核能電廠為目前世界上對其安全性最要求的設施之一，但我們依舊不可忽略其可能造成的事故風險，因為如發生嚴重事故將會大幅度的影響周圍居民及環境，其影響甚鉅。為降低風險，電廠之結構可靠度有其考量的必要。其中，電廠反應器壓力槽（Reactor Pressure Vessel, RPV）包含許多重要組件並具有爐心及內部組件結構支撐功能，其有效燃料位置所對應之腹帶區（Beltline region）較其他組件會承受更高的溫度、壓力以及輻射劑量，正常運轉下材料因為高溫還能保有足夠的韌性，如因失水事故引發注水使得溫度驟降，容易引發材料脆化進而威脅其結構可靠度，因此在此區域內材料裂紋的檢測與其成長的評估就成為分析電廠安全性中不可忽視的部分。直至今日，機率破壞力學（Probabilistic Fracture Mechanics, PFM）因可考慮如裂紋深度與密度分佈、腹帶區焊道與板材之材質、中子照射量（Neutron fluence）等參數之不確定性，越發受到核能發電業者及管制單位之重視與使用，例如美國的核能管制委員會（Nuclear Regulatory Commission, NRC）目前就是以其境內橡樹嶺國家實驗室所開發的機率破壞力學分析軟體FAVO0R做為評估美國境內反應器壓力槽在遭受壓力熱震（Pressurized Thermal Shock, PTS）暫態下之結構損壞風險與管制法規制定之主要分析工具。

為了理解壓力熱震暫態下的溫度分佈行為對電廠反應器壓力槽可能的影響，本論文中計算流體力學（Computational Fluid Dynamics, CFD）之運算結果被用來與機率破壞力學結合，藉以探討以往機率破壞力學為求保守可能會產生的過多的安全裕度（Safety Margin）；並且使用機器學習領域中簡單而強大的貝氏推論法（Bayesian inference）來結合機率破壞力學，藉以達成探討分析目標電廠之非破壞檢測數據、檢測各式之不確定性對於電廠結構可靠度評估結果的影響，並探討分析結果之現象與合理性。而根據計算結果，將貝氏推論結合機率破壞力學分析可更好的反應出非破壞檢測出的裂紋對於機率的影響，進一步使的裂紋分佈更加貼近分析目標。

Nuclear power plants are among the safest and most secure facilities in the world. We cannot rule out the possibility of an accident, it may affect people and the environment. To minimize the likelihood of an accident, the structural integrity of a nuclear power plant is necessary. The reactor pressure vessel (RPV) contains the reactor core and other key reactor internals, which is exposed under high neutron fluence during normal operation. And the neutron irradiation may cause the vessel material to become more brittle especially for the materials in the beltline region corresponding to the reactor core. Therefore, crack detection and propagation analysis of RPV is one of the most serious issue for the safety of nuclear power plant. Nowadays, the probabilistic fracture mechanics (PFM) technique is widely used in evaluating the structural integrity of RPVs. The probabilistic fracture mechanics code, FAVOR , which was developed by the Oak Ridge National Laboratory (ORNL) in the United States, is employed by the Nuclear Regulatory Commission as a technical analysis tool the rules of pressurized thermal shock (PTS).

In order to understand the possible influence of temperature distribution behavior under pressure thermal shock transients on probability failure mechanics, the computational fluid dynamics (CFD) calculation results are used in combination with probability failure mechanics, which may reduce some safety margin. The Bayesian inference which combines new inspection results as well as uncertainties is used to develop the posterior vessel-specific flaw distributions. Then, the distributions are used for PFM analysis to investigate the effects of updated flaw characteristics on the fracture probability of RPV subjected to pressurized thermal shocks. Considering the updated flaws based on the inspection data, the analyzed results could be more plant-specific to predict the fracture risks of RPVs during operation. According to the analysis results, the Bayesian inference which combines the PFM technique can make the flaw distributions to become more representative.

