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Monte Carlo methods

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Proceedings Papers

*Proc. ASME*. PVP2017, Volume 1B: Codes and Standards, V01BT01A027, July 16–20, 2017

Paper No: PVP2017-65430

Abstract

Partial safety factor (PSF) is a reliability approach for considering the variance of parameters in flaw assessment procedure in major fitness-for-service (FFS) codes, such as recent API579 and BS7910 codes, but is still not adopted in Chinese FFS code GB/T 19624-2005. This study investigated the derivation method for PSFs based on GB/T 19624 procedure. The limit state equations for PSFs calculation were proposed based on GB/T 19624 level 2 failure assessment diagram (FAD). The distribution of random variables was determined according to China’s domestic features. The first order reliability method (FORM) and second order reliability method (SORM) were employed as reliability analysis methods, and the calculated results were both compared with that simulated using Monte Carlo method. The PSFs at different target reliability levels were established and compared with that in API 579 and BS 7910. The method proposed in this study provides a basis for introducing PSF approach into Chinese FFS code.

Proceedings Papers

*Proc. ASME*. PVP2016, Volume 8: Seismic Engineering, V008T08A013, July 17–21, 2016

Paper No: PVP2016-63064

Abstract

This paper describes the application of Monte Carlo method for the quantitative seismic risk assessment (QSRA) of process plants. Starting from the seismic hazard curve of the site where the plant is located, the possible chains of accidents are modelled using a sequence of propagation levels in which Level 0 is represented by the components directly damaged by the earthquake whereas the subsequent Levels represent the resulting consequence propagation. In greater detail all units damaged by energy and materials releases from level 0 units are included in level 1 and so forth, so that referring to process units belonging to a generic i-th Level, they are damaged by level (i-1) units and damage units of level (i+1). The sequence of levels represents the damage propagation across the plant through any multiple interacting sequences of accidents. For each unit a damage (DM) - loss of containment (LOC) matrix is generated allowing to estimate the amount of energy and material releases as well as resulting physical effects based on which the scenario at i-th level is generated. The process stops when no further damage propagation is allowed.

Proceedings Papers

*Proc. ASME*. PVP2015, Volume 1A: Codes and Standards, V01AT01A051, July 19–23, 2015

Paper No: PVP2015-45730

Abstract

This paper describes the safety factors used for fracture assessments of pipes having circumferential surface flaws. The “Fitness-For-Services Codes (the FFS Codes)” of the Japan Society of Mechanical Engineers (JSME) restrict the depths of flaws according to their angles to prevent pipes with flaws from being fracture. Past restrictions were determined based on deterministic evaluations. In fracture assessments of pipes having flaws, however, the effects of measurement errors in flaw dimensions and of variations on material strength must be taken into account. Thus, we evaluated the effects of such variations on fracture assessments of pipes having flaws, and examined safety factors for giving failure probability (or reliability) equal, irrespective of the cracking angles. We found out that failure probability is heavily dependent on the measurement accuracy of flaw depths and material strength (flow stress). In view of this finding, we examined and proposed a simple approach which meets the target reliability without conducting complex evaluations by the Monte Carlo method and reliability evaluation methods (e.g., first-order second-moment method (FOSM)).

Proceedings Papers

*Proc. ASME*. PVP2013, Volume 2: Computer Technology and Bolted Joints, V002T02A028, July 14–18, 2013

Paper No: PVP2013-97646

Abstract

We investigated the transitional size of metal clusters where the electronic effect and the size effect on the ground state structure become weaker. Identification of a transitional size cluster provides the means to efficiently determine the ground state structure of large clusters using density functional theory. Beyond the critical size of clusters, geometrical effects become important and the putative global minimum obtained from an empirical method can be used to determine the true ground state structure where the size effect on structures is less significant. We identified the lowest-energy structure using a first principles method in combination with the global search algorithm. We then used the similarity function to quantify structural difference and similarity between the global minimum obtained from an empirical method and the true ground state structure. Two structures become similar beyond a certain critical size. To investigate low-lying structures of metal clusters, we used a Monte Carlo simulated annealing method which employs the Aggregate-Volume-Bias Monte Carlo (AVBMC) algorithm. Incorporated in the Monte Carlo method is an Embedded Atom Method (EAM) potential developed by the authors.

