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NASA NTRS2001

Probabilistic Structural Analysis and Reliability Using NESSUS With Implemented Material Strength Degradation Model

This project included both research and education objectives. The goal of this project was to advance innovative research and education objectives in theoretical and computational probabilistic structural analysis, reliability, and life prediction for improved reliability and safety of structural components of aerospace and aircraft propulsion systems. Research and education partners included Glenn Research Center (GRC) and Southwest Research Institute (SwRI) along with the University of Texas at San Antonio (UTSA). SwRI enhanced the NESSUS (Numerical Evaluation of Stochastic Structures Under Stress) code and provided consulting support for NESSUS-related activities at UTSA. NASA funding supported three undergraduate students, two graduate students, a summer course instructor and the Principal Investigator. Matching funds from UTSA provided for the purchase of additional equipment for the enhancement of the Advanced Interactive Computational SGI Lab established during the first year of this Partnership Award to conduct the probabilistic finite element summer courses. The research portion of this report presents the cumulation of work performed through the use of the probabilistic finite element program, NESSUS, Numerical Evaluation and Structures Under Stress, and an embedded Material Strength Degradation (MSD) model. Probabilistic structural analysis provided for quantification of uncertainties associated with the design, thus enabling increased system performance and reliability. The structure examined was a Space Shuttle Main Engine (SSME) fuel turbopump blade. The blade material analyzed was Inconel 718, since the MSD model was previously calibrated for this material. Reliability analysis encompassing the effects of high temperature and high cycle fatigue, yielded a reliability value of 0.99978 using a fully correlated random field for the blade thickness. The reliability did not change significantly for a change in distribution type except for a change in distribution from Gaussian to Weibull for the centrifugal load. The sensitivity factors determined to be most dominant were the centrifugal loading and the initial strength of the material. These two sensitivity factors were influenced most by a change in distribution type from Gaussian to Weibull. The education portion of this report describes short-term and long-term educational objectives. Such objectives serve to integrate research and education components of this project resulting in opportunities for ethnic minority students, principally Hispanic. The primary vehicle to facilitate such integration was the teaching of two probabilistic finite element method courses to undergraduate engineering students in the summers of 1998 and 1999.

Bast, Callie C.↗

NASA NTRS2010

Probabilistic Design and Analysis Framework

PRODAF is a software package designed to aid analysts and designers in conducting probabilistic analysis of components and systems. PRODAF can integrate multiple analysis programs to ease the tedious process of conducting a complex analysis process that requires the use of multiple software packages. The work uses a commercial finite element analysis (FEA) program with modules from NESSUS to conduct a probabilistic analysis of a hypothetical turbine blade, disk, and shaft model. PRODAF applies the response surface method, at the component level, and extrapolates the component-level responses to the system level. Hypothetical components of a gas turbine engine are first deterministically modeled using FEA. Variations in selected geometrical dimensions and loading conditions are analyzed to determine the effects of the stress state within each component. Geometric variations include the cord length and height for the blade, inner radius, outer radius, and thickness, which are varied for the disk. Probabilistic analysis is carried out using developing software packages like System Uncertainty Analysis (SUA) and PRODAF. PRODAF was used with a commercial deterministic FEA program in conjunction with modules from the probabilistic analysis program, NESTEM, to perturb loads and geometries to provide a reliability and sensitivity analysis. PRODAF simplified the handling of data among the various programs involved, and will work with many commercial and opensource deterministic programs, probabilistic programs, or modules.

Strack, William C.↗

DOE OSTI2024

Yet Another Discriminant Analysis (YADA): A Probabilistic Model for Machine Learning Applications

This paper presents a probabilistic model for various machine learning (ML) applications. While deep learning (DL) has produced state-of-the-art results in many domains, DL models are complex and over-parameterized, which leads to high uncertainty about what the model has learned, as well as its decision process. Further, DL models are not probabilistic, making reasoning about their output challenging. In contrast, the proposed model, referred to as Yet Another Discriminate Analysis(YADA), is less complex than other methods, is based on a mathematically rigorous foundation, and can be utilized for a wide variety of ML tasks including classification, explainability, and uncertainty quantification. YADA is thus competitive in most cases with many state-of-the-art DL models. Ideally, a probabilistic model would represent the full joint probability distribution of its features, but doing so is often computationally expensive and intractable. Hence, many probabilistic models assume that the features are either normally distributed, mutually independent, or both, which can severely limit their performance. YADA is an intermediate model that (1) captures the marginal distributions of each variable and the pairwise correlations between variables and (2) explicitly maps features to the space of multivariate Gaussian variables. Numerous mathematical properties of the YADA model can be derived, thereby improving the theoretic underpinnings of ML. Validation of the model can be statistically verified on new or held-out data using native properties of YADA. However, there are some engineering and practical challenges that we enumerate to make YADA more useful.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

