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All sourcesNASA NTRSDOE OSTI
NASA NTRS2009

The Integrated Medical Model: A Risk Assessment and Decision Support Tool for Space Flight Medical Systems

The Integrated Medical Model (IMM) is a decision support tool that is useful to mission planners and medical system designers in assessing risks and designing medical systems for space flight missions. The IMM provides an evidence based approach for optimizing medical resources and minimizing risks within space flight operational constraints. The mathematical relationships among mission and crew profiles, medical condition incidence data, in-flight medical resources, potential crew functional impairments, and clinical end-states are established to determine probable mission outcomes. Stochastic computational methods are used to forecast probability distributions of crew health and medical resource utilization, as well as estimates of medical evacuation and loss of crew life. The IMM has been used in support of the International Space Station (ISS) medical kit redesign, the medical component of the ISS Probabilistic Risk Assessment, and the development of the Constellation Medical Conditions List. The IMM also will be used to refine medical requirements for the Constellation program. The IMM outputs for ISS and Constellation design reference missions will be presented to demonstrate the potential of the IMM in assessing risks, planning missions, and designing medical systems. The implementation of the IMM verification and validation plan will be reviewed. Additional planned capabilities of the IMM, including optimization techniques and the inclusion of a mission timeline, will be discussed. Given the space flight constraints of mass, volume, and crew medical training, the IMM is a valuable risk assessment and decision support tool for medical system design and mission planning.

Kerstman, Eric↗
DOE OSTI2022

Modeling Predictive Maintenance for NuScale’s Condensate and Feedwater System Using EMRALD

Event Modeling Risk Assessment using Linked Diagrams (EMRALD) is used to model of NuScale’s feedwater and condenser system to capture component degradation and repairing process. To model the reliability of the components, a three-stage failure rate is used contrary to a constant rate leading to component failure. Results from EMRALD will be used to compare different maintenance strategies to reduce system downtime. The outcome of the project will assist in maximizing remaining useful life of the overall system and increasing the availability and revenue of the plant.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI1994

Nuclear Safety [Vol. 35, No. 1, January-June 1994]

Nuclear Safety is a review journal that covers significant developments in the field of nuclear safety. Its scope includes the analysis and control of hazards associated with nuclear energy, operations involving fissionable materials, and the products of nuclear fission and their effects on the environment. Primary emphasis is on safety in reactor design, construction, and operation; however, the safety aspects of the entire fuel cycle, including fuel fabrication, spent-fuel processing, nuclear waste disposal, handling of radioisotopes, and environmental effects of these operations, are also treated. Table of Contents for this issue follows. GENERAL SAFETY CONSIDERATIONS: THE CHERNOBYL ACCIDENT: 1 Chernobyl Accident Management Actions, A. R Sich 25 The IAEA-ASSET Approach to Avoiding Accidents is to Recognize the Precursors to Prevent Incidents, F. Reisch; ACCIDENT ANALYSIS: 36 A Review of the Available Information on the Triggering Stage of a Steam Explosion, D. F. Fletcher; 58 Analysis and Modeling of Flow-Blockage-Induced Steam Explosion Events in the High-Flux Isotope Reactor, R. P. Taleyarkhan, V. Georgevich, C. W. Nestor, U. Gat, B. L. Lepard, D. H. Cook, J. Freels, S. J. Chang, C. Luttrell, R. C. Gwaltney, and J. Kirkpatrick; 74 An Analysis of Disassembling the Radial Reflector of a Thermionic Space Nuclear Reactor Power System, M. S. El-Genk and D. V. Paramonov; CONTROL AND INSTRUMENTATION: 86 Standards for High-Integrity Software, D. R. Wallace, D. R. Kuhn, L M. Ippolito, and L. Beltracchi; DESIGN FEATURES: 98 Adoption of New Design Features for the Next Generation Nuclear Power Reactors, L. S. Tong; 114 Review of Nuclear Piping Seismic Design Requirements, G. C. Slagis and S. E. Moore; ENVIRONMENTAL EFFECTS: 128 PC-Based Probabilistic Safety Assessment Study for a Geological Waste Repository Placed in a Bedded Salt Formation, S. A. Khan; OPERATING EXPERIENCES: 142 Managing Aging in Nuclear Power Plants: Insights from NRC’s Maintenance Team Inspection Reports, A. Fresco and M. Subudhi; 150 Reactor Shutdown Experience, Compiled by J. W. Cletcher; 153 Selected Safety-Related Events, Compiled by G. A. Murphy; RECENT DEVELOPMENTS: 158 Reports, Standards, and Safety Guides, D. S. Queener; 168 Proposed Rule Changes as of Dec. 31, 1993; ANNOUNCEMENTS: 177 Symposium on Radioactive and Mixed Waste—Risk as a Basis for Waste Classification; 177 1995 Incineration Conference 178 Ninth Power Plant Dynamics Control and Testing Symposium; 174 The Authors