Proceedings Papers

*Proc. ASME*. PVP2011, Volume 2: Computer Technology and Bolted Joints, 295-299, July 17–21, 2011

Paper No: PVP2011-57748

Abstract

The study of metal clusters has attracted much attention in recent years. Noble metal nanoparticles are of particular interest since their chemical, thermodynamic, electronic, and optical properties make them interesting candidates as building blocks of nanostructure materials. Delineation of these properties requires a complete and definitive characterization of the cluster’s geometrical structure. To find the ground state structure for a cluster, the potential-energy surface (PES) needs to be searched. In this paper, we proposed an efficient hierarchical search method to determine a ground state structure of copper clusters using an effective Monte Carlo simulated annealing method, which employs the Aggregate-Volume-Bias Monte Carlo (AVBMC) algorithm. Incorporated in the Monte Carlo method, is an efficient Embedded Atom Method (EAM) potential developed by the authors.

Proceedings Papers

*Proc. ASME*. PVP2010, ASME 2010 Pressure Vessels and Piping Conference: Volume 6, Parts A and B, 1187-1191, July 18–22, 2010

Paper No: PVP2010-25968

Abstract

A multiple integral representation has been developed to analytically model the probability of failure of reactor vessel. The probability of fracture is a basic methodology for projecting for the life of a new vessel as well as to estimate the remaining life of an existing vessel. The integral representation for the probability of fracture calculation is based on the number count of critical cracks across the whole section of a vessel, based on a given calibrated crack distribution function, obtained by experimental examination of the vessel cross section. Multiple integral is implemented because of the degraded, or variable, fracture toughness and other factors representing the variable facture toughness. For example, the nuclear reactor vessel that is subjected to neutron radiation, will increase the reactor vessel steel brittleness. The effect of neutron irradiation can be calibrated by its increase in ductile-brittle transition temperature (DBTT) in fracture toughness versus temperature curve. Higher DBTT implies a decrease in fracture toughness and an increase in the chance of vessel fracture in brittle fracture mode. The extent of degradation that the High Flux Isotope Reactor (HFIR) vessel has experienced is characterized by its probability of fracture in this paper. The fracture probabilities under the accident pressure conditions against possible HFIR operating life are calculated for the safety analysis of the reactor vessel. Conventional numerical methods of fracture probability calculation such as that adopted by the NRC-sponsored PRAISE CODE and the FAVOR CODE developed in this Laboratory are based on Monte Carlo simulation. Heavy computations are required. The present method of Probability Integral has been used to verify numerical results of approximately 8–10 reports on HFIR remaining-life calculations by Cheverton using FAVOR CODE for the installation of HFIR new cold neutron source. The numerical result based on the method of Probability Integral confirms almost exactly as compared with that obtained by Monte Carlo Method adopted by FAVOR CODE. This Method of Probability Integral, because of its analytical structure, shows the clear physical interpretation of the fracture probability. It provides simple and expedient procedure to obtain numerical values of fracture probability. Moreover, it retains all possible features that the Monte Carlo Method of simulation can accomplish.

Proceedings Papers

*Proc. ASME*. PVP2010, ASME 2010 Pressure Vessels and Piping Conference: Volume 2, 355-358, July 18–22, 2010

Paper No: PVP2010-25735

Abstract

The structural stability and energetics for small copper and gold clusters Cu n and Au n (n = 21–56) were investigated using an effective Monte Carlo simulated annealing method, which employs the Aggregate-Volume-Bias Monte Carlo (AVBMC) algorithm. Incorporated in the Monte Carlo method, is an efficient Embedded Atom Method (EAM) potential developed by the authors. In general agreement with previous empirical studies, the lowest-energy copper structures adapt a single icosahedral structural motif, with pentagonal bipyramid geometry as the building block. However, contrary to studies that describe gold as less symmetric, this work demonstrates that gold clusters adapt both an icosahedral and icositetrahedral structural motifs with many clusters having symmetric geometries.