摘要 i
ABSTRACT ii
致謝 iii
目錄 iv
表目錄 vi
圖目錄 vii
第一章 緒論 1
1.1研究背景與目的 1
1.2 文獻回顧 6
第二章 基礎理論 13
2.1 破壞力學概述 13
2.2 機率破壞力學 16
2.3 計算流體力學概述 21
2.4 貝氏推論 24
2.5 機率分佈函數 26
2.5.1 均勻分佈（Uniform distribution）: U(a,b) 26
2.5.2 常態分佈（Normal distribution）: N(μ,σ) 27
2.5.3 對數常態分佈（Lognormal distribution）: Λ(μlog,σlog) 28
2.5.4 偉伯分佈（Weibull distribution）: W(a,b,c) 30
2.5.5 邏輯分佈（Logistic distribution）: L(α,β) 31
2.5.6 指數分佈（Exponential distribution）: f(x | λ) 32
2.5.7卜松分佈（Poisson distribution）: P(k | λ) 33
2.5.8伽瑪分佈（Poisson distribution）: G(α,β) 34
第三章 壓力熱震事件概述 36
3.1 一次側管路破裂事故 （LOCA） 36
3.2 一次側系統安全釋壓閥卡開 （SO-1） 38
3.3 二次側系統安全釋壓閥卡開 （MSLB&SO-2） 38
第四章 FAVOR分析架構 39
4.1 FAVOR程式資料流程簡介 39
4.2 熱預應力（WPS）之判定與分析流程 42
第五章 化學組成不確定性分析 45
5.1 腹帶區模型化學參數與所選用之壓力熱震暫態介紹 45
5.2化學組成對PFM影響分析 51
第六章 腹帶區溫度不均勻性分析 58
6.1計算流體力學模型與網格不準度 58
6.2腹帶區模型與所選用之壓力熱震暫態介紹 63
6.3溫度不均勻性分析結果與對PFM之影響 68
第七章 非破壞檢測裂紋分析 75
7.1非破壞檢測分析流程 75
7.2貝氏推論流程 79
7.3考量非破壞檢測之PFM分析結果 84
第八章 結論 94
參考文獻 95
附錄A : 貝氏推論MATLAB code 104
附錄B : 所修改的VFLAW主程式碼 110

[1] World Nuclear Association, World Nuclear Performance Report 2019, International Atomic Energy Agency（IAEA）, England, 2019.
[2] T. L. Dickson, P. T. Williams, S. Yin, “Fracture Analysis of Vessels - Oak Ridge FAVOR, v04.1, Computer Code: Theory and Implementation of Algorithms, Methods, and Correlations,” NUREG/CR-6854, Oak Ridge National Laboratory, 2004.
[3] J. Medoff, "Status Report: Intergranular Stress Corrosion Cracking of BWR Core Shrouds and Other Internal Components," US NRC, NUREG-1544, March 1996.
[4] S. Bush, A. Chockie, "An Overview of Stress Corrosion in Nuclear Reactors from the Late 1950s to the 1990s," Swedish Nuclear Power Inspectorate, SKI Report 96:24, February 1996.
[5] W. J. Dircks, “NRC Staff Evaluation of Pressurized Thermal Shock,” United States Nuclear Regulatory Policy Issue, SECY 82-465, 1982.
[6] W. Bilanin, BWR Vessel and Internals Project - BWR Reactor Pressure Vessel Shell Weld Inspection Recommendations（BWRVIP-05）, TR-105697, Electric Power Research Institute（EPRI）, 1995.
[7] United States Nuclear Regulatory Commission, “Boiling Water Reactor Licensees Use of the BWRVIP-05 Report to Request Relief from Augmented Examination Requirements on Reactor Pressure Vessel Circumferential Shell Welds,” Generic Letter 98-05, 1998.
[8] K. Wichman, B. Elliot, C. Carpenter, “U.S. Nuclear Regulatory Commission’s Review of the Impact of Inservice Inspection of BWR Reactor Pressure Vessel Welds on Vessel Failure,” SMiRT 15, Seoul, South Korea, August 15-20, 1999.