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI1994

Nuclear Safety [Vol. 35, No. 1, January-June 1994]

Nuclear Safety is a review journal that covers significant developments in the field of nuclear safety. Its scope includes the analysis and control of hazards associated with nuclear energy, operations involving fissionable materials, and the products of nuclear fission and their effects on the environment. Primary emphasis is on safety in reactor design, construction, and operation; however, the safety aspects of the entire fuel cycle, including fuel fabrication, spent-fuel processing, nuclear waste disposal, handling of radioisotopes, and environmental effects of these operations, are also treated. Table of Contents for this issue follows. THE CHERNOBYL ACCIDENT: 1 Chernobyl Accident Management Actions, A. R Sich; GENERAL SAFETY CONSIDERATIONS: 25 The IAEA-ASSET Approach to Avoiding Accidents is to Recognize the Precursors to Prevent Incidents, F. Reisch; ACCIDENT ANALYSIS: 36 A Review of the Available Information on the Triggering Stage of a Steam Explosion, D. F. Fletcher; 58 Analysis and Modeling of Flow-Blockage-Induced Steam Explosion Events in the High-Flux Isotope Reactor, R. P. Taleyarkhan, V. Georgevich, C. W. Nestor, U. Gat, B. L. Lepard, D. H. Cook, J. Freels, S. J. Chang, C. Luttrell, R. C. Gwaltney, and J. Kirkpatrick; 74 An Analysis of Disassembling the Radial Reflector of a Thermionic Space Nuclear Reactor Power System, M. S. El-Genk and D. V. Paramonov; CONTROL AND INSTRUMENTATION: 86 Standards for High-Integrity Software, D. R. Wallace, D. R. Kuhn, L M. Ippolito, and L. Beltracchi; DESIGN FEATURES: 98 Adoption of New Design Features for the Next Generation Nuclear Power Reactors, L. S. Tong; 114 Review of Nuclear Piping Seismic Design Requirements, G. C. Slagis and S. E. Moore; ENVIRONMENTAL EFFECTS: 128 PC-Based Probabilistic Safety Assessment Study for a Geological Waste Repository Placed in a Bedded Salt Formation, S. A. Khan; OPERATING EXPERIENCES: 142 Managing Aging in Nuclear Power Plants: Insights from NRC’s Maintenance Team Inspection Reports, A. Fresco and M. Subudhi; 150 Reactor Shutdown Experience, Compiled by J. W. Cletcher; 153 Selected Safety-Related Events, Compiled by G. A. Murphy; RECENT DEVELOPMENTS: 158 Reports, Standards, and Safety Guides, D. S. Queener; 168 Proposed Rule Changes as of Dec. 31, 1993; ANNOUNCEMENTS: 177 Symposium on Radioactive and Mixed Waste—Risk as a Basis for Waste Classification; 177 1995 Incineration Conference 178 Ninth Power Plant Dynamics Control and Testing Symposium; 174 The Authors

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2020

Angular Correlation Function Estimators Accounting for Contamination from Probabilistic Distance Measurements