A comparison of probabilistic generative frameworks for molecular simulations

Generative artificial intelligence is now a widely used tool in molecular science. Despite the popularity of probabilistic generative models, numerical experiments benchmarking their performance on molecular data are lacking. Here, in this work, we introduce and explain several classes of generative models, broadly sorted into two categories: flow-based models and diffusion models. We select three representative models: neural spline flows, conditional flow matching, and denoising diffusion probabilistic models, and examine their accuracy, computational cost, and generation speed across datasets with tunable dimensionality, complexity, and modal asymmetry. Our findings are varied, with no one framework being the best for all purposes. In a nutshell, (i) neural spline flows do best at capturing mode asymmetry present in low-dimensional data, (ii) conditional flow matching outperforms other models for high-dimensional data with low complexity, and (iii) denoising diffusion probabilistic models appear the best for low-dimensional data with high complexity. Our datasets include a Gaussian mixture model and the dihedral torsion angle distribution of the Aib9 peptide, generated via a molecular dynamics simulation. We hope our taxonomy of probabilistic generative frameworks and numerical results may guide model selection for a wide range of molecular tasks.

Artificial intelligence↗

DOE OSTI2025

Probabilistic flux limiters

The stable numerical integration of shocks in compressible flow simulations relies on the reduction or elimination of Gibbs phenomena (unstable, spurious oscillations). A popular method to virtually eliminate Gibbs oscillations caused by numerical discretization in under-resolved simulations is to use a flux limiter. A wide range of flux limiters have been studied in the literature, with recent interest in their optimization via machine learning methods trained on high-resolution datasets. The common use of flux limiters in numerical codes as plug-and-play blackbox components makes them key targets for design improvement. Even for deterministic dynamical models, numerical uncertainty is introduced via coarse-graining required by insufficient computational power to solve all scales of motion. Conventional flux limiters are deterministic and lack the capacity to address uncertainties, both aleatoric (inherent randomness) and epistemic (modeling uncertainty due to limited knowledge), which arise in coarse-grained numerical simulations. Here, we introduce a conceptually distinct type of flux limiter that is designed to handle the effects of randomness in the model and uncertainty in model parameters. Unlike traditional single-function flux limiters, these new probabilistic flux limiters incorporate multiple flux limiting functions, each applied with a learned probability drawn from high-resolution data to mitigate the effects of uncertainty in numerical simulations. This approach departs from traditional single-function limiters by explicitly modeling and incorporating uncertainty into the shock capturing process. Using the example of Burgers' equation as a testbed, we show that a machine learned, probabilistic flux limiter may be used in a shock capturing code to more accurately capture shock profiles. In particular, we show that our probabilistic flux limiter outperforms standard limiters and can be successively improved upon (up to a point) by expanding the set of probabilistically chosen flux limiting functions.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

UNDERSTANDING THE SEMI-PROBABILISTIC APPROACHES IN STRUCTURAL RELIABILITY USED TO SET DESIGN RELIABILITY TARGETS FOR GRAPHITE COMPONENTS USING ASME BPVC METHODS

Graphite is a quasi-brittle material, resulting in random variability in tensile strength distributions. To account for the random variability in strength, HHA-3000 of the ASME BPVC provides two semi-probabilistic methods for qualifying nuclear graphite components in the design stage, the simplified and full assessments. The full and simplified assessments apply statistical methods to engineering-based design problems. This is often referred to as reliability-based design. Reliability-based design (RBD) is a method to develop reliable designs by accounting for uncertainties and result in small chances of failure when also considering safety factors. RBDs provide reliability targets using semi-probabilistic approaches. RBD is implemented in ASME BPVC HHA-3000 for nuclear graphite components, but is not specific to that application. There has been much confusion around the methods implemented in ASME BPVC HHA-3000 for qualifying nuclear graphite components. To address the confusion, this paper takes a hierarchical approach. First, the general RBD framework is presented. Then, the semi-probabilistic methods and the underlying assumptions implemented in the assessments are presented. The semi-probabilistic methods are separated from the engineering modifications that have been made to the assessments. After building the framework and underlying assumptions, the specific methods in the full and simplified assessments are explained in three steps: inputs, methods, outputs. The methods are applied to an H-451 reflector block. Tensile strength properties for other graphite grades are provided.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2025

Day-Ahead Probabilistic Forecasting of Net-Load and Demand Response Potentials with High Penetration of Behind-the-Meter Solar-plus-Storage

