Learning with a probabilistic teacher
Learning scheme for solving unsupervised learning problems with correct estimate convergence and for state estimates of Gauss-Markov sequences with additive and multiplicative observed noise
Agrawala, A. K.↗
SEARCH · Search NASA
Results for “PROBABILISTIC ESTIMATION”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Reliability analysis of composite structures
A probabilistic static stress analysis methodology has been developed to estimate the reliability of a composite structure. Closed form stress analysis methods are the primary analytical tools used in this methodology. These structural mechanics methods are used to identify independent variables whose variations significantly affect the performance of the structure. Once these variables are identified, scatter in their values is evaluated and statistically characterized. The scatter in applied loads and the structural parameters are then fitted to appropriate probabilistic distribution functions. Numerical integration techniques are applied to compute the structural reliability. The predicted reliability accounts for scatter due to variability in material strength, applied load, fabrication and assembly processes. The influence of structural geometry and mode of failure are also considerations in the evaluation. Example problems are given to illustrate various levels of analytical complexity.
Kan, Han-Pin↗
Automatic probabilistic knowledge acquisition from data
A computer program for extracting significant correlations of attributes from masses of data is outlined. This information can then be used to develop a knowledge base for a probabilistic expert system. The method determines the best estimate of joint probabilities of attributes from data put into contingency table form. A major output from the program is a general formula for calculating any probability relation associated with the data. These probability relations can be utilized to form IF-THEN rules with associated probability, useful for expert systems.
Gevarter, W. B.↗
Probabilistic reasoning for intelligent wind shear avoidance
A computerized intelligent cockpit aid can increase flight-crew awareness of wind shear, improving avoidance decisions. The FAA Windshear Training Aid (WTA) provides guidelines for assessing the risk of wind shear encounter from meteorological evidence. Use of these guidelines in the cockpit is complicated by uncertainty surrounding meteorological knowledge of wind shear. Bayesian network representation is discussed as a means of modeling this uncertain knowledge in a computer. A probabilistic model of the WTA guidelines using Bayesian network representation is presented. This model combines evidence from sources of varying reliability and incorporates results from meteorological studies of wind shear. The probabilistic model can provide flight crews with meaningful estimates of risk to aid their decisions, using evidence from a variety of sources and a base of meteorological knowledge.
Stratton, D. Alexander↗
Assessment and Control of Spacecraft Charging Risks on the International Space Station
The International Space Station (ISS) operates in the F2 region of Earth's ionosphere, orbiting at altitudes ranging from 350 to 450 km at an inclination of 51.6 degrees. The relatively dense, cool F2 ionospheric plasma suppresses surface charging processes much of the time, and the flux of relativistic electrons is low enough to preclude deep dielectric charging processes. The most important spacecraft charging processes in the ISS orbital environment are: 1) ISS electrical power system interactions with the F2 plasma, 2) magnetic induction processes resulting from flight through the geomagnetic field and, 3) charging processes that result from interaction with auroral electrons at high latitude. Recently, the continuing review and evaluation of putative ISS charging hazards required by the ISS Program Office revealed that ISS charging could produce an electrical shock hazard to the ISS crew during extravehicular activity (EVA). ISS charging risks are being evaluated in an ongoing measurement and analysis campaign. The results of ISS charging measurements are combined with a recently developed model of ISS charging (the Plasma Interaction Model) and an exhaustive analysis of historical ionospheric variability data (ISS Ionospheric Specification) to evaluate ISS charging risks using Probabilistic Risk Assessment (PRA) methods. The PRA combines estimates of the frequency of occurrence and severity of the charging hazards with estimates of the reliability of various hazard controls systems, as required by NASA s safety and risk management programs, to enable design and selection of a hazard control approach that minimizes overall programmatic and personnel risk. The PRA provides a quantitative methodology for incorporating the results of the ISS charging measurement and analysis campaigns into the necessary hazard reports, EVA procedures, and ISS flight rules required for operating ISS in a safe and productive manner.