[9] M. T. EricksonKirk et al., “Technical Basis for Revision of the Pressurized Thermal Shock Screening Limit in the PTS Rule （10 CFR 50.61）,” NUREG-1806, United States Nuclear Regulatory Commission, 2007.
[10] M. T. EricksonKirk, T. L. Dickson, “Recommended Screening Limits for Pressurized Thermal Shock,” NUREG-1874, Oak Ridge National Laboratory, 2010.
[11] United States Nuclear Regulatory Commission, “Alternative Fracture Toughness Requirements for Protection against Pressurized Thermal Shock Events,” 10 CFR 50.61a.
[12] G. Stevens, M. Kirk, M. Modarres, “Technical Basis for Regulatory Guidance on the Alternate Pressurized Thermal Shock Rule, Final Report,” NUREG-2163, United States Nuclear Regulatory Commission, 2018.
[13] 10CFR50.61, "Fracture Toughness Requirements for Protection against Pressurized Thermal Shock Events, " Code of Federal Regulations.
[14] W. E. Vesely, E. K. Lynn, F. F. Goldberg, “The Octavia Computer Code: PWR Reactor Pressure Vessel Failure Probabilities Due to Operationally Caused Pressure Transients,” NUREG-0258, United States Nuclear Regulatory Commission, 1978.
[15] Integrity of Reactor Pressure Vessels in Nuclear Power Plants: Assessment of Irradiation Embrittlement Effects in Reactor Pressure Vessel Steels, International Atomic Energy Agency（IAEA）, England, 2009.
[16] 10CFR50.55a, "Final rule : The ASME Code for Operation and Maintenance of Nuclear Power Plants for construction, in-service inspection, and in-service testing. , " Code of Federal Regulations.
[17] W. C. Arcieri et al., “RELAP5 Thermal Hydraulic Analysis to Support PTS Evaluations for the Oconee-1, Beaver Valley-1, and Palisades Nuclear Power Plants.” NUREG/CR-6858, U. S. NRC, September, 2004.
[18] Y. H. J. Chang, A. Mosleh, and K. Almenas, “Thermal Hydraulic Uncertainty Analysis in Pressurized Thermal Shock Risk Assessment.” NUREG/CR-6899, U. S. NRC, November, 2004.
[19] E. D. Eason, G. R. Odette, R. K. Nanstad, and T. Yamamoto, “A Physically Based Correlation of Irradiation-Induced Transition Temperature Shifts for RPV Steels” ORNL/TM-2006/530, November, 2007.
[20] S. T. Wood, C. L. Smith, K. J. Kvarfordt, and S.T. Beck, “Systems Analysis Programs for Hands-on Integrated Reliability Evaluations (SAPHIRE) Vol. 1 Summary Manual.” NUREG/CR-6952, U. S. NRC, September, 2008.
[21] O. J. V. Chapman, RR-PRODIGAL: A Model for Estimating the Probabilities of Defects in Reactor Pressure Vessel Welds, United States Nuclear Regulatory Commission, 1998.
[22] F. A. Simonen, S. R. Doctor, G. J. Schuster, P. G. Heasler, “A Generalized Procedure for Generating Flaw-related Inputs for the FAVOR Code,” NUREG/CR-6817, Pacific Northwest National Laboratory, 2003.
[23] S. J. Gershman, David M. Blei, “A Tutorial on Bayesian Nonparametric Models,” Journal of Mathematical Psychology, Volume 56, Issue 1, February 2012, pp. 1-12.
[24] Subhashis Ghosal, Aad van der Vaart, Fundamentals of Nonparametric Bayesian Inference, Cambridge University Press, 2017.
[25] R. D. Cheverton, D. G. Ball, “A Deterministic and Probabilistic Fracture-Mechanics Code for Applications to Pressure Vessels,” NUREG/CR-3618, United States Nuclear Regulatory Commission, 1984.
[26] D. L. Stevens et al., “VISA – A Computer Code for Predicting the Probability of Reactor Pressure Vessel Failure,” NUREG/CR-3384, Pacific Northwest Laboratory, 1983.
