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Monte Carlo sampling
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Series: ASM Handbook
Volume: 11A
Publisher: ASM International
Published: 30 August 2021
DOI: 10.31399/asm.hb.v11A.a0006803
EISBN: 978-1-62708-329-4
..., and the most frequently used simulation methods, standard Monte Carlo sampling, Latin hypercube sampling, and discrete probability distribution sampling. Further, the article discusses methods developed to analyze the results of probabilistic methods and covers the use of epistemic and aleatory sampling...
Abstract
This article provides an outline of the issues to consider in performing a probabilistic life assessment. It begins with an historical background and introduces the most common methods. The article then describes those methods covering subjects such as the required random variable definitions, how uncertainty is quantified, and input for the associated random variables, as well as the characterization of the response uncertainty. Next, it focuses on specific and generic uncertainty propagation techniques: first- and second-order reliability methods, the response surface method, and the most frequently used simulation methods, standard Monte Carlo sampling, Latin hypercube sampling, and discrete probability distribution sampling. Further, the article discusses methods developed to analyze the results of probabilistic methods and covers the use of epistemic and aleatory sampling as well as several statistical techniques. Finally, it illustrates some of the techniques with application problems for which probabilistic analysis is an essential element.
Series: ASM Handbook Archive
Volume: 11
Publisher: ASM International
Published: 01 January 2002
DOI: 10.31399/asm.hb.v11.a0003514
EISBN: 978-1-62708-180-1
..., the response surface method, and Monte Carlo sampling. A brief introduction to importance sampling, time-variant reliability, system reliability, and risk analysis and target reliabilities is also provided. The article examines the various application problems for which probabilistic analysis is an essential...
Abstract
This article describes the historical background, uncertainties in structural parameters, classifications, and application areas of probabilistic analysis. It provides a discussion on the basic definition of random variables, some common distribution functions used in engineering, selection of a probability distribution, the failure model definition, and a definition of the probability of failure. The article also explains the solution techniques for special cases and general solution techniques, such as first-second-order reliability methods, the advanced mean value method, the response surface method, and Monte Carlo sampling. A brief introduction to importance sampling, time-variant reliability, system reliability, and risk analysis and target reliabilities is also provided. The article examines the various application problems for which probabilistic analysis is an essential element. Examples of the use of probabilistic analysis are presented. The article concludes with an overview of some of the commercially available software programs for performing probabilistic analysis.
Image
in Crystallographic Analysis by Electron Backscatter Diffraction in the Scanning Electron Microscope
> Materials Characterization
Published: 15 December 2019
Fig. 6 Monte Carlo electron trajectory simulations for 20 kV primary beam energy. (a) Simulations of the backscattered electron energy distributions for normal (black curve) and tilted (blue curve) incidence of the electron beam. Note that the energy distribution for the tilted sample
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Image
Published: 01 December 1998
Fig. 2 Monte Carlo simulations of the interaction volume of a 20 keV primary electron beam in an iron sample. (a) Electron trajectories. (b) Sites of K-shell ionizations and production of characteristic x-rays. Source: Ref 1
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Image
Published: 15 December 2019
Fig. 6 Monte Carlo simulations of a focusing guide performance. (a) Unfocused beam at the sample position. (b) Incident beam through a 2.4 m (7.9 ft) focusing square guide with logarithmic spiral curvature. Note that the color scale at right is much higher (500 versus 200 counts).
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Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005534
EISBN: 978-1-62708-197-9
... experiments in which there is a large sample size ( Ref 38 ). Additional recommendations for experimental designs for use in computer-based experimentation are given here. Monte Carlo Sampling The Monte Carlo sampling method is popular in industry. Because it involves repeated sampling of the output...
Abstract
This article presents an approach to manage the uncertainty present in materials design. It describes inductive and deductive approaches to deal with uncertainty. The article focuses on providing an understanding of the opportunities for managing uncertainty and the decisions that influence the accuracy of the results. A design of experiments (DOE) represents a sequence of experiments to be performed, expressed in terms of factors set at specified levels. The article discusses the two types of DOEs: the full factorial design and the fractional factorial design. It explains the factors to be considered when selecting a procedure for propagating uncertainty. The article lists the categories of the popular types of uncertainty propagation methods, including simulation-based methods, local expansion methods, and numerical integration-based methods.
