in this thesis contribute to fatigue crack growth prediction in different aspects. The probabilistic model with a closed-form solution enables an accurate and computationally efficient prediction considering the material uncertainty. The probabilistic model with a hybrid prior model increases the robustness of fatigue crack growth prediction by integrating multi-source prior. The probabilistic model considering the crack location randomness can be used for the prediction of crack growth when the crack location is random to some degree.
Probabilistic Models for Fatigue CrackGrowth PredictionbyTeng WangB.Eng., Hefei University of Technology, 2016M.Eng., Shandong University, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)June 2023© Teng Wang, 2023The following individuals certify that they have read, and recommend tothe College of Graduate Studies for acceptance, the dissertation entitled:PROBABILISTIC MODELS FOR FATIGUE CRACK GROWTH PRE-DICTIONsubmitted by Teng Wang in partial fulfilment of the requirements of thedegree of Doctor of PhilosophyDr. Zheng Liu, School of EngineeringSupervisorDr. Min Liao, National Research Council CanadaCo-SupervisorDr. Rudolf Seethaler, School of EngineeringSupervisory Committee MemberDr. Ahmad Al-Dabbagh, School of EngineeringSupervisory Committee MemberDr. Lukas Bichler, School of EngineeringUniversity ExaminerDr. Lalita Udpa, Michigan State UniversityExternal ExamineriiAbstractFatigue cracks pose a critical threat to the structural integrity of mechanicaland structural components. In damage tolerance design philosophy, it isallowed that an in-service component has growing fatigue cracks, as long asthe cracks can be reliably detected during scheduled inspections, and willnot cause the failure of the component in a catastrophic manner. In thiscontext, the accurate prediction of fatigue crack growth is crucial to ensurethe reliability of in-service components while reducing maintenance costs.This thesis research aims to develop a set of probabilistic models for fatiguecrack growth prediction considering a variety of uncertainty factors.First, to model the material uncertainty of individual structure, a prob-abilistic model for the fatigue crack growth prediction is proposed based ona closed-form solution. In this model, the stress intensity factor is relatedto the crack length and load condition through the finite element model.The material parameters and model error are regarded as random variables.Based on conjugate Bayesian analysis, a closed-form solution is derived toupdate the posterior distribution of the material parameters according tothe crack growth observation. Given the posterior distribution, a modifiedParis–Erdogan equation is adopted to predict the crack growth in a proba-bilistic view. The effectiveness of the model is validated based on the fatigueexperimental data of a set of aluminum alloy specimens.Second, to consider multi-source prior knowledge, a probabilistic modelfor the fatigue crack growth prediction with hybrid prior is proposed. Thismodel comprises two inference steps. In the first inference step, a set ofcandidate priors are input. Then, the Monte Carlo integration is adoptedin the calculation of the posterior belief of each candidate prior. In thesecond inference step, the particle filter is extended to conduct the Bayesianinference with a hybrid prior. Numerical studies on the spur gear toothshow integrating multiple priors can increase the robustness of the fatiguecrack growth prediction.Last but not least, to consider crack location randomness on a pipecomponent, a probabilistic model is proposed for the fatigue crack predic-tion based on the localization of the right and left crack tips. A real-worldiiiapplication of pipeline fatigue crack prediction is specified. In this model,the guided-wave sensor network is adopted for the crack tip localizationwhere the probabilistic of detection and measurement accuracy are calcu-lated first. Then, the finite element model is established for stress intensityfactor calculation. To conduct a Monte Carlo analysis, a data-driven modelis built as the surrogate of the finite element model. Finally, the crackgrowth is predicted based on the state space model, with the findings fromthe guided-wave sensor network.The research outcomes in this thesis contribute to fatigue crack growthprediction in different aspects. The probabilistic model with a closed-formsolution enables an accurate and computationally efficient prediction consid-ering the material uncertainty. The probabilistic model with a hybrid priormodel increases the robustness of fatigue crack growth prediction by inte-grating multi-source prior. The probabilistic model considering the cracklocation randomness can be used for the prediction of crack growth whenthe crack location is random to some degree.ivLay SummaryFatigue cracks pose a serious threat to the structural integrity of mechani-cal and structural components. Accurate prediction of fatigue crack growthis crucial to ensure the reliability of in-service components while reducingmaintenance costs. This thesis research aims to develop a set of proba-bilistic models for fatigue crack growth prediction considering a variety ofuncertainty factors.First, a closed-form solution is proposed for the fatigue crack growth pre-diction considering the material variance of individual structures. Second,to consider multi-source prior knowledge, a probabilistic prediction modelwith hybrid prior is derived. Last, a probabilistic model is proposed for thefatigue crack growth prediction considering crack location randomness.vPrefaceThis thesis is based on the research work conducted in the School of Engi-neering at The University of British Columbia, Okanagan Campus, underthe supervision of Dr. Zheng Liu and the co-supervision of Dr. Min Liao. Iconducted the majority of the research, including the literature review, prob-lem formulation, algorithm development, and paper writing. Dr. Zheng Liuand Dr. Min Liao helped me with the research ideas and suggestions toimprove the quality of the manuscripts.Chapter 2 is based on the following published papers:• Teng Wang, Zhila Bahrami, Guillaume Renaud, Chunsheng Yang,Min Liao and Zheng Liu. “A probabilistic model for fatigue crackgrowth prediction based on closed-form solution.” in Structures, vol.44, pp. 1583-1596, 2022.• Teng Wang, Zheng Liu, Xiaoli Zhao, Nezih Mrad. “Bayesian-basedmethod for the remaining useful life and reliability prediction of steelstructure,” in 2020 Asia-Pacific International Symposium on AdvancedReliability and Maintenance Modeling (APARM-Vancouver), IEEE,Aug. 20-23, 2020.Chapter 3 is based on the following published paper:• Teng Wang, Junchi Bin, Guillaume Renaud, Min Liao, GuoliangLu and Zheng Liu. “Probabilistic method for fatigue crack growthprediction with hybrid prior.” in International Journal of Fatigue, vol.157, pp. 1-11, 2022.Chapter 4 is based on the following paper:• Teng Wang, Ke Feng, Jiatong Lin, Min Liao, Chunsheng Yang andZheng Liu. “Condition monitoring of pipeline towards digital twinecosystem: A case study.” in Structural Health Monitoring, In revision.Other publications I authored or co-authored are summarized as follows:vi• Teng Wang, Zheng Liu, Min Liao, Nezih Mrad, and Guoliang Lu.“Probabilistic analysis for remaining useful life prediction and relia-bility assessment.” in IEEE Transactions on Reliability, vol. 71, no.3, pp. 1207-1218, 2022.• Teng Wang, Zheng Liu, and Nezih Mrad. “A probabilistic frameworkfor remaining useful life prediction of bearings.” in IEEE Transactionson Instrumentation and Measurement, vol. 70, p.3503412, 2021.• Teng Wang, and Zheng Liu. “Chapter 10 - Digital twin and its appli-cation for the maintenance of aircraft.” in Handbook of NondestructiveEvaluation 4.0, 2021, pp. 1-19.• Teng Wang, Zheng Liu, Min Liao, Nezih Mrad. “Life prediction foraircraft structure based on Bayesian inference: towards a digital twinecosystem,” in 2020 Annual Conference of the PHM Society (PHM-Nashville). Nov. 09-13, 2020.• Teng Wang, Zheng Liu, and Guoliang Lu.“Bearing condition moni-toring based on the indicator generated in time-frequency domain.” in2019 Annual Conference of the PHM Society (PHM-Scottsdale). vol.11. no. 1. 2019.• Zhila Bahrami, Ran Zhang, Teng Wang and Zheng Liu. “An end-to-end framework for shipping container corrosion defect inspection.”in IEEE Transactions on Instrumentation and Measurement, vol. 71,p.5020814, 2022.• Ran Zhang, Zhila Bahrami, Teng Wang and Zheng Liu, “An adap-tive deep learning framework for shipping container code localizationand recognition,” in IEEE Transactions on Instrumentation and Mea-surement, vol. 70, pp. 1-13, 2021.• Zhao, Xiaoli, Minping Jia, Junchi Bin, Teng Wang and Zheng Liu.“Multiple-order graphical deep extreme learning machine for unsu-pervised fault diagnosis of rolling bearing.” in IEEE Transactions onInstrumentation and Measurement, vol. 70, p.1-12, 2020.• Xiaoli Zhao, Zheng Liu, Teng Wang, Junchi Bin and Minping Jia.“Unsupervised fault diagnosis of machine via multiple-order graph-ical deep extreme learning machine,” in 2020 Asia-Pacific Interna-tional Symposium on Advanced Reliability and Maintenance Modeling(APARM-Vancouver), IEEE, Aug. 20-23, 2020.vii• Zhila Bahrami, Ran Zhang, Rakiba Rayhana, TengWang, and ZhengLiu, “Optimized deep neural network architectures with anchor boxoptimization for shipping container corrosion inspection,” in 2020 IEEESymposium Series on Computational Intelligence (SSCI-Canberra), Dec.01-04, 2020.• Meng, Kang, Teng Wang, Shashank Pant, Marc Genest, Zheng Liu.“Fault detection and diagnosis for PZT sensors with electro-mechanicalimpedance technique by using one-dimensional convolutional autoen-coder.” in International Society for Optics and Photonics, Mar. 22-25,2021.• Rakiba Rayhana, Zhila Bahrami, Teng Wang, Zheng Liu, Angie Wu,Xiangjie Kong. “Environment prediction to enhance the navigationsystem of water pipeline inspection platforms.” in 2021 IEEE 30thInternational Symposium on Industrial Electronics (ISIE). Jun. 20-23, 2021.viiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xxChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivations . . . . . . . . . . . . . . . . . . 11.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Principle of Fatigue Crack Growth . . . . . . . . . . . 41.2.2 Stress Intensity Factor Calculation . . . . . . . . . . . 61.2.3 Uncertainty Analysis . . . . . . . . . . . . . . . . . . . 71.2.4 Fatigue Crack Growth Prediction Models . . . . . . . 101.3 Research Gaps and Objectives . . . . . . . . . . . . . . . . . 161.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 18Chapter 2: Probabilistic Model for Fatigue Crack GrowthPrediction Based on Closed-Form Solution . . . . 192.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 20ix2.2.1 Fatigue Crack Growth Modeling and Closed-Form So-lution Derivation . . . . . . . . . . . . . . . . . . . . . 202.2.2 Fatigue Crack Growth Prediction Using the Closed-Form Solution . . . . . . . . . . . . . . . . . . . . . . 292.3 Experiment Validation and Discussion . . . . . . . . . . . . . 312.3.1 Experiment I: Validation of Confidence Interval Cal-culation . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.2 Experiment II: Validation of Fatigue Crack GrowthPrediction . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 382.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Chapter 3: Probabilistic Model for Fatigue Crack GrowthPrediction with Hybrid Prior . . . . . . . . . . . . 433.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.1 Fatigue Crack Growth Modeling and Model Calibra-tion with Non-Destructive Inspection Data . . . . . . 443.2.2 Bayesian Inference for Posterior Belief of Prior Distri-bution . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.3 Bayesian Inference with Hybrid Prior . . . . . . . . . 503.3 Experiment Validation and Discussion . . . . . . . . . . . . . 533.3.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . 533.3.2 Experimental Results and Analysis . . . . . . . . . . . 553.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Chapter 4: Probabilistic Model for Fatigue Crack GrowthPrediction Considering Crack Location Random-ness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . 664.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3.1 Physics-Based Model . . . . . . . . . . . . . . . . . . . 674.3.2 Non-Destructive Inspection Signal Processing . . . . . 714.3.3 Finite Element Analysis . . . . . . . . . . . . . . . . . 764.3.4 Surrogate Model . . . . . . . . . . . . . . . . . . . . . 784.3.5 Load Condition Analysis . . . . . . . . . . . . . . . . . 814.3.6 Bayesian Inference . . . . . . . . . . . . . . . . . . . . 864.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93x4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Chapter 5: Conclusions . . . . . . . . . . . . . . . . . . . . . . . 965.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 Limitations and Future Work . . . . . . . . . . . . . . . . . . 97Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100xiList of TablesTable 2.1 Predicted posterior distributions with parameters (µlogC ,µm, σ2logC , σ2m, ρ) at P1, P2, and P3 using HMC,NUTS, Gibbs and the closed-form solution, respec-tively. P1, P2, and P3 are inferred based on 12, 33,and 74 crack length observations of specimen 1 from 9mm to 11.2 mm, 15.4 mm, and 23.6 mm, respectively. 38Table 2.2 Prediction performance and running time comparisonfor Virkler’s dataset. Specimens 1, 17, 33, 49, and68 are tested respectively, as they have different crackgrowth speeds. . . . . . . . . . . . . . . . . . . . . . . 41Table 2.3 Prediction performance comparison in the discussion. . 41Table 3.1 Material properties and geometry parameters [1]. . . . 54Table 3.2 Candidate joint priors. . . . . . . . . . . . . . . . . . . 55Table 3.3 Ground-truth for specimen A-F. . . . . . . . . . . . . 56Table 3.4 Likelihood of each prior. . . . . . . . . . . . . . . . . . 58Table 3.5 Posterior belief of each prior. . . . . . . . . . . . . . . 59Table 3.6 CRA of different methods for the RUL prediction. . . 61Table 3.7 RMSE of different methods for the RUL prediction. . . 63Table 4.1 Prediction performance of ANN. (h, s) denotes the ANNwith h hidden layers and each hidden layer has s neurons. 80Table 4.2 Prediction performance of KNN. . . . . . . . . . . . . 81Table 4.3 Weights and parameters of the first five componentsin the Dirichlet process mixture model. . . . . . . . . . 85Table 4.4 Prediction performance for Pipe A, Pipe B and Pipe C. 91xiiList of FiguresFigure 1.1 Typical cyclic load: ground-air-ground (GAG) loadhistory used in the fatigue tests of the aircraft struc-ture [2]. Image reproduced with permission by Elsevier. 1Figure 1.2 Fatigue damage of propeller shaft for illustration [3].Image reproduced with permission by Elsevier. . . . . 2Figure 1.3 Three types of SIF on a cracked body [4]. Imagereproduced with permission by CRC Press. . . . . . . 4Figure 1.4 Typical fatigue rate curve [5]. Image reproduced withpermission by Cambridge University Press. . . . . . . 5Figure 1.5 Two-dimensional stress field near the crack tip [5].Image reproduced with permission by Cambridge Uni-versity Press. . . . . . . . . . . . . . . . . . . . . . . . 8Figure 1.6 (a) Typical 8-node singular element. (b) 8-node sin-gular elements around the crack tip. © Teng Wang . 9Figure 1.7 Crack tip yield zones [6]. Image reproduced with per-mission by SPIE. . . . . . . . . . . . . . . . . . . . . 12Figure 2.1 A comparison between the actual da/dn and the da/dnpredicted using Paris-Erdogan equation (FCG datafrom the Virkler’s dataset [7]) [8]. Image reproducedwith permission by Elsevier. . . . . . . . . . . . . . . 21Figure 2.2 Crack growth trajectories of the 68 specimens [8]. Im-age reproduced with permission by Elsevier. . . . . . 32Figure 2.3 (a) Geometry of the tested specimens. (b) Corre-sponding FE model [8]. Image reproduced with per-mission by Elsevier. . . . . . . . . . . . . . . . . . . . 33Figure 2.4 SIF range calculated from the FE model and the ana-lytical model respectively [8]. Image reproduced withpermission by Elsevier. . . . . . . . . . . . . . . . . . 34xiiiFigure 2.5 (a) Prior distribution assigned for the FCG predic-tion. (b) Posterior distribution at P1. (c) Posteriordistribution at P2. (d) Posterior distribution at P3.In particular, ∆K is in MPa ·m0.5 and da/dn is inmm/cycle [8]. Image reproduced with permission byElsevier. . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.6 (a) At each specific time, R follows lognormal dis-tribution. (b) Predicted distribution of future crackgrowth speed R at P1. (c) Predicted distribution offuture crack growth speed R at P2. (d) Predicteddistribution of future crack growth speed R at P3 [8].Image reproduced with permission by Elsevier. . . . . 36Figure 2.7 (a-1) (b-1) and (c-1) show the possible degradationpaths calculated from the Gibbs sampler at P1, P2and P3. (a-2) (b-2) and (c-2) show the 95% CI calcu-lated from the proposed model at P1, P2 and P3 [8].Image reproduced with permission by Elsevier. . . . . 37Figure 2.8 (a-1,2,3) Posterior distribution sampled via Gibbs atP1, P2 and P3. (b-1,2,3) Posterior distribution wassampled via HMC at P1, P2 and P3. (c-1,2,3) Pos-terior distribution sampled via NUTS at P1, P2 andP3 [8]. Image reproduced with permission by Elsevier. 39Figure 2.9 Predicted results of the closed-form solution, Gibbs,HMC and NUTS. (a)-(e) are prediction results forspecimens 1, 17, 33, 49 and 68 respectively [8]. Imagereproduced with permission by Elsevier. . . . . . . . . 40Figure 2.10 Change of correlation coefficient ρ with the increaseof the number of observations [8]. Image reproducedwith permission by Elsevier. . . . . . . . . . . . . . . 41Figure 3.1 Overview of the proposed methodology [9]. Imagereproduced with permission by Elsevier. . . . . . . . . 44Figure 3.2 Sample hybrid prior distribution. Each candidateprior πj , j = 1, ..., p is denoted as a bar with lengthP (πj |D). Connecting all the bars results in a bar witha length of 1 [9]. Image reproduced with permissionby Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . 51xivFigure 3.3 Illustration of the CDF method where s is taken as 8for example. 0.5 is the random value generated froma uniform distribution between 0 and 1 [9]. Imagereproduced with permission by Elsevier. . . . . . . . . 52Figure 3.4 (a) FE model of a single spur gear tooth. (b) Re-sponse surface of SIF range which is a function re-garding crack length and torque [9]. Image repro-duced with permission by Elsevier. . . . . . . . . . . . 54Figure 3.5 Location of π1, π2, π3 and π4 in three-dimensionalspace. Each distribution is represented by 10000 sam-ples from it [9]. Image reproduced with permission byElsevier. . . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 3.6 Crack growth paths of gear A-F with measurementnoise simulated from equation (3.4) [9]. Image repro-duced with permission by Elsevier. . . . . . . . . . . . 57Figure 3.7 (a-1)(a-2) and (a-3) show the initialization of particlesat load cycle 5.0 · 107, 5.5 · 107 and 6.0 · 107 respec-tively for specimen A. (b-1)(b-2) and (b-3) show theinitialization of particles at load cycle 5.0·107, 5.5·107and 6.0 · 107 respectively for specimen B [9]. Imagereproduced with permission by Elsevier. . . . . . . . . 60Figure 3.8 Posterior distribution of specimens A and B at differ-ent load cycles [9]. Image reproduced with permissionby Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 3.9 Predicted RUL distribution of specimens A and Bwith hybrid prior. The histogram shows the distribu-tion of particles. The red line is the smoothing fittingcurve of the histogram. (a-1,2,3) are results for spec-imen A at load cycle 5.0 · 107, 5.5 · 107 and 6.0 · 107respectively. (b-1,2,3) are results for specimen B atload cycle 5.0·107, 5.5·107 and 6.0·107 respectively [9].Image reproduced with permission by Elsevier. . . . . 62Figure 3.10 Two schemes for comparison [9]. Image reproducedwith permission by Elsevier. . . . . . . . . . . . . . . 64xvFigure 3.11 An illustrative example. (a) Scheme 2 does not takeinto account the information that candidate priorscome from different information sources. (b) Pro-posed method considers the group-level difference (in-dicated using color) of particles by the Monte Carlointegration [9]. Image reproduced with permission byElsevier. . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 4.1 Overview of the proposed method. © Teng Wang . . 66Figure 4.2 Experimental setup for pipe fatigue test under four-point bending load [10]. Image reproduced with thepermission from the Fraunhofer Institute for CeramicTechnologies and Systems (IKTS), Dresden, Germany. 67Figure 4.3 Diagram of the experiment configuration. Image re-produced with the permission from the Fraunhofer In-stitute for Ceramic Technologies and Systems (IKTS),Dresden, Germany. . . . . . . . . . . . . . . . . . . . 68Figure 4.4 Guided-wave sensor network deployment. Image re-produced with permission from the Fraunhofer Insti-tute for Ceramic Technologies and Systems (IKTS),Dresden, Germany. . . . . . . . . . . . . . . . . . . . 69Figure 4.5 (a) Cross section of the pipe with a through-thicknesscrack. (b) Polar coordinate system adopted in thisstudy. © Teng Wang . . . . . . . . . . . . . . . . . . 70Figure 4.6 (a) The layout of the guide-wave sensors for acquir-ing the defect sensory data. (b) The positions of thepurple dotted line and the green dotted line in thecross section of pipe. © Teng Wang . . . . . . . . . . 72Figure 4.7 The schematic diagram of the TFM [11]. Image re-produced under the Creative Commons AttributionAttribution 4.0 International License. . . . . . . . . . 73Figure 4.8 (a) FMC image when no crack is detected. (b) FMCimage when the crack is just detected. (c) FMC imagewhere the defect area is enlarged due to the growthof the crack. © Teng Wang . . . . . . . . . . . . . . . 74Figure 4.9 (a) The relationship between θ˜ and θ collected dur-ing the test of pipes. (b) The estimated POD of thesensor network. © Teng Wang . . . . . . . . . . . . . 75xviFigure 4.10 FE model of the pipeline. The global model is builtusing ANSYS®. The submodel is extracted and ana-lyzed with Franc 3D®. © Teng Wang . . . . . . . . . 78Figure 4.11 Comparison of SIF factors calculated from the closed-form solution and FE model. © Teng Wang . . . . . 79Figure 4.12 Training samples and validation samples generatedfrom the FE model. © Teng Wang . . . . . . . . . . 80Figure 4.13 (a-1) (b-1) (c-1) are the relationship between ∆KLand θL, θR and θL − θR, respectively. (a-2) (b-2) (c-2) are the relationship between ∆KR and θL, θR andθL − θR, respectively. © Teng Wang . . . . . . . . . 81Figure 4.14 ANN for SIFs prediction. © Teng Wang . . . . . . . 82Figure 4.15 The prediction error of the surrogate model.© TengWang . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 4.16 (a) Nominal force range and actual force range mea-sured at the hydraulic unit. (2) Frequency histogramof the actual force range. © Teng Wang . . . . . . . 83Figure 4.17 (a)(b)(c) and (d) are the generated Markov chains forµ1, τ1, w1, and α, respectively. © Teng Wang . . . . 86Figure 4.18 2000 mixtures and their mean values calculated fromthe posterior distributions of µu, τu, wu and α.© TengWang. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 4.19 (a) The weights of normal distribution components.(b) The mixture of the first five normal distributioncomponents. © Teng Wang . . . . . . . . . . . . . . 87Figure 4.20 Prediction result for pipe A: (a) Predicted right cracktip. (b) Predicted left crack tip. (c) Predicted cracklength. (d) Prior and posterior distributions aftereach update. © Teng Wang . . . . . . . . . . . . . . 91Figure 4.21 Prediction result for pipe B: (a) Predicted right cracktip. (b) Predicted left crack tip. (c) Predicted cracklength. (d) Prior and posterior distributions aftereach update. © Teng Wang . . . . . . . . . . . . . . 92Figure 4.22 Prediction result for pipe C: (a) Predicted right cracktip. (b) Predicted left crack tip. (c) Predicted cracklength. (d) Prior and posterior distributions aftereach update. © Teng Wang . . . . . . . . . . . . . . 93Figure 4.23 (a), (b) and (c) are the reliabilities of pipe A, B and Crespectively. The dotted line indicates the inspectiontime. © Teng Wang . . . . . . . . . . . . . . . . . . . 