[27] F. A. Simonen et al., “VISA-II – A Computer Code for Predicting the Probability of Reactor Pressure Vessel Failure,” NUREG/CR-4486, Pacific Northwest Laboratory, 1986.
[28] R. Labbens, A. Pellissier-Tanon, and J. Heliot, “Practical Method for Calculating Stress Intensity Factors Through Weight Functions,” ASTM STP-590, Mechanics of Crack Growth, American Society for Testing and Materials, (1976) 368-384.
[29] J. Heliot, R. C. Labbens, and A. Pellissier-Tanon, “Semi-Elliptical Cracks in the Meridonal Plane of a Cylinder Subjected to Stress Gradients, Calculation of Stress Intensity Factors by the Boundary Integral Equations Method,” XIth National Symposium on Fracture Mechanics, Blacksburg, VA, 1978.
[30] S. S. Tang et al., “The Effect of Adjusted Reference Temperature on the Probability of Failure in Boiling Water Reactor Vessel Welds,” Fracture, design analysis of pressure vessels, heat exchangers, piping components and fitness for service , 1999, pp. 121-126.
[31] Nmki Soneda, Takeo Onchi, “Benchmarking Studies of Probabilistic Fracture Mechanics Analysis Code, PROFMAC-II, for Assessing Pressurized Thermal Shock Events of Reactor Pressure Vessel Integrity Issues,” Journal of Nuclear Science and Technology, 33（1）, January 1996, pp. 87-98.
[32] G. Yagawa, Y. Kanto, S. Yoshimura, H. Machida, and K. Shibata, “Probabilistic Fracture Mechanics Analysis of Nuclear Structural Components: A Review of Recent Japanese Activities,” Nuclear Engineering and Design 207, (2001) 269-286.
[33] K. Shibata, D. Kato, and Y. Li, “Development of a PFM Code for Evaluating Reliability of Pressure Components Subject to Transient Loading,” Nuclear Engineering and Design 208, (2001) 1-13.
[34] Y. Li, D. Kato, K. Shibata, and K. Onizawa, “Improvements to a Probabilistic Fracture Mechanics Code for Evaluating the Integrity of an RPV Under Transient Loading,” International Journal of Pressure Vessels and Piping 78, (2001) 271-282.
[35] C. B. Buchalet, H. W. Bamford, Stress intensity factor solution for continuous surface flaws in reactor pressure vessels. Mechanics of crack growth, ASTM STP 590, 385-402, 1976.
[36] Levy N, Rice JR, Surface cracks in elastic plates and shells, Brown university, 1976.
[37] J. C. Newman, I. S. Raju, Stress-intensity factor equations for cracks in three-dimensional finite bodies subjected to tension and bending loads. NASA technical memorandum 85793, NASA, 1984.
[38] T. L. Dickson, “A Fracture Mechanics Analysis Code for Nuclear Reactor Pressure Vessel – Release 9401,” ORNL/NRC/LTR/94/1, United States Nuclear Regulatory Commission, 1994.
[39] N. A. Palm, B. A. Bishop, C. L. Boffess, “Risk-informed Extension of the Inservice Inspection Interval for Pressurized Water Reactor Vessel from 10 to 20 years,” Nuclear Engineering and Design, August 2008, pp. 2027-2037.
[40] H. W. Chou et al., “Probabilistic Fracture Analysis for Boiling Water Reactor Pressure Vessels Subjected to Low Temperature Over-Pressure Event,” ASME 2010 Pressure Vessels & Piping Conference, January 2011, pp. 157-164.
[41] H. W. Chou et al., “Demonstration of Structural Integrity of Boiling Water Reactor Pressure Vessels Under Ultimate Response Guideline Operation,” Nuclear Technology, 2020, pp. 1-13.
[42] V. F. González-Albuixech et al., “Comparison of PTS analyses of RPVs based on 3D-CFD and RELAP5,” Nuclear Engineering and Design 291, 168-178, 2015.
[43] J. C. Cheng et al., “CFD simulation of a four-loop PWR at asymmetric operation conditions,” Nuclear Engineering and Design 300, 591-600, 2016.