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005401
EISBN: 978-1-62708-196-2
... with a strong initial texture showing periods of rapid and slow growth. Source: Ref 22 Texture-controlled grain growth during beta annealing of titanium alloys has been simulated using both the Monte Carlo (Potts) and phase-field modeling approaches. In the work of Ivasishin et al. ( Ref 24 , 25...
Abstract
The modeling and simulation of texture evolution for titanium alloys is often tightly coupled to microstructure evolution. This article focuses on a number of problems for titanium alloys in which such coupling is critical in the development of quantitative models. It discusses the phase equilibria, crystallography, and deformation behavior of titanium and titanium alloys. The article describes the modeling and simulation of recrystallization and grain growth of single-phase beta and single-phase alpha titanium. The deformation- and transformation-texture evolution of two-phase (alpha/beta) titanium alloys are also discussed.
Series: ASM Handbook
Volume: 14A
Publisher: ASM International
Published: 01 January 2005
DOI: 10.31399/asm.hb.v14a.a0004027
EISBN: 978-1-62708-185-6
... the grayness. These involve probabilistic methods such as Monte Carlo-Potts (MC-P) or cellular automata (CA) in which the consequences of using certain selection rules for the evolution of a microstructure can be computed and pictorially represented for comparison with observed microstructures. Local...
Abstract
The systematic study of microstructural evolution during deformation under hot working conditions is important in controlling processing variables to achieve dimensional accuracy. This article explains the microstructural features that need to be modeled and provides an outline of the principles and achievements of each of the various microstructural models, including black-box modeling, gray-box modeling, white-box modeling, and hybrid modeling.
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005428
EISBN: 978-1-62708-196-2
... with the total boundary area. H may represent stored energies that arise in the case of deformed structures and so provide a driving force for recrystallization. Dynamics Microstructural evolution is simulated by using a Monte Carlo method to sample different states of the system. The method...
Abstract
The misorientation of a boundary of a growing grain is defined not only by its crystallography but also by the crystallography of the grain into which it is growing. This article focuses on the Monte Carlo Potts model that is typically used to model grain growth, Zener-Smith pinning, abnormal grain growth, and recrystallization. It introduces the basics of the model, providing details of the dynamics, simulation variables, boundary energy, boundary mobility, pinning systems, and stored energy. The article explains how to incorporate experimental parameters and how to validate the model by comparing the observed behavior quantitatively with theory. The industrial applications of the model are also discussed. The article also provides a wide selection of the algorithms for implementing the Potts model, such as boundary-site models, n -fold way models, and parallel models, which are needed to simulate large-scale industrial applications.
Series: ASM Handbook
Volume: 11
Publisher: ASM International
Published: 15 January 2021
DOI: 10.31399/asm.hb.v11.a0006770
EISBN: 978-1-62708-295-2
... accelerating voltages and lower atomic number compositions, the interaction area can be substantially larger. Fig. 8 Monte-Carlo-predicted interaction volume Often it is not possible in failure investigations to polish the sample. Many times, the desire is to examine the surface of a sample...
Abstract
X-ray spectroscopy is generally accepted as the most useful ancillary technique that can be added to any scanning electron microscope (SEM), even to the point of being considered a necessity by most operators. While “stand-alone” x-ray detection systems are used less frequently in failure analysis than the more exact instrumentation employed in SEMs, the technology is advancing and is worthy of note due to its capability for nondestructive analysis and application in the field. This article begins with information on the basis of the x-ray signal. This is followed by information on the operating principles and applications of detectors for x-ray spectroscopy, namely energy-dispersive spectrometers, wavelength-dispersive spectrometers, and handheld x-ray fluorescence systems. The processes involved in x-ray analysis in the SEM and handheld x-ray fluorescence analysis are then covered. The article ends with a discussion on the applications of x-ray spectroscopy in failure analysis.
Book: Fatigue and Fracture
Series: ASM Handbook
Volume: 19
Publisher: ASM International
Published: 01 January 1996
DOI: 10.31399/asm.hb.v19.a0002369
EISBN: 978-1-62708-193-1
.... If the values are selected at random from the distribution of possible values, the approach is called “Monte Carlo analysis” ( Ref 57 , 58 ). The probability of failure is equal to the fraction of outcomes that have shorter than desired lifetimes. Monte Carlo methods have become quite popular due to the ready...