94xviiGlossaryARIMA Autoregressive Integrated Moving AverageANN Artificial Neural NetworkBEM Boundary Element MethodBM Brownian MotionBGPD Bivariate Gaussian Prior DistributionCI Confidence IntervalCGL Crack Growth LifeCDF Cumulative Distribution FunctionCRA Cumulative Relative AccuracyEIFS Equivalent Initial Flaw SizeFCG Fatigue Crack GrowthFEM Finite Element ModelFE Finite ElementFMC Full Matrix CaptureGAG Ground-Air-GroundHMC Hamiltonian Monte CarloKNN K-Nearest NeighborMCMC Markov Chain Monte CarloMV Median ValueMLV Most Likely ValuexviiiMA Measurement AccuracyMAG Metal Active GasMAE Mean Absolute ErrorNUTS No-U-Turn SamplerNDI Non-Destructive InspectionPCE Polynomial Chaos ExpansionPOD Probability Of DetectionPF Particle FilterPDF Probability Density FunctionPBPD Posterior Belief Of Prior DistributionRMSE Root Mean Square ErrorRUL Remaining Useful LifeRA Relative AccuracyROI Region Of InterestSIF Stress Intensity FactorSIS Sequential Importance SamplingTFM Total Focus MethodXFEM Extended Finite Element MethodxixAcknowledgementsFrom the bottom of my heart, I would like to express my gratitude to thosewho supported and helped me to complete my Ph.D. journey.First and foremost, I would like to express my sincere gratitude to mysupervisor Dr. Zheng Liu for his dedicated support and guidance. He con-tinuously provided encouragement and was always willing and enthusiasticto assist in any way he could throughout my Ph.D. program. With his helpand guidance, I could be involved in so many advanced research projectsand build my expertise in this research field. It was a great honour for meto work under his supervision.Second, I must express my gratitude to my co-supervisor Dr. Min Liao.His guidance and advice carried me through all the stages of my researchjourney. Especially, I am grateful for his helpful suggestions and construc-tive criticism throughout the National Research Council of Canada (NRC)collaborative projects. Without his guidance and persistent help, the com-pletion of this dissertation would not have been possible.I am also grateful to Dr. Rudolf Seethaler and Dr. Ahmad Al-Dabbaghto be on my supervisory committee. I would like to thank Dr. Lukas Bichlerto serve as the university examiner. I want to thank Dr. Lalita Udpafrom Michigan State University for her willingness to serve as my externalexaminer. Thank you all for letting my defense be an enjoyable moment,and for your brilliant comments and suggestions.I would like to thank my colleagues at the Intelligent Sensing, Diagnosticsand Prognostics Research Lab (ISDPRL). Their encouragement helped meto go through the tough period during my Ph.D. journey. I really enjoy mystay with them in Canada.Last but not least, I want to present my greatest gratitude to my parentsand sister, for their moral encouragement, financial assistance as well astheir spiritual support. None of my dreams could be achieved without theirunconditional love and support.xxChapter 1Introduction1.1 Background and MotivationsMore than 90% of mechanical and structural components are subjected tocyclically fluctuating load [12]. The cyclic fluctuation of the load is mainlycaused by the change in external loads. For example, the engagement anddisengagement of the gear tooth will produce a cyclic load on the gear toothroot [13]. The pressure variation of gas or oil will induce the cyclic load onpipeline [14]. Wind gusts will produce cyclic load on aircraft [15]. Fig. 1.1shows a typical cyclic load which is the ground-air-ground (GAG) load his-tory used in the fatigue tests of aircraft structure [2]. It can be seen thatthe cyclic load fluctuates around the mean value. During the service of themechanical or structural components, the cyclic load may cause materialfatigue.Flight loadingMean-to-meanGAG cycleGroundmeanGroundloadingGroundloadingOnce per flightpeak-to-peakGAG cycleOne flightFigure 1.1: Typical cyclic load: ground-air-ground (GAG) load history usedin the fatigue tests of the aircraft structure [2]. Image reproduced withpermission by Elsevier.1Material fatigue is the weakening of a material caused by the cyclic loadthat results in progressive and localized structural damage and the growthof crack [16]. Material fatigue can be categorized as low cycle fatigue andhigh cycle fatigue. In the low cycle fatigue, the strain level is higher thanthe yield strain of the material, and the cumulative damage is considered tobe strain-controlled. In this case, the number of cycles to failure is less thanabout 104 cycles [12]. In high cycle fatigue, the stress levels are much belowthe yield stress of the material. The number of cycles to failure is greaterthan 104 cycles. This study takes high cycle fatigue as the main focus.Under the cyclic load, the fatigue crack may initiate and grow with eachloading cycle. During the service of the mechanical or structural compo-nents, the crack will continue to grow until it reaches a critical size, produc-ing rapid propagation and complete fracture of the structure. An illustrativeexample is shown in Fig. 1.2. Due to the cyclic load, a crack was initiatedfrom one single point at the shaft surface. There are semi-elliptical shapedbeach marks on the surface that gives the position of the crack front at differ-ent stages during the crack propagation. The final fracture of the propellershaft occurred when the crack reached a critical size.Figure 1.2: Fatigue damage of propeller shaft for illustration [3]. Imagereproduced with permission by Elsevier.In modern industry, the damage tolerance design approach is widelyadopted to guarantee the reliability of in-service components while reduc-ing maintenance costs. The damage tolerance design approach explicitlytakes the fatigue crack growth (FCG) process into account. In this designapproach, it is allowed that an in-service component has growing fatiguecracks, as long as the cracks will not cause the failure of the component ina catastrophic manner. In this context, the accurate prediction of the FCG2is highly needed.Besides, it is worth mentioning that the FCG behaviors in brittle ma-terials, such as glass and ceramics, are quite distinct from those in ductilematerials, such as aluminum and titanium [17]. In brittle material, thecrack growth rate displays a markedly higher sensitivity to the stress inten-sity factor in crack tips. Brittle cracking can lead to the complete fracture ofmaterial very rapidly once the critical load for crack growth is reached. Asa result, brittle fracture generally occurs in two stages: (i) initiation of thecrack and (ii) rapid propagation of the crack leading to a complete fracture.In contrast, the FCG in a ductile material is more predictable, as there is astable crack growth phase before the final fracture. Hence, ductile materialis preferred in the damage-tolerance design [18]. In this regard, the FCG ina ductile material is focused on in this thesis.Existing prediction methods can be roughly classified into data-drivenmethods [19–21] and model-based methods [22–24]. This study focuses onthe model-based method which describes the crack growth using a specificmathematical model. This mathematical model is generally derived basedon the physics of FCG. In the model, the growth of a crack is related to var-ious physical variables including load condition, structural geometry, cracklength, material parameters, etc. Based on the experimental study, Parisand Erdogan [25] showed that the cyclic crack growth under fatigue loadingcan be characterized in terms of cyclic SIF range. This paved the way forthe estimation of the cyclic life of components. Subsequently, various SIFapproaches were proposed for the FCG prediction [6, 26, 27].FCG is a typical stochastic process and is affected by various uncertaintyfactors. For example, the external load may be uncertain. The geometry ofthe component may be with error due to the manufacturing process. Thematerial property of each individual component is also different to somedegree. In this context, a probabilistic model was developed for the FCGprediction considering the stochastic property and randomness in the FCGprocess [28].This study is motivated to develop a set of probabilistic models for theFCG prediction considering a variety of uncertainty factors. In particular,different Bayesian inference approaches are developed to reduce these uncer-tainty factors based on real-world crack growth observation. To be specific,in Chapter 2, a conjugate Bayesian analysis approach is developed to reducethe uncertainty of material parameters in the FCG model. In Chapter 3,Bayesian inference is extended for the FCG prediction with hybrid prior.In Chapter 4, a probabilistic model is developed for the FCG predictionconsidering crack location randomness.31.2 Related Work1.2.1 Principle of Fatigue Crack GrowthA crack is a discontinuity, internal or external, in the material with zero tipradius. The discontinuity in the geometry introduced by the crack gives riseto a singularity of the stress and strain fields near the crack tip. Based ondifferent loading conditions that a crack can experience, Irwin [29] identifiedthree types of stress intensity factor (SIF) as shown in Fig. 1.3. Model I,KI , is the opening model which tends to open the crack. Model II, KII , isthe sliding model and refers to in-plane shear loading. Model III, KIII , isthe tearing model and corresponds to out-of-plane shear.FFFF FFFigure 1.3: Three types of SIF on a cracked body [4]. Image reproducedwith permission by CRC Press.For the crack under model I loading condition, Paris and Erdogan [25]did a wide range of experiments considering specimens with cracks undercyclic load and obtained the typical fatigue rate curve. A typical fatiguerate curve is shown in Fig. 1.4. Specifically, a is the crack length, n is theload cycles that are imposed on the structure, and ∆K is the model I SIFrange. In Fig. 1.4, there are three regions where the fatigue crack showsdifferent growth behaviors. ∆Kth is the threshold SIF value below whichthere is no FCG or the growth is too small to be measured. Kc is thefracture toughness. Fracture toughness is the critical SIF of a sharp crackwhere propagation of the crack suddenly becomes rapid and unlimited.When multi-model SIFs are involved in crack growth, both crack growthspeed and crack growth orientation are determined by the combination ofeach component. Specifically, the crack growth rate can be calculated usingequivalent SIF Keq which is determined by integrating model I, model IIand model III SIFs. For example, Tanaka [30] proposed a model for cal-4culating the range of equivalent SIF for the FCG prediction under modelI and model II loading conditions. Their model was derived from the in-finitesimal dislocation theory [31], and was validated with a set of aluminumsheet specimens with inclined cracks. Biner [32] proposed a model for theequivalent SIF calculation, based on the sum of the individual strain energyrelease rates [29] under model I and model II loading condition and em-ployed the model to study the FCG rates in steel. Liu and Mahadevan [33]proposed a universal model for the equivalent SIF calculation under modelI, model II and model III loading conditions. Their model was derivedfrom the critical plane approach and was validated with the FCG data ofa wide range of metals. Similarly, the crack propagation angle can also becalculated considering the component of each model based on the specificcriterion. Commonly-used criteria for crack propagation angle calculationinclude minimum strain energy density [34], maximum strain energy releaserate [35], and maximum tangential stress [36]). So far, various models andcriteria have been developed for FCG prediction under multi-model SIFs.However, there is no single criterion which gives satisfactory predictionsunder all conditions. Experimental results which do not agree with thepredictions can often be found in the literature [37].log △Klog da/dn mm/cycle△Kth△KcRegion I Region IIRegion IIIFigure 1.4: Typical fatigue rate curve [5]. Image reproduced with permissionby Cambridge University Press.51.2.2 Stress Intensity Factor CalculationSIF is a critical parameter which describes the stress states near the crackfront fields. To determine the crack growth rate and crack growth direction,the SIF of the crack tip should be calculated.In general, the SIF can be obtained using either analytical or numericaltechniques [38]. Specifically, analytical techniques (such as weight func-tion method [39], Green’s function method [40], and integral transformmethod [41]) are usually adopted when the structural component is withtypical geometry and load condition. For example, Punit et al. [42] uti-lized analytical techniques in the remaining useful life prediction of thecracked pipes under bending load. Zongchen et al. [43] proposed an an-alytical method to evaluate the SIF of circumferential surface cracks in steelpipe for the FCG prediction. Numerical techniques such as the finite elementmethod (FEM), boundary element method (BEM), and extended finite el-ement method (XFEM), have been very widely applied for calculating theSIF in practical engineering. It is because of its applicability to a widerange of problems, irrespective of the complexity of loading and geometryof components. As a result, most existing studies [2, 13, 44, 45] rely onnumerical techniques in the calculation of SIF. For example, Aleksandar etal. [2] adopted the XFEM for the FCG prediction in light aircraft wing spars.Bowen et al. [46] used the FEM in the curved crack growth prediction undervariable amplitude load. Fuqiong et al. [1] utilized the FEM in the crackgrowth prediction in spur gear tooth.In particular, the finite element (FE) method is the most widely used nu-merical technique in the calculation of SIF. In the FE analysis, the structuralcomponent is discretized into a number of elements of finite dimensions [47].The elements are interconnected at their nodal points. Field variables (suchas displacements in the case of displacement FE formulation) are associatedwith each of these nodes. Usually, a large number of element choices areavailable for discretization. However, the conventional elements (such asthe constant strain triangles, linear strain triangles, 4-node quadrilaterals,8-node quadrilaterals) have difficulty in simulating the stress singularity,which motivates the invention of the special element called the “singular”element [5].Fig. 1.5 shows the stress field near the crack tip in 2D problem [48]where σθ,θ and σr,r are normal stresses and σr,θ denotes shear stress. Theapproximate calculation of σθ,θ, σr,r and σr,θ are given by,6σθ,θ =1√2πrcosθ2[KIcos2 θ2− 32KIIsinθ], (1.1)σrr =1√2πrcosθ2[KI(1 + sin2 θ2) +32KIIsinθ − 2KIItanθ2], (1.2)σr,θ =12√2πrcosθ2[KIsinθ +KII(3cosθ − 1)]. (1.3)From the above equations, it can be seen a singularity of r−1/2. It meansthe stress in the crack tip will be infinite. The conventional element typewith a polynomial shape function has difficulty in simulating such singu-larity. Thus, a special element called the “singular” element is invented tocreate the r−1/2 singularity. Fig. 1.6 (a) shows a typical 8-node singularelement for illustration. In a singular element, the element node is removedto the quarter of the element edge, when the point approaches the crack tipas shown in Fig. 1.6 (b). This kind of singular element is able to reproducethe r−1/2 singularity near the crack tip in FE formulation [49].Several methods are available for calculating or estimating the SIF withthe FEM, such as the J-integral and its variants (e.g. M-integral) [50], thestiffness derivative methods [51], and the displacements-based methods [52].Among them, the J-integral and its variants, which measure energy flux intothe crack-tip region, are widely used due to their high accuracy [50].1.2.3 Uncertainty AnalysisThe process of FCG prediction is affected by many uncertainty factors.These uncertainty factors appear at the different stages of the analysis, andmay combine in nonlinear, nested or iterative manner. It is necessary to ana-lyze these uncertainty factors to make a reliable prediction. The uncertaintyanalysis generally comprises uncertainty quantification, uncertainty propa-gation and uncertainty calibration. Specifically, uncertainty quantificationaims to quantify the uncertainty factor using a probabilistic distribution or anon-probabilistic convex model. Uncertainty propagation studies the effectof model inputs’ uncertainties on the model output. Uncertainty calibrationattempts to update variables’ uncertainties based on real-world observation.The detailed descriptions are given as follows.7xyrθσθθσrrσrθσrθFigure 1.5: Two-dimensional stress field near the crack tip [5]. Image repro-duced with permission by Cambridge University Press.Uncertainty quantificationThe uncertainty quantification can be conducted based on either probabilis-tic analysis or non-probabilistic analysis.In the probabilistic analysis, specific probabilistic distributions (such asGaussian distribution, lognormal distribution, exponential distribution, andGamma distribution) are assigned to uncertainty factors [53]. The distribu-tion parameters of these distributions are evaluated using maximum likeli-hood estimation or Bayesian inference.The non-probabilistic analysis was developed in the early 1990s [54], andhas been applied in various applications for reliability analysis [55, 56]. Thenon-probabilistic analysis uses a convex set to quantify the uncertainty do-main, which is very effective for structural uncertainty analysis with limitedor poor-quality experimental data [57]. Compared to the precise probabilitydistribution, the variation bounds are much easier to be obtained, since itgenerally only needs a small number of experimental samples or just theexperience of engineers. It should be pointed out that the non-probabilisticconvex model is not a single model. It represents a class of models suchas interval model, parallelepiped model, ellipsoid model, and hybrid model.For example, Lei wang et al. [56] utilized an interval model for describingthe uncertainty of material parameters and the critical crack length for FCGprediction. In their work, the uncertainty domain of variables was described8Crack tipll/4l/2hh/4h/2Crack tip(a) (b)Figure 1.6: (a) Typical 8-node singular element. (b) 8-node singular ele-ments around the crack tip. © Teng Wangby the shape and orientation of a multidimensional interval. Xu Bai etal. [58] used the ellipsoid model to consider a set of uncertainty factors inthe FCG prediction of a new titanium alloy used in the deep-sea mannedcabin. In their work, a minimum volume ellipsoid was constructed to encloseall the uncertainty sample data to describe the uncertainty domain.It is worth mentioning that, in the non-probabilistic model, the uncer-tainty factors are regarded as bounded variables, which brings difficultiesin uncertainty calibration using Bayesian inference approaches. Hence, thisthesis mainly focuses on the probabilistic model, which can not only quan-tify the uncertainty but also calibrate the uncertainty given the real-worldcrack growth observation.Uncertainty propagationThe objective of uncertainty propagation is to evaluate the uncertainty in themodel output given the uncertainty in the model inputs. Commonly used un-certainty propagation methods include the first-order reliability method [56],second-order reliability method [59], Monte Carlo simulation [60], etc.Among them, the first-order reliability method and second-order reliabil-ity method utilize the first-order and second-order Taylor series to approx-imate the model output. Even though these methods are computationallyefficient, they require the explicit expression between the model inputs andthe output, which is usually unavailable.In comparison, Monte Carlo simulation treats the model as a black boxand can propagate the uncertainty without an explicit expression of the9model inputs and output. In Monte Carlo simulation, the model output isapproximated using a set of Monte Carlo samples. Theoretically, a good ap-proximation accuracy can be achieved with sufficient Monte Carlo samples.Nevertheless, the Monte Carlo simulation requires a high computational ex-pense in the model output approximation.Uncertainty calibrationUncertainty calibration attempts to update variables’ uncertainties basedon real-world observation. Bayesian inference is commonly used to achievethis end. A general expression of Bayesian inference is given as,p(θ|D) = p(D|θ)π(θ)∫Θ p(D|θ)π(θ)dθ∝ p(D|θ)π(θ), (1.4)where θ = {θ1, ..., θj} is a set of unknown parameters in the FCG model. Dis the real-world observation. p(θ|D) is the posterior distribution of θ giventhe observation D and prior distribution π(θ). The posterior distributioncan be derived using either the analytical method or the numerical method.To derive the posterior distribution analytically, it is required that thelikelihood functions and prior distributions should be conjugated [61]. Hence,a closed-form solution of the posterior distribution can be obtained basedon conjugate Bayesian analysis [62, 63].In the case that the likelihood functions and prior distributions are notconjugate, there is no closed-form solution for the posterior distribution.Thus, Monte Carlo [64] techniques are usually adopted to approximate theposterior distribution using a set of samples. Commonly-used MC tech-niques include Metropolis-Hastings sampler [65], particle filter (PF) [66],Hamiltonian Monte Carlo (HMC) [67], no-U-turn sampler (NUTS) [67], etc.1.2.4 Fatigue Crack Growth Prediction ModelsThe FCG can be predicted by building a mathematical equation in terms ofstress, material properties, geometry, crack length, etc. Existing predictionmodels can be classified into deterministic models and probabilistic models.In the deterministic model, the related factors in the equation are treated asdeterministic and should be determined from the handbook, FE analysis orexperimental test. In comparison, the probabilistic model considers specificuncertainty factors in the equation and infers these factors based on the real-world crack growth observation. A detailed description of the deterministicand probabilistic model is given in the following.10Deterministic Model for FCG PredictionAs to the deterministic model, a pioneer work is conducted by Paris P.C. and F. Erdogan [25]. They showed that the cyclic crack growth rateunder fatigue loading can be characterized in terms of cyclic SIF range∆K = Kmax−Kmin, whereKmax andKmin are the maximum and minimumSIF in a load cycle respectively. In Paris-Erdogan equation, the crack growthrate dadn is related to the SIF range ∆K by linear data fitting. The Paris-Erdogan equation is applicable in the mid-region (region II) of the fatiguerate curve. In this region, the relation between dadn and SIF range ∆K follows,dadn= Cp(∆K)mp (1.5)where Cp and mp are related to the material property of the structuralcomponent. The value of Cp and mp can be acquired from the materialexperiment. Specifically, one can obtain the crack growth paths of the samebatch of specimens. The da/dn versus ∆K can be calculated subsequently.By fitting the log-log plot of da/dn versus ∆K, one can obtain the value ofCp and mp.For different applications, many variants of Paris-Erdogan equation weredeveloped to consider more factors in the FCG. For example, Walker equa-tion [26] was developed to consider the factor of stress ratio asdadn= Cw(∆K(1−Rr)1−γw )mw (1.6)where Rr is the stress ratio. Apart from Cw and mw, γw is the third curvefitting parameter. The value of Cw, mw and γw can be acquired from theexperiment that is similar to the experiment depicted in the Paris-Erdoganequation. Specifically, the value of γw is the one that best consolidates thedata along a single straight line on the log-log plot of da/dn versus ∆K(1−R)1−γw .If the value of γw is equal to 1, the stress ratio has no effect on the fatiguecrack growth.Since the point of transition from region II to region III depends on thefracture toughness of the material, Forman et al. [68] proposed another formof the equation which considers the instability of the crack growth when theSIF approaches its critical value as,dadn=Cf (∆K)mf(1−Rr)Kc −∆K (1.7)11where Rr is the stress ratio, Kc is the fracture toughness as shown in Fig. 1.4.Cf and mf are fitting parameters. Kc can be acquired from the experi-ment. Commonly-used testing method for acquiring Kc includes single edgenotched beam (SENB) test and indentation microfracture (IM) test. Thefitting parameters, Cf and mf can be obtained by fitting the fatigue crackpaths of the same batch of specimens, which is similar to the experimentdepicted in the Paris-Erdogan equation.Wheeler equation [6] considers any retardation effects by including a re-tardation parameter. Thus, the retardation of crack growth due to overloadscan be modeled. Wheeler equation can be applied to any FCG equation ofthe form da/dn = f(∆K). It considers any retardation effects by includinga retardation parameter ϕR which is calculated by,ϕR =[riaOL + rOL − ai ]λ for ai + ri < aOL + rOL1 for ai + ri ≥ aOL + rOL(1.8)where aOL is the crack length at which the overload is applied, ai is the cur-rent crack length, rOL is the size of the plastic zone produced by the overloadat aOL, and ri is the size of the plastic zone produced at the current cracklength ai, and λ is the curve fitting parameter. Fig. 1.7 shows an illustrationof the crack tip yield zones. The precondition of using the Wheeler equa-tion is knowing the size of the plastic zone in the crack tip, which can becalculated using either empirical equation [69] or FE analysis [46].Figure 1.7: Crack tip yield zones [6]. Image reproduced with permission bySPIE.Later