With the advent of surveys containing millions to billions of galaxies, it is imperative to develop analysis techniques that utilize the available statistical power. In galaxy clustering, even small sample contamination arising from distance uncertainties can lead to large artifacts, which the standard estimator for two-point correlation functions does not account for. We first introduce a formalism, termed decontamination, that corrects for sample contamination by utilizing the observed cross-correlations in the contaminated samples; this corrects any correlation function estimator for contamination. Using this formalism, we present a new estimator that uses the standard estimator to measure correlation functions in the contaminated samples but then corrects for contamination. We also introduce a weighted estimator that assigns each galaxy a weight in each redshift bin based on its probability of being in that bin. We demonstrate that these estimators effectively recover the true correlation functions and their covariance matrices. Our estimators can correct for sample contamination caused by misclassification between object types as well as photometric redshifts; they should be particularly helpful for studies of galaxy evolution and baryonic acoustic oscillations, where forward modeling the clustering signal using the contaminated redshift distribution is undesirable.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

SSRS (Stochastic Soaring Raptor Simulator)

SSRS (Stochastic Soaring Raptor Simulator) is a generalizable, probabilistic, and predictive tool for wind energy developers, ecologists, wildlife managers and industry consultants to estimate the potential for soaring raptors to interact with operating wind turbines, without the need for site-specific data collection. Rapid expansion of wind energy development across the world has exposed the risk of turbine collisions for birds and bats. The risk to obligate soaring raptors such as golden eagles is of particular concern due to their small population and influence on ecological balance. Golden eagles rely heavily on updrafts to subsidize their flight, putting them in direct conflict with operational wind turbines that utilize the same wind resource. Understanding the behavior of soaring raptors with varying atmospheric conditions is crucial for predicting and mitigating the risk of turbine collision. This software contains a predictive movement model that simulates individual flight paths of golden eagles during updraft-subsidized long-distance flight, including migration. For a given set of atmospheric conditions, the model simulates thousands of eagles at turbine-scale spatial resolution (50m) to produce a relative presence density map. The simulated eagles rely on updrafts to pursue uninterrupted directional flight with minimal energy expenditure, following fluid-flow principles. The simulator includes a stochastic model of eagle behavior and a systematic method of accounting for spatiotemporal variations in atmospheric conditions. This framework only requires publicly available atmospheric data to estimate orographic and thermal updrafts, ensuring general usability.

Sandhu, Rimple↗
NASA NTRS1997

Shape Optimization by Bayesian-Validated Computer-Simulation Surrogates

A nonparametric-validated, surrogate approach to optimization has been applied to the computational optimization of eddy-promoter heat exchangers and to the experimental optimization of a multielement airfoil. In addition to the baseline surrogate framework, a surrogate-Pareto framework has been applied to the two-criteria, eddy-promoter design problem. The Pareto analysis improves the predictability of the surrogate results, preserves generality, and provides a means to rapidly determine design trade-offs. Significant contributions have been made in the geometric description used for the eddy-promoter inclusions as well as to the surrogate framework itself. A level-set based, geometric description has been developed to define the shape of the eddy-promoter inclusions. The level-set technique allows for topology changes (from single-body,eddy-promoter configurations to two-body configurations) without requiring any additional logic. The continuity of the output responses for input variations that cross the boundary between topologies has been demonstrated. Input-output continuity is required for the straightforward application of surrogate techniques in which simplified, interpolative models are fitted through a construction set of data. The surrogate framework developed previously has been extended in a number of ways. First, the formulation for a general, two-output, two-performance metric problem is presented. Surrogates are constructed and validated for the outputs. The performance metrics can be functions of both outputs, as well as explicitly of the inputs, and serve to characterize the design preferences. By segregating the outputs and the performance metrics, an additional level of flexibility is provided to the designer. The validated outputs can be used in future design studies and the error estimates provided by the output validation step still apply, and require no additional appeals to the expensive analysis. Second, a candidate-based a posteriori error analysis capability has been developed which provides probabilistic error estimates on the true performance for a design randomly selected near the surrogate-predicted optimal design.