Koontz, Steve↗
Assessment and Control of Spacecraft Charging Risks on the International Space Station
Electrical interactions between the F2 region ionospheric plasma and the 160V photovoltaic (PV) electrical power system on the International Space Station (ISS) can produce floating potentials (FP) on the ISS conducting structure of greater magnitude than are usually observed on spacecraft in low-Earth orbit. Flight through the geomagnetic field also causes magnetic induction charging of ISS conducting structure. Charging processes resulting from interaction of ISS with auroral electrons may also contribute to charging albeit rarely. The magnitude and frequency of occurrence of possibly hazardous charging events depends on the ISS assembly stage (six more 160V PV arrays will be added to ISS), ISS flight configuration, ISS position (latitude and longitude), and the natural variability in the ionospheric flight environment. At present, ISS is equipped with two plasma contactors designed to control ISS FP to within 40 volts of the ambient F2 plasma. The negative-polarity grounding scheme utilized in the ISS 160V power system leads, naturally, to negative values of ISS FP. A negative ISS structural FP leads to application of electrostatic fields across the dielectrics that separate conducting structure from the ambient F2 plasma, thereby enabling dielectric breakdown and arcing. Degradation of some thermal control coatings and noise in electrical systems can result. Continued review and evaluation of the putative charging hazards, as required by the ISS Program Office, revealed that ISS charging could produce a risk of electric shock to the ISS crew during extra vehicular activity. ISS charging risks are being evaluated in ongoing ISS charging measurements and analysis campaigns. The results of ISS charging measurements are combined with a recently developed detailed model of the ISS charging process and an extensive analysis of historical ionospheric variability data, to assess ISS charging risks using Probabilistic Risk Assessment (PRA) methods. The PRA analysis (estimated frequency of occurrence and severity of the charging hazards) are then used to select the hazard control strategy that provides the best overall safety and mission success environment for ISS and the ISS crew. This paper presents: 1) a summary of ISS spacecraft charging analysis, measurements, observations made to date, 2) plans for future ISS spacecraft charging measurement campaigns, and 3) a detailed discussion of the PRA strategy used to assess ISS spacecraft charging risks and select charging hazard control strategies
Koontz, Steve↗
Characterizing Epistemic Uncertainty for Launch Vehicle Designs
NASA Probabilistic Risk Assessment (PRA) has the task of estimating the aleatory (randomness) and epistemic (lack of knowledge) uncertainty of launch vehicle loss of mission and crew risk, and communicating the results. Launch vehicles are complex engineered systems designed with sophisticated subsystems that are built to work together to accomplish mission success. Some of these systems or subsystems are in the form of heritage equipment, while some have never been previously launched. For these cases, characterizing the epistemic uncertainty is of foremost importance, and it is anticipated that the epistemic uncertainty of a modified launch vehicle design versus a design of well understood heritage equipment would be greater. For reasons that will be discussed, standard uncertainty propagation methods using Monte Carlo simulation produce counter intuitive results, and significantly underestimate epistemic uncertainty for launch vehicle models. Furthermore, standard PRA methods, such as Uncertainty-Importance analyses used to identify components that are significant contributors to uncertainty, are rendered obsolete, since sensitivity to uncertainty changes are not reflected in propagation of uncertainty using Monte Carlo methods. This paper provides a basis of the uncertainty underestimation for complex systems and especially, due to nuances of launch vehicle logic, for launch vehicles. It then suggests several alternative methods for estimating uncertainty and provides examples of estimation results. Lastly, the paper describes how to implement an Uncertainty-Importance analysis using one alternative approach, describes the results, and suggests ways to reduce epistemic uncertainty by focusing on additional data or testing of selected components.
Novack, Steven D.↗
Characterizing Epistemic Uncertainty for Launch Vehicle Designs
NASA Probabilistic Risk Assessment (PRA) has the task of estimating the aleatory (randomness) and epistemic (lack of knowledge) uncertainty of launch vehicle loss of mission and crew risk and communicating the results. Launch vehicles are complex engineered systems designed with sophisticated subsystems that are built to work together to accomplish mission success. Some of these systems or subsystems are in the form of heritage equipment, while some have never been previously launched. For these cases, characterizing the epistemic uncertainty is of foremost importance, and it is anticipated that the epistemic uncertainty of a modified launch vehicle design versus a design of well understood heritage equipment would be greater. For reasons that will be discussed, standard uncertainty propagation methods using Monte Carlo simulation produce counter intuitive results and significantly underestimate epistemic uncertainty for launch vehicle models. Furthermore, standard PRA methods such as Uncertainty-Importance analyses used to identify components that are significant contributors to uncertainty are rendered obsolete since sensitivity to uncertainty changes are not reflected in propagation of uncertainty using Monte Carlo methods.This paper provides a basis of the uncertainty underestimation for complex systems and especially, due to nuances of launch vehicle logic, for launch vehicles. It then suggests several alternative methods for estimating uncertainty and provides examples of estimation results. Lastly, the paper shows how to implement an Uncertainty-Importance analysis using one alternative approach, describes the results, and suggests ways to reduce epistemic uncertainty by focusing on additional data or testing of selected components.