Abstract
There are two parts to deal with uncertainty in fatigue design: determining the distributions of possible values for all uncertain inputs and calculating the probability of failure due to all the uncertain inputs. This article discusses the sources of uncertainty in a fatigue analysis, such as the material properties, distribution of applied stress levels within a given environment, environments or loading intensities, and modeling or prediction. It presents a probabilistic approach for analyzing the uncertainties and determining the level of reliability (probability of failure).
Series: ASM Handbook
Volume: 22B
Publisher: ASM International
Published: 01 November 2010
DOI: 10.31399/asm.hb.v22b.a0005505
EISBN: 978-1-62708-197-9
... design quality (reliability and robustness). Many methods have been developed for stochastic sampling, including Monte Carlo methods ( Ref 31 , 32 ), structural reliability analysis methods ( Ref 33 , Ref 34 , Ref 35 ), sensitivity-based methods, based on Taylor's expansion ( Ref 36 , 37...
Abstract
The process of optimization involves choosing the best solution from a pool of potential candidate solutions. This article provides a description of some classes of problems and the optimization methods that solve them. These problems include the deterministic single-objective problem, the deterministic multiobjective problem, and the nondeterministic, stochastic optimization problem. The article presents several complementary approaches to solve a wide variety of single-objective and multiobjective mechanical engineering applications. Multiobjective optimization study and stochastic optimization studies are also discussed.
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005432
EISBN: 978-1-62708-196-2
... that distinguish CA simulations (and other representational simulations such as Monte Carlo simulations) from mean field analytical models and models employing a homogeneous effective medium. This article examines how CA can be applied to the simulation of static and dynamic recrystallization. It describes...
Abstract
This article examines how cellular automaton (CA) can be applied to the simulation of static and dynamic recrystallization. It describes the steps involved in the CA simulation of recrystallization. These include defining the CA framework, generating the initial microstructure, distributing nuclei of recrystallized grains, growing the recrystallized grains, and updating the dislocation density. The article concludes with information on the developments in CA simulations.
Book Chapter
Series: ASM Handbook
Volume: 14B
Publisher: ASM International
Published: 01 January 2006
DOI: 10.31399/asm.hb.v14b.a0005169
EISBN: 978-1-62708-186-3
...; radius of curvature after HT hot isostatic pressing MDOL maximum HV Knoop hardness Monte Carlo R springback; reduction; plastic horsepower MDRX Monte Carlo step Hz Rockwell hardness; requires mg multidisciplinary optimization R0 strain anisotropy ratio; thermal Mg multidisciplinary design I scale...
Series: ASM Handbook
Volume: 10
Publisher: ASM International
Published: 15 December 2019
DOI: 10.31399/asm.hb.v10.a0006637
EISBN: 978-1-62708-213-6
... the critical angle, those particles have transverse kinetic energy exceeding the collective string potential; the collective steering effect subsequently disappears. Channeling was first predicted in 1912 ( Ref 11 ). In 1963, channeling was accidently observed through Monte Carlo simulations ( Ref 12 , 13...
Abstract
This article provides a detailed account of the basic concepts of Rutherford backscattering spectrometry (RBS). It begins with a description of the principles of RBS, as well as the effect of channeling in conjunction with backscattering measurements and the effect of energy loss under this condition. This is followed by a section on equipment used in RBS analysis. Channel-energy conversion, energy-depth conversion, and separation of the dechanneling background are then discussed as the main steps of RBS data analysis. The article also discusses the applications of RBS—including composition of bulk samples, thin-film composition and layer thickness, impurity profiles, damage depth profile, and surface peak—as well as the various codes developed to simulate it.
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005410
EISBN: 978-1-62708-196-2
.... Fig. 4 Dependence on the nominal concentration x Zr 0 of the cluster size distribution of an aluminum solid solution at 500 °C. At this temperature, the solubility limit is x Zr e = 0.0548 % . Symbols correspond to atomic simulations (kinetic Monte Carlo) ( Ref 18...