Patera, Anthony T.↗
NASA NTRS2001

Portfolios in Stochastic Local Search: Efficiently Computing Most Probable Explanations in Bayesian Networks

Portfolio methods support the combination of different algorithms and heuristics, including stochastic local search (SLS) heuristics, and have been identified as a promising approach to solve computationally hard problems. While successful in experiments, theoretical foundations and analytical results for portfolio-based SLS heuristics are less developed. This article aims to improve the understanding of the role of portfolios of heuristics in SLS. We emphasize the problem of computing most probable explanations (MPEs) in Bayesian networks (BNs). Algorithmically, we discuss a portfolio-based SLS algorithm for MPE computation, Stochastic Greedy Search (SGS). SGS supports the integration of different initialization operators (or initialization heuristics) and different search operators (greedy and noisy heuristics), thereby enabling new analytical and experimental results. Analytically, we introduce a novel Markov chain model tailored to portfolio-based SLS algorithms including SGS, thereby enabling us to analytically form expected hitting time results that explain empirical run time results. For a specific BN, we show the benefit of using a homogenous initialization portfolio. To further illustrate the portfolio approach, we consider novel additive search heuristics for handling determinism in the form of zero entries in conditional probability tables in BNs. Our additive approach adds rather than multiplies probabilities when computing the utility of an explanation. We motivate the additive measure by studying the dramatic impact of zero entries in conditional probability tables on the number of zero-probability explanations, which again complicates the search process. We consider the relationship between MAXSAT and MPE, and show that additive utility (or gain) is a generalization, to the probabilistic setting, of MAXSAT utility (or gain) used in the celebrated GSAT and WalkSAT algorithms and their descendants. Utilizing our Markov chain framework, we show that expected hitting time is a rational function - i.e. a ratio of two polynomials - of the probability of applying an additive search operator. Experimentally, we report on synthetically generated BNs as well as BNs from applications, and compare SGSs performance to that of Hugin, which performs BN inference by compilation to and propagation in clique trees. On synthetic networks, SGS speeds up computation by approximately two orders of magnitude compared to Hugin. In application networks, our approach is highly competitive in Bayesian networks with a high degree of determinism. In addition to showing that stochastic local search can be competitive with clique tree clustering, our empirical results provide an improved understanding of the circumstances under which portfolio-based SLS outperforms clique tree clustering and vice versa.

Mengshoel, Ole J.↗
NASA NTRS2016

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.↗
NASA NTRS2016

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.↗
DOE OSTI2022

Development of a leading simulator/trailing simulator methodology as part of an integrated safety-security analysis for nuclear power plants

Nuclear power plant (NPP) risk assessment is broadly separated into disciplines of nuclear safety, security, and safeguards. Different analysis methods and computer models have been constructed to analyze each of these as separate disciplines. However, due to the complexity of NPP systems, there are risks that can span all these disciplines and require consideration of safety-security (2S) interactions which allows a more complete understanding of the relationship among these risks. In this work, a novel leading simulator/trailing simulator (LS/TS) method is introduced to integrate multiple generic safety and security computer models into a single, holistic 2S analysis. A case study is performed using this novel method to determine its effectiveness. The case study shows that the LS/TS method avoided introducing errors in simulation, compared to the same scenario performed without the LS/TS method. A second case study is then used to illustrate an integrated 2S analysis which shows that different levels of damage to vital equipment from sabotage at a NPP can affect accident evolution by several hours.