Novack, Steven D.↗
Dynamic Medical Risk Assessment Supported by Inference Networks
The Human Research Program's next generation risk model, the Medical Extensible Dynamic Probabilistic Risk Assessment Tool (MEDPRAT), provides researchers with estimates of astronaut medical and health risk. MEDPRATs Susceptibility Inference Network (SIN) facilitates the dynamic component of this tool. The SIN provides MEDPRAT with a "memory" that allows the system to use knowledge of what simulation events have occurred to alter the representative probability that future simulation events will occur during a given trial. The SIN allows for the medical events being simulated to be related to, and influence the likelihood of one another, providing a more robust risk estimate. We present initial work of our efforts to mathematically quantify and represent these dependent relationships between medical events.
McIntyre, L.↗
Introduction to the IMPACT Probabilistic Risk and Tradespace Analysis Tool for Medical System Design
Background: Probabilistic risk analysis (PRA) is a method for estimating risk in complex engineered systems that, at a basic level, focuses on what can go wrong and the likelihood and consequences of those occurrences. NASA has used PRA as an integral component of medical system risk estimation and design for spaceflight. IMPACT (Informing Mission Planning via Analysis of Complex Tradespaces) is a novel tool to meet these goals for exploration missions. Overview: IMPACT performs hundreds of thousands of Monte Carlo simulations of missions to build aggregate pictures of medical risk. These simulations are based on 120 possible medical conditions (the IMPACT Condition List) selected in a consensus-based process because they are of highest likelihood and/or consequence for exploration spaceflight. The conditions are then tied to clinical capabilities which can be used for management (e.g., inserting an IV) and then to over 600 specific resources needed to deliver a capability (e.g., an angiocath or an ultrasound). Different mission profiles can be simulated with user-specified inputs such as mission duration, destination, number of crew and pre-existing medical conditions, and EVA frequency. While the IMPACT evidence base is designed for exploration environments, these user inputs allow the tool to be used across a broad range of missions. IMPACT’s primary outcome metrics include loss of crew life (LOCL, a measure of in-flight mortality due to medical conditions), need for evacuation (RTDC, return to definitive care), and crew disability (TTL, task time lost based on how medical conditions impact the ability to perform over 1000 specific exploration mission crew tasks). In addition to modeling medical risk, IMPACT also accepts user-specified constraints, such as limitations of mass or volume, and will output a recommended clinical capability set and specific medical resources that meet the mission constraints. Discussion: This abstract will provide an introduction to IMPACT and describe the nature of the underlying medical evidence. It will also detail potential use cases for how this tool can be utilized by NASA or commercial spaceflight providers.
Ben Easter↗
IMPACTing Medical System Design with a Risk Analysis Tool [“IMPACT” sur la Conception du Système Médical avec un Outil d'Analyse des Risques]
Background: Following the success of Artemis I, NASA is preparing for human extended duration missions. Ongoing efforts are focused on mitigating mission-related risks, including those affecting crew health and performance. Communication latency, logistics of resupply and time frame of medical evacuation are barriers to provision of healthcare for these missions, especially with respect to constraints in mass, volume, and crew training. An in-depth assessment of medical risks, capabilities and resources for a specific mission design is necessary to determine an optimal balance that maximizes likelihood of mission success. Overview: IMPACT (Informed Mission Planning via Analysis of Complex Tradespaces) is a dynamic tool designed to estimate medical risk and outcomes for a specific mission design. In its current iteration, a list of medical conditions selected based on likelihood of occurrence and/or consequence was linked to a set of clinical capabilities and resources necessary for diagnosis and management. A probabilistic risk analysis tool was then used to identify and estimate the likelihood and consequence of risks through the following outcome metrics: loss of crew life (inflight mortality due to medical conditions), need for medical evacuation (return to definitive care), and crew disability (task time affected based on how medical conditions influence the ability to perform specific exploration mission crew tasks). Finally, the model’s optimization algorithm provides recommendations for medical capabilities that maximize risk mitigation relative to mass and volume constraints. In the Spring of 2023, IMPACT was utilized to estimate outcome metrics for a design reference mission that would be representative of an extended duration Artemis mission. Notional data generated were then used to determine a recommended set of medical capabilities and resources relative to user-defined mass and volume constraints. A multidisciplinary team has also been updating IMPACT to strengthen the model’s fidelity. Figure 1 shows how updates to outcome metric inputs for the conditions resulted in different capability and resource allocation recommendations. Discussion: This presentation will discuss the IMPACT tool and share the latest data generated for a representative extended duration Artemis mission. Efforts to improve the fidelity of data generated by the model’s algorithm will also be discussed.