Elber [70] introduced the concept of crack closure and suggestedthat crack growth rate is a function of ∆Keff where ∆Keff = Kmax−Kopen12and Kopen is opening SIF. Hence, the crack growth rate in terms of theeffective SIF range ∆Keff can be calculated as,dadn= Cp(∆Keff )mp = Cp(U∆K)mp (1.9)where Cp andmp are Paris-Erdogan equation constants and U = Kmax−KopenKmax−Kmin .The U can be determined experimentally by fitting the ∆Keff∆K with stressratio [71].Nasgro equation [72] takes into account the effect of a mean stress, thefracture toughness of the material and the threshold value of the SIF range,and the contribution of crack closure. The Nasgro equation is as follows,dadn= Cn(1− f1−Rr )mn(1− ∆Kth∆K )q(1− ∆Kmax∆Kc )p(1.10)where Cn, mn, q and p are the material parameters obtained by experi-ments, Kc is the critical SIF, and f =KopenKmaxrepresents the contribution ofcrack closure. The value of Kc and f can be determined experimentally asaforementioned.Apart from the models above, there are several other deterministic mod-els developed to date, such as Dover [73] and Suliva [74]. However, nosingle model is applicable to all materials nor considers the influence of allparameters at a time [75].Probabilistic Model for FCG PredictionIn the probabilistic model, the FCG prediction is conducted consideringvarious uncertainty factors. The uncertainty source in the FCG predic-tion can be basically divided into natural uncertainty, data uncertainty andmodel uncertainty [76]. Specifically, natural uncertainty refers to the naturalvariability or fluctuations in the environment, test procedure, instruments,observer, etc. Data uncertainty refers to the uncertainty induced by fieldinspections such as crack detection uncertainty, and crack measurement un-certainty. Model uncertainty is induced from the models which are used inthe FCG prediction. These models include the FE model, surrogate model,crack growth model, etc.In the past decades, many studies were conducted to evaluate these un-certainty factors and integrate them into the FCG prediction. As a pioneerwork, Virkler et al. [7] found fatigue cracks experimental data display a largescatter in the crack size at a given cycle, as well as the number of cycles13to reach a given crack size. This large scatter occurs even under a care-fully controlled environment. Thus, two material parameters C and m inthe Paris-Erdogan equation are usually treated as the uncertain variables inthe FCG prediction. For instance, Zhao et al. [1] developed an integratedmethod for the FCG prediction considering a time-varying operating con-dition of spur gear, where a polynomial chaos expansion (PCE) collocationmethod is used to update the posterior distribution of logC and m. YumeiYe et al. [45] conducted a numerical case study for the FCG prediction of aload-bearing frame, where the material parameters are modeled as randomvariables. D. Cristiani [77] conducted the FCG prediction for an aeronauti-cal structure, in which the posterior distribution of the material parametersis updated based on surrogate modeling and particle filter (PF).Due to the complex operating environment, the external load conditionof the structure components can also be uncertain. You Ling et al. [78]proposed an FCG prediction mode which integrates an autoregressive in-tegrated moving average (ARIMA) model to characterize the uncertaintyin the external load condition. In their method, future load condition ispredicted from the past load observation. Then, the predicted load condi-tion and the prediction uncertainty are fed into a modified Paris-Erdoganequation for the FCG prediction.The concept of an equivalent initial flaw size (EIFS) was proposed in [79]to bypass short crack growth analysis and make direct use of a long crackgrowth law for total fatigue life prediction (including both the short crackgrowth and long crack growth regime). To consider the uncertainty in EIFS,You Ling [78] treated EIFS as a normal-distributed random variable, andupdated it based on the Bayesian inference.By noticing the irregularity in an individual test, the authors in [80]claimed high-frequency randomness in the crack propagation. Many stud-ies [1, 78] assign an uncertainty variable to take account of this high-frequencyrandomness. Usually, this uncertainty variable is assumed with a lognormaldistribution, where the distribution parameters can be obtained by fittingthe residual error between the crack growth model and experimental data [1].Due to the manufacturing error, the geometry of mechanical or struc-tural components may be uncertain. Minor changes in geometric featuresmay lead to a significant alteration in stress distributions, which means apotential influence on the FCG. Chenzhao Li et al. [81] proposed an FCGprediction method for aircraft wings where the thickness of the leading edgeis considered as an uncertain variable. Dianyin Hu et al. [82] consideredthe geometry uncertainty in the remaining useful life prediction of turbinedisc where 15 random geometric features were integrated into the prediction14model.FE analysis is usually adopted in the calculation of SIF. The discretiza-tion error of the FE analysis will also induce uncertainty in crack growthprediction. Theoretically, an infinitesimally small mesh size will lead to theexact solution but this is difficult to implement in practice. I. Babusˇka etal. [83] developed an uncertainty estimation method for FE analysis in anasymptotic form for h −→ 0 where h measures the size of the elements.This method requires the FE model to be convergent and the domain to beuniformly meshed [84].For the health monitoring of structural components, non-destructive in-spection (NDI) is commonly adopted for damage detection and measure-ment. The inspection error from the NDI tool will also bring uncertainty tothe FCG prediction. The NDI uncertainty can be characterized by severalmetrics such as the probability of detection (POD), flaw size measurementaccuracy and false call probability [85]. For example, the flaw size measure-ment accuracy is usually modeled using regression analysis [76]. The PODcurve for a particular NDI method can be obtained by introducing flaws ofvarious sizes into test specimens and calculating the ratio of the number offlaws detected to the number of flaws existing [86]. False call probabilitycan be obtained by performing NDI on unflawed components and calculat-ing the ratio of the number of flaws detected to the number of componentsinspected [85].Even though many sources of uncertainty factors existed in the FCGprediction model, it is very difficult or impossible to model all of them dur-ing the prediction, due to the prohibitive computational cost induced by thehigh-dimensional calculation. Fortunately, not every uncertainty factor willsignificantly affect the prediction result. In other words, some uncertaintyfactors have a negligible effect on the result. Thus, sensitivity analysis isusually adopted to select the most significant uncertainty factors for model-ing. Commonly-used sensitivity analysis methods can be classified as localsensitivity analysis methods and global sensitivity analysis methods. In par-ticular, the local sensitivity analysis method focuses on the local impact offactors on the model and is considered as a particular case of the one-factor-at-a-time approach, because all other factors are held constant when one isvaried. Global sensitivity analysis methods vary all model inputs simulta-neously, and the parametric sensitivities are calculated over the entire rangeof each model input. For example, Chenzhao Li et al. [81] conducted globalsensitivity analysis by Sobol indices [87] to assess the importance of 15 un-certainty factors in the FCG prediction of aircraft wings. Sankararamanet al. [76, 88] comprehensively reviewed various uncertainty factors in the15FCG prediction, and utilized first-order effect index, a local sensitivity anal-ysis method, and total effects index, a global sensitivity analysis method, toassess their contributions to the prediction result.Once the most significant uncertainty factors are identified, Bayesian in-ference can be employed to update the posterior distribution of these factorsbased on prior knowledge and real-world observation.It is worth mentioning that apart from modeling each individual un-certainty factor in the physics-based model, some researchers attemptedto describe the FCG behavior as a stochastic process, such as Wiener pro-cess [89, 90] and Gamma process [91, 92]. In the Wiener process, the stochas-tic dynamics of the degradation process are characterized by the Brownianmotion (BM) [89]. The conventional Wiener process model comprises theinitial degradation state of the concerned system, a drift coefficient cap-turing the rate of degradation, and a diffusion coefficient multiplied by astandard BM. To capture the stochastic property of the FCG, Naipeng Liet al. [90] proposed a Wiener-process-model-based method for crack growthprediction considering unit-to-unit variability. The Gamma process is acontinuous-time stochastic process described by shape function and scaleparameter. By assuming that the first time for a specimen to reach a cracklength a is a Gamma process, Murizio et al. [92] presented a Gamma processmodel for the analysis of crack growth data. Since the stochastic processmodels do not consider the physics of FCG, they are usually used whenthe physical information (such as load condition, structural geometry, andmaterial parameters) is unknown.1.3 Research Gaps and ObjectivesEven though various FCG prediction methods have been proposed in theliterature, there still exist some research gaps.First of all, in existing methods, the inference of material parametersmainly relies on Monte Carlo algorithms, such as Markov chain MonteCarlo (MCMC) [65] and PF [93]. As in every update at least thousandsof stochastic samples should be calculated according to the Monte Carlomechanism, Monte Carlo-based methods are computationally expensive.Moreover, due to the stochastic property of the sampling process, MonteCarlo-based method suffers from the convergence problem [66, 94], i.e., theMonte Carlo samples can not converge to the target distribution. At last,the implementation of the Monte Carlo algorithm requires selecting relevanthyper-parameters (such as the length of the Markov chain, the percentage16of the burning period, the Gelamn-Rubin coefficient, and particle number)manually, but there is no unified standard for the selection of these param-eters.Second, existing methods conduct the FCG prediction only based onsingle prior input [23, 75, 92]. However, in practical engineering, there isusually multi-source information that can be used for prior distribution set-tings. For instance, the stress on a specific structure can be estimated fromeither operating parameters [95] or FE software [13]. The material test prop-erty of an in-service structure can be acquired from either the specimen-levelmaterial test [96] or the field data from other similar structures [97]. Thus,how to conduct prediction with multi-prior knowledge should be furtherexplored.Third, existing methods for the FCG prediction are generally conductedwith an initial flaw with the assumption that the crack location is known.However, in real-world applications, the crack location is also random tosome degree due to the impact of microstructure flaws, residual stress, ge-ometry variation, and so on. Hence, the crack location randomness shouldbe considered in the FCG prediction.These research gaps plague the implementation of FCG prediction ingeneral practical engineering, To address the research gaps identified above,the detailed research objectives of this thesis are as follows.• This research aims to propose a closed-form solution for the FCGprediction considering the randomness of material parameters. Thisclosed-form solution enables the posterior distribution of the materialparameter logC and m in the Paris-Erdogan equation to be calculateddirectly. Thus, future crack growth behavior can be predicted.• This research aims to propose a probabilistic FCG prediction model,which enables the input of hybrid prior. As a result, in a real-worldscenario, any promising prior knowledge can be integrated, increasingthe robustness of the prediction.• This research aims to present a probabilistic FCG prediction model,considering the crack location randomness. This probabilistic modelis updated based on the crack tip findings from the non-destructiveinspection data and can be used for the prediction of the crack withdifferent locations.171.4 Thesis OutlineThis thesis is organized into five chapters. The summary of each chapter isdescribed as follows:• Chapter 1 first gives the background information on material fatigueand the FCG prediction. In addition, this chapter presents a com-prehensive literature review on the related work, and identifies theresearch gaps, and brings up the objectives of this research.• Chapter 2 presents the probabilistic model for the FCG predictionusing a closed-form solution. First, the FCG modeling is presentedunder the framework of linear elastic fracture mechanics. Then, thederivation of the closed-form solution is illustrated, followed by theFCG prediction using the closed-form solution. In Section 2.2.2, thedeveloped model is validated on the experimental FCG data of a setof aluminum alloy specimens in Virkler’s dataset. Finally, Section 2.4summarizes this chapter.• Chapter 3 shows a new probabilistic prediction method which enablesthe input of hybrid prior. The FCG modeling and model calibrationwith NDI data are described first. Then, the Bayesian inference for theposterior belief of prior distribution (PBPD) and Bayesian inferencewith hybrid prior are illustrated. Section 3.3 presents the numericalstudies on the spur gear. Section 2.4 summarizes this chapter.• In Chapter 4, an FCG prediction model is proposed considering thecrack location randomness. A real-world application of the pipelineFCG prediction is specified. The concrete methodology is elaboratedin Section 4.3, followed by the discussion in Section 4.4. Finally, thesummary is presented in Section 4.5.18Chapter 2Probabilistic Model forFatigue Crack GrowthPrediction Based onClosed-Form Solution2.1 OverviewThis chapter puts forward a probabilistic model for the FCG predictionbased on closed-form solution 1. The theoretical basis of the proposed modelis conjugate Bayesian analysis [62, 63]. Conjugate Bayesian analysis focuseson the pairs of likelihood functions and prior distributions with tractableanalytical solutions. These families of distributions are known as conju-gate distributions which were first discussed and formalized by Raiffa andSchlaifer [61]. Conjugate Bayesian analysis uses conjugate distributions inBayesian inference. This strategy allows the Bayesian inference to be de-rived in a closed form. Even though closed-form solution is hard to bederived, and often with complicated formulas, it has two main merits overMonte Carlo-based methods: (1) closed-form solution is easy to be codedin a computer, and the running of closed-form solution is very efficient. (2)closed-form solution provides a deterministic result. No hyper-parametersshould be set or tuned to achieve a satisfying inference result. In this regard,a closed-form solution for the probabilistic FCG prediction is attractive.In the proposed method, on the basis of FE analysis, the SIF is related tothe crack length and operating loading condition. The material parametersand model error are regarded as random variables. Then, the crack growthobservation is used to update the posterior distribution of material param-1The closed-form solution refers to any formula that can be evaluated in a finite numberof elementary operations (e.g., + - × ÷), and functions (e.g., n-th root, exponent, loga-rithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, orintegral [98].19eters, characterizing the unit-to-unit variability of crack growth behaviorin individual structures. Based on the inference result, a modified form ofthe Paris-Erdogan equation is used to predict the crack growth behavior ina probabilistic view where the prediction confidence interval is considered.Finally, the crack growth life can be calculated based on a specific failurethreshold. The main contribution of this chapter is proposing a closed-formsolution for updating the well-known Paris-Erdogan equation and predictingcrack propagation more efficiently and accurately. The theoretical interpre-tation and comprehensive investigation of the proposed model are presented,demonstrating its superiority.In the rest of this chapter. Section 2.2 presents the developed probabilitymodel, followed by the experiment validations and discussions in Section 2.3.2.2 Methodology2.2.1 Fatigue Crack Growth Modeling and Closed-FormSolution DerivationGiven a set of input parameters, the crack growth model describes the crackgrowth in the critical area of the structure. The crack growth life of thestructure is defined when the crack length reaches a threshold size. Thisthreshold size can be user-specified or it can be taken to correspond to themaximum crack size that the component can withstand at the maximumstress in the load profile [5]. Most of the existing crack growth models arebuilt on the Paris-Erdogan equation as,dadn= C(∆K(a, L))m (2.1)where a is the crack length, and n is the number of load cycles that the struc-ture experienced. Parameters C and m are material-dependent constants.Many studies [1, 80] show that for a batch of structures with the same ge-ometry and material, even under the same load condition, their degradationbehaviors display a large difference. This difference exists even under acarefully controlled environment, demonstrating the stochastic property ofC and m in nature. In other words, for each individual structure, its Cand m are unique. Thus, C and m are treated as random variables in thischapter. The Bayesian inference is adopted to infer the C and m for the in-dividual structure given its degradation observations. Thus, the unit-to-unitvariability of the FCG can be characterized.20∆K is the so-called SIF range, which is a function of stress level (∆σ),geometry factor (Y ), and crack size (a), i.e., ∆K = ∆σY√πa. It describesthe stress range in the region near the crack tip. Given a structure witha specific geometry and crack growth path, ∆K is related to fatigue cracksize a and load condition L, namely ∆K(a, L). In real-world applications,the calculation of ∆K(a, L) generally relies on FE analysis. Under theframework of linear elastic mechanics, ∆K(a, L) has a linear relation withthe L. Thus, for the crack growth prediction of a specific structure, therelation between ∆K, a, and L can be obtained by running the FE modelwith a baseline load condition Lbase only. The remaining work is merelyto multiply the ∆K(a, Lbase) with a ratio L/Lbase. Thus, ∆K(a, L) withrespect to different crack sizes a and different load conditions L [1] can beobtained. Note that the running of the FE model is not required during theFCG prediction. The response surface ∆K(a, L) is calculated in advanceusing the FE analysis. As a result, the ∆K with specific a and L can beacquired in real-time during the FCG prediction.Crack length (mm)da/dn (mm/cycle)Predicted da/dnActual da/dnCrack length (mm)da/dN (mm/cycle)Predicted da/dNActual da/dNCrack length (mm)da/dN (mm/cycle)Predicted da/dNActual da/dNFigure 2.1: A comparison between the actual da/dn and the da/dn predictedusing Paris-Erdogan equation (FCG data from the Virkler’s dataset [7]) [8].Image reproduced with permission by Elsevier.Moreover, many studies [80, 99] found that there is a high-frequency ran-domness between pre icted da/dn and actual da/dn as the example shownin Fig. 2.1. Thus, a model error ϵ is added to the original Paris-Erdoganequation to account for this high-frequency randomness. By considering themodel error ϵ, the Paris-Erdogan equation is modified as [99],dadn= C(∆K(a, L))mϵ (2.2)where ϵ is assumed as a lognormal random variable as logϵ ∼ N (0, σ2) [80].The distribution of ϵ can be determined by the least-square regression in thelog-log scale of the Paris-Erdogan equation based on the training data.21Taking the natural logarithm for (2.2), one obtains,logRi = logC +mlog∆Ki + logϵ (2.3)where i is the time point index, Ri and ∆Ki are shorthands ofdadn at cracklength ai, and ∆K(ai, Li) at time i, respectively.Then, given the crack length observation a1, ..., ak+1, the crack growthrates R1, R2, ..., Rk can be calculated using the secant method [100]. Sincethe logarithmic form of model error logϵ follows N (0, σ2), the likelihoodfunction of logC and m given R1, R2, ..., Rk can be formulated as,L(logC,m|R1, R2, ..., Rk) =(1√2πσ)k · exp{−k∑i=1(logRi − logC −mlog∆Ki)22σ2}. (2.4)In the conjugate Bayesian analysis, the proportional relationship is con-sidered to simplify the derivation and makes the analytical solution tractable [63].At a specific inspection time index, k and σ are both constants. Given acertain observation R1, R2, ..., Rk, the likelihood of (logC,m) is proportionalto the last factor in (2.4) as [63],L(logC,m|R1, R2, ..., Rk) ∝exp{−k∑i=1(logRi − logC −mlog∆Ki)22σ2} (2.5)where ∝ denotes proportional.In this chapter, for the initial prior distribution setting, it is assumed thatlogC and m follow Gaussian distributions as logC ∼ N (µ1, σ21) and m ∼N (µ2, σ22). The reason for this prior distribution choice is two-fold: (1) Manyexisting studies [1, 45, 80, 99, 101, 102] validate the Gaussian distributionis a good prior for the material parameters logC and m. (2) Gaussiandistribution belongs to the family of conjugate prior [103], which enablesthe derivation of the closed-form solution. Moreover, in this chapter, it isassumed that the logC andm are independent in the initial prior distributionsetting. This independence assumption can simplify the derivation of theclosed-form solution in the following. However, it is worth mentioning thatsome studies [1, 101, 102] assume (logC,m) follows the bivariate Gaussianprior distribution, considering the correlation between logC and m. Thisissue will be discussed in Discussion 2.3.3.22Since it is assumed that logC ∼ N (µ1, σ21) and m ∼ N (µ2, σ22), theirjoint density function can be written as,p(logC,m) = p(logC)p(m)=12πσ1σ2exp{−(logC − µ1)22σ21} · exp{−(m− µ2)22σ22}. (2.6)Similarly, once the initial prior distribution is specified, p(logC,m) is pro-portional to the last two factors in equation (2.6) as [63],p(logC,m) ∝exp{−(logC − µ1)22σ21} · exp{−(m− µ2)22σ22}. (2.7)According to Bayesian inference, the posterior distribution is propor-tional to the product of prior distribution and likelihood function as,p(logC,m|R1, R2, ..., Rk) ∝L(logC,m|R1, R2, ..., Rk)p(logC,m).(2.8)Meanwhile, based on conjugate Bayesian inference, the factor of the poste-rior distribution that is proportional to logC and m must be proportionalto the product of the factors in the prior and likelihood functions that arethemselves proportional to logC and m [63, 103]. Therefore, equation (2.5)and equation (2.7) can be substituted into equation (2.8) as,p(logC,m|R1, R2, ..., Rk) ∝exp{−k∑i=1(logRi − logC −mlog∆Ki)22σ2}×exp{−(logC − µ1)22σ21} × exp{−(m− µ2)22σ22}·(2.9)By completing the square in each factor in equation (2.9), one can obtain,23p(logC,m|R1, R2, ..., Rk) ∝exp{−12[σ21σ22σ2σ21σ22k(logC)2 +m2k∑i=1(log∆Ki)2− 2logCk∑i=1logRi − 2mk∑i=1logRilog∆Ki+ 2mlogCk∑i=1log∆Ki) +σ2σ21σ2σ22σ21((logC)2 − 2µ1logC)+σ2σ21σ2σ21σ22(m2 − 2µ2m)]}.(2.10)Note that from equation (2.9) to equation (2.10), the factors which donot comprise the parameters of interest, logC and m, are dropped. Thissimplification method holds in the proportional equation [63, 97, 103].Subsequently, the factors in equation (2.10) can be re-organized in termof logC and m as,p(logC,m|R1, R2, ..., Rk) ∝exp{−12[(logC)2(kσ21σ22 + σ2σ22σ¯2)+m2(k∑i=1(log∆Ki)2σ21σ22 + σ2σ21σ¯2)− 2logC(k∑i=1logRiσ21σ22 + µ1σ2σ22σ¯2)]}− 2m(k∑i=1logRi(log∆Ki)σ21σ22 + µ2σ2σ21σ¯2)+ 2mlogC(k∑i=1log∆Kiσ21σ22σ¯2)]}(2.11)where σ¯2 = σ2σ2logCσ2m.24On the other hand, according to the conjugate Bayesian analysis, if theGaussian distribution is selected as the prior and likelihood function, theposterior distribution is also a Gaussian distribution [103]. Thus, theoreti-cally, the posterior distribution of logC and m should be a two-dimensionalGaussian distribution. According to this, the posterior distribution of logCandm is a two-dimensional Gaussian distribution, asN (µlogC , µm, σ2logC , σ2m, ρ).Here, note that µlogC , σlogC , µm and σm differ from µ1, σ1, µ2 and σ2, dueto the input of observations from the inspected structure. Thus the prob-ability distribution of the posterior distribution of logC and m is a typicaltwo-dimensional Gaussian distribution as [53],p(logC,m|R1, R2, ..., Rk) =12πσlogCσm√1− ρ2×exp{− 12σ2logCσ2m(1− ρ2)[σ2m(logC − µlogC)2−2σlogCσmρ(logC − µlogC)(m− µm) + σ2logC(m− µm)2]}.