42 ENGINEERING↗
DOE OSTI2022

Game-Theoretic Strategies for Cyber-Physical Infrastructures Under Component Disruptions

In this work, networked infrastructures of recursively defined systems composed of discrete cyber and physical components are considered. The components of basic systems at the finest levels can be disrupted by cyber or physical means, and can be reinforced to survive at certain costs. A problem of ensuring the infrastructure performance is formulated as a game between a provider and an attacker, who probabilistically choose components to reinforce and attack, respectively. The disruptions of this infrastructure are characterized using the aggregate failure correlation function that specifies the conditional failure probability of the infrastructure given the failure of an individual system at that level. The survival probabilities of basic systems satisfy simple product-form, first-order differential equations expressed in terms of the multiplier functions. The utility functions of the provider and attacker are composed of the reward and cost terms, both expressed in terms of the component reinforcement and attack probabilities. The Nash equilibrium of this game is characterized, along with the sensitivity functions of the survival probabilities of basic systems that highlight their dependence individually on the cost-benefit terms, the correlation functions, and the multiplier functions. These results are illustrated using simplified models of a distributed cloud servers infrastructure, a 5G data network infrastructure, a high performance computing federation, and a smart energy grid infrastructure.

42 ENGINEERING↗
NASA NTRS2009

Probabilistic Estimation of Critical Flaw Sizes in the Primary Structure Welds of the Ares I-X Launch Vehicle

The primary structure of the Ares I-X Upper Stage Simulator (USS) launch vehicle is constructed of welded mild steel plates. There is some concern over the possibility of structural failure due to welding flaws. It was considered critical to quantify the impact of uncertainties in residual stress, material porosity, applied loads, and material and crack growth properties on the reliability of the welds during its pre-flight and flight. A criterion--an existing maximum size crack at the weld toe must be smaller than the maximum allowable flaw size--was established to estimate the reliability of the welds. A spectrum of maximum allowable flaw sizes was developed for different possible combinations of all of the above listed variables by performing probabilistic crack growth analyses using the ANSYS finite element analysis code in conjunction with the NASGRO crack growth code. Two alternative methods were used to account for residual stresses: (1) The mean residual stress was assumed to be 41 ksi and a limit was set on the net section flow stress during crack propagation. The critical flaw size was determined by parametrically increasing the initial flaw size and detecting if this limit was exceeded during four complete flight cycles, and (2) The mean residual stress was assumed to be 49.6 ksi (the parent material s yield strength) and the net section flow stress limit was ignored. The critical flaw size was determined by parametrically increasing the initial flaw size and detecting if catastrophic crack growth occurred during four complete flight cycles. Both surface-crack models and through-crack models were utilized to characterize cracks in the weld toe.

Pai, Shantaram S.↗
NASA NTRSDate not supplied

Dynamic Assurance of Autonomous Systems through Ground Control Software

Assurance cases are being increasingly acknowledged as a way to build trust in complex systems with autonomous capabilities [1]. An assurance case is a comprehensive, defensible, and valid justification that a system will function as intended for the specific mission and operating environment. Such justifications for systems with autonomous capabilities are often based on various probabilistic quantifications [2]. Due to the dynamic nature of the environmental conditions in which these systems operate, as well as the changing nature of the autonomous systems themselves, these probabilistic quantifications cannot be simply estimated once during design time. Rather, they need to be continually evaluated during systems operations to ensure that the assurance case justifications are valid. We refer to the assurance case that combines both the static and dynamic elements as a Dynamic Assurance Case (DAC). Such complex systems with autonomous capabilities are often deployed with a Ground Control Software (GCS) component to enable remote operation. Whether the system is composed of a single unit or a fleet of units, deployed distributed or in remote environments, GCS acts as a window into the behavior of the deployed system. It receives telemetry from the system, issues commands to the system and provides various functionalities to visualize the system performance. We propose a dynamic assurance framework where the GCS acts as a relay between the autonomous system and its DAC. GCS can be used to measure both unit-specific as well as system-wide probabilistic quantifications using the incoming telemetry. We embed these quantifications throughout the DAC as variables that can be updated by external sources. We use the GCS to periodically update these variables, which allows us to continually evaluate the formally defined assurance case justifications. We demonstrate our dynamic assurance framework in the NASA Ames project Troupe1 that aims at developing a fleet of rovers capable of au- tonomously mapping their environment. The rovers work cooperatively, each collecting data for different parts of the environment. Each rover runs an identical core Flight System (cFS) [4] application. Troupe1 uses OpenC3 Cosmos [5] as the ground system, and AdvoCATE [3] to capture the system DAC. We show how we can measure both rover-specific and system-wide quantifications in Cosmos using its Ruby scripting editor and pass them into the DAC modelled in AdvoCATE. Then, we show how these incoming variables can be embedded in different parts of the DAC and how effects of their updates can be observed