Abstract
This article describes the results obtained by Volmer, Weber, Farkas, Becker, and Doring, which constitute the classical nucleation theory. These results are the predictions of the precipitate size distribution, steady-state nucleation rate, and incubation time. The article reviews a nucleating system as a homogeneous phase using the classical nucleation theory, along with heterophase fluctuations that led to the formation of precipitates. It discusses the gas cluster dynamics using the kinetic approach to describe nucleation. The article presents key parameters, such as cluster condensation and evaporation rates, to describe the time evolution of the system. The predictions and extensions of the classical nucleation theory are discussed. The article also provides the limitations of classical nucleation theories in cluster dynamics.
Series: ASM Handbook
Volume: 10
Publisher: ASM International
Published: 15 December 2019
DOI: 10.31399/asm.hb.v10.a0006660
EISBN: 978-1-62708-213-6
... that have lost very little energy. These electrons have been shown to exit the sample very close to the initial beam position, resulting in spatial resolutions in typical transition metals to be better than 0.1 μm for beam voltages up to 20 kV ( Ref 10 ). Monte Carlo electron trajectory simulations...
Abstract
The electron backscatter diffraction (EBSD) technique has proven to be very useful in the measurement of crystallographic textures, orientation relationships between phases, and both plastic and elastic strains. This article focuses on backscatter diffraction in a scanning electron microscope and describes transmission Kikuchi diffraction. It begins with a discussion on the origins of EBSD and the collection of EBSD patterns. This is followed by sections providing information on EBSD spatial resolution and system operation of EBSD. Various factors pertinent to perform an EBSD experiment are then covered. The article further describes the processes involved in sample preparation that are critical to the success or usefulness of an EBSD experiment. It also discusses the applications of EBSD to bulk samples and the development of EBSD indexing methods.
Series: ASM Handbook
Volume: 10
Publisher: ASM International
Published: 15 December 2019
DOI: 10.31399/asm.hb.v10.a0006668
EISBN: 978-1-62708-213-6
... of the SEM compared with other common microscopy and microanalysis techniques. The following sections cover the critical issues regarding sample preparation, the physical principles regarding electron beam-sample interaction, and the mechanisms for many types of image contrast. The article also presents...
Abstract
This article provides detailed information on the instrumentation and principles of the scanning electron microscope (SEM). It begins with a description of the primary components of a conventional SEM instrument. This is followed by a discussion on the advantages and disadvantages of the SEM compared with other common microscopy and microanalysis techniques. The following sections cover the critical issues regarding sample preparation, the physical principles regarding electron beam-sample interaction, and the mechanisms for many types of image contrast. The article also presents the details of SEM-based techniques and specialized SEM instruments. It ends with example applications of various SEM modes.
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005434
EISBN: 978-1-62708-196-2
.... The article mainly focuses on phenomena and modeling approaches that are specific to VPP; however, most of the complementary information needed on general or specialized topics of interest (nucleation, microstructure evolution, Monte Carlo methods, etc.) is found in the cited references or in other articles...
Abstract
This article focuses on transport phenomena and modeling approaches that are specific to vapor-phase processes (VPP). It discusses the VPP for the synthesis of materials. The article reviews the basic notions of molecular collisions and gas flows, and presents transport equations. It describes the modeling of vapor-surface interactions and kinetics of hetereogeneous processes as well as the modeling and kinetics of homogenous reactions in chemical vapor deposition (CVD). The article provides information on the various stages of developing models for numerical simulation of the transport phenomena in continuous media and transition regime flows of VPP. It explains the methods used for molecular modeling in computational materials science. The article also presents examples that illustrate multiscale simulations of CVD or PVD processes and examples that focus on sputtering deposition and reactive or ion beam etching.
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005422
EISBN: 978-1-62708-196-2
... of boundary junctions on grain growth. Various models have been used for the simulation of grain-boundary migration and related phenomena, in particular, grain growth and recrystallization, notably Monte Carlo ( Ref 1 , Ref 2 , Ref 3 , Ref 4 ), phase field ( Ref 5 , 6 ), and network models ( Ref 7...
Abstract
This article reviews network models and their applications for the simulation of various physical phenomena related to grain-boundary migration. It discusses the steps involved in the implementation of two and three-dimensional network models, namely, acquisition and discretization of the microstructure, formulation of the equation of motion, and implementation of the topological transformations. The article presents examples that illustrate the simulation of physical phenomena to demonstrate the predictive power and flexibility of network models.