(2.12)Similarly, by dropping factors which are independent to logC and m,equation (2.12) can be re-written in the proportional form as,p(logC,m|R1, R2, ..., Rk) ∝exp{− 12σ2logCσ2m(1− ρ2)[σ2m(logC − µlogC)2−2σlogCσmρ(logC − µlogC)(m− µm) + σ2logC(m− µm)2]}.(2.13)Then, equation (2.13) can be re-organized in terms of logC and m as,p(logC,m|R1, R2, ..., Rk) ∝exp{−12[(logC)2(1σ2logC(1− ρ2)) +m2(1σ2m(1− ρ2)− 2logC( µlogCσ2logC(1− ρ2)− µmρσlogCσm(1− ρ2))− 2m( µmσ2m(1− ρ2)− µlogCρσlogCσm(1− ρ2))+ 2mlogC(−ρσlogCσm(1− ρ2))]}.(2.14)25By matching coefficients in the right hand of equation (2.11) and equa-tion (2.14), equations (2.15)-(2.19) can be obtained as,kσ21σ22 + σ2σ22σ¯2=1σ2logC(1− ρ2), (2.15)k∑i=1(log∆Ki)2σ21σ22 + σ2σ21σ¯2=1σ2m(1− ρ2),(2.16)k∑i=1logRiσ21σ22 + µ1σ2σ22σ¯2=µlogCσ2logC(1− ρ2)− µmρσlogCσm(1− ρ2) ,(2.17)k∑i=1logRi(log∆Ki)σ21σ22 + µ2σ2σ21σ¯2=µmσ2m(1− ρ2)− µlogCρσlogCσm(1− ρ2) ,(2.18)k∑i=1log∆Kiσ21σ22σ¯2=−ρσlogCσm(1− ρ2) .(2.19)Substitute equation (2.15) and equation (2.19) into equation (2.17), andequation (2.16) and equation (2.19) into equation (2.18). Then, it can beobtained that two equations regarding µlogC and µm, from which the ex-plicit expression of µlogC and µm can be calculated. Subsequently, throughstep-by-step substitution, the explicit expression of all the other parametersin the posterior distribution can be obtained. Without going into details(readers can refer to reference [103] for more mathematical skills in conju-gate Bayesian derivation), the derivation result is given in equations (2.20).Readers can refer to reference [97] for a similar derivation result. It is worthmentioning that even though the equations (2.20) are complicated in termsof algebra, they are very easy to be coded on the computer and calculated inreal time. Thus, the outcome in equations (2.20) is significant, because it en-ables a real-time and deterministic Bayesian inference by solely five explicitexpressions. No hyperparameter is included in this inference process.26µlogC =A−BC, µm =D − EC, σ2logC =σ¯2σ22k∑i=1(log∆Ki)2σ22 + σ2F,σ2m =σ¯2σ21kσ21 + σ2F, ρ =−σ1σ2k∑i=1log∆Ki√kσ21 + σ2√σ22k∑i=1(log∆Ki)2 + σ2.(2.20)whereA = (k∑i=1σ21logRi + µ1σ2)[k∑i=1(log∆Ki)2σ22 + σ2],B = (k∑i=1σ21log∆Ki)[k∑i=1(logRi)(log∆Ki)σ22 + µ2σ2],C = (kσ21 + σ2)[k∑i=1(log∆Ki)2σ22 + σ2]− [k∑i=1(log∆Ki)σ22][k∑i=1(log∆Ki)σ21],D = (kσ21 + σ2)[k∑i=1(logRi)(log∆Ki)σ22 + µ2σ2],E = (k∑i=1σ22log∆Ki)(k∑i=1σ21logRi + µ1σ2),F = (kσ21 + σ2)[k∑i=1(log∆Ki)2σ22 + σ2]− (k∑i=1log∆Ki)2σ21σ22.So far, it can be concluded that in (2.3), (logC,m) follows N (µlogC , µm,σ2logC , σ2m, ρ) as posterior distribution. The following proposition is given,Proposition 1. If two random variables x and y are correlated Gaussianvariables as x ∼ N (µx, σ2x) and y ∼ N (µy, σ2y) with correlation coefficientρxy, then their linear combination v = a1x+ a2y is also a Gaussian variablewith the mean value: µv = a1µx + a2µy, and the variance: σ2v = a21σ2x +a22σ2y + 2Cov(x, y)a1a2 = a21σ2x + a22σ2y + 2ρxyσxσya1a2.Based on Proposition 1, it can be concluded that {logC +mlog∆Ki}follows Gaussian distribution with mean value {µlogC + µmlog∆Ki}, and27variance {σ2logC + σ2mlog∆K2i + 2ρσlogCσmlog∆Ki}. Moreover, recall thatlogϵ follows N (0, σ2) and is independent of logC and m. The followingproposition is given,Proposition 2. If two random variables x and y are independent Gaussianvariables as x ∼ N (µx, σ2x) and y ∼ N (µy, σ2y), then v = x + y is also aGaussian variable with the mean value: µv = µx + µy, and the variance:σ2v = σ2x + σ2y .Based on Proposition 2, it can be concluded thatlogRi =logC +mlog∆Ki + logϵ∼N (µ3, σ23)(2.21)whereµ3 = µlogC + µmlog∆Ki,σ23 = σ2logC + σ2mlog∆K2i + 2ρσlogCσmlog∆Ki + σ2.(2.22)Thus, Ri follows lognormal distribution [104] with probability density func-tion (PDF) as,f(Ri) =1σ3√2πRiexp{−(logRi − µ3)22σ23}, Ri > 0. (2.23)Note that, to calculate equation (2.20), the values of µ1, σ1, µ2, σ2, σ, whichrepresent the prior knowledge of the FCG in a batch of structures, shouldbe known. In other words, µ1, σ1, µ2, σ2, σ represent the population char-acteristic of FCG in a batch of structures. Given the training data whichrecords the load and crack growth history of other structures in the samebatch, the prior distribution parameters µ1, σ1, µ2, σ2, σ can be estimatedas follows,1. For each structure in the database, calculate the ∆Ki regarding cracklength ai and load condition Li via FE analysis.2. For each structure in the database, calculate the Ri regarding cracklength ai via the secant method [100].3. In equation (2.3), first ignore logϵ and fit the rest terms using linearregression. Thus, the value (logC,m) can be obtained from each struc-ture in the training data set. As aforementioned, logC and m followprior distribution N (µ1, σ1) and N (µ2, σ2) respectively. Thus, µ1, σ1,µ2, σ2 can be estimated from the training data.284. Besides, for each structure, a series of fitting residuals can be obtained.Note that logϵ is a zero-mean Gaussian variable N (0, σ2). To estimateσ, the fitting residuals from each structure are first collected. Obvi-ously, these residuals are the samples from N (0, σ2). Thus, σ2 is thevariance of these residuals.So far, FCG modeling has been introduced. The FCG prediction basedon the closed-form solution will be presented in the next section.2.2.2 Fatigue Crack Growth Prediction Using theClosed-Form SolutionSince R = dadn is in differential form, and equation (2.2) can not be integrated,the first-order Euler method [105] is needed for crack growth prediction. Letus denote the initial crack length as a0 and the incremental crack length as∆a. The experienced load cycles when crack length reaches aN = a0+N∆acan be calculated as,n(aN ) =N∑i=1∆aR˜i(2.24)where N is the number of incremental steps from crack length a0 to a0 +N∆a, R˜i is the estimated crack growth speed at crack length ai = a0 +i∆a. Hence, the crack growth life (CGL) can be predicted, given a pre-specified crack size threshold. In the proposed closed-form solution, forthe point-wise prediction, R˜i is chosen as the median value (MV) of thelognormal distribution, i.e., exp{µ3(i)}. This choice will be discussed inDiscussion 2.3.3.Besides, the 95% confidence interval (CI) of the prediction result is alsoan important reference in the prediction. Note that random variable nshown in equation (2.24) is not a standard distribution (Gaussian, Gamma,etc). Thus, the explicit cumulative distribution function of n can not bederived to calculate the CI. However, an approximate estimation is givenin this study. Given logRi ∼ N (µ3(i), σ3(i)), the 95% CI for logRi canbe calculated as [µ3(i) − 1.96σ3(i), µ3(i) + 1.96σ3(i)]. Thus, the confidenceinterval for Ri is [exp(µ3(i)− 1.96σ3(i)), exp(µ3(i) + 1.96σ3(i))]. Then, theCI for the prediction result is,29CI95% ≈[N∑i=1∆aexp{µ3(i)− 1.96σ3(i)} ,N∑i=1∆aexp{µ3(i) + 1.96σ3(i)} ].(2.25)Note that equation (2.25) is an approximation as it is impractical to calculatethe exact value of CI. To validate the CI result calculated using equation(2.25), another approximation method is designed based on the MCMCalgorithm for comparison. This method is used to generate a set of valuesof random variable n shown in equation (2.24) for calculating the CI. Thiscomparison result can intuitively show the difference between CI calculatedfrom the closed-form solution and the MCMC method. The design of thiskind of MCMC method is given as follows.In equation (2.3), it can be seen that, Ri is related to C, m, ∆Ki, ϵ,where ∆Ki is deterministic, C, m and ϵ are random variables. As ϵ followsindependent identically distributed (i.i.d.) lognormal distribution, samplingϵ is easy and independent from sampling C and m. Without going intodetails about sampling ϵ, let us focus on the sampling for C and m. InSection 2.2.1, It has been derived that (logC,m) follows binormal distribu-tion N (µlogC , µm, σ2logC , σ2m, ρ) as given in equations (2.12) and (2.20). As aresult, sampling C and m can become very effective using a simple MCMCsampler, Gibbs sampler. Note that using the Gibbs sampler for samplingC and m is also based on the closed-form solution given in equation (2.12)and equations (2.20). Knowing the explicit distribution function improvesthe sampling efficiency dramatically [64] (this improvement is validated inSection 2.3.2).Gibbs sampler sweeps through each variable to sample from its condi-tional distribution with the remaining variables fixed to their current values.For the case in this chapter, given the initial sample (C(0),m(0)), Gibbs sam-pler samples (C,m) as follows,logC(1) ∼ p(logC|m(0))m(1) ∼ p(m|logC(1))logC(2) ∼ p(logC|m(1))m(2) ∼ p(m|logC(2))...(2.26)30where p(logC|m) and p(m|logC) are conditional distributions. Previousstudies show that the conditional distribution of multivariate Gaussian dis-tribution is still Gaussian distribution (strict mathematical proof can befound in [106]) as,p(logC|m) =N (µlogC + ρσlogC(m− µm)/σm, σ2logC(1− ρ2)),p(m|logC) =N (µm + ρσm(logC − µlogC)/σlogC , σ2m(1− ρ2)).(2.27)By combining equation (2.26) and equation (2.27), a set of samples for(logC,m) can be generated and used subsequently to generate samples forR˜i using equation (2.3). Then, all the samples of R˜i are used for calculatingn in equation (2.24). As a result, the CI can be approximated through afrequentist view.2.3 Experiment Validation and DiscussionIn this section, on the basis of Virkler’s dataset [7], the performance ofthe proposed model is validated and discussed. The tested structures weremade of 2024-T3 aluminum alloy. The reason for choosing the 2024-T3aluminum alloy is owing to its popularity in use for aging aircraft made in theseventies. The tests were run at room temperature in random order on a 20kip electrohydraulic closed loop system operated in load control with loadscontrolled within 0.2 percent. The loads were applied as a sinusoidal wavewith a maximum of 5.25 kip, and stress ratio R=0.2. The sinusoidal loadcycles were applied at a frequency of 20 Hz. Crack growth was monitoredwith a zoom stereo microscope (set at 150x) mounted on a digital traversingsystem which has a resolution of 0.001 mm. The crack length was recordedfrom 9 mm to 49.8 mm. Totally, 68 specimens were tested. Fig. 2.2 showsthe crack growth trajectories of these tested specimens. In this chapter, thefailure threshold is set as 49.8 mm, tested specimens are numbered accordingto their crack growth life (CGL).The geometry of the tested specimens and the corresponding FE modelare shown in Fig. 2.3. The material properties were set as, Young’s Modu-lus=72 GPa, Poisson ratio=0.33, according to the experimental result fromreference [107]. In the FE model, the entire structure was modeled withcoarse mesh, and the mesh was refined at the critical area. In this ex-periment, since the load amplitude is constant, ∆K is only the function31n (cycles)0 0.5 1 1.5 2 2.5 3 3.55101520253035404550a (mm) Specimen 1Specimen 68N (cycles)0 0.5 1 1.5 2 2.5 3 3.55101520253035404550Crack (mm) ×105Figure 2.2: Crack growth trajectories of the 68 specimens [8]. Image repro-duced with permission by Elsevier.of crack length. Fig. 2.4 shows the calculated ∆K corresponding to cracklength. In Fig. 2.4, the FE solution is acquired via ANSYS APDL® whichuses M-integral [108], a variant of J-integral, in the calculation of SIF. Theanalytical solution is calculated according to the equation in [100]. The sim-ilarity between the FE solution and the analytical solution is measured viacumulative relative accuracy (CRA) [109] as,CRA =1NN∑i=1(1− |∆Kf (ai)−∆Ka(ai)|∆Ka(ai)) (2.28)where ∆Kf (ai) and ∆Ka(ai) are the ∆K calculated from the FE solutionand the analytical solution respectively at the crack length ai, i = 1, ..., N .It is shown that the CRA achieves 99.84%. From Fig. 2.4, it can be seen thatthe FE result is very close to the analytical result. Based on ∆K calculatedfrom FE analysis, in the following, Section 2.3.1 and 2.3.2 are organized tovalidate the proposed model for CI calculation and FCG prediction, respec-tively. Moreover, Section 2.3.3 presents some necessary discussions regardingthe proposed model.32(a)(b)Details of mesh in crack tip630.31 0.5622112.25 2.251.05 0.650.851.01 1.010.050.100.007Figure 2.3: (a) Geometry of the tested specimens. (b) Corresponding FEmodel [8]. Image reproduced with permission by Elsevier.2.3.1 Experiment I: Validation of Confidence IntervalCalculationIn Experiment I, specimen 1 is taken for investigation. Specimens 2-68 aretreated as training data. First, the related parameters for prior distribu-tion are estimated according to the procedure given in Section 2.2.1. Theresult shows µ1 = −16.22, σ1 = 0.40, µ2 = 2.87, σ2 = 0.16, σ = 0.22. Forillustration, three times of prediction, P1, P2, and P3, are conducted when55638, 111327, and 166902 load cycles are imposed respectively. For speci-men 1, the crack length equals 11.2 mm, 15.4 mm and 23.6 mm at P1, P2,and P3 respectively. Fig. 2.5 shows the distribution of (logC,m) calculated33a (mm)28.2 28.3 28.4 28.5 28.615.85Analytical solutionFE solution (M-integral)5 10 15 20 25 30 35 40 45 5030 25 20 15 10 5 △K (MPa m0 .5 )15.815.7515.7Figure 2.4: SIF range calculated from the FE model and the analyticalmodel respectively [8]. Image reproduced with permission by Elsevier.from the proposed model. Specifically, Fig. 2.5 (a) shows the initial priordistribution. Figs. 2.5 (b)(c)(d) show the posterior distribution at P1, P2and P3 respectively. It can be seen that the more observations are fed, themore convergent the distribution of (logC,m) becomes. It is because theuncertainty decreases with the increase of observations.Besides, Fig. 2.6 (a) shows the distribution of R calculated via equation(2.23) at a specific time. Figs. 2.6 (b)(c) and (d) are the predicted distribu-tions of R at different crack lengths. It can be seen that along the increaseof crack length, the predicted distribution of R becomes dispersed. It is be-cause the longer future we predict, the more uncertainty we have. Besides,it can be seen from Fig. 2.6 (d) that the fluctuation of observations will alsoincrease the prediction uncertainty.The 95% CI intervals calculated from the proposed model at P1, P2, andP3 are given in Figs. 2.7 (a-2)(b-2) and (c-2) respectively. For comparison,Figs. 2.7 (a-1)(b-1) and (c-1) give the CI generated from the Gibbs samplerwhich is designed in Section 2.2.2. Specifically, the designed Gibbs samplergenerates 5000 possible degradation trajectories in each prediction. It canbe seen that the 95% CI calculated from the proposed model generally fitsthe shape/region of these possible degradation trajectories in Figs. 2.7 (a-1)(b-1) and (c-1), which validates the proposed model for calculating the95% CI.2.3.2 Experiment II: Validation of Fatigue Crack GrowthPredictionIn Experiment II, the performance of the proposed model for the FCG pre-diction is validated. For comparison, two state-of-the-art MCMC methods,HMC and NUTS, are also implemented. Besides, the Gibbs sampler de-34边框2，线型23.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -153.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -153.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -153.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -15(a) (b)(c) (d)-16.222.87-16.302.86-16.332.94-16.653.11P1P2 P3 Maximum probabilityMaximum probabilityMaximum probabilityMaximum probabilitylog C log Clog C log Cmmmm0.511.520.511.520.511.520.511.52Figure 2.5: (a) Prior distribution assigned for the FCG prediction. (b)Posterior distribution at P1. (c) Posterior distribution at P2. (d) Posteriordistribution at P3. In particular, ∆K is in MPa · m0.5 and da/dn is inmm/cycle [8]. Image reproduced with permission by Elsevier.signed in Section 2.2.2 is also used for comparison.Specifically, HMC uses Hamiltonian dynamics to sample a target dis-tribution. HMC can avoid the random walk behavior that plagues manyMCMC methods and thus converges to the target distribution more quicklythan other common MCMC methods. NUTS is an extension of HMC anddoes not require setting any parameters manually. In this study, the HMCand NUTS are implemented according to reference [67] where the pseu-docode and open-source package can be found. The third one is the Gibbssampler. Its implementation is mentioned in Section 2.2.2.Fig. 2.8 shows an example of sampling result when the posterior dis-tributions at P1, P2, and P3 are sampled 5000 times respectively. It canbe seen that the general shape of these samples is consistent with the re-sults calculated from the proposed closed-form solution shown in Figs. 2.5(b)(c) and (d). Moreover, Table. 2.1 summarizes the parameters of pos-terior distributions at P1, P2, and P3, calculated by HMC, NUTS, Gibbsand closed-form solution respectively. It can be seen that the closed-form35)}()(exp{ 233 iiMLV  9 13 17 21 25 29 32 a (mm)10 0 1 2 3 4 5 6 7 8 9 R (mm/cycle)0 2 4 6 8 10 R (mm/cycle) ×10-3 0 500 1000 1500 2000 2500 3000 3500 4000 4500 PDF9 13 17 21 25 29 32 a (mm)10 0 1 2 3 4 5 6 7 8 9 ×10-3 R (mm/cycle)10 0 1 2 3 4 5 6 7 8 9 ×10-3 R (mm/cycle)9 13 17 21 25 29 32 a (mm)(a) (b)(c) (d)P1P2 P3ObservationObservationObservation0408012016004080160120P304080120160)}(exp{ 3 iMV ×10-3 Figure 2.6: (a) At each specific time, R follows lognormal distribution. (b)Predicted distribution of future crack growth speed R at P1. (c) Predicteddistribution of future crack growth speed R at P2. (d) Predicted distributionof future crack growth speed R at P3 [8]. Image reproduced with permissionby Elsevier.solution and MCMC samplers can generate similar posterior distributions,which demonstrates the effectiveness of the closed-form solution.To evaluate different methods for the remaining useful life (RUL) predic-tion, specimens 1, 17, 33, 49 and 68 are tested, as they have different crackgrowth speeds. Fig. 2.9 shows the prediction result of different methods. Itcan be seen that the predicted RUL gradually converges to the ground-truthRUL with the increase of loading cycles. It is because more crack lengthobservations are available during the crack growth. Hence, the inference re-sult approaches the ground truth gradually. It is worth mentioning that theRUL prediction results for specimen 68 (shown in Fig. 2.9(e)) have a largediscrepancy. The reason is that specimen 68 has the longest crack growthtrajectory and highest RUL (shown in Fig. 2.2), which is more difficult tobe predicted in the early stage.Table 2.2 summarizes the prediction results of different methods wherethe prediction performance is indicated using root mean square error (RMSE),CRA and the convergence of absolute error. These indicators are commonly360 0.5 1 1.5 2 2.5 3 3.5 40102030405060×1050 0.5 1 1.5 2 2.5 3 3.5 40102030405060×1050 0.5 1 1.5 2 2.5 3 3.5 40102030405060×1050 0.5 1 1.5 2 2.5 3 3.5 40102030405060×1050 0.5 1 1.5 2 2.5 3 3.5 40102030405060×1050 0.5 1 1.5 2 2.5 3 3.5 40102030405060×105Threshold Boundary for 95% CI Predicted crack growth path Ground truth (a-2) (b-2) (c-2)(a-1) (b-1) (c-1)P1 P2 P3P1 P2 P3a (mm)a (mm)a (mm)a (mm)a (mm)a (mm)n (cycles) n (cycles) n (cycles)n (cycles) n (cycles) n (cycles)Figure 2.7: (a-1) (b-1) and (c-1) show the possible degradation paths cal-culated from the Gibbs sampler at P1, P2 and P3. (a-2) (b-2) and (c-2)show the 95% CI calculated from the proposed model at P1, P2 and P3 [8].Image reproduced with permission by Elsevier.used in evaluating the performance of the RUL prediction algorithm [109].Specifically,RMSE =√√√√ 1TT∑u=1(RULp(nu)−RULt(nu))2, (2.29)CRA =1TT∑u=1(1− |RULp(nu)−RULt(nu)|RULt(nu)), (2.30)where RULp(nu) and RULt(nu) are the predicted and ground-truth RUL atloading cycle nu,u = 1, ..., T and T is the number of predictions conducted.RMSE and CRA describe the prediction accuracy of different methods. Alower RMSE and a higher CRA mean a more accurate prediction perfor-mance. Moreover, the convergence of absolute error is calculated as:Convergence =√(xc − nf )2 + y2c (2.31)where nf is the number of loading cycles when the first RUL prediction isconducted, and (xc, yc) denotes the centroid of the area under the absolute37Table 2.1: Predicted posterior distributions with parameters (µlogC , µm,σ2logC , σ2m, ρ) at P1, P2, and P3 using HMC, NUTS, Gibbs and the closed-form solution, respectively. P1, P2, and P3 are inferred based on 12, 33,and 74 crack length observations of specimen 1 from 9 mm to 11.2 mm, 15.4mm, and 23.6 mm, respectively.Distribution Method µlogC µm σ2logC σ2m ρHMC -16.287 2.872 0.091 0.019 -0.970P1 NUTS -16.283 2.870 0.088 0.019 -0.969Gibbs -16.320 2.880 0.071 0.015 -0.969Closed-form -16.303 2.869 0.074 0.016 -0.971HMC -16.323 2.956 0.081 0.016 -0.990P2 NUTS -16.319 2.954 0.085 0.017 -0.991Gibbs -16.348 2.954 0.051 0.010 -0.985Closed-form -16.325 2.944 0.072 0.014 -0.990HMC -16.671 3.129 0.068 0.012 -0.995P3 NUTS -16.666 3.126 0.068 0.012 -0.995Gibbs -16.647 3.114 0.045 0.008 -0.993Closed-form -16.650 3.115 0.055 0.010 -0.994error curve AE(nu) = |RULp(nu)−RULt(nu)|, u = 1, 2, ..., T . A lower con-vergence of absolute error means a faster convergence speed of the predictionvalue to the ground truth. More details of the convergence of absolute errorcan be found in [109]. Besides, the average running/inference time for eachprediction is also compared on the computer with a 2.80 GHz CPU and 8.00GB RAM.From Table 2.2 it can be seen that the proposed model can not only givemore accurate prediction results but also converge to the ground truth faster.More importantly, it dramatically decreases the running time of prediction.The inference and prediction only cost 0.08 s. Here, note that the Gibbssampler designed in Section 2.2.2 also outperforms HMC and NUTS. Thereason is that the explicit distribution function of {logC,m} is derived inthis study, which makes the sampling very efficient.2.3.3 DiscussionIn the proposed model, there still exist two issues that are needed to bediscussed further.First, recall that in prior distribution setting,m and logC are assumed as383.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -15P1log CmP2Initial sample . Samples First 50 samplesP3P1 P2 P3P1 P3(a-1)P23.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -15log Cm(a-2)3.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -15log Cm(a-3)3.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -15log Cm(b-3)3.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -15log Cm(b-2)3.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -15log Cm(b-1)3.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -15log Cm(c-1)3.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -15log Cm(c-2)3.33.12.92.72.5-17.5 -17 -16.5 -16 -15.5 -15log Cm(c-3)Directly posteriorFigure 2.8: (a-1,2,3) Posterior distribution sampled via Gibbs at P1, P2 andP3. (b-1,2,3) Posterior distribution was sampled via HMC at P1, P2 andP3. (c-1,2,3) Posterior distribution sampled via NUTS at P1, P2 and P3 [8].Image reproduced with permission by Elsevier.independent Gaussian variables, i.e., logC ∼ N (µ1, σ21) and m ∼ N (µ2, σ22)in this study. However, some studies [1, 101, 102] assume (logC,m) followsthe bivariate Gaussian prior distribution, given the fact that logC and mare correlated in nature. Here, the first objective of this section is to com-pare the proposed model with these studies which assume (logC,m) followsthe bivariate Gaussian prior distribution (BGPD). The solution under thisassumption is implemented according to Ref. [1] where m and logC are as-sumed as dependent Gaussian variables and estimated by fitting the FCGdata. Since there is no closed-form solution available under this assump-tion, the Gibbs sampler is no longer available. Instead, HMC and NUTSare taken for posterior sampling respectively under this assumption.Second, recall that in equation (2.24), the R˜i is chosen as the MV oflognormal distribution, i.e., exp{µ3(i)}, for point-wise prediction. However,the MV of lognormal distribution is not the point with the highest prob-ability. The point with the highest probability, the so-called most likelyvalue (MLV), is exp{µ3(i) − σ23(i)}, as previously shown in Fig. 2.6 (a).Here, the second objective of this section is to compare the proposed model,39×105(a)(b)(c)(d)(e)×105×105×105×105N (cycle)N (cycle)N (cycle)N (cycle)N (cycle)RUL(cycle)RUL(cycle)RUL(cycle)RUL(cycle)RUL(cycle)Closed-form Gibbs NUTSHMC Ground truth统一字体1.75 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.246201.9 2 2.1 2.2 2.3 2.4 052 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.546202 2.1 2.2 2.3 2.4 2.5 2.6846202.5 2.6 2.7 2.8 2.9 3 3.10510×104×104×104×104×104(a)(b)(c)(d)(e)n (cycles)n (cycles)n (cycles)n (cycles)n (cycles)RUL(cycles)RUL(cycles)RUL(cycles)RUL(cycles)RUL(cycles)Closed-form NUTS Ground truthGibbs HMC1.75 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2×1054620×1041.9 2 2.1 2.2 2.3 2.4 05×1042 2.1 2.2 2.3 2.4 2.5×105×1054620×1042 2.1 2.2 2.3 2.4 2.5 2.6862042.5 2.6 2.7 2.8 2.9 3 3.10510×104×104×105×105Figure 2.9: Predicted results of the closed-form solution, Gibbs, HMC andNUTS. (a)-(e) are prediction results for specimens 1, 17, 33, 49 and 68respectively [8]. Image reproduced with permission by Elsevier.which uses MV for point-wise prediction, with the model which uses MLVfor point-wise prediction.The same experimental configuration presented in Section 2.3.2 is usedin this subsection. Table 2.3 summarizes the prediction performance of thesemethods.From Table 2.2 and 2.3, it can be seen that the BGPD-based HMC andthe BGPD-based NUTS achieve a comparable performance with the origi-nal HMC and NUTS. This means considering the correlation of m and logCin prior distribution does not improve the prediction accuracy dramatically.Due to this, the proposed model also outperforms BGPD-based methods.The reason is that even though the correlation between m and logC in theprior distribution is not considered, m and logC become correlated auto-40Table 2.2: Prediction performance and running time comparison forVirkler’s dataset. Specimens 1, 17, 33, 49, and 68 are tested respectively, asthey have different crack growth speeds.Method RMSE CRA Convergence Running timeHMC 4850 79.56% 19470 593.59 sec.NUTS 4796 79.60% 19577 10.62 sec.Gibbs 4454 80.69% 19396 0.53 sec.Closed-form 4094 80.89% 19284 0.08 sec.Table 2.3: Prediction performance comparison in the discussion.Method RMSE CRA ConvergenceBGPD-based HMC 4384 80.17% 19580BGPD-based NUTS 4379 80.22% 19613MLV-based solution 5368 66.62% 28024Closed-form 4094 80.89% 19284matically in the posterior distribution as they are updated according tothe same observation. This phenomenon can be observed from Fig. 2.5where m and logC are without correlation in prior distribution (as shown inFig. 