Irfan Sljivo↗
DOE OSTI2022

Improved Distributed-memory Triangle Counting by Exploiting the Graph Structure

Graphs are ubiquitous in modeling complex systems and representing interactions between entities to uncover structural information of the domain. Traditionally, graph analytics workloads are challenging to efficiently scale (both strong and weak cases) on distributed memory due to the irregular memory-access driven nature (with little or no computations) of the methods. The structure of graphs and their relative distribution over the processing elements poses another level of complexity, making it difficult to attain sustainable scalability across platforms. In this paper, we discuss enhancements to TriC, a distributed-memory implementation of graph triangle counting using Message Passing Interface (MPI), which was featured in the 2020 Graph Challenge competition. We have made some incremental enhancements to TriC, primarily adopting a user-defined buffering strategy to overcome the startup problem for large graphs (by fixing the memory for intermediate data), and experimenting with probabilistic data structures such as bloom filter to improve the query response time for assessing edge existence, at the expense of increasing the overall false positive rate. These adjustments have led to a modest improvements in most cases, as compared to the previous version.

Graph Analytics, HPC↗
DOE OSTI2020

An Integrated Platform for Wind Plant Operations: From Atmosphere to Electrons to the Grid

Under the Atmosphere to Electrons to Grid project (A2e2g), the National Renewable Energy Laboratory (NREL) is developing an optimization platform for wind plant operations. The platform merges forecasting tools with aerodynamic and economic models in order to identify optimal operating schedules and controller functions that will maximize a wind plant's value streams for energy and other grid services. In this presentation, we will describe functionalities of the platform. We will also discuss the implications the use of the platform might have for understanding and valuing the capabilities of wind resources for power system operations.

A2E2G↗
NASA NTRS1986

Expected performance of m-solution backtracking

This paper derives upper bounds on the expected number of search tree nodes visited during an m-solution backtracking search, a search which terminates after some preselected number m problem solutions are found. The search behavior is assumed to have a general probabilistic structure. The results are stated in terms of node expansion and contraction. A visited search tree node is said to be expanding if the mean number of its children visited by the search exceeds 1 and is contracting otherwise. It is shown that if every node expands, or if every node contracts, then the number of search tree nodes visited by a search has an upper bound which is linear in the depth of the tree, in the mean number of children a node has, and in the number of solutions sought. Also derived are bounds linear in the depth of the tree in some situations where an upper portion of the tree contracts (expands), while the lower portion expands (contracts). While previous analyses of 1-solution backtracking have concluded that the expected performance is always linear in the tree depth, the model allows superlinear expected performance.

Nicol, D. M.↗
NASA NTRS1989

An approximate methods approach to probabilistic structural analysis

A major research and technology program in Probabilistic Structural Analysis Methods (PSAM) is currently being sponsored by the NASA Lewis Research Center with Southwest Research Institute as the prime contractor. This program is motivated by the need to accurately predict structural response in an environment where the loadings, the material properties, and even the structure may be considered random. The heart of PSAM is a software package which combines advanced structural analysis codes with a fast probability integration (FPI) algorithm for the efficient calculation of stochastic structural response. The basic idea of PAAM is simple: make an approximate calculation of system response, including calculation of the associated probabilities, with minimal computation time and cost, based on a simplified representation of the geometry, loads, and material. The deterministic solution resulting should give a reasonable and realistic description of performance-limiting system responses, although some error will be inevitable. If the simple model has correctly captured the basic mechanics of the system, however, including the proper functional dependence of stress, frequency, etc. on design parameters, then the response sensitivities calculated may be of significantly higher accuracy.

Mcclung, R. C.↗