Proceedings Papers

*Proc. ASME*. PVP2010, ASME 2010 Pressure Vessels and Piping Conference: Volume 3, 735-742, July 18–22, 2010

Paper No: PVP2010-25942

Abstract

Deterministic Fracture Mechanics (DFM) assessments of structural components (e.g. pressure vessels and piping used in the nuclear industry) containing defects can usually be carried out using the R6 procedure. The aim of such an assessment is to demonstrate that there are sufficient safety margins on the applied loads, defect size and fracture toughness for the safe continual operation of the component. To ensure a conservative assessment is made, a lower-bound fracture toughness, and upper-bound defect sizes and applied loads are used. In some cases, this approach will be too conservative and will provide insufficient safety margins. Probabilistic Fracture Mechanics (PFM) allow a way forward in such cases by allowing for the inherent scatter in material properties, defect size and applied loads explicitly. Basic Monte Carlo Methods (MCM) allow an estimate of the probability of failure to be calculated by carrying out a large number of fracture mechanics assessments, each using a random sample of the different random variables (loads, defect size, fracture toughness etc). The probability of failure is obtained by counting the proportion of simulations which lead to assessment points that lie outside the R6 failure assessment curve. This approach can give good results for probabilities greater than 10 −5 . However, for smaller probabilities, the calculation may be inefficient and a very large number of assessments may be necessary to obtain an accurate result, which may be prohibitive. Engineering Reliability Methods (ERM), such as the First Order Reliability method (FORM) and the Second Order Reliability Method (SORM), can be used to estimate the probability of failure in such cases, but these methods can be difficult to implement, do not always give the correct result, and are not always robust enough for general use. Advanced Monte Carlo Methods (AMCM) combine the two approaches to provide an accurate and efficient calculation of probability of failure in all cases. These methods aim to carry out Importance Sampling so that only assessment points that lie close to or outside the failure assessment curve are calculated. Two methods are described in this paper: (1) orthogonal sampling, and (2) spherical sampling. The power behind these methods is demonstrated by carrying out calculations of probability of failure for semi-elliptical, surface breaking, circumferential cracks in the inside of a pressure vessel. The results are compared with the results of Basic Monte Carlo and Engineering Reliability calculations. The calculations use the R6 assessment procedure.

Proceedings Papers

*Proc. ASME*. PVP2010, ASME 2010 Pressure Vessels and Piping Conference: Volume 3, 13-18, July 18–22, 2010

Paper No: PVP2010-25173

Abstract

Initial geometric imperfections have a significant effect on the load carrying capacity of asymmetrical cylindrical pressure vessels. This research paper presents a comparison of a reliability technique that employs a Fourier series representation of random asymmetric imperfections in a defined cylindrical pressure vessel subjected to external pressure. Evaluations as prescribed by the ASME Boiler and Pressure Vessel Code, Section VIII, Division 2 rules are also presented and discussed in light of the proposed reliability technique presented herein. The ultimate goal of the reliability type technique is to statistically predict the buckling load associated with the cylindrical pressure vessel within a defined confidence interval. The example cylindrical shell considered in this study is a fractionating tower for which calculations have been performed in accordance with the ASME B&PV Code. The maximum allowable external working pressure of this tower for the shell thickness of 0.3125 in. is calculated to be 15.1 psi when utilizing the prescribed ASME B&PV Code, Section VIII, Division 1 methods contained within example L-3.1. The Monte Carlo method as developed by the current authors and published in the literature is then used to calculate the maximum allowable external working pressure. Fifty simulated shells of geometry similar to the example tower are generated by the Monte Carlo method to calculate the nondeterministic buckling load. The representation of initial geometric imperfections in the cylindrical pressure vessel requires the determination of appropriate Fourier coefficients. The initial functional description of the imperfections consists of an axisymmetric portion and a deviant portion that appears in the form of a double Fourier series. Multi-mode analyses are expanded to evaluate a large number of potential buckling modes for both predefined geometries and the associated asymmetric imperfections as a function of position within a given cylindrical shell. The method and results described herein are in stark contrast to the dated “knockdown factor” approach currently utilized in ASME B&PV Code.