2.5 (a)), but they become correlated in posterior distribution (as shownin Fig. 2.5 (b)(c)(d)). To make this point clear, Fig. 2.10 shows the changeof correlation coefficient ρ, which is calculated from equation (2.20), withthe increase of the number of observations. Even though it is set ρ = 0 inthe initial phase, ρ converges to −1 very fast. Thus, the improvement ofconsidering this correlation in prior distribution is limited.Number of observationρ = - 0.75ρ= - 0.8ρ0 5 10 15 20 25 30-1-0.8-0.6-0.4-0.20Figure 2.10: Change of correlation coefficient ρ with the increase of thenumber of observations [8]. Image reproduced with permission by Elsevier.Besides, the proposed model also outperforms MLV-based solution, in-41dicating using the MV for prediction is more effective than using the MLV.It is because lognormal distribution is skewed as shown in Fig. 2.6(a). Com-pared with the MV, using the MLV for point-wise prediction will bring alarge prediction error when the real crack growth speed R lies far from MLV.2.4 SummaryThis chapter proposed a probabilistic model for the FCG prediction basedon closed-form solution. FE analysis is adopted to link the SIF with cracklength and operating load condition. The material parameters and modelerror are treated as random variables. A closed-form solution is derived inthis study for updating the well-known Paris-Erdogan equation, and pre-dicting the FCG. Compared with existing methods, the proposed model iseasy to be implemented. Besides, it provides a deterministic result withoutany hyper-parameters. The proposed model is validated on the experimentalFCG data of a set of aluminum alloy specimens in Virkler’s dataset. The re-sults show the proposed model can not only predict FCG accurately but alsorun faster than Monte Carlo methods. The proposed model is applicable forthe FCG prediction considering the uncertainty of material parameters andmodel error.42Chapter 3Probabilistic Model forFatigue Crack GrowthPrediction with Hybrid Prior3.1 OverviewThis chapter proposes a probabilistic model for the FCG prediction withhybrid prior. Existing methods [23, 45, 75, 76, 92, 110, 111] conduct theFCG prediction only based on single prior input. However, in practicalengineering, there is usually multi-source information that can be used forthe prior distribution setting. For instance, the stress on a specific structurecan be estimated from either operating parameters [95] or FE software [13].The material property of an in-service structure can be acquired from eitherthe specimen-level material test [100] or the field data from other similarstructures [97]. In this context, this chapter studies how to conduct Bayesianinference based on multi-source prior knowledge and proposes a probabilisticmethod for the FCG prediction, which enables the input of hybrid prior.Fig. 3.1 shows the overview of the proposed methodology, which com-prises two inference steps. Bayesian inference A is built based on the conceptof posterior belief of prior distribution (PBPD) [112], which quantifies thebelief of different choices of prior distribution using Bayes theory. Specifi-cally, multi-source priors are input into Bayesian inference A. Meanwhile,subjective belief is also assigned for each prior. Then, the Monte Carlo in-tegration is adopted in the calculation of PBPD for each candidate prior.Bayesian inference B is built based on PF. Specifically, PF is extended tosimultaneously sample each candidate prior according to their PBPD. Thenthe posterior distribution is calculated based on importance sampling andthe inverse cumulative distribution function (CDF) method. Finally, theFCG can be predicted with hybrid prior. The main contribution of thisstudy is summarized as,• Based on the concept of PBPD, this chapter proposes a probabilistic43FCG prediction method, which enables the input of hybrid prior. Asa result, in a real-world scenario, any promising prior knowledge canbe integrated, increasing the robustness of the FCG prediction.• To tackle high-dimensional uncertainty, this chapter introduces theMonte Carlo integration in the calculation of the PBPD in the high-dimensional space.• This chapter extends the PF for the Bayesian inference with hybridprior, where each prior distribution can be sampled simultaneouslyaccording to their PBPD.Bayesian inference AObservationBayesian inference BPosteriordistributionFCG predictionCandidate priors &Subjective beliefsCandidate priors &Posterior beliefsMC integration PFFigure 3.1: Overview of the proposed methodology [9]. Image reproducedwith permission by Elsevier.The rest of this chapter is organized as follows. Section 3.2 presents theproposed methodology. Based on a numerical study, Section 3.3 validates theproposed model with necessary discussion. Finally, Section 3.4 summarizesthis study.3.2 Methodology3.2.1 Fatigue Crack Growth Modeling and ModelCalibration with Non-Destructive Inspection DataWithout loss of generality, the Paris-Erdogan equation is adopted in thischapter as,dadn= C∆Km. (3.1)In equation (3.1), a is the crack length, and n is the number of the imposedload cycles. Parameters C and m are material-dependent constants. ∆K is44the so-called SIF range. It describes the stress state in the region near thecrack tip. For a specific structure, ∆K is a function of crack length a, loadL and geometry. In real-world applications, FE analysis is widely used forthe calculation of ∆K [1, 45, 99, 113], as the analytical solutions are difficultto be obtained or do not exist at all.Similarly, in this chapter, the stress analysis is constrained in the frame-work of linear elastic mechanics. As a result, ∆K(a, L) has a linear relationwith the load L [1]. Thus, for FCG prediction of a specific structure, therelation between ∆K, a, and L can be obtained by running the FE modelonce for a baseline load Lb only. The remaining work is merely to multiplya ratio between the new and baseline loads. Subsequently, a surface of ∆Kfor different crack sizes and different load conditions can be obtained [1].Moreover, many studies found that there is a high-frequency randomnessin crack growth [99, 100]. Thus, a random error ϵ is added to the originalParis-Erdogan equation to account for this high-frequency randomness. Byconsidering the model error ϵ, the Paris-Erdogan equation is modified as,dadn= C(∆K(a, L))mϵ (3.2)where ϵ is assumed as a lognormal random variable with zero mean aslogϵ ∼ N (0, σ2) [99]. In the following numerical simulation, logϵ is sam-pled from the distribution N (0, 0.222) at each iteration step to simulate thehigh-frequency randomness in real-world crack growth. The value of 0.22was estimated from Virkler’s data set for aluminum alloys 2024-T3 [7]. Thisvalue is taken in the following numerical study to simulate the FCG. It isworth mentioning that in the FCG prediction of this study, ϵ is assumed un-known, and the original Paris-Erdogan equation in equation (3.1) is adoptedfor FCG prediction. ϵ is the unavoidable model error of the original Paris-Erdogan equation.Since the Paris-Erdogan equation is in differential form and can not beintegrated, the first-order Euler method [105] is used to discretize it as:a(0) = a0a((i+ 1)∆n) = a(i∆n) + ∆n(da(i∆n)dn)(3.3)where a0 is the initial crack length, and ∆n is the incremental loading cycles.Both the crack length and SIF range will increase during the service of thestructure. Finally, the structural fatigue life is the RUL till a exceeds a pre-defined threshold or maximum stress intensity factor Max K(a, L) exceedsthe fracture toughness [5].45For an in-service structure, non-destructive inspection (NDI) is com-monly used for the measurement of fatigue cracks. Commonly-used NDItechniques include eddy current, ultrasonic, acoustic emission, X-ray radio-graphy, etc. A comprehensive review of these NDI techniques can be foundin [95]. To consider the measurement uncertainty, the measured crack sizecan be generally modeled by regression analysis [76] as,am = β0 + a× β1 + ξm, ξm ∼ N (0, σ2m) (3.4)where regression coefficient β0 and β1 are constants. ξm is a zero-meannormal random variable with the standard deviation σm. In real-world ap-plications, for a specific NDI technique, the values of β0, β1 and σm areusually estimated from laboratory tests where the true size a and measuredsize am are known [85, 114]. Specifically, β0 and β1 can be obtained byfitting a and am using linear regression. σm is the variance of the fittingresiduals.Thus, the likelihood function Li(am(n)|a(n)) can be derived as,Li(am(n)|a(n)) = 1√2πσ2mexp[−(am(n)− β0 − β1a(n))22σ2m]. (3.5)Based on observations am(n1), ..., am(nk), the random parameters in theFCG can be inferred as,P (θ|am(n1), ..., am(nk)) ∝∫Li(am(n1), ..., am(nk)|a(n1), ..., a(nk), θ)P (a(n1), ..., a(nk)|θ)π(θ)dθ(3.6)where θ denotes the unknown parameter vector in the crack growth modeland π(θ) is the prior in Bayesian inference. In the following example, thematerial parameters C, m and load condition L are treated as fixed but un-known parameters, that is θ = {C,m,L}. The observation {am(n1), ..., am(nk)}is denoted as D for shorthand. It is worth mentioning that the FCG modelabove is presented for the sake of illustration. The proposed methodologycan be adopted to other prediction models, depending on specific practicalapplication.463.2.2 Bayesian Inference for Posterior Belief of PriorDistributionBayesian inference plays an important role in inferring uncertain parametersbased on prior knowledge and real-world observation. However, in practicalengineering, there is usually multi-source information that can be used asprior knowledge. For example, the prior knowledge can come from exper-imental results, historical records, expert knowledge, simulation software,etc.To tackle this problem, a new probabilistic model is proposed in thischapter. It enables the input of hybrid prior. For each uncertain parameterθi, i = 1, ..., z, a set of marginal candidate priors can be assumed for it. Byconsidering the joint distribution, the candidate priors {π1, ..., πp} for theunknown parameter vector θ = {θ1, ..., θz} can be obtained, where p is thenumber of candidate priors.Let µ(πi), denotes our subjective belief regarding each candidate priorπi, i = 1, ..., p. µ(πi) is a function regarding πi and called subjective beliefin the following. In real-world application, µ(πi) can be set based on theexperience or prior knowledge of each prior. If there is no preference for anyprior distribution, µ(πi), i = 1, ..., p, can be set as 1/p for every prior. Theoverall hybrid prior can be formulated as,π(θ) =p∑i=1µ(πi)πi (3.7)wherep∑i=1µ(πi) = 1. (3.8)Intuitively, µ(πi) denotes the probability that the unknown parametervector θ comes from the prior πi. Then, the joint probability of πi and θ isP (πi, θ) = µ(πi)πi(θ). (3.9)On the other hand, based on Bayes theory, the posterior distribution fora specific prior πj can be derived as,P{πj |D} =∫P (D|πj , θ)P (πj , θ)dθp∑i=1∫P (D|πi, θ)P (πi, θ)dθ.(3.10)47Substitute equation (3.9) into equation (3.10). It can be obtained that,P{πj |D} =∫P (D|πj , θ)µ(πj)πj(θ)dθp∑i=1∫P (D|πi, θ)µ(πi)πi(θ)dθ=µ(πj)∫P (D|πj , θ)πj(θ)dθp∑i=1µ(πi)∫P (D|πi, θ)πi(θ)dθ.(3.11)Define the likelihood of prior πj as,Li{πj |D} =∫P (D|πj , θ)πj(θ)dθ. (3.12)Thus, equation (3.11) can be re-written as,P{πj |D} = µ(πj)Li{πj |D}p∑i=1µ(πi)Li{πi|D}.(3.13)Note thatp∑i=1µ(πi)Li{πi|D} is a constant. Thus, equation (3.13) can besimplified as,P{πj |D} ∝ µ(πj)Li{πj |D}. (3.14)Intuitively, from the Bayesian view, µ(πj) is the prior belief regarding priorπj . P{πj |D} is the posterior distribution regarding πj given observation D.To calculate the posterior distribution P{πj |D}, the integration in equa-tion (3.12) should be solved. However, there is no analytical solution forequation (3.12). In this study, the Monte Carlo integration is introduced toapproximate it. Monte Carlo integration is a technique for numerical inte-gration using random samples. It randomly chooses samples at which theintegrand is evaluated. Assume θi, i = 1, ..., s are independent identicallydistributed (i.i.d.) samples from a distribution with PDF q(θ) where s isthe number of samples. Equation (3.12) can be estimated as,Lˆi{πj |D} = 1ss∑i=1P (D|πj , θi)πj(θi)q(θi). (3.15)48The proof that equation (3.15) is the unbiased estimation of equation (3.12)is given as follows.First, denote the random variable Y as,Y =P (D|πj , θ)πj(θ)q(θ). (3.16)Random variable Y is a function regarding θ. Thus, the following equa-tions hold,E[Lˆi(πj |D)] = E[1ss∑i=1P (D|πj , θi)πj(θi)q(θi)]=1ss∑i=1E[P (D|πj , θi)πj(θi)q(θi)]=1ss∑i=1E[Y ]=1ss∑i=1∫[P (D|πj , θ)πj(θ)q(θ)]q(θ)dθ=1ss∑i=1∫P (D|πj , θ)πj(θ)dθ=∫P (D|πj , θ)πj(θ)dθ.(3.17)Hence E[Lˆi(πj |D)] = Li{πj |D}, and equation (3.15) is the unbiased estima-tion of equation (3.12).Moreover, if samples θi, i = 1, ..., s are generated from prior πj (that isθi ∼ πj and q(θk) = πj(θk)), equation (3.15) can be simplified as,Lˆi{πj |D} = 1ss∑i=1P (D|πj , θi)πj(θi)πj(θi)=1ss∑i=1P (D|πj , θi).(3.18)Namely, to calculate Lˆi{πj |D}, a set of samples θi, i = 1, ..., s shouldbe generated from πj . Lˆi{πj |D} equals the average of the likelihood value49of each sample. For the FCG prediction considered in this study, D equals{am(n1), ..., am(nk)}. The likelihood of sample θi = {Ci,mi, Li} is calcu-lated as,Li(θi|D) = P (am(n1), ..., am(nk)|θi) =P (am(n1), ..., am(nk)|a(n1), ..., a(nk), Ci,mi, Li)P (a(n1), ..., a(nk)|Ci,mi, Li).(3.19)It is easy to understand equation (3.18) from an intuitive view. If a prioris promising, the samples of it tend to have a large likelihood value. OnceLi{πj |D} is calculated, the PBPD in equation (3.14) can be obtained.Here, it is worth mentioning that the Monte Carlo integration presentedhere differs from the PF, which will be presented in the next section. In theMonte Carlo integration, the exact likelihood value of each sample is calcu-lated. In contrast, in the PF, the relative value of each sample is calculated.Thus, we can conduct normalization, resampling, and roughening phase inthe PF (see Section 3.2.3 for details), but not in Monte Carlo integration.3.2.3 Bayesian Inference with Hybrid PriorIn Section 3.2.2, the concept and the calculation method of PBPD have beenpresented. Based on PBPD, this section is organized to introduce how toconduct Bayesian inference with hybrid prior π(θ).As illustrated in Section 3.2.1, the posterior distribution of unknownparameter vector θ is,P (θ|am(n1), ..., am(nk)) ∝∫P (am(n1), ..., am(nk)|θ)π(θ). (3.20)As π is treated as a hybrid prior in this chapter, equation (3.20) does nothave an analytical solution. Thus, the PF is extended in this chapter forcalculating equation (3.20). Compared with other traditional inference al-gorithms (such as Kalman filter [115] and extended Kalman filter [116]), PFcan handle non-Gaussian distributions and non-linear relationships. Thus,it is adequate for the Bayesian inference with hybrid prior.In PF, the importance sampling algorithm [64] is employed to approxi-mate the posterior distribution. Specifically, a set of particles θi, i = 1, ..., sare generated from the importance PDF q(θ) to approximate equation (3.20)as,50P (θ|am(n1), ..., am(nk)) ≈s∑i=1wiδ(θ − θi) (3.21)where δ(·) is the Dirac delta function, and wi is the weight of particle θi.The wi is calculated and normalized as [117],wi =P (am(n1), ..., am(nk)|θi)π(θi)q(θi), wi = wi/s∑i=1wi. (3.22)When the importance PDF q(θi) is chosen as π(θi), equation (3.22) is sim-plified as,wi = P (am(n1), ..., am(nk)|θi), wi = wi/s∑i=1wi. (3.23)Thus, the calculation of particle weight can be summarised as follows: (1)Sample the hybrid prior to generating s particles. (2) Calculate the likeli-hood of each particle. (3) Normalize the weight of particles. In this proce-dure, the main challenge lies in how to sample the hybrid prior.For this issue, a multinomial resampling is designed as follows, (1) Denoteeach prior πj , j = 1, ..., p as a bar with length P (πj |D) and connect allthe bars as shown in Fig. 3.2. (2) Generate a random variable from auniform distribution between 0 and 1. (3) Find the bar in which the variablelies, and then find the corresponding prior distribution. (4) Sample thecurrent particle from this prior distribution. By repeating the steps (1)-(4),s samples can be generated from the hybrid prior based on the PBPD.0 1P{π1|D}... ...P{πp|D}Figure 3.2: Sample hybrid prior distribution. Each candidate prior πj , j =1, ..., p is denoted as a bar with length P (πj |D). Connecting all the barsresults in a bar with a length of 1 [9]. Image reproduced with permission byElsevier.Moreover, one problem in the PF is particle degeneracy, i.e., some par-ticles have negligible weights. Especially for the Bayesian inference with51hybrid prior, particle degeneracy is a critical problem. The main reason isthat if a candidate prior is with a small PBPF, the particles tend to havea small weight. The solution for this problem is conducting a resamplingstep to eliminate particles with a small weight. In this chapter, the inverseCDF [66] method is used for this purpose.s = 80.5Probability densityCDFPDFFigure 3.3: Illustration of the CDF method where s is taken as 8 for example.0.5 is the random value generated from a uniform distribution between 0 and1 [9]. Image reproduced with permission by Elsevier.Specifically, the CDF of particles is constructed from the PDF of particlesas shown in Fig. 3.3. Next, a random value is generated from a uniformdistribution between 0 and 1 (e.g., 0.5 in Fig. 3.3). Finally, the particle whichhas the closest CDF value to the random value is selected. By repeatingthis process s times, s new particles can be generated to replace the previousparticles.Another problem in the PF is particle impoverishment [94]. That is thesamples that approach the ground truth will be duplicated lots of timesduring the resampling step. As a result, many particles in the particle poolhave the same value. To address this problem, a roughening phase [118]is added. Specifically, the random noise ∆θ = {∆θ1, ...,∆θz} is added toθ = {θ1, ..., θz} as,θ = θ +∆θ (3.24)52where∆θ = {∆θ1, ...,∆θi, ...,∆θz}, ∆θi ∼ N (0, κ · Di). (3.25)In particular, Di is the maximum difference of θi between the particles inthe particle set, and κ is a constant tuning parameter which is set as 0.03in this study. The selection of κ is a compromise. A large value of κ wouldblur the distribution, but a small value of κ would not solve the particleimpoverishment issue [118].3.3 Experiment Validation and Discussion3.3.1 Experiment SetupThe proposed method is verified by a numerical study. The crack propaga-tion at the root of a spur gear is taken as an example. ANSYS® is usedto build a FE model for a single spur gear tooth with a crack at the root,as shown in Fig. 3.4 (a). The geometry parameters and material propertiesare listed in Table 3.1. To simulate the load during the gear mesh, a con-centrated load was imposed on the outer point of a single engagement [113].In practical engineering, this load is determined from the torque which ismeasured on the propeller shaft [113]. Specifically, the minimum load is zero(when the gear tooth leaves mesh). The maximum load is T/r (when thegear is in single engagement), where r is the radius of the outer cycle of thegear and T is the torque on the gear. The load ratio on the gear tooth iszero.The crack at the gear tooth root starts with an initial length of 0.1mm, and the tooth is considered to be failed when the crack size reaches5.0 mm, which is about 80% of the circular thickness of the tooth. SinceKI dominates the fracture behavior and the values of KII and KIII arenegligible [1], only KI is considered in crack growth prediction. Fig. 3.4(b) shows the calculated response surface of the SIF which is a functionregarding crack length and load. Under the framework of linear elasticmechanics, ∆K has a linear relationship with the load L, i.e., the torque onthe gear. For the online prediction, this response surface is modeled as,∆K = (∆Kb) · L/Lb (3.26)where L is the load condition of the inspected gear, Lb is the baseline loadcondition which is set as 30 N ·m and ∆Kb is the SIF range under baseline53Table 3.1: Material properties and geometry parameters [1].Young’smodulus(Pa)Poisson’sratioModule(mm)Pitchdiameter(mm)Base circlediameter(mm)Outer circlediameter(mm)Pressureangle(degree)Teeth No.2.07e11 0.30 3.20 60.8 57.1 67.2 20.0 19load. ∆Kb is acquired by running the FE model under baseline load Lb.The ANSYS® is used to generate a set of samples, (∆Kb, a), under baselineload. Then the quintic equation is used to fit these samples as,∆Kb = A1 · a4 +A2 · a3 +A3 · a2 +A4 · a+A5 (3.27)where A1, A2, ..., A5 are the fitting parameters of the quintic equation andcalculated using the built-in function in MATLAB. The results show A1 =1.1963, A2 = 7.6026, A3 = 14.1081, A4 = 6.8270, A5 = 10.2811.Initial crackCrack growth path(a) (b)Model I SIF (MPa mm^0.5)300 240 180 120 60 06 4 2 00Crack length (mm) Torque (N·m)5001000150020002500Figure 3.4: (a) FE model of a single spur gear tooth. (b) Response surface ofSIF range which is a function regarding crack length and torque [9]. Imagereproduced with permission by Elsevier.In this example, multiple priors are assigned for parameters (C,m,L).Specifically, since C and m are correlated, two two-dimensional Gaussiandistributions are assumed to (m, logC) as 2,2Without loss of generality, the priors and ground truth in the following are set forillustration.54Table 3.2: Candidate joint priors.Candidate π1 π2 π3 π4(logC,m,L) N1 ∗ N3 N1 ∗ N4 N2 ∗ N3 N2 ∗ N4(m, logC) ∼ N1([1.44−23.12],[0.04 −0.98 · 10−1−0.98 · 10−1 0.25]), (3.28)(m, logC) ∼ N2([2.21−24.18],[0.09 −11.64 · 10−2−11.64 · 10−2 0.16]). (3.29)In real-world applications, these priors can be acquired from differentsources, e.g., experimental material tests, historical records of other gears inthe same batch, and expert estimation. As to load condition, two candidateone-dimensional distributions are assumed to L as, N3(µ = 45 N ·m,σ =5 N · m) and N4(µ = 110 N · m,σ = 10 N · m). Similarly, these pri-ors can be acquired from different sources such as expert estimation, dif-ference sensor data, simulation software and the load from other similarequipment. Table 3.2 summarises the candidate joint priors which are de-noted as π1-π4, respectively. Fig. 3.5 shows the location of each joint priorin three-dimensional space. In this experiment, six individual gears A-Fare simulated with randomly set parameters. Table 3.3 shows the groundtruth of unknown parameters for each specimen. These ground-truth pa-rameters are diverse and close to different candidate priors, which simulatethe uncertainty in the individual specimen.It is assumed the crack length is measured by NDI (e.g. eddy currentinspection). The regression coefficients in equation (3.4) are set as β0 = 0,β1 = 1 and σm = 0.1 to account for the error in NDI. Fig. 3.6 shows thecrack growth paths of these specimens with the measurement error of NDIas shown in equation (3.4). It is assumed that three times of inspectionsare conducted with the non-destructive tool at load cycles 5.0 ·107, 5.5 ·107,6.0 · 107, respectively.3.3.2 Experimental Results and AnalysisTable 3.4 exhibits the calculated likelihood of each prior under different loadcycles. From Table 3.4 it can be seen that given the same observation, thelikelihood values of each prior are diverse. Take specimen A for example.55log CLmπ1π2π3π4Figure 3.5: Location of π1, π2, π3 and π4 in three-dimensional space. Eachdistribution is represented by 10000 samples from it [9]. Image reproducedwith permission by Elsevier.Table 3.3: Ground-truth for specimen A-F.No. A B C D E FlogC -23.50 -24.18 -23.21 -23.88 -22.98 -23.88m 1.43 2.00 1.33 1.90 1.39 1.89L 120.00 45.00 100.00 40.00 90.00 43.00At 5.0 · 107 cycle, the likelihood of π2 achieves the largest value, while thelikelihood of π1 achieves the lowest value. This indicates prior π2 is themost promising prior assumption for specimen A, while π1 is the worst one.Moreover, the likelihood of prior distribution also changes regarding differentload cycles. Take specimen B for instance. At 5.0 · 107 load cycle, π2 is themost promising prior distribution, while at 5.5 ·107 load cycle, π3 is the mostpromising one. This is because the likelihood of each prior adjusts when newobservation is fed.Based on the subjective belief (0.25 of each) and the likelihood of eachcandidate prior, the posterior belief of prior distribution (PBPD) can becalculated as shown in Table 3.5. These posterior beliefs will influencethe initialization of particles at different load cycles as aforementioned inFig. 