[44] G. L. Chen et al., “Challenge Analysis and Schemes Design for the CFD Simulation of PWR,” Science and Technology of Nuclear Installations, 2017.
[45] Michal Jaros et al., “Computational Fluid Dynamics Study of Pressurized Thermal Shock Transients in the Reactor Pressure Vessel,” 26th International Conference Nuclear Energy for New Europe, 2018.
[46] Pressurized Thermal Shock in Nuclear Power Plants: Good Practices for Assessment, International Atomic Energy Agency（IAEA）, IAEA-TECDOC-1627, 2010.
[47] MILOUDI, S., Etude du Dommage d’Irradiation dans les Aciers de Cuve des Réacteurs à Eau Pessurisée, Université d’Orsay, Paris, 1997.
[48] Guohua Wu et al., “Framework for fault diagnosis with multi-source sensor nodes in nuclear power plants based on a Bayesian network,” Annals of Nuclear Energy, Volume 122, December 2018, pp. 297-308.
[49] J. L. Beck, L. S. Katafygiotis, “Updating Models and Their Uncertainties. I: Bayesian Statistical Framework,” The American Society of Civil Engineers, Vol. 124, Issue 4, April 1998.
[50] L. S. Katafygiotis, J. L. Beck, “Updating Models and Their Uncertainties. II: Model Identifiability,” The American Society of Civil Engineers, Vol. 124, Issue 4, April 1998.
[51] Enrique Castillo, José María Menéndez, Santos Sánchez‐Cambronero, “Traffic Estimation and Optimal Counting Location Without Path Enumeration Using Bayesian Networks,” Computer‐Aided Civil and Infrastructure Engineering, February 2008.
[52] B. A. Zárate et al., “Bayesian model updating and prognosis of fatigue crack growth,” Engineering Structures, Volume 45, December 2012, pp. 53-61.
[53] George Deodatis, Hiroo Asada, Seiichi Ito, “Reliability of aircraft structures under non-periodic inspection: a Bayesian approach,” Engineering Fracture Mechanics, Volume 53, Issue 5, March 1996, pp. 789-805.
[54] R.A. Heller, G.H. Stevens, “Bayesian Estimation of Crack Initiation Times from Service Data,” Journal of aircraft, 1978.
[55] Tao Yin, Heung‐Fai Lam, Heung‐Ming Chow, “A Bayesian Probabilistic Approach for Crack Characterization in Plate Structures,” Computer‐Aided Civil and Infrastructure Engineering, Volume25, Issue5, 2010.
[56] H. F. Lam et al., “Application of the spatial wavelet transform and Bayesian approach to the crack detection of a partially obstructed beam,” Thin-Walled Structures, Volume 43, Issue 1, January 2005, pp. 1-21.
[57] Boris A. Zárate et al., “Bayesian model updating and prognosis of fatigue crack growth,” Engineering Structures, Volume 45, December 2012, pp. 53-61.
[58] J.M. Karandikar, N.H. Kim, T.L. Schmitz, “Prediction of remaining useful life for fatigue-damaged structures using Bayesian inference,” Engineering Fracture Mechanics, Volume 96, December 2012, pp. 588-605.
[59] C. E. Inglis, “Stresses in a plate due to the presence of cracks and sharp corners,” Transactions of the Royal Institute of Naval Architects, 60, 1913, pp.219-241.
[60] A. A. Griffith, “The Phenomena of Rupture and Flow in Solids,” Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Vol. 221, 1921, pp. 163-198.
[61] C. Sundararajan, “Probabilistic Assessment of Pressure Vessel and Piping Reliability,” J. Pressure Vessel Technology, February 1986, pp. 1-13.
[62] W. O. Shabbits, W.H. Pryle, E.T. Wessel, “Heavy Section Fracture Toughness Properties of A533, Grade B, Class-1 Steel Plate and Submerged Arc Weldments,” HSST Technical Report 6, WCAP-7414, 1969.
[63] H. W. Chou et al., “Theoretical Basis and Application of Probabilistic Fracture Mechanics Analysis Computer Code for Reactor Pressure Vessels,” Institute of Nuclear Energy Research.