Proceedings Papers

*Proc. ASME*. PVP2010, ASME 2010 Pressure Vessels and Piping Conference: Volume 3, 1031-1040, July 18–22, 2010

Paper No: PVP2010-25888

Abstract

The assessment of fatigue crack growth due to turbulent mixing of hot and cold coolants presents significant challenges, in particular to determine the thermal loading spectrum. Thermal striping is defined as a random temperature fluctuation produced by incomplete mixing of fluid streams at different temperatures, and it is essentially a random phenomenon in a temporal sense. The objective of this work is to develop a stochastic model to assess thermal fatigue crack growth in mixing tees, based on the power spectral density (PSD) of the temperature fluctuation at the inner pipe surface. Based on the analytical solution for temperature distribution through the wall thickness, obtained by means of Hankel transform, a frequency temperature response function is proposed, in the framework of single-input, single-output (SISO) methodology from random noise/signal theory under sinusoidal input. For the elastic thermal stresses distribution solutions, the magnitude of the frequency response function is first derived and checked against the prediction by FEA. The frequency response of the stress intensity factor (SIF) is obtained by a polynomial fitting of the stress profiles through the wall thickness at various instants of time. The variability in load is given by the statistical properties of thermal spectrum. The temperature spectrum is assumed to be given as a stationary normalized Gaussian narrow-band stochastic process, with constant PSD for a defined range of frequencies. The connection between SIF’s PSD and temperature’s PSD is assured with SIF frequency response function modulus. The frequency of the peaks of each magnitude for K I , which is supposed to be a stationary narrow-band Gaussian process, is characterized by the Rayleigh distribution, and, consequently, the expected value of crack growth rate in respect to cycles is obtained. The probabilities of failure are estimated by mean of the Monte Carlo methods considering a limit state function, which is based on the developed stochastic model. The results of the stochastic approach of thermal fatigue crack growth in mixing tees is completed with probabilistic input to account for the variability in the material characteristics, and finally an application is given to obtain the probability of mixing tees piping failure as function of time reference period.

Proceedings Papers

*Proc. ASME*. PVP2008, Volume 7: Operations, Applications and Components, 285-291, July 27–31, 2008

Paper No: PVP2008-61600

Abstract

The goal of this work was to measure the temporally varying heat flux and surface temperature of a pipe calorimeter in a pool fire, and assess its uncertainty. Three large-scale fire tests were conducted at the Sandia National Laboratories outdoor fire test facility. In each test a cylindrical calorimeter was suspended above a water pool with JP8 fuel floating on top. The calorimeter was a 2.4 m diameter, 4.6 m long, and 2.5 cm wall thickness pipe with end-caps suspended 1 m above the 7.2 m diameter pool. 58 thermocouples were attached to the calorimeter interior surface and backed with 8 cm of insulation. The Sandia One-Dimensional Direct and Inverse Thermal (SODDIT) code was used to determine the calorimeter external surface heat flux and temperature from the measured interior surface temperature versus time. To determine the uncertainty of the SODDIT results, a simulation of the calorimeter in a fire similar to the experiments was performed using the Container Analysis Fire Environment (CAFE) computer code. In this code, a Computational Fluid Dynamics (CFD) fire model applies a temporally and spatially varying heat flux to the exterior surface of a Finite Element (FE) calorimeter model. Flux is similar but not identical to the flux in the experiment. The FE model calculates the internal calorimeter surface temperature, which is used by SODDIT to calculate heat flux which was compared to the applied values. The absorbed heat flux and surface temperature at one calorimeter location was calculated by SODDIT and then compared to the CAFE applied heat flux and surface temperature. From this comparison a base case uncertainty due to inherent inverse calculation errors and frequency smoothing methods is presented. Uncertainties in temperature measurements, calorimeter material properties and wall thickness were applied to the SODDIT calculation and iterated using the Monte Carlo method to determine the overall heat flux and surface temperature uncertainty. The total absorbed heat flux uncertainty at the one studied location is ±4.8 kW/m 2 at 95% confidence. The outer surface temperature uncertainty for all data at the one studied location is ±6.6°C at 95% confidence. For all 58 measurement locations, the overall combined total absorbed heat flux uncertainty is ±13.8 kW/m 2 at 95% confidence, surface temperature uncertainty is ±7.6°C. These uncertainties are valid only when the calorimeter temperature is not within the Curie temperature range of 999 to 1037K.