3.2.Fig. 3.7 shows an illustrative example, where the initialization of parti-cles for specimen A-B is exhibited. It can be seen that for specimen A, moreparticles are generated from π2 and π3. For specimen B, more particles aregenerated from π3. This phenomenon is consistent with the PBPD of spec-56A B DCFEThresholdCrack length (mm)Load cycleFigure 3.6: Crack growth paths of gear A-F with measurement noise simu-lated from equation (3.4) [9]. Image reproduced with permission by Elsevier.imen A and specimen B. Concretely, as shown in Table 3.5, for specimen Aat load cycle 6.0·107, P (π2|D) and P (π3|D) are comparable, and larger thanP (π1|D) and P (π4|D). Similarly, for specimen B, P (π3|D) is the highest.Moreover, Fig. 3.8 shows the posterior distributions of specimen A-B atdifferent load cycles. From Fig. 3.8 it can be seen that the hybrid priordistribution converges to different posterior distributions for each specimen.Specifically, in specimen A, most particles locate at the π2 area, and becomeconvergent. In contrast, in specimen B, particles gradually converge at π3area. This phenomenon indicates the inference result converges to the mostpromising area based on different observations.Fig. 3.9 shows the fatigue life prediction of specimen A and B withhybrid prior at different load cycles. From Fig. 3.9, it can be seen that atthe initial phase (as shown in Fig. 3.9 (a-1) and Fig. 3.9 (b-1)), the predictionresult tends to have multiple peaks. This is because the inference result isinfluenced by multiple candidate priors. Then, when more observations arefed, the inference result converges to a single peak (as shown in Fig. 3.9(a-3) and Fig. 3.9 (b-3)). The reason is that the most promising prior isdistinguished with more observations.For benchmark, five methods (I-V) are constructed for comparison. Specif-ically, method I-IV conduct the RUL prediction using single candidate priorπ1, π2, π3 and π4 respectively. Method V conducts the RUL predictionbased on the hybrid prior distribution. However, in Method V, the belief ofeach prior distribution is fixed as the same (0.25 of each). That is the belief57Table 3.4: Likelihood of each prior.Specimen A Specimen BLoadcycleLi(π1|D) Li(π2|D) Li(π3|D) Li(π4|D) Li(π1|D) Li(π2|D) Li(π3|D) Li(π4|D)5.0 · 107 3.1 · 10−4 2.0 · 10−1 5.5 · 10−2 1.2 · 10−2 1.2 · 10−2 1.9 · 10−1 1.4 · 10−1 1.6 · 10−25.5 · 107 4.8 · 10−6 3.2 · 10−3 8.1 · 10−4 7.2 · 10−5 2.3 · 10−3 1.8 · 10−3 4.8 · 10−2 4.7 · 10−36.0 · 107 2.4 · 10−27 1.5 · 10−3 8.9 · 10−4 2.5 · 10−4 2.6 · 10−3 8.1 · 10−4 6.7 · 10−2 2.8 · 10−3Specimen C Specimen DLoadcycleLi(π1|D) Li(π2|D) Li(π3|D) Li(π4|D) Li(π1|D) Li(π2|D) Li(π3|D) Li(π4|D)5.0 · 107 4.5 · 10−2 1.5 · 10−1 1.7 · 10−1 1.7 · 10−2 6.2 · 10−1 5.1 · 10−2 2.6 · 10−1 1.7 · 10−25.5 · 107 2.9 · 10−2 2.8 · 10−1 2.3 · 10−1 2.5 · 10−2 6.0 · 10−1 1.5 · 10−1 5.0 · 10−1 3.8 · 10−26.5 · 107 6.7 · 10−2 7.1 · 10−1 6.1 · 10−1 6.3 · 10−2 1.0 · 100 4.1 · 10−1 1.3 · 100 1.0 · 10−1Specimen E Specimen FLoadcycleLi(π1|D) Li(π2|D) Li(π3|D) Li(π4|D) Li(π1|D) Li(π2|D) Li(π3|D) Li(π4|D)5.0 · 107 7.0 · 10−4 2.2 · 10−1 6.3 · 10−2 1.2 · 10−2 1.1 · 10−1 1.1 · 10−1 2.1 · 10−1 1.8 · 10−25.5 · 107 8.0 · 10−5 1.9 · 10−2 5.1 · 10−3 7.6 · 10−4 6.6 · 10−2 2.3 · 10−1 2.6 · 10−1 2.6 · 10−26.5 · 107 6.7 · 10−5 1.7 · 10−2 8.2 · 10−3 1.6 · 10−3 7.6 · 10−2 3.0 · 10−1 5.7 · 10−1 4.4 · 10−2of each candidate prior will not update based on new observations. TheRUL prediction is conducted after each inspection. The prediction perfor-mance of each method is indicated by the RMSE and CRA. These indicatorsare commonly used in evaluating the accuracy of the RUL prediction algo-rithm [109]. Specifically, RMSE and CRA are calculated as,RMSE =√√√√1kk∑i=1(RULp(ni)−RULt(ni))2, (3.30)CRA =1kk∑i=1(1− |RULp(ni)−RULt(ni)|RULt(ni)), (3.31)where RULp(ni) and RULt(ni) are the predicted and ground-truth RULat loading cycle ni, i = 1, ..., k. RMSE and CRA describe the predictionaccuracy of different methods. In particular, a higher CRA and a lowerRMSE mean a more accurate prediction performance.Table 3.6 and Table 3.7 show the CRA and RMSE of different meth-ods. From Table 3.6 and Table 3.7, It can be observed that none of the sixmethods is absolutely superior for all the prediction cases. The reason forthis phenomenon is two-fold: First, it is difficult that a method can consis-tently outperform all the other methods in each individual prediction, due tovarious sparsity, randomness and nonlinearity. For example, the simulatedcrack growth process includes randomness and nonlinearity; the inspectionis sparse and comes with noise; the inference algorithm also induces ran-58Table 3.5: Posterior belief of each prior.Specimen A Specimen BLoadcycleP (π1|D) P (π2|D) P (π3|D) P (π4|D) P (π1|D) P (π2|D) P (π3|D) P (π4|D)5.0 · 107 1.2 · 10−1 7.5 · 10−1 2.1 · 10−1 0.4 · 10−1 0.3 · 10−1 5.3 · 10−1 3.9 · 10−1 0.4 · 10−15.5 · 107 2.2 · 10−4 9.3 · 10−1 0.7 · 10−1 1.3 · 10−3 3.8 · 10−3 0.5 · 10−1 9.4 · 10−1 0.1 · 10−16.0 · 107 3.7 · 10−28 9.6 · 10−1 0.4 · 10−1 2.2 · 10−4 1.6 · 10−4 6.1 · 10−4 1.0 · 100 4.7 · 10−4Specimen C Specimen DLoadcycleP (π1|D) P (π2|D) P (π3|D) P (π4|D) P (π1|D) P (π2|D) P (π3|D) P (π4|D)5.0 · 107 1.2 · 10−1 3.9 · 10−1 4.5 · 10−1 4.5 · 10−2 6.5 · 10−1 5.3 · 10−2 2.7 · 10−1 1.8 · 10−25.5 · 107 1.6 · 10−2 5.1 · 10−1 4.7 · 10−1 5.1 · 10−3 7.3 · 10−1 1.5 · 10−2 2.5 · 10−1 1.3 · 10−36.0 · 107 1.6 · 10−3 5.5 · 10−1 4.4 · 10−1 4.9 · 10−4 6.8 · 10−1 5.8 · 10−3 3.1 · 10−1 1.2 · 10−4Specimen E Specimen FLoadcycleP (π1|D) P (π2|D) P (π3|D) P (π4|D) P (π1|D) P (π2|D) P (π3|D) P (π4|D)5.0·107 2.4 · 10−3 7.4 · 10−1 2.1 · 10−1 4.1 · 10−2 2.5 · 10−1 2.5 · 10−1 4.7 · 10−1 4.0 · 10−25.5 · 107 1.3 · 10−5 9.3 · 10−1 7.1 · 10−2 2.0 · 10−3 8.2 · 10−2 2.9 · 10−1 6.2 · 10−1 5.3 · 10−36.0 · 107 5.2 · 10−8 9.6 · 10−1 3.6 · 10−2 2.0 · 10−4 1.4 · 10−2 1.9 · 10−1 7.9 · 10−1 5.2 · 10−4domness and nonlinearity. Second, the ground-truth of each specimen isdiverse and close to different priors, resulting in the diverse performance ofeach method in different cases.Besides, It can be seen that for specimen D, method III significantly out-performs others with the highest CRA and lowest RMSE. This phenomenonis not surprising. Specifically, among a set of candidate priors, if a specificprior is very close to the ground truth than others, the prediction based onthis single prior will be more accurate than the prediction based on a hy-brid prior. It is because integrating other unpromising candidate priors willincrease the prediction error. However, in practical engineering, which prioris the most promising one is unknown. Integrating all the candidate priorscan increase the robustness of prediction and avoid the large prediction errorcaused by poor prior choice. This strategy is important, as a large predic-tion error may mislead the maintenance action and induce a catastrophicresult.Since this study focuses on the robustness of the FCG prediction, theaverage CRA and average RMSE are taken as evaluation criteria. It canbe seen that the proposed method achieves the best prediction performancewith the highest average CRA, 0.932 and the lowest RMSE, 7.03 ∗ 106. Thereason why the proposed method outperforms method I-IV is that integrat-ing multiple promising priors can increase the robustness of FCG predic-tion. Moreover, the proposed method also outperforms method V in termsof average CRA and RMSE, because of adjusting the PBPD of each priordistribution in the hybrid prior.59Specimen A(a-1) (a-2) (a-3)(b-1) (b-2) (b-3)log CLlog CLlog CLlog CLlog CLlog CL150100500-20 -22 -24 -26 0 1 2 3 4150100500-20 -22 -24 -26 0 1 2 3 4150100500-20 -22 -24 -26 0 1 2 3 4150100500-20 -22 -24 -26 0 1 2 3 4150100500-20 -22 -24 -26 0 1 2 3 4150100500-20 -22 -24 -26 0 1 2 3 4m m mmmmFigure 3.7: (a-1)(a-2) and (a-3) show the initialization of particles at loadcycle 5.0·107, 5.5·107 and 6.0·107 respectively for specimen A. (b-1)(b-2) and(b-3) show the initialization of particles at load cycle 5.0 · 107, 5.5 · 107 and6.0 · 107 respectively for specimen B [9]. Image reproduced with permissionby Elsevier.3.3.3 DiscussionIn the proposed method, the observations are used twice as shown in Fig. 3.1.Specifically, the observations are used in Bayesian inference A to calculatethe PBPD of each candidate prior. Besides, the observations are also fedinto Bayesian inference B for inferring posterior distribution. Thus, the priordistribution of Bayesian inference B is also affected by the observations. Inthis regard, this design is a form of empirical Bayesian analysis [119]. Inempirical Bayesian analysis, the prior distribution is also affected by theobservations. Even though this design philosophy has been widely used inBayesian analysis with various applications, the debate between the classicBayesian analysis and empirical Bayesian analysis has been going on foryears. Readers can refer to [119] for more details about this debate. In thisdiscussion, the motivation for adopting empirical Bayesian analysis in thisstudy is presented. This motivation is explained by comparing the proposedmethod with two schemes which only comprise Bayesian inference A and Brespectively as shown in Fig. 3.10.For scheme 1 shown in Fig. 3.10(a), given a set of candidate priors withsubjective beliefs, the Bayesian inference A is used to calculate the PBPD of60mπ1π2π3π4Specimen ASpecimen Blog CLlog CLlog CLmlog CLlog CLmlog CLUpdate at 5.0·107 cycleUpdate at 5.5·107 cycleUpdate at 6.0·107 cycleUpdate at 5.0·107 cycleUpdate at 5.5·107 cycleUpdate at 6.0·107 cycle(a-1) (a-2) (a-3)(b-1) (b-2) (b-3)log C-20 -22 -24 -26 0 1 2 3 4L1501005000 1 2 3 4L150100500150100500-20 -22 -24 -26-20 -22 -24 -26 m0 1 2 3 4150100500-20 -22 -24 -26 0 1 2 3 4150100500-20 -22 -24 -26-20 -22 -24 -26-20 -22 -24 -260 1 2 3 40 1 2 3 40 1 2 3 4150100500150100500m mmmFigure 3.8: Posterior distribution of specimens A and B at different loadcycles [9]. Image reproduced with permission by Elsevier.Table 3.6: CRA of different methods for the RUL prediction.Method I II III IV V ProposedSpec. A 0.923 0.965 0.891 0.969 0.972 0.974Spec. B 0.933 0.640 0.973 0.853 0.732 0.981Spec. C 0.890 0.875 0.771 0.927 0.866 0.942Spec. D 0.724 0.393 0.978 0.739 0.801 0.765Spec. E 0.862 0.960 0.862 0.945 0.969 0.971Spec. F 0.943 0.630 0.913 0.902 0.976 0.958Aver. 0.879 0.744 0.8982 0.889 0.886 0.932each candidate on the basis of observations. Then the posterior distributioncan be obtained by sampling all the priors according to their PBPD asaforementioned in Fig. 3.2. In this scenario, the promising prior can besampled more frequently than the unpromising one. However, scheme 1cannot guarantee the posterior distribution converges to the ground truth.In the extreme case, the posterior distribution can only converge to the mostpromising prior when its PBPD approaches 100%.For scheme 2, given a set of candidate priors, the Bayesian inference Bis used to calculate the posterior distribution using PF, where each prior issampled with equal probability. In this scenario, scheme 2 is the same asthe method V that is constructed in Section 3.3.2. Even though scheme 2can guarantee the posterior distribution converges to the ground truth, it61illustrate specimen B,D,FRUL (cycles) RUL (cycles) RUL (cycles)RUL (cycles) RUL (cycles) RUL (cycles)FrequenceFrequenceFrequenceFrequenceFrequenceFrequence(a-1) (a-2) (a-3)(b-1) (b-2) (b-3)Ground-truth Ground-truth Ground-truthGround-truthGround-truthGround-truth9008007006005004003002001005.5 6 6.5 7 7.5 8 8.5×1079008007006005004003002001009008007006005004003002001007006005004003002001000.5 0.7 0.9 1.1 1.3 1.55.5 6 6.5 7 7.5 8 8.5 5.5 6 6.5 7 7.5 8 8.5×107 ×107 ×107×1081400120010008006004002007006005004003002001000.5 0.7 0.9 1.1 1.3 1.5×1080.5 0.7 0.9 1.1 1.3 1.5×108800Figure 3.9: Predicted RUL distribution of specimens A and B with hybridprior. The histogram shows the distribution of particles. The red line is thesmoothing fitting curve of the histogram. (a-1,2,3) are results for specimenA at load cycle 5.0·107, 5.5·107 and 6.0·107 respectively. (b-1,2,3) are resultsfor specimen B at load cycle 5.0 · 107, 5.5 · 107 and 6.0 · 107 respectively [9].Image reproduced with permission by Elsevier.has two inherent weaknesses: (1) Scheme 2 does not take into account theinformation that candidate priors come from different information sources.From the perspective of Monte Carlo, scheme 2 overlooks the informationthat particles come from different groups. An illustrative example is shownin Fig. 3.11(a) where the particles from different priors are not distinguished.However, in practical engineering, when candidate priors come from differ-ent sources, the reliabilities of different sources (such as estimations fromdifferent experts, different experimental tests and simulation software) areusually distinguished. It is necessary to consider this group-level differenceof particles. (2) Scheme 2 treats each candidate prior fairly and does notenable the assignment of subjective belief regarding each candidate prior.In real-world applications, it is necessary to assign a higher subjective beliefto more reliable/promising prior.Given the weaknesses of scheme 1 and scheme 2, it is motivated to com-bine them together, resulting in an empirical Bayesian framework. Com-pared with scheme 1, the proposed method can guarantee the posterior62Table 3.7: RMSE of different methods for the RUL prediction.Method I II III IV V ProposedSpec. A 5.58 ∗ 106 2.44 ∗ 106 7.69 ∗ 106 2.17 ∗ 106 2.00 ∗ 106 1.97 ∗ 106Spec. B 5.85 ∗ 106 2.73 ∗ 107 2.24 ∗ 106 1.11 ∗ 107 2.03 ∗ 107 2.04 ∗ 106Spec. C 1.18 ∗ 107 1.33 ∗ 107 2.41 ∗ 107 9.03 ∗ 106 1.42 ∗ 107 7.74 ∗ 106Spec. D 2.77 ∗ 107 5.91 ∗ 107 2.24 ∗ 106 2.54 ∗ 107 2.33 ∗ 107 2.37 ∗ 107Spec. E 1.04 ∗ 107 3.00 ∗ 106 1.04 ∗ 107 4.15 ∗ 106 2.33 ∗ 106 2.27 ∗ 106Spec. F 6.58 ∗ 106 3.47 ∗ 107 8.18 ∗ 106 9.25 ∗ 106 3.30 ∗ 106 4.43 ∗ 106Aver. 1.13 ∗ 107 2.33 ∗ 107 9.14 ∗ 106 1.02 ∗ 107 1.09 ∗ 107 7.03 ∗ 106distribution converges to the ground truth. Compared with scheme 2, theproposed method considers the group-level difference of particles as shownin Fig. 3.11(b). Mathematically, this consideration is implicitly included inthe Monte Carlo integration for particles with the same prior. Moreover,the proposed method also enables the assignment of subjective beliefs forcandidate priors, to distinguish their reliability before inference.3.4 SummaryThis study proposed a new FCG prediction method which enables the inputof hybrid prior. Two inference steps are included in the proposed method.In the first inference step, the observation is used to adjust the posteriorbelief of each candidate prior. In particular, the Monte Carlo integrationis adopted for calculating the likelihood of each prior. Then the posteriorbelief of each prior distribution can be acquired through Bayesian inference.In the second inference step, PF is extended to sample the hybrid prior,where the posterior is approximated based on importance sampling andinverse CDF method. The effectiveness of the proposed method is verifiedby a numerical study, where the crack propagation at the root of a spurgear is studied. The results show integrating multiple prior distributionscan increase the robustness of the FCG prediction. The proposed method isapplicable in practical engineering when multi-source prior knowledge (suchas expert experience, experimental investigation, software simulation, andsensor data) is available in the FCG prediction.63Bayesian inference AObservationPosteriordistributionFCG predictionCandidate priors &Subjective beliefsCandidate priors &Posterior beliefsMC integrationBayesian inference BObservationPosteriordistributionFCG predictionCandidate priorsPF(a) Scheme 1 for comparison.(b) Scheme 2 for comparison.Figure 3.10: Two schemes for comparison [9]. Image reproduced with per-mission by Elsevier.(a) (b)Prior 1Prior 2Prior 3(b)Prior 1Prior 2Prior 3Figure 3.11: An illustrative example. (a) Scheme 2 does not take into ac-count the information that candidate priors come from different informationsources. (b) Proposed method considers the group-level difference (indicatedusing color) of particles by the Monte Carlo integration [9]. Image repro-duced with permission by Elsevier.64Chapter 4Probabilistic Model forFatigue Crack GrowthPrediction ConsideringCrack Location Randomness4.1 OverviewExisting methods for the FCG prediction are generally conducted with aninitial flaw with the assumption that the crack location is known. However,in real-world application, the crack location is also random to some degreedue to the impact of microstructure flaw, residual stress, geometry variation,and so on. To tackle this issue, this chapter proposes a probabilistic modelfor the FCG prediction considering crack location randomness. In particular,the crack location randomness is quantified based on the localization of theright and left crack tips using Bayesian inference with a modified Paris-Erdogan equation. Fig. 4.1 shows the overview of the proposed method.Specifically, the guided-wave sensor network is adopted as the NDI tech-nique. At each update time, the guided-wave sensor network will be trig-gered to collect the sensory data from the inspected structure. These sensorydata go through a signal processing model which localizes the left and rightcrack tips. In particular, the probability of detection (POD) and measure-ment accuracy (MA) of the guided-wave sensor network are also quantified.The findings of crack tips are then fed into Bayesian inference to infer theposterior distribution of the random variables of individual structure witha pre-specified prior. Here, the Bayesian inference is specified as the statespace model which comprises a state transition function and a measurementfunction. The measurement function is derived based on the crack tip find-ings, while the state transition function is used to estimate the FCG state.On the other hand, a FE model is built to calculate the SIF on the cracktips under a specific loading condition. Then, a surrogate model is estab-653Future load conditionMaterial parameters Stress intensityfactorPhysics-based modelThe methodology for the implementation.Signal processing modelUpdate timeSurrogate modelFinite element modelDiagnosis&PrognosisNext update timeGuided-wavesensor network 2.2 Implementation of the pipeline digital twin Prior distributionLoad conditionanalysisMeasurement accuracyProbability of detectionCrack tip localizationUserMeasurement functionState transitionfunctionBayesian inferenceIf updatingUpdate Self-evolving InteractionTraining samplesFigure 4.1: Overview of the proposed method. © Teng Wanglished based on the FE model, to avoid the prohibitive computational costof the FE model. A specific data-driven model is used to predict future loadcondition based on historical load observation. The predicted future loadcondition is subsequently combined with the data-driven surrogate modelfor calculating the SIF on each crack tip, given the crack configuration. Asa result, the defect diagnosis and prognosis can be realized with the physics-based model. The next update time will be scheduled for reactivating theguided-wave sensor network.4.2 Experiment SetupThe application of pipeline FCG prediction is specified in this chapter. Threepipes (Pipe A, Pipe B, Pipe C) under cyclic load are tested in this study [10].The cyclic load is simulated from the four-point bending machine as shownin Fig. 4.2. In real-world application, the cyclic load is generally caused bypressure variation [120]. The operational load/force is measured at the hy-draulic unit which applies the load over the steel beam and the two loadingpins into the pipe. Fig. 4.3 shows the diagram of these pipes under consid-eration. These pipes are made of the material of S235 with a length of 3000mm, and external diameter of 197.3 mm. The wall thickness is 5 mm.In practical engineering, since most pipelines are connected by girthwelding, an off-center girth weld line is created in the pipeline by metal-66active-gas (MAG) welding. Due to the weld, it is expected that a circumfer-ential crack arises in the welded joint and grows along the weld. For somecritical applications such as nuclear piping system, it is necessary to deployin-situ sensors in vulnerable spots to acquire the defect sensory data. Forthis result, the guided-wave sensor network is deployed to acquire the defectinformation near the weld line in this chapter. Its location relative to thepipe can be found in Fig. 4.3 and Fig. 4.4. It can be seen from Fig. 4.3 thatthe measurement setup comprises three independent rings that are mountedover the circumference of the pipe.Figure 4.2: Experimental setup for pipe fatigue test under four-point bend-ing load [10]. Image reproduced with the permission from the FraunhoferInstitute for Ceramic Technologies and Systems (IKTS), Dresden, Germany.4.3 Methodology4.3.1 Physics-Based ModelAs the through-thickness fatigue crack is focused on in this study, thephysics-based model is specified as the FCG model. Without loss of gener-ality, the well-known Paris-Erdogan equation [1] is adopted as,dadn= C∆Km (4.1)673005005002483197.32940620 85014705Welding seam Sensor ring 1Sensor ring 2Sensor ring 3Figure 4.3: Diagram of the experiment configuration. Image reproducedwith the permission from the Fraunhofer Institute for Ceramic Technologiesand Systems (IKTS), Dresden, Germany.where a is the crack length, and n is the load cycles imposed on the pipe.Parameters C and m are material-dependent constants. ∆K is the so-calledSIF range. It describes the stress range in the region near the crack tip as,∆K = Kmax −Kmin (4.2)where Kmax and Kmin are the maximum and minimum SIF induced by theapplied load in one cycle. The SIF K is related to structure geometry, cracklength, crack shape and stress level as,K = σY√πa (4.3)where Y is the correction factor of stress intensity, which reflects the geom-etry of the component and crack, and σ is the stress level.Since the Paris-Erdogan equation is in differential form and cannot beintegrated, the first-order Euler method [105] is used to discretize it as,a(i+1)∆n = ai∆n +∆n[C(∆K)m] (4.4)where ∆n is the incremental cycle interval and ai∆n is the crack length afteri∆n cycles.Previous studies [121, 122] found the fronts of through-thickness crackturn to be straight, and their extensions run through the center of the pipecross section as shown in Fig. 4.5 (a). Thus, the location and shape ofthe through-thickness crack can be indicated by two crack tips on the pipe68Figure 4.4: Guided-wave sensor network deployment. Image reproducedwith permission from the Fraunhofer Institute for Ceramic Technologies andSystems (IKTS), Dresden, Germany.external surface. Two equations in the form of (4.1) are needed to describethe crack propagation, one for each crack front as,daLdn= C(∆KL)m,daRdn= C(∆KR)m,(4.5)where ∆KL and ∆KR are the ranges of the SIF at the left and right cracktips.However, the Paris-Erdogan equation requires the semi-crack length,meaning that the asymmetrical splitting of the crack makes little sense.For this issue, an averaged semi-crack prediction model is adopted as,a(i+1)∆n = ai∆n + C[(∆KLi∆n)m + (∆KRi∆n)m]∆n2. (4.6)To be intuitive, the FCG prediction is conducted in the polar coordinatesystem shown in Fig. 4.5 (b). Under the polar coordinate system, (4.6) canbe written as,θL(i+1)∆n − θR(i+1)∆n =θLi∆n − θRi∆n + C[(∆KLi∆n)m + (∆KRi∆n)m]∆n2∗ 180πr(4.7)where r is the external radius of pipe. θRi∆n and θLi∆n are the locations ofthe right tip and left tip at load cycle i∆n in the polar system. ∆KLi∆n690°30°60°90°120°150°180°210°240°270°300°330°Left Right(a) (b)Figure 4.5: (a) Cross section of the pipe with a through-thickness crack. (b)Polar coordinate system adopted in this study. © Teng Wangand ∆KRi∆n are the SIF ranges at locations θLi∆n and θRi∆n respectively. Thecrack growth for each crack tip is calculated as,θL(i+1)∆n = θLi∆n + C[(∆KLi∆n)m]∆n2∗ 180πr, (4.8)θR(i+1)∆n = θRi∆n − C[(∆KRi∆n)m]∆n2∗ 180πr. (4.9)Note that the growths of left and right crack tips are dependent. The changeof θRi∆n will affect the values of both ∆KRi∆n and ∆KLi∆n, and vice versa.It is notable that the variance of material parameters m and C in theParis-Erdogan equation is well-known [1, 8, 22]. Many studies showed fa-tigue crack experimental data display a large scatter in the crack size ata given cycle and the number of cycles to reach a given crack size. Thislarge scatter occurs even under a carefully controlled environment, which iscaused by the variance of material parameters m and C. Thus, m and Care considered as statistically correlated random variables in this study. Aprior distribution is assigned to m and C. It is worth mentioning that theresidual stress in the welding area is not modeled. Since the residual stresswill change the value of m and C in the Paris-Erdogan equation [42], theinfluence of residual stress is considered by tracking the change of m and Cduring the FCG prediction.Reference [123] estimated the m and C of the S235, which is the ma-terial of the tested pipeline in this study. The evaluation result showed70m = 1.23, logC = −15.75. However, it is difficult to construct the priordistribution based on the single-point estimation.For this problem, a hierarchical prior distribution is adopted. It is as-sumed that both m and logC follow independent Gaussian distributions asm ∼ N (µm, σm) and logC ∼ N (µlogC , σlogC). Note that even though manystudies show m and logC are highly related, it is assumed that they areindependent in the initial assumption due to the lack of their correlation in-formation in this study. However, m and logC will correlate in the posteriordistribution as they are updated based on the same observation [8].Specifically, it is set that µm = 1.23 and µlogC = −15.75 according to[123]. Since there is no prior knowledge regarding σm and σlogC , they areassigned with a hyper prior respectively as σm ∼ HN (µhm = 0, σhm = 0.1)and σlogC ∼ HN (µhlogC = 0, σhlogC = 0.15) where HN (µh, σh) stands forthe half-normal distribution with mean value µh and standard deviation σh.Thus, σm and σlogC can be estimated from the observation. Here σhm andσhlogC are empirically set. In this study, σhlogC is given with a higher valuethan σhm, since logC usually has a higher variance than m.4.3.2 Non-Destructive Inspection Signal ProcessingNDI signal processing is essential for localizing the crack tips and predict-ing the FCG. In this study, the guided-wave sensor network is deployed toacquire the defect sensory data. The type of sensor is a piezoelectric trans-ducer with the model DuraAct P876-SP1. The 10mm x 10mm x 0.2 mmlarge ceramic plates are embedded in a layer of polyimide under compressivestress. This means that the converters are flexible and can adjust accordingto pipe geometry without breaking. Fig. 4.6 (a) shows the layout of thesesensors. In particular, the purple dotted line and the green dotted line cor-respond to the top and the bottom of the pipe, respectively, as shown inFig. 