K A Shair↗
IMPACTing Medical System Design with a Risk Analysis Tool
Background: Following the success of Artemis I, NASA is preparing for human extended duration missions. Ongoing efforts are focused on mitigating mission-related risks, including those affecting crew health and performance. Communication latency, logistics of resupply and time frame of medical evacuation are barriers to provision of healthcare for these missions, especially with respect to constraints in mass, volume, and crew training. An in-depth assessment of medical risks, capabilities and resources for a specific mission design is necessary to determine an optimal balance that maximizes likelihood of mission success. Overview: IMPACT (Informed Mission Planning via Analysis of Complex Tradespaces) is a dynamic tool designed to estimate medical risk and outcomes for a specific mission design. In its current iteration, a list of medical conditions selected based on likelihood of occurrence and/or consequence was linked to a set of clinical capabilities and resources necessary for diagnosis and management. A probabilistic risk analysis tool was then used to identify and estimate the likelihood and consequence of risks through the following outcome metrics: loss of crew life (inflight mortality due to medical conditions), need for medical evacuation (return to definitive care), and crew disability (task time affected based on how medical conditions influence the ability to perform specific exploration mission crew tasks). Finally, the model’s optimization algorithm provides recommendations for medical capabilities that maximize risk mitigation relative to mass and volume constraints. In the Spring of 2023, IMPACT was utilized to estimate outcome metrics for a design reference mission that would be representative of an extended duration Artemis mission. Notional data generated were then used to determine a recommended set of medical capabilities and resources relative to user-defined mass and volume constraints. A multidisciplinary team has also been updating IMPACT to strengthen the model’s fidelity. Figure 1 shows how updates to outcome metric inputs for the conditions resulted in different capability and resource allocation recommendations. Discussion: This presentation will discuss the IMPACT tool and share the latest data generated for a representative extended duration Artemis mission. Efforts to improve the fidelity of data generated by the model’s algorithm will also be discussed.
K A Shair↗
Reliability and Confidence Interval Analysis of a CMC Turbine Stator Vane
High temperature ceramic matrix composites (CMC) are being explored as viable candidate materials for hot section gas turbine components. These advanced composites can potentially lead to reduced weight, enable higher operating temperatures requiring less cooling and thus leading to increased engine efficiencies. However, these materials are brittle and show degradation with time at high operating temperatures due to creep as well as cyclic mechanical and thermal loads. In addition, these materials are heterogeneous in their make-up and various factors affect their properties in a specific design environment. Most of these advanced composites involve two- and three-dimensional fiber architectures and require a complex multi-step high temperature processing. Since there are uncertainties associated with each of these in addition to the variability in the constituent material properties, the observed behavior of composite materials exhibits scatter. Traditional material failure analyses employing a deterministic approach, where failure is assumed to occur when some allowable stress level or equivalent stress is exceeded, are not adequate for brittle material component design. Such phenomenological failure theories are reasonably successful when applied to ductile materials such as metals. Analysis of failure in structural components is governed by the observed scatter in strength, stiffness and loading conditions. In such situations, statistical design approaches must be used. Accounting for these phenomena requires a change in philosophy on the design engineer s part that leads to a reduced focus on the use of safety factors in favor of reliability analyses. The reliability approach demands that the design engineer must tolerate a finite risk of unacceptable performance. This risk of unacceptable performance is identified as a component's probability of failure (or alternatively, component reliability). The primary concern of the engineer is minimizing this risk in an economical manner. The methods to accurately determine the service life of an engine component with associated variability have become increasingly difficult. This results, in part, from the complex missions which are now routinely considered during the design process. These missions include large variations of multi-axial stresses and temperatures experienced by critical engine parts. There is a need for a convenient design tool that can accommodate various loading conditions induced by engine operating environments, and material data with their associated uncertainties to estimate the minimum predicted life of a structural component. A probabilistic composite micromechanics technique in combination with woven composite micromechanics, structural analysis and Fast Probability Integration (FPI) techniques has been used to evaluate the maximum stress and its probabilistic distribution in a CMC turbine stator vane. Furthermore, input variables causing scatter are identified and ranked based upon their sensitivity magnitude. Since the measured data for the ceramic matrix composite properties is very limited, obtaining a probabilistic distribution with their corresponding parameters is difficult. In case of limited data, confidence bounds are essential to quantify the uncertainty associated with the distribution. Usually 90 and 95% confidence intervals are computed for material properties. Failure properties are then computed with the confidence bounds. Best estimates and the confidence bounds on the best estimate of the cumulative probability function for R-S (strength - stress) are plotted. The methodologies and the results from these analyses will be discussed in the presentation.