Proceedings Papers

*Proc. ASME*. PVP2008, Volume 6: Materials and Fabrication, Parts A and B, 1137-1142, July 27–31, 2008

Paper No: PVP2008-61421

Abstract

Stress corrosion cracking (SCC) has been observed at some piping joints made by Austenitic stainless steel in BWR plants. In JAEA, we have been developing probabilistic fracture mechanics (PFM) analysis methods for aged piping based on latest aging knowledge and an analytical code, PASCAL-SP. PASCAL-SP evaluates the failure probability of piping at aged welded joints under SCC by a Monte Carlo method. We proposes a simplified probabilistic model which can be applied to the failure probability analysis based on PFM for welded joint of piping considering the uncertainty of welding residual stress. And the probabilistic evaluation model is introduced to PASCAL-SP. A parametric PFM analysis concerning uncertainties of residual stress distribution using PASCAL-SP was performed. The PFM analysis showed that the uncertainties of residual stress distribution largely influenced break probability. The break probability increased with increasing the uncertainties of residual stress.

Proceedings Papers

*Proc. ASME*. PVP2005, Volume 3: Design and Analysis, 465, July 17–21, 2005

Paper No: PVP2005-71309

Abstract

This paper presents the second of a series of solutions to the buckling of imperfect cylindrical shells subjected to an axial compressive load. In particular, the current problem reviewed is the case of a homogeneous cylindrical shell with random axisymmetric imperfections. The problem solution for the determination of the critical buckling load utilizes a statistical approach to define the random imperfections as opposed to the deterministic methods most often employed in the pressure vessel industry. The imperfections are treated as a random function of the axial (i.e., longitudinal) position on the shell. The Monte Carlo technique is utilized to create a large sample of random shell geometries from which to eventually calculate a critical buckling load for each randomly generated shell geometry. Having matched or predefined the statistical parameters (including the co-variance) of interest as determined from actual manufacturing statistics to the Monte Carlo simulation of shell geometries, the reliability of the critical buckling load is then calculated for the set of cylindrical shells with the random axisymmetric imperfections. The ASME Boiler and Pressure Vessel Code Section VIII fabrication tolerances as supplemented by ASME Code Case 2286-1 are reviewed and addressed in light of the findings of the current study and resulting solutions with respect to the critical buckling loads. The method and results described herein are in stark contrast to the “knockdown factor” approach currently utilized in ASME Code Case 2286-1. Recommendations for further study of the imperfect cylindrical shell are also outlined in an effort to improve on the current design rules regarding column buckling of large diameter shells designed in accordance with ASME Section VIII, Divisions 1 and 2 and ASME STS-1 in combination with the suggestions contained within Code Case 2286-1.

Proceedings Papers

*Proc. ASME*. PVP2006-ICPVT-11, Volume 1: Codes and Standards, 553-561, July 23–27, 2006

Paper No: PVP2006-ICPVT-11-93469

Abstract

A statistical assessment model for structural integrity of steam generator tubes was proposed using Monte Carlo method. The growth of flaws in steam generator tubes was predicted using statistical approaches. The statistical parameters that represent the characteristics of flaw growth and initiation were derived from in-service inspection (ISI) non-destructive evaluation (NDE) data. Based on the statistical approaches, flaw growth models were proposed and applied to predict distribution of flaw size at the end of cycle (EOC). Because NDE measurement results differ from that of real ones in steam generator tubes, a simple method for predicting the physical number of flaws from periodic in-service inspection data was proposed. The probabilistic flaw growth rate was calculated from the in-service non-destructive inspection data. And the statistical growth of flaw was simulated using the Monte Carlo method. Probabilistic distributions of the flaw size and the probability of burst were obtained from numerously repeated simulations using the proposed assessment model.