4.6 (b).Each time when the sensor network is activated, sensors 1-15 are excitedconsecutively, and all sensors are receiving. As a result, full-matrix capture(FMC) data can be obtained. These data are processed based on the totalfocusing method (TFM) [11]. TFM is the systematic application of the basicfocusing principle of the phased array in a defined region of interest (ROI)in an inspected part. The ROI is segmented into a grid of positions, or“pixels”, and TFM is applied to every pixel in that grid. The schematicdiagram of the TFM applied on FMC is shown in Fig. 4.7.71992LoadCircumferential axis [m]Length axis [m]1 1.5 2 2.50 0.2 0.4 0.6BottomBottomTopTop(a) (b)Sensor 1Sensor 15Figure 4.6: (a) The layout of the guide-wave sensors for acquiring the defectsensory data. (b) The positions of the purple dotted line and the greendotted line in the cross section of pipe. © Teng WangThe definition of TFM is as follows,Ip (xp, zp) =N∑g=1N∑j=1Agj (tgj (xp, zp)) (4.10)where (xp, zp) denotes the coordinate of scatter p, Agj represents an ampli-tude intensity of scatter p, which is excited by sensor namely g and receivedby the corresponding sensor j. The time of flight is tgj (xp, zp), it starts fromthe sensor g, then passes through the scatter p, and finally arrives at thesensor j. The sensor number applied for the TFM is N . The time of flighttgj (xp, zp) can be expressed as,tgj (xp, zp) =√(xg − xp)2 + z2p +√(xj − xp)2 + z2pc(4.11)where c indicates the ultrasonic velocity and Cartesian coordinates of thesensors (sender and receiver) are (xg, 0) and (xj , 0), respectively. t0 (zp) =zp/c is defined as the normal incidence one-fight time, which is from ithultrasonic sensor to the scatter p of the single medium. The time of flightcan be rewritten as:tgj (xp, zp) =√t20 (zp) +(xg − xp)2c2+√t20 (zp) +(xj − xp)2c2. (4.12)Fig. 4.8 (a)(b) and (c) show the FMC images for Pipe A at differentstages. Fig. 4.8 (a) is the FMC image where no crack is detected. Fig. 4.8721012... ...X1 g j N... ... ... Z0p(xp, zP)(xg, 0) (xj, 0)Figure 4.7: The schematic diagram of the TFM [11]. Image reproducedunder the Creative Commons Attribution Attribution 4.0 International Li-cense.(b) is the FMC image when the crack is just detected. Fig. 4.8 (c) is theFMC image where the defect area is enlarged due to the growth of the crack.In particular, the weld seam position on the image is highlighted using a reddashed line. We can see the occurrence of crack is reflected in the TFMresult, and its size is growing with the increase of loading cycles. To getthe point-wise estimation of the left and right crack tips, a high-pass filteris adopted to convert the FMC data into a binary image. In particular,the Ip (xp, zp) above the threshold 80% ∗ Imax will be classified as the defectarea where Imax is the maximum value in the ROI of the FMC image. Asa result, the left/right crack tip can be localized as the left/right end of thedefect area. Meanwhile, the crack length is recognized as the length betweenthe right and left crack tips.The performance of NDI techniques (guided-wave sensor network in thischapter) is affected by many uncertain factors. Several probabilistic metricshave been developed to evaluate the performance of NDI in the literature. Inthis study, MA is used to quantify the uncertainty in the measurement fromthe sensor network. For modeling the MA, a regression-type expression [85]is adopted as,θ˜ = βθ0 + θ × βθ1 + ξθm, ξθm ∼ N (0, (σθ)2) (4.13)where θ˜ is the crack tip location measured from the guided-wave sensornetwork, and θ is the actual crack tip location. In the experiment, the actual73932Weld seam0 0.1 0.2 0.3 0.4 0.5edFixture induced Fixture inducedLength axis (m)(a) (b) (c)Circumferential axis (m)Circumferential axis (m)Circumferential axis (m)Length axis (m)Length axis (m)Weld seam Weld seamWeld seam21.91.81.71.61.51800160014001200100080021.91.81.71.61.51.41.21.00.8×1041.21.00.80.6×1040 0.1 0.2 0.3 0.4 0.5ed 0 0.1 0.2 0.3 0.4 0.5ed1.91.81.71.61.52Figure 4.8: (a) FMC image when no crack is detected. (b) FMC image whenthe crack is just detected. (c) FMC image where the defect area is enlargeddue to the growth of the crack. © Teng Wangcrack tip is measured using the tapeline aided with plump bob. The tapelineis with the resolution of 1 mm. βθ0 and βθ1 are the regression coefficients, ξθmrepresents a normal variable with zero mean and standard deviation σθ.For a specific NDT technique, its performance is usually estimated fromlaboratory tests where the θ˜ and θ are known. Note that (4.13) comprisesthree unknown parameters as βθ0 , βθ1 and σθξ , which has a high risk of over-fitting with the limited samples of θ˜ and θ. Thus, it is assumed that βθ1 = 1in this study, and only estimate βθ0 and σθϵ from the laboratory test. Fig. 4.9shows the relationship between θ˜ and θ for pipe A, B and C. The estimationresult based on the method of maximum likelihood shows βθ0 = 3.92 andσθ = 10.63.In addition to MA, the POD is also an important performance indicatorfor the NDI techniques. It expresses the probability of detecting a flaw of agiven size. Traditionally, the POD curve for a particular NDI method canbe obtained by introducing flaws of various sizes into test specimens. Then,the POD can be obtained by calculating the ratio of the number of flawsdetected to the number of flaws actually existing. However, this strategyrequires a large number of tests to be conducted in the lab, which will induceprohibitive cost. For this issue, an alternative strategy proposed in [85] isadopted in this case study. Reference [85] revealed that the POD and sizeMA are related. The POD can be derived once the crack length measurementaccuracy is known. According to this, crack length measurement accuracyis modeled as,a˜ = βa0 + a× βa1 + ξam, ξam ∼ N (0, (σa)2) (4.14)74Reference line x=yInspection resultPODActual crack tip location (degree)Measured crack tip location (degree)Crack length a (mm)(a) (b)Actual crack length (mm)a100 140 180 220 260 1001401802202600 0.5 1 1.5 2 2.5 3 3.5 4 10.90.80.70.60.50.4Figure 4.9: (a) The relationship between θ˜ and θ collected during the testof pipes. (b) The estimated POD of the sensor network. © Teng Wangwhere a is the actual crack length, a˜ is the measured crack length, βa0 and βa1are constants. ξam represents a normal variable with zero mean and standarddeviation σa. Similar to (4.13), βa1 = 1 is set to avoid over-fitting. Parameterβa0 and σa are estimated by fitting a˜ and a using the method of maximumlikelihood.On the other hand, the probability that a crack with length a is detectedcan be expressed as,POD(a) = P (a˜ > 0|a). (4.15)Intuitively, (4.15) shows POD(a) is the probability that a positive cracklength is measured given the actual crack length a. Combining (4.15) and(4.14). the following equation can be obtained,POD(a) = P (βa0 + a× βa1 + ϵam > 0|a)= P (−ϵam < βa0 + a× βa1 )= P (ϵam > −(βa0 + a× βa1 )).(4.16)Since ϵam follows a zero-mean normal distribution, the following equation75holds,POD(a) = P (ϵam > −(βa0 + a× βa1 ))= P (ϵam < βa0 + a× βa1 )= Φ(βa0 + a× βa1σa).(4.17)The estimated POD of the sensor network is shown in Fig. 4.9(b). It canbe seen that the POD approaches 100% when the crack length is larger than3 mm. Since the pipes considered in this study are with a wall thickness of5 mm, it is reasonable to believe the deployed sensor network can detect allkinds of through-thickness cracks in the pipe.The NDT results will be subsequently fed into the Bayesian inference forupdating the digital twin. Given θ˜R and θ˜L, the measured locations of theright and left crack tips, the likelihood of the actual crack tip locations θRand θL can be derived from (4.13) as,Li(θR) =1√2πσθexp(−12(θ˜R − βθ0 − θAσθ)), (4.18)Li(θL) =1√2πσθexp(−12(θ˜L − βθ0 − θAσθ)). (4.19)Note that existing studies usually use the measured crack length forBayesian inference. This strategy is justified for pro-notched specimen.However, in practical engineering, the cracks may with the same length butin different locations. Thus, this study conducts Bayesian inference basedon the location of crack tips. Also, the crack length can be calculated giventhe left and right crack tips. Moreover, in this case study, the MA and PODof pipe A, pipe B and pipe C are obtained by cross validation. That is whenpipe A is tested, the MA and POD of the sensor network are estimated bypipe B and pipe C, and vice versa.4.3.3 Finite Element AnalysisIn the FE analysis, the physical entity is discretized into a number of ele-ments of finite dimensions. The elements are interconnected at their nodalpoints. Field variables (such as displacements in the case of displacementfinite element formulation) are associated with each of these nodes.76Recall that the predictive models in (4.8) and (4.9) comprise ∆KL and∆KR which are the SIF ranges in the left and right crack tips respectively.In general, the SIF range is calculated using either closed-form solution orFE model.Even though several closed-form solutions were proposed for calculatingthe SIF range of through-thickness crack for the pipeline under bendingload, all of them are built based on a pro-notched symmetrical crack withthe assumption that the left crack tip and right crack tip are symmetricalwith respect to the vertical axis. This assumption is idealistic. In a real-world scenario, it is hard to guarantee the crack is initialized exactly inthe stress concentration point due to the material microstructure effect andresidual stress. See Fig. 4.5 (b) for example, if the crack is not initializedright underneath the pipeline (θ = 180◦), the SIF range in the left and rightcrack tips will be different, causing different crack propagation rates in twotips. Thus, the though-thickness crack can be asymmetrical. No closed-formsolution is available for calculating θL and θR in the literature.In this context, the SIF range is calculated from the FE model as shownin Fig. 4.10. First, the full model of the pipeline is established usingANSYS® where the tetrahedron method is adopted for meshing. The meshsize for the global model is 20 mm. The area of interest is re-fined using themesh with the size of 5 mm. Then, the submodel is extracted and analyzedwith Franc 3D®. The through-thickness crack is inserted using Franc 3D®where 0.05 ∼ 0.5 mm mesh is used to discretize the crack front. The SIF inthe crack front is calculated using M-integral due to its high accuracy [124].To validate the accuracy of the developed FE model. The SIF calcu-lated from the FE model and the closed-form solution reported in the lit-erature [125] are compared. Fig. 4.11 shows the comparison result. Thesimilarity between FE solution and closed-form solution is measured viaCRA [109], which achieves 95.14%.This result validates the effectiveness of the developed FE model. It isworth mentioning that the closed-form solution is only available for sym-metric through-thickness crack. In contrast, the FE model can be used inevaluating crack with any shape and location, which provides a more generalsolution in the SIF calculation. Moreover, in this study, the stress analysisis constrained in the framework of linear elastic mechanics. As a result, ∆Khas a linear relation with the load. By following this convention, θL and θRare obtained under base loading condition. Given a new loading condition,the SIFs in both crack tips are calculated by multiplying the derived ∆Kwith a ratio between the new loading condition and the baseline loadingcondition.77452 METHODOLOGIES2.3 FCG prediction towards digital twin ecosystem: a case studyThe FE model is built based on commercial software Ansys and Franc3D：Pipeline database -- finite-element modelFigure 4.10: FE model of the pipeline. The global model is built us-ing ANSYS®. The submodel is extracted and analyzed with Franc3D®. © Teng Wang4.3.4 Surrogate ModelFor the probabilistic analysis, the θL and θR should be calculated at leastthousands of times in each update based on the Monte Carlo analysis. Thus,using the FE model in the probabilistic analysis is computationally pro-hibitive and prevents the real-time update of the digital twin ecosystem.For this issue, a physics-informed data-driven model is built in this studyas a surrogate of the FE model for calculating ∆KL and ∆KR. Specifi-cally, the crack with different θL and θR at the welding seam is repeatedlycreated. Then ∆KL, ∆KR can be obtained from the FE analysis under abaseline loading condition. Finally, a data-driven model is adopted to buildthe mapping relationship of {θL, θR} −→ {∆KL,∆KR}.To achieve this end, the first step is generating the training samplesfrom the FE analysis. It is well-known that the data-driven model has abetter performance in interpolation than extrapolation. Thus, to ensure theaccuracy of the data-driven model, the training samples should be close tothe testing samples. In this case study, given the experimental configurationin Fig. 4.2 and polar coordinate system in Fig 4.5(b), the following points arenoticed first: (1) It is more likely that two crack tips are located around thestress concentration point (i.e., θ = 180◦). Thus, the following constraintis given: θL, θR ∈ [90◦, 270◦]; (2) It is intuitive that θL > θR; (3) Sincea large open angle of the through-thickness crack leads the total failure ofthe pipeline, the open angle of the through-thickness crack is constrainedas θL − θR ≤ 170◦. Constraints (1)-(3) indicate the uncertainty region78SIF (MPa mm^0.5)Open angle (degree)Open angleFE left crack tipFE right crack tipClosed-form solution 30 40 50 60 70 80 90 100 110200040006000800010000Figure 4.11: Comparison of SIF factors calculated from the closed-formsolution and FE model. © Teng Wangof two crack tips shown in the center of Fig. 4.12. As aforementioned, itis expected that the data-driven model does the interpolation during theprediction. Thus, the training samples are generated by grid sampling tooccupy the region. Besides, some samples are arbitrarily generated in thehigh-probability area in which a through-thickness crack may lie. Thesearbitrarily generated samples are used as validation samples to quantify theprediction error of the data-driven model. Totally, 218 samples are generatedby repeatedly running the FE model with different through-thickness cracks.168 samples (grid samples) are used for training and 50 samples (arbitrarilygenerated samples) are used for validation.Fig. 4.13 shows the testing samples generated from the FE software.It can be seen that both ∆KL and ∆KR are highly related to θL, θR andθL−θR in a nonlinear manner. To model their relationship, two data-drivenmodels are investigated.The first one is the artificial neural network (ANN), as it is known tobe a universal function approximator. Fig. 4.14 shows the diagram of ANNwith three hidden layers. In this study, the ANN architecture is made upof a three-neuron input layer, h hidden layers, and two output layers. TheANNs with h = 1, 2, 3 are tested in this study. Each hidden layer comprisess neuron. The ANNs with s = 50, 10 are tested in this study. The activa-tion function is ReLU. The loss function ruling the backpropagation is themean absolute error (MAE). An adaptive moment estimation optimizationalgorithm is used for the weights and biases updating. The training phaseis stopped when the prediction error on the validation set increases for eightconsecutive training epochs. The MAE of ANN prediction on the testingdata set is shown in Table 4.1.79* Training samples Validation samples Left tip (degree)0 50 100 150 200 250 300 3500Right tip (degree)50100150200250300350Figure 4.12: Training samples and validation samples generated from theFE model. © Teng WangTable 4.1: Prediction performance of ANN. (h, s) denotes the ANN with hhidden layers and each hidden layer has s neurons.ANN (1,50) (1,100) (2,50) (2,100) (3,50) (3,100)MAE 2216 1108 850 623 676 643The second one is the k-nearest neighbor (KNN). The KNN predictsthe testing sample by averaging its k-nearest neighbors. The reason forselecting KNN in this study is two-fold. First, compared to other data-driven models, KNN has an edge in interpretability. Second, the trainingsamples are generated by grid sampling to occupy the region in which testingsamples may lie. Hence, the distances between each testing sample and itsk-nearest training samples will be very small. It is effective to predict thetesting sample based on its k-nearest neighbors. In this study, the KNNwith k= 1, 2, ..., 6 are tested respectively. The MAE of KNN prediction onthe testing data set is shown in Table 4.2.From Table 4.1 and Table 4.2, KNN outperforms ANN. The possible rea-son is that ANN is suffered from overfitting when handling low-dimensionaldata. In contrast, the grid-sampling strategy makes sure the training sam-ples occupy the whole uncertainty region. Thus, it is effective to predict thetesting sample based on its KNN. In particular, the KNN achieves the best80Table 4.2: Prediction performance of KNN.KNN k=1 k=2 k=3 k=4 k=5 k=6MAE 258 125 128 110 113 131θR (Degree)△KL (MPa mm^0.5)△KR (MPa mm^0.5)θL (Degree)θL (Degree)(a-2)(a-1)△KL (MPa mm^0.5)△KR (MPa mm^0.5)(b-1)(b-2)θR (Degree)(c-2)θL -θR (Degree)(c-1)θL -θR (Degree)△KL (MPa mm^0.5)△KR (MPa mm^0.5)100 140 180 220 260 12×1031086420100 140 180 220 260 12×103108642080 120 160 200 240 280 12×103108642080 120 160 200 240 280 12×10310864200 40 80 120 160 12×10310864200 40 80 120 160 12×1031086420Figure 4.13: (a-1) (b-1) (c-1) are the relationship between ∆KL and θL,θR and θL − θR, respectively. (a-2) (b-2) (c-2) are the relationship between∆KR and θL, θR and θL − θR, respectively. © Teng Wangresult when k equals 4. Hence, the KNN with k=4 is taken as the surrogatemodel.Fig. 4.15 shows the histogram of prediction error. The Chi-square goodness-of-fit test is conducted, showing the prediction error follows a Gaussian dis-tribution at a significance level of 5%. Thus, the Gaussian distribution isadopted to fit the prediction error. In this study, this prediction error is alsopropagated into the final FCG prediction result.4.3.5 Load Condition AnalysisFor the FCG prediction, it is essential to know its future operational con-dition. In this study, the future operational condition is predicted from the81HiddenHiddenHiddenOutputHiddenInput OutputθL -θR (Degree)θL -θRθRθL... ... ...Figure 4.14: ANN for SIFs prediction. © Teng WangN (26.93, 167.062)Prediction error (Mpa mm^0.5)FrequencyPrediction error (Mpa mm^0.5)Frequency N (-24.02, 144.692)Prediction error (MPa mm^0.5)Frequency-600 -400 -200 0 200 400 6000510152025Figure 4.15: The prediction error of the surrogate model. © Teng Wangobservation of past operational condition. Thus, the loading uncertaintycan be quantified and propagated into the FCG prediction result. In thepipeline test, the nominal load imposed on the pipeline is a sinusoidal wavewith maximum force Fmax = 130 kN, minimum force Fmin = 5 kN and forcerange ∆F = 125 kN. However, due to the inaccuracy of the bending ma-chine, there is a difference between the nominal load and the actual imposedload, which is treated as loading uncertainty.The actual operational load is measured at the hydraulic unit whichapplies the load into the pipeline. Since the operation load follows a generalsinusoidal shape in this study, the force ranges are obtained by extractingthe peak and valley of each cycle. Fig. 4.16 (a) shows the nominal forcerange and actual force range for a specific time slot during the experiment.Fig. 4.16 (b) shows the frequency histogram of the actual force range. Sincethe nominal load is constant, it is reasonable to assume the variability ofthe actual force range is time-invariant.82Force range △F (kN) Cycle number(a) (b)FrequencyActual force range △F (kN) Nominal force range Actual force range0 400 800 1200 1600 2000126 124 122 120 120 121 122 123 124 125 126020406080100120Figure 4.16: (a) Nominal force range and actual force range measured at thehydraulic unit. (2) Frequency histogram of the actual force range. © TengWangFrom Fig. 4.16 (b), it can be seen that the force range distribution ismultimodal, which may be caused by the coupled errors in the force gen-eration and transmission. The Chi-square goodness-of-fit test is conducted,showing the force range distribution does not follow a Gaussian distributionat a significance level of 5%. To model this complicated uncertainty factor,the Dirichlet process [126] is introduced in this study. The Dirichlet processis a probability distribution over the space of distributions, i.e. each drawfrom a Dirichlet process is itself a distribution. A brief introduction to theDirichlet process is given in the following.In general, a probability distribution P on a set Ω is a measure. Itassigns measure one to the entire space as P (Ω) = 1. A Dirichlet processP ∼ DP (α, P0) is a measure that has the property that, for every finitedisjoint partition S1, ..., Sl of Ω,(P (S1), ..., P (Sl)) ∼ Dir(αP0(S1), ..., αP0(Sl)) (4.20)where P0 is the base probability measure on the space Ω, and α > 0 isthe precision parameter that controls how close samples from the Dirichletprocess are to the base measure P0. When α → ∞, samples from theDirichlet process approach the base measure P0.Samples (i.e., distributions) from a Dirichlet process are discrete withprobability one. Sampling a Dirichlet process P ∼ DP (α, P0) can be for-malized as,83P =∞∑u=1wuδeu (4.21)where e1, e2, ..., eu ∼ P0 and δ is the Dirac delta function. In particular wuis the weight signed to δeu and is calculated as,wu =βu, u = 1βuu−1∏v=1(1− βv), u > 1(4.22)where β1, β2, ... are samples from the Beta distribution Beta(1, α). Thestick-breaking process gives a good metaphor for constructing the weightswu and samples eu above. Readers can refer to [126] for more details of theDirichlet process.For modeling the complex uncertainty factor, it is effective to use a mix-ture model. Many studies have shown that mixtures of normal distributionsprovide a simple and effective basis for probabilistic density modeling as,f =η∑u=1wuN (µu, τu) (4.23)where η is the number of components. The integration of Dirichlet processenables η →∞, which gives a very high flexibility for modeling the complexuncertainty factor. This integration is realized by constructing a Dirichletprocess mixture model [127]. A Dirichlet process mixture model uses compo-nent densities from a parametric family and represents the mixture weightsas a Dirichlet process. In this case study, to quantify the actual force rangeuncertainty, the following Dirichlet process mixture model is constructed as,∆F ∼η∑u=1wuN (µu, τu) (4.24)where ∆F is generated from η normal distributions N (µu, τu) and η →∞.The mixture weight wu is calculated using (4.22).Since α > 0, a week prior is assigned to α as α ∼ Gamma(1, 1). More-over, µu and τu are also assigned with week priors as µu ∼ Uniform(∆Fmin,∆Fmax)and τu ∼ Uniform(0,∆Fmax − Fmin), where ∆Fmax and ∆Fmin are themaximum force range and minimum force range respectively.84Table 4.3: Weights and parameters of the first five components in the Dirich-let process mixture model.Component u = 1 u = 2 u = 3 u = 4 u = 5wu 0.328 0.207 0.208 0.138 0.113µu 123.690 121.357 122.518 121.992 122.320τu 0.358 0.274 0.537 0.690 2.873The above Dirichlet process mixture model is inferred with the actual op-erational load shown in Fig. 4.16. The posterior distribution of parametersµu, τu, wu, u = 1, ..., η and α are inferred using MCMC method [64]. Specif-ically, MCMC is the combination of the Markov chain and Monte Carlo.Markov chain is a mathematical system that is consisted of a set of statesand transition probabilities from one state to another. The main idea of theMonte Carlo algorithm is to use a set of random samples to approach a tar-get distribution. MCMC aims to construct a Markov chain that satisfies thedetailed balance condition so that the sampling process will be a state moveprocess on this chain. In this study, an effective MCMC method, NUTS [22],is adopted to approximate the posterior distribution of µu, τu, wu, α. In par-ticular, 2000 samples are generated to approach the posterior distribution ofeach parameter. For illustration, the generated Markov chain for µ1, τ1, w1, αare shown in Fig. 4.17 (a) (b) (c) and (d), respectively. From these Markovchains, the probability distributions of µ1, τ1, w1, α can be estimated. Basedon the probability distributions of µu, τu, wu, u = 1, ..., η and α, the mixturescan be calculated based on (4.24) from the Monte Carlo view.Fig. 4.18 shows the 2000 mixtures and their mean value calculated fromthe posterior distribution of µu, τu, wu, α. It can be seen the mean value ofmixtures is flexible and describes the uncertainty of the actual operationalload very well.Fig. 4.16 (a) shows the weights wu of normal distribution components.It can be found that the normal distribution components have negligibleweights when u > 5. Thus, the first five normal distribution componentsare integrated as a mixture for describing the operational load uncertainty,as shown in Fig. 4.16 (b). Table. 4.3 summarizes the weights and parametersof the first five components. These weights and parameters are estimated asthe mean value of their Markov chains. Ultimately, the quantified loadinguncertainty is integrated into the prognostic result.85872(a) (b)(c) (d)MCMC sample MCMC sample MCMC sample MCMC sample 0 400 800 1200 1600 2000123.55123.6123.65123.7123.75123.80 400 800 1200 1600 20000.250.30.350.40.450.50 400 800 1200 1600 200000.511.522.530 500 1000 1500 20000.10.20.30.40.5Figure 4.17: (a)(b)(c) and (d) are the generated Markov chains for µ1, τ1,w1, and α, respectively. © Teng Wang4.3.6 Bayesian InferenceIn this chapter, the Bayesian inference is built on the basis of state spacemodel. In the following, the basics of the state space model are introducedfirst, followed by the implementation of the state space model for the FCGprediction considering crack location randomness.The general state space model is formally expressed as,xi = f(xi−1, γi−1, ωi−1), (4.25)yi = h(xi, ξi, vi), (4.26)where f(·) and h(·) are the state transition and the measurement functions,respectively. xi is the estimated system state and yi is the measurement attime index i. The state transition function expresses a one-step transitionof the state using a first-order Markov process. γi and ξi are the vector ofmodel coefficients at time i in state transition function and measurementfunction, respectively. ωi and vi are the process noises related to the statetransition and measurement function at time index i, respectively. Moredetails for the state space model can be found in [93].86Force range △F (kN) Cycle number(a)(b)FrequencyActual force range △F (kN) Nominal force range Actual force rangeActual force range △F (kN) ProbabilityMean value of mixtures 2000 mixtures Actual force range △F (kN) ProbabilityMean value of mixtures 2000 mixtures 119 120 121 122 123 124 125 1260 0.1 0.2 0.3 0.4 0.5 119 120 121 122 123 124 125 1260 0.1 0.2 0.3 0.4 0.5 Figure 4.18: 2000 mixtures and their mean values calculated from the pos-terior distributions of µu, τu, wu and α. © Teng Wang.Actual force range △F (kN) (b)Probability Component index Weight (a)Mixture 1st Component 2nd Component3rd Component 4th Component 5th Component0 2 4 6 8 10 12 14 16 18 20 00.10.20.3119 120 121 122 123 124 125 12600.10.20.30.4Figure 4.19: (a) The weights of normal distribution components. (b) Themixture of the first five normal distribution components. © Teng WangIn this case study, the crack growth in both right and left tips should beestimated, two state transition functions are needed as shown in (4.8) and(4.9). In particular, θLi∆n in (4.8) and θRi∆n in (4.9) correspond to xi in (4.25)respectively in each state transition function. Besides, γi equals the {C,m}estimated at i∆n in (4.8) and (4.9). Since no noise is modeled during thestate transition, ω equals zero in this case study.Similarly, two measurement functions for localizing both crack tips areneeded. They have been derived as shown in (4.18) and (4.19), respectively.In particular, θ˜Li∆n/θ˜Ri∆n and θLi∆n/θRi∆n correspond to yi and xi in (4.26)respectively.For convenience, θ˜Li∆n/θ˜Ri∆n and θLi∆n/θRi∆n are denoted as L˜i/R˜i and87Li/Ri for shorthand in the following. As a result, the transition PDFp(Li,Ri|Li−1,Ri−1) and state PDF p(Li−1,Ri−1|L˜1:i−1, R˜1:i−1) at cycle i−1can be obtained. Then, the prior state PDF at i is predicted using theChapman-Kolmogorov equation:p(Li,Ri|L˜1:i−1, R˜1:i−1)=∫ ∫p(Li,Ri|Li−1,Ri−1)p(Li−1,Ri−1|L˜1:i−1, R˜1:i−1)dLi−1dRi−1.(4.27)Given new measured data, the posterior state PDF can be generatedfrom the prior state PDF as,p(Li,Ri|L˜1:i, R˜1:i) = p(L˜i, R˜i|Li,Ri)p(Li,Ri|L˜1:i−1, R˜1:i−1)p(Li−1,Ri−1|L˜1:i−1, R˜1:i−1). (4.28)However, equation (4.28) only has a conceptual solution and cannot besolved analytically. Thus, PF is adopted to approximate the state PDF,using a set of particles {Lji ,Rji}, j = 1, ..., s from the importance PDF asq(Li,Ri|Li−1,Ri−1, L˜1:i, R˜1:i). s is the number of particles. The systemstate is approximated as,p(Li,Ri|L˜1:i, R˜1:i) ≈s∑j=1wji δ(Li − Lji ) ∗ δ(Ri −Rji ) (4.29)where δ(·) is the Dirac delta function, and wji is the weight of {Lji ,Rji}. wjiis updated and normalized as,wji = wji−1p(L˜i, R˜i|Li−1,Ri−1), wji = wji /s∑j=1wji . (4.30)One problem in the PF is particle degeneracy [66], i.e., after a few itera-tions, some particles have negligible weights. The solution for this problemis conducting a resampling step to eliminate particles with small weights. Inthis study, the inverse CDF [66] method is used for resampling. After theresampling step, s new particles will be generated to replace the previousparticles.Another problem in the PF is particle impoverishment [94]. That is thesamples that approach the ground truth will be duplicated lots of times88during the resampling step. As a result, many particles in the particlepool have the same value. To address this problem, a roughening phase isadded [9]. Specifically, the random noise ∆γ = {∆logC,∆m} is added toγ = {logC,m} as γ = γ +∆γ. In particular, ∆logC ∼ N(0, κ · DlogC) and∆m ∼ N(0, κ · Dm), where DlogC and Dm are the maximum difference oflogC and m respectively between the particles in the particle set. κ is aconstant tuning parameter set as 0.03 in this study. The selection of κ isa compromise. A large value of κ would blur the distribution, but a smallvalue of κ would not solve the particle impoverishment issue [118].In each time step, the system state {Li,Ri, ai} is calculated with themean value as,µ(Li) = 1ss∑j=1wjiLji , (4.31)µ(Ri) = 1ss∑j=1wjiRji , (4.32)µ(ai) =1ss∑j=1wji [Lji −Rji ]180rπ, (4.33)and the standard deviation as,σ(Li) = 1ss∑j=1wji [Lji − µ(Li)]2, (4.34)σ(Ri) = 1ss∑j=1wji [Rji − µ(Ri)]2, (4.35)σ(ai) =1ss∑j=1wji [(Lji −Rji )180rπ− µ(ai)]2. (4.36)Based on the system state model, the FCG prediction can be realized.Fig. 4.20, Fig. 4.21 and Fig. 4.22 show the prediction results for Pipe A, Pipe89B and Pipe C, respectively. Denote the loading cycle when the initial crackis detected by the sensor network as 0. For pipe A, the sensor network wasactivated at loading cycles 0, 600, 800, 1100, 1200, and 1400, respectively.For pipe B, the sensor network was activated at loading cycles 0, 5000,10000, 12500, 15000, 17500, 17800, and 17937, respectively. For pipe C, thesensor network was activated at 0, 5000, and 8387, respectively. In eachfigure, subfigures (a) (b) and (c) show the prediction result of the rightcrack tip, the left crack tip and the crack length, respectively. Subfigure(d) shows the posterior distribution of logC,m where U1, U2,..., representthe particles after update 1, 2, ... which are triggered by activating thesensor network. Moreover, the ground-truth values of the right crack tip,the left crack tip and the crack length measured using a tapeline are alsolabeled in these figures. The following results can be seen: (1) Even withthe same experiment configuration, the location of crack tips for Pipe A,Pipe B and Pipe C are different, which is possibly caused by the residualstress induced by the welding process. (2) Every time when the sensornetwork is activated, the prediction result will be calibrated accordingly.The calibration improves the prediction accuracy and narrows the confidenceinterval of the prediction result. (3) The posterior distributions of {logC,m}become convergent gradually. It is because the sensor network obtains thedefect information from the individual physical pipeline, characterizing theunit-to-unit variance of the structure degradation.Table 4.4 summarizes the prediction accuracy for Pipe A, Pipe B, andPipe C. The RMSE and CRA [109] are employed to quantify the predictionaccuracy of right crack tip, left crack tip and crack length. Specifically,RMSE and CRA are calculated as,RMSE =√√√√1kk∑i=1(Yˆ (ni)− Y (ni))2, (4.37)CRA =1kk∑i=1(1− |Yˆ (ni)− Y (ni)|Y (ni)), (4.38)where Yˆ (ni) and Y (ni) are the predicted value and actual value at loadingcycle ni, i = 1, ..., k, respectively. RMSE and CRA describe the predictionaccuracy of different methods. In particular, a higher CRA and a lowerRMSE mean a more accurate prediction performance.It can be seen that the prognostics for Pipe C achieves the worst result,as the sensor network was only activated three times. In comparison, seven90Table 4.4: Prediction performance for Pipe A, Pipe B and Pipe C.Right tip (degree) Left tip (degree) Crack length (mm)PipeIDRMSE CRA RMSE CRA RMSE CRAPipe A 13.931 0.920 9.997 0.963 29.300 0.801Pipe B 15.593 0.920 26.913 0.896 49.079 0.777Pipe C 42.344 0.750 18.662 0.936 98.560 0.468times of activation for Pipe A and Pipe B were conducted, respectively,which results in a better prediction performance. In particular, the predic-tion for Pipe A outperforms Pipe B. The reason is that the sensor networkinspections for Pipe A are conducted more uniformly along the x-axis. It iseasy to understand that if the current inspection is very close to the previ-ous one, less new information will be obtained. In this regard, the updateshould be carefully scheduled to reduce uncertainties.91Load cyclesRight tip (degree)Load cyclesLeft tip (degree)Load cyclesCrack length (mm)mlogC(a) (b)(c) (d)98% CI Mean value Ground-truth0 200 400 600 800 1000 1200 1400 1600 0 200 400 600 800 1000 1200 1400 160060801001201401601802001802002202402602803000 200 400 600 800 1000 1200 1400 1600-500501001502002503003501.0 1.2 1.4 1.6 1.8 -16.4-16.2-16.0-15.8-15.6-15.4-15.2Figure 4.20: Prediction result for pipe A: (a) Predicted right crack tip. (b)Predicted left crack tip. (c) Predicted crack length. (d) Prior and posteriordistributions after each update. © Teng WangThe reliability of the pipeline can also be predicted given a specificthreshold λ. This threshold size can be user-specified or it can be taken9194Load cyclesRight tip (degree)Load cyclesLeft tip (degree)Load cyclesCrack length (mm)mlogC(a) (b)(c) (d)98% CI Mean value Ground-truth0 5000 10000 150001601802002202402602803001.0 1.1 1.2 1.3 1.4 1.5 1.6-16.75-16.50-16.25-16.00-15.75-15.50-15.25-10001002003000 5000 10000 150000 5000 10000 150006080100120140160180200400Figure 4.21: Prediction result for pipe B: (a) Predicted right crack tip. (b)Predicted left crack tip. (c) Predicted crack length. (d) Prior and posteriordistributions after each update. © Teng Wangto correspond to the maximum crack size that the component can with-stand at the maximum stress in the load profile [5]. For illustration, thefailure of the pipe is defined when the through-thickness crack exceeds 310mm which is about half of the pipe circumference. Then, the reliability ofthe pipe along with loading cycles can be predicted as,Ri =∫ΘF(λ− ai) · p(ai|L˜1:i−1, R˜1:i−1))dai (4.39)where F{x} = 1 when x > 0 and F{x} = 0 when x ≤ 0.Fig. 4.23 shows the reliabilities of Pipe A, Pipe B and Pipe C, respectivelywith respect to the loading cycles. It can be seen that even though the sameprior was assigned, the reliabilities of Pipe A, Pipe B and Pipe C are diverse.Specifically, for Pipe A, its reliability rapidly decreases after 1400 load cycles.Nevertheless, for Pipe B and C, the rapid decreases occur after 12000 and9000 load cycles, respectively. This phenomenon shows a large unit-to-unit9296Load cyclesRight tip (degree)Load cyclesLeft tip (degree)Load cyclesCrack length (mm)mlogC(a) (b)(c) (d)98% CI Mean value Ground-truth0 2000 4000 6000 8000 10000 180 200 220 240 260 280 300 320 340 0 2000 4000 6000 8000 10000 -100 0 100 200 300 400 500 0 2000 4000 6000 8000 10000 50 100 150 200 250 0.8 1.0 1.2 1.4 1.6 1.8 -16.4 -16.2 -16.0 -15.8 -15.6 -15.4 -15.2 Figure 4.22: Prediction result for pipe C: (a) Predicted right crack tip. (b)Predicted left crack tip. (c) Predicted crack length. (d) Prior and posteriordistributions after each update. © Teng Wangvariance between the tested pipes. Hence, it is necessary to update theprediction result based on the NDI findings from the individual pipe.4.4 DiscussionCompared to the existing study, the presented work considers crack locationrandomness in the FCG prediction. In particular, a surrogate model is builtto calculate the stress intensity factor given the crack with different size andlocation. The training of this surrogate model is challenging, as the possiblecrack configurations (length and location) are diverse. To guarantee predic-tion accuracy, the surrogate model should be trained with a large numberof training samples. In the presented work, to generate training samplesfor this surrogate model, FE analysis is conducted 218 times with differentcombinations of crack lengths and locations. This computational load is stillacceptable, as the area of interest (i.e., the weld line of the pipe) is relatively93Load cyclesReliability(a)Load cycles(b)Load cycles(c)ReliabilityReliability0 4000 8000 12000 16000 0 4000 8000 1200010.90.80.70.610.90.80.70.60 2000 4000 6000 8000 1000010.90.80.70.6Figure 4.23: (a), (b) and (c) are the reliabilities of pipe A, B and C respec-tively. The dotted line indicates the inspection time. © Teng Wangsmall. For the application, where the crack can possibly occur in a large areaof interest (for example pipeline without weld line), the diversity of the pos-sible crack configurations will also increase. The heavy computational loadin FE analysis will be caused by generating sufficient training samples forthe surrogate model. Hence, high-performance computing equipment maybe needed to conduct the FE analysis.Moreover, in order to localize the crack tip, accurate nondestructive testtechniques are needed. For example, 40 guided-wave sensors were deployedin this study to obtain the FMC data for crack tip localization. It willbe a challenge to consider the crack location randomness, if low-accuracynondestructive test techniques are adopted (for example, nondestructive in-spection test with several guided-wave sensors [128]).4.5 SummaryThis chapter presented a probabilistic model for the FCG prediction con-sidering the crack location randomness. A real-world application of pipelinefatigue crack prediction is specified. In order to localize the crack tips, theguided-wave sensor network is deployed to acquire the FMC data. Thesedata are processed based on the TFM, where the left and right crack tipscan be localized. To calculate the SIF in each crack tip, the FE analysis isconducted first. Then, a surrogate model is built to speed up the calcula-tion of the SIF for the probabilistic FCG prediction. Moreover, the Dirichletprocess is adopted to predict future load conditions based on historical loadobservation. The Paris-Erdogan equation is extended to predict the locationof left and right crack tips under the polar coordinate system. Finally, theFCG prediction result can be obtained through a state-space model. The94effectiveness of the proposed method is validated based on the real-worldtest of three pipes under a four-point bending load, showing its potential forreal-world pipeline health management. The proposed model is applicablefor FCG prediction considering the crack location randomness95Chapter 5ConclusionsIn this thesis, a set of probabilistic models are proposed for the FCG pre-diction considering a variety of uncertainty factors. Based on a closed-formsolution, a probabilistic model is proposed for the FCG prediction consid-ering the material uncertainty of individual structure. In this model, thematerial parameters logC and m in the Paris-Erdogan equation, and modelerror are regarded as random variables. A closed-form solution is derivedto update the posterior distribution of material parameters according to thecrack growth observation. Finally, a modified Paris-Erdogan equation isadopted to predict the crack growth in a probabilistic view. The effective-ness of the model is verified using the Virkler’s dataset.To consider multi-source prior knowledge, a probabilistic model for theFCG prediction with hybrid prior is proposed. In this model, the MonteCarlo integration is introduced to calculate the posterior belief of each can-didate prior. Then, the PF is extended to conduct Bayesian inference withthe hybrid prior. A numerical study on the spur gear tooth shows integratingmultiple priors can increase the robustness of the FCG prediction.To consider crack location randomness, a probabilistic model is proposedfor the FCG prediction based on the localization of right and left crack tips.Specifically, the guided-wave sensor network is adopted for the crack tiplocalization where the POD and MA are derived, A surrogate model is builtto output the SIF for each crack tip given the crack configuration. Finally,the crack is predicted based on the state space model with the findingsfrom the guided-wave sensor network. The developed model is validated topredict the through-thickness crack growth along the weld line of a set ofpipes.To sum up, the above presented research outcomes can help structuralengineers conduct accurate FCG prediction considering various uncertaintyfactors in the real-world scenario. With the accurate FCG prediction re-sult, the appropriate maintenance activities (such as inspection, repair, andreplacement) can be further conducted to prevent the catastrophic conse-quence of the machinery and structures.965.1 ContributionsThe main contributions of this thesis research are summarized as follows.• A closed-form solution is provided for updating the well-known Paris-Erdogan equation, considering the uncertainty of material parametersand the model error. Hence, the FCG can be predicted more efficientlyand accurately.• Based on the concept of PBPD, this thesis proposes a probabilisticFCG prediction model which enables the input of hybrid prior. As aresult, in a real-world scenario, any promising prior knowledge can beintegrated, increasing the robustness of the FCG prediction.• This study develops a Monte Carlo method for calculating the PBPD,which enables the posterior belief of the high-dimensional prior distri-bution to be calculated.• This thesis extends the PF for the Bayesian inference with hybridprior, where each prior distribution can be sampled simultaneouslyaccording to their posterior belief.• The thesis presents a probabilistic model for the FCG prediction con-sidering the crack location randomness. Compared to the existingstudies, this work focuses on a real-world structure without any pre-notch. To consider the crack location randomness, the state spacemodel is modified to predict the change of crack tips based on thespecifically designed signal process model and surrogate model.• A probabilistic model for pipeline health management is developedin this thesis. It can detect, localize, quantify and predict the fatiguecrack in the region of interest. The real-time reliability of the inspectedpipeline can be calculated to facilitate maintenance activities.5.2 Limitations and Future WorkThe limitations and the potential future work of this research are discussedand listed as below.• In regard to the probabilistic model based on closed-form solution,only the material parameters and model error are considered as ran-dom variables. The main reason for this consideration is material pa-rameters and model error are the most important variables reflecting97the inherent stochastic property of the FCG. The proposed methodmay not be applicable in the industrial applications where there ex-ist other significant uncertainties (e.g., the uncertainties induced bymeasurement error, data sparsity, SIF calculation). For example, inpractical engineering, if the measurement accuracy from the NDI toolis low, the measurement uncertainty can not be neglected. Integratingmore uncertainties will further increase the complexity of the deriva-tion of closed-form solution. However, theoretically, the derivationis still feasible as long as the prior and likelihood distributions areconjugate. Based on the limitation above, the future work will bededicated to enhancing the closed-form solution to consider more un-certainty factors in practical engineering. Moreover, the closed-formsolution is derived based on the Paris-Erdogan equation. As the Paris-Erdogan equation was developed to describe the stable stage of crackgrowth, the closed-form solution is only applicable for the FCG predic-tion when the fatigue rate curve in the region II (as shown in Fig. 1.4).The closed-form solution for the FCG prediction when the fatigue ratecurve in the region I and III should be further studied.• In regard to the probabilistic model with hybrid prior, each candidateprior is fused in the Bayesian analysis. However, if a prior is far fromthe ground-truth value, fusing it into the hybrid prior will disturb theconvergence of posterior distribution. A possible solution is to directlyreject the candidate prior with low posterior belief, which should bestudied further.• In regard to the probabilistic model considering crack location ran-domness, the pipeline FCG prediction in the weld line is simplified asa one-dimensional problem where the crack can only occur and growalong the weld line. The proposed method is not applicable for two-dimensional or three-dimensional problem where the possible region ofFCG is in a two-dimensional or three-dimensional space. For exam-ple, in a real-world application, the crack may occur in the non-weldarea of pipeline due to the impact of microstructure flaw, geometryvariation, etc. In this case, it will be challenging for building the sur-rogate model to output the SIF of the crack tip/front considering thecrack location randomness, as the possible crack configuration (length,shape, and location) will increase significantly. Hence, future workwill be dedicated to studying two-dimensional and three-dimensionalFCG prediction considering crack location randomness. Moreover, the98developed probabilistic model does not explicitly model the residualstress in the weld line for FCG prediction. Instead, the residual stressis implicitly considered by tracking the change of material parametersm and C during the FCG prediction. In this regard, future work willalso be dedicated to explicitly modeling the residual stress to furtherincrease the accuracy of the model..99Bibliography[1] F. Zhao, Z. Tian, E. 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Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Liu, Zheng (Professor of engineering)
Liao, Min
Probabilistic models for fatigue crack growth prediction
Text