[64] K. O. Bowman, P. T. Williams, “Technical Basis for Statistical Models of Extended KIC and KIa Fracture Toughness Databases for RPV Steels,” ORNL/NRC/LTR-99/27, Oak Ridge National Laboratory, Oak Ridge, February, 2000.
[65] S. V. Patankar, D. B. Spalding, A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional Parabolic Flows. International Journal of Heat and Mass Transfer, 15, 1787-1806, 1976.
[66] J. Dacles-Mariani et al., "Numerical/Experimental Study of a Wingtip Vortex in the Near Field", AIAA Journal, 33(9), pp. 1561-1568, 1995.
[67] P. R. Spalart, and S. R. Allmaras, "A One-Equation Turbulence Model for Aerodynamic Flows", AIAA Paper 92-0439, 1992.
[68] C. H. Kang, et al., " Methodology Development of CFD/PFM for PTS Analysis on Nuclear Reactor Safety," Nuclear Energy Agency, 2017.
[69] A. Rukhin et al., “A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications,” NIST Special Publication, 800-22, 2001.
[70] W. C. Arcieri, R. M. Beaton, C. D. Fletcher and D. E. Bessette, “RELAP5 Thermal Hydraulic Analysis to Support PTS Evaluations for the Oconee-1, Beaver Valley-1, and Palisades Nuclear Power Plants,” NUREG/CR-6858, United States Nuclear Regulatory Commission, 2004.
[71] T. L. Dickson, S. Yin and P. T. Williams, “Electronic Archive of the Results of Pressurized Thermal Shock Analysis for Beaver Valley, Oconee, and Palisades Reactor Pressure Vessels Generated with the 06.1 version of FAVOR,” ORNL/ United States Nuclear Regulatory Commission /LTR-07/04, Oak Ridge National Laboratory, 2007.
[72] D. L. Whitehead et al., “Beaver Valley Pressurized Thermal Shock （PTS） Probabilistic Risk Assessment （PRA）,” ADAMS #ML042880454, United States Nuclear Regulatory Commission, 2004.
[73] A. J. Brothers, S. Yukawa, “The Effect of Warm Prestressing on Notch Fracture Strength,” Journal of Basic Engineering, Volume 85, Issue 1, March 1963, pp.97-101.
[74] M. T. Kirk, “Inclusion of Warm Prestressing Effects in Probabilistic Fracture Mechanics Calculation Performed to Assess the Risk of RPV Failure Produced by Pressurized Thermal Shock Events: An Opinion,” NATO Advanced Research Workshop – Scientific Fundamentals for the Life Time Extension of Reactor Pressure Vessels, Kiev, Ukraine, 2002.
[75] T. L. Dickson, S. Yin and P. T. Williams, “Electronic Archive of the Results of Pressurized Thermal Shock Analysis for Beaver Valley, Oconee, and Palisades Reactor Pressure Vessels Generated with the 06.1 version of FAVOR,” ORNL/ United States Nuclear Regulatory Commission /LTR-07/04, Oak Ridge National Laboratory, 2007.
[76] D. L. Whitehead et al., “Beaver Valley Pressurized Thermal Shock （PTS） Probabilistic Risk Assessment （PRA）,” ADAMS #ML042880454, United States Nuclear Regulatory Commission, 2004.
[77] S. Kliem, T. Sühnel, U. Rohde, T. Höhne, H.-M. Prasser, F.-P. Weiss “ Experiments at the mixing test facility rocom for menchmarking of cfd-codes,” Forschungszentrum Rossendorf, Institute of Safety Research P.O.B. 510119, D-01314 Dresden, Germany, 2008.
[78] The American Society of Mechanical Engineers, Standard for verification and validation in computational fluid dynamics and heat transfer, ASME V&V 20–2009, 2009.
[79] P.J. Roache, K. Ghia, F. White, Editorial policy statement on the control of numerical accuracy, ASME Journal of Fluids Engineering 108 (1), 1986.