Murthy, Pappu L. N.↗
Assessment of Crew Time for Maintenance and Repair Activities for Lunar Surface Missions
NASA is currently evaluating different methods to predict how much time crewmembers will spend conducting repair and maintenance activities on future space missions. As mission scope and spacecraft architectures change, understanding how crew repair and maintenance timelines are impacted by mission operations and technology changes is vital for future mission planning. Past work has been done using historical International Space Station (ISS) data to accurately predict crew habitation and operation timelines, resulting in the development of NASA’s Exploration Crew Time Model (ECTM). However, understanding crew maintenance and repair requirements has posed a unique challenge due to the complexity of available datasets, the probabilistic nature of sub-system failures, and the impacts of reliability growth on failure rates. This paper presents a methodology to collect and condition empirical repair and maintenance time data from available datasets, to extrapolate from that data to estimate projected maintenance and repair times for a lunar Surface Habitat (SH), and to assess how uncertainty in repair time could impact utilization time on the lunar surface. NASA ISS maintenance and crew time data are logged into two central databases: the Maintenance Data Collection (MDC) and the Operations Planning Timeline Integration System (OPTimIS). Separately, each of these two datasets capture only portions of the complete set of data required to generate an accurate assessment of crew time spent on maintenance activities at a sub-system level. To create a more useful crew time estimate for maintenance timelines, the authors developed a methodology to capture relevant data from each set and combine and utilize that data by linking crew time requirements to specific components. The authors compare the failure logs in the MDC to crew activity logs pulled from OPTimIS and then process the data to estimate required repair time for each failure and repair event. The entire maintenance activity dataset is then categorized based on the class of failed component to ensure a significant sample size for each class and accurate crew time estimates for any components lacking relevant data. This resultant component repair time data can be used in the future to generate Mean Time to Repair (MTTR) estimates and confidence intervals for each class of component based on a probabilistic distribution of documented maintenance events. These improved MTTR values can then be applied to candidate element sub-system architectures, along with component Mean Time Between Failure (MTBF) data to generate distributions for potential required system crew repair time estimates for a given mission. The authors applied these modeling methods to a case study of a crewed mission to the planned SH and produced expected corrective maintenance crew time distributions. The results produced an expected corrective maintenance crew time at over 24 hours per mission, and a maintenance crew time distribution that reflects the importance of planning for sufficient maintenance requirements each mission. Repair time distributions can then be used to develop more accurate crew schedules and to assess potential available utilization time.