The goal of this project is to develop advanced methods for day-ahead net-load forecasting, by leveraging the state-of-the-art machine learning techniques. The developed models produce both point and probabilistic forecasts for a variety of use cases, and are versatile to work with different types of data sets. The innovation lies in the novel design of the architectures, leveraging the most recent advances in machine learning that have not been explored in power systems, accompanied by techniques in the broader artificial intelligence fields such as fuzzy systems. This project has achieved the following accomplishments: (1) preprocessing of over 10 data sets covering varying geographical regions, time horizons, and system levels, which form a robust foundation for training and evaluating forecasting models across a wide range of realistic grid scenarios; (2) development of an interactive web app that enables exploratory analysis of load and generation data, and supports better understanding of data trends, anomalies, and correlations, facilitating model development and stakeholder engagement; (3) implementation of over 10 benchmark models for point and probabilistic forecasting, which include a mix of conventional machine learning methods and state-of-the-art deep learning approaches, providing a comprehensive baseline for performance comparison and validation of the proposed models; (4) development of a fuzzy system based gradient boosting model, tailored for small (less than 3 years) data sets, which achieves a mean absolute percentage error (MAPE) of 4% for point forecasting and a 20% improvement in average pinball loss for probabilistic forecasting; (5) development of a Transformer (a state-of-the-art deep learning architecture) based neural network model, tailored for large (3 years or more) data sets, which achieves a MAPE of 2% for point forecasting and a 20% improvement in average pinball loss for probabilistic forecasting; (6) development of a methodology for quantifying DR potential, and extensions of the previous models for multi-target forecasting of net load and DR potential, which achieve a MAPE of 10% for DR potential.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2025

Practical Probabilistic Programming

Recent advances in probabilistic programming languages (PPLs) have provided the capability for exact inference: computing a closed-form probability distribution for a given probabilistic program. In particular, the new language Roulette uses a language oriented programming (LOP) approach, wherein analysts build new programming languages on top of a set of primitives provided by Roulette, which then translates these structures into a weighted model counting problem which can be solved by automated reasoning tools. However, because Roulette provides few convenience features, developing these new languages is challenging even for expert users. We developed a standard library of common probability functions for Roulette with the goal of improved usability. This included approximation of continuous probability density functions using discrete probability mass functions. We demonstrated this approach by modeling a cosmic ray striking a RAM controller. We found that Roulette provides a powerful interface for highly expressive probabilistic programs to be generated. In collaboration with the NNSA Advanced Simulation and Computing program, which resulted in development of a tool called Circulette, we were able to model complex circuits expressed in Verilog using probabilistic programs with an expressivity not previously possible. Our research question that motivated the development of a Roulette standard library was to determine whether non-experts could use a PPL to model relevant problems regarding radiation effects on microelectronics. This standard library improved the expressivity of Roulette by implementing common probability density functions, mathematical operators on distributions, and support for empirical distributions. While Roulette is a powerful modeling language, the untyped, LOP approach makes error messages difficult to understand and requires expert aid. We recommend further research on Roulette, especially with its error messages, to enable improved usability. At the same time, this project demonstrated that for users familiar with Roulette and the LOP approach, Roulette provides powerful new capabilities that can be integrated with other Sandia modeling capabilities.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1985

Probabilistic Finite Element Development

The probabilistic finite element computer program known as Numerical Evaluation of Stochastic Structures Under Stress (NESSUS) is being developed for the analysis of critical structural components for reusable space propulsion systems. First year efforts involve the formulation of the probabilistic analysis strategy and the development of a probabilistic linear analysis code. The ultimate goal of the 3-year program is the development of a finite element code capable of performing nonlinear dynamic analysis of structures having stochastic material properties, geometry, and boundary conditions and subjected to random loading. Three levels of sophistication are envisioned for the stochastic description of the structural problem, namely: (1) homogeneous random variable for stiffness, mass, damping, and external loading; (2) stochastic characterization of variables at the element level, with specified interelement correlations; and (3) stochastic interpolation of variables within a finite element. Two alternative probabilistic analysis methods will be developed, allowing for all three levels of modeling sophistication.

Nagtegaal, J.↗

NASA NTRS1986

Probabilistic structural analysis methods for space propulsion system components

The development of a three-dimensional inelastic analysis methodology for the Space Shuttle main engine (SSME) structural components is described. The methodology is composed of: (1) composite load spectra, (2) probabilistic structural analysis methods, (3) the probabilistic finite element theory, and (4) probabilistic structural analysis. The methodology has led to significant technical progress in several important aspects of probabilistic structural analysis. The program and accomplishments to date are summarized.

Chamis, C. C.↗

NASA NTRS1987

Probabilistic structural analysis methods for space propulsion system components

Chamis, Christos C.↗

NASA NTRS1988

Probabilistic methods for structural response analysis

This paper addresses current work to develop probabilistic structural analysis methods for integration with a specially developed probabilistic finite element code. The goal is to establish distribution functions for the structural responses of stochastic structures under uncertain loadings. Several probabilistic analysis methods are proposed covering efficient structural probabilistic analysis methods, correlated random variables, and response of linear system under stationary random loading.

Wu, Y.-T.↗

NASA NTRS1989

Probabilistic structural analysis of space propulsion system turbine blades

Probabilistic loads and stress analysis methods are presently applied to the second-stage blade of the SSME's high-pressure fuel turbopump, in order to illustrate how loads and stress responses can be quantified probabilistically and how the sensitivity of such component- and engine-level independent variables as engine inlet temperatures and local seal wear within the turbopump can be determined. Attention is given to a method for the derivation of probabilistic pressure, temperature, and centrifugal steady-state load descriptions of this second-stage blade from the top of the airfoil to its intersection with the firtree. Dependent-load random variables