Crew Time↗
Assessment of Crew Time for Maintenance and Repairs Activities for Lunar Surface Missions
NASA is currently evaluating different methods to predict how much time crewmembers will spend conducting repair and maintenance activities on future space missions. As mission scope and spacecraft architectures change, it will be necessary to understand how crew repair and maintenance timelines are impacted by mission operations and technology changes. Past work has been done using historical ISS data to accurately predict crew habitation and operation timelines, resulting in the development of NASA’s Exploration Crew Time Model (ECTM). However, understanding crew maintenance and repair requirements has posed a unique challenge due to the complexity of available datasets, the probabilistic nature of sub-system failures, and the impacts of reliability growth on failure rates. This paper presents a methodology to collect and condition empirical repair and maintenance time data from available data sets, to extrapolate from that data to estimate projected maintenance and repair times for a lunar Surface Habitat, and to assess how uncertainty in repair time could impact utilization time on the lunar surface. NASA International Space Station (ISS) maintenance and crew time data are logged into two central databases, the Maintenance Data Collection (MDC) and the Operations Planning Timeline Integration System (OPTimIS) respectively. Separately, each of these two datasets capture only portions of the complete set of data required to generate an accurate assessment of crew time spent on maintenance activities at a sub-system level. MDC provides a detailed catalog of failure events and an overview of the failure’s required maintenance and OPTimIS provides a description of crew activities and crew time durations dedicated to maintenance. To create a more useful crew time estimate for maintenance timelines, the authors developed a methodology to capture relevant data from each set and combine and utilize that data by linking crew time requirements to specific components. The authors compare the failure logs in the MDC to crew activity logs pulled from OPTimIS and then process the data to estimate required repair times for each failure event. Data is also classified by the outcome of each repair event, whether the failed component was replaced or whether it was repaired in place. The entire maintenance activity dataset is then categorized based on the class of failed component to allow for a statistically significant sample size for each class and to provide accurate crew time estimates for any components lacking relevant data. This resultant component repair time data can be used in the future to generate Mean Time To Repair (MTTR) estimates and confidence intervals for each class of component based on a probabilistic distribution of documented maintenance events. These improved MTTR values can then be applied to candidate element sub-system architectures, along with component Mean Time Between Failure (MTBF) data to generate distributions for potential required system crew repair time estimates for a given mission. Repair time distributions can then be used to develop more accurate crew schedules and to assess potential available utilization time.
Crew Time↗
Dynamic Adjustment of Simulation Parameters for Efficient Vehicle Prognostics
Model-based diagnostics and prognostics rely on state estimation and uncertainty management algorithms to produce useful information for system operators and maintainers. This information enables more informed operational decisions, condition-based maintenance, and overall mission safety assurance. Typically, uncertainty is associated with vehicle state-of-health estimation and prediction results because of modeling errors, internal or external sources of noise, and sensor inaccuracy. Probabilistic uncertainty management methods including Sequential Monte Carlo simulation are commonly used to reason about state-of-health estimates and predictions in the presence of these sources of uncertainty. However, such algorithms can be computationally expensive as they require a very large number of samples to obtain a sufficiently accurate quantification of the end of life probability distribution. As a result, highly mobile autonomous systems that leverage the prognostic results for mission-level replanning are often constrained in their processing capability because of these computationally expensive simulation approaches. Therefore, in this paper, we investigate algorithmic methods for dynamically adjusting simulation time step as well as number of samples to achieve highly efficient prognostic results while maintaining results accuracy. Results obtained from simulated flight experiments of an electric unmanned aerial vehicle are presented to verify the efficacy of such algorithms.
Prognostics↗
Advanced probabilistic methods for quantifying the effects of various uncertainties in structural response
The effects of actual variations, also called uncertainties, in geometry and material properties on the structural response of a space shuttle main engine turbopump blade are evaluated. A normal distribution was assumed to represent the uncertainties statistically. Uncertainties were assumed to be totally random, partially correlated, and fully correlated. The magnitude of these uncertainties were represented in terms of mean and variance. Blade responses, recorded in terms of displacements, natural frequencies, and maximum stress, was evaluated and plotted in the form of probabilistic distributions under combined uncertainties. These distributions provide an estimate of the range of magnitudes of the response and probability of occurrence of a given response. Most importantly, these distributions provide the information needed to estimate quantitatively the risk in a structural design.
Nagpal, Vinod K.↗
Probabilistic fracture finite elements
The Probabilistic Fracture Mechanics (PFM) is a promising method for estimating the fatigue life and inspection cycles for mechanical and structural components. The Probability Finite Element Method (PFEM), which is based on second moment analysis, has proved to be a promising, practical approach to handle problems with uncertainties. As the PFEM provides a powerful computational tool to determine first and second moment of random parameters, the second moment reliability method can be easily combined with PFEM to obtain measures of the reliability of the structural system. The method is also being applied to fatigue crack growth. Uncertainties in the material properties of advanced materials such as polycrystalline alloys, ceramics, and composites are commonly observed from experimental tests. This is mainly attributed to intrinsic microcracks, which are randomly distributed as a result of the applied load and the residual stress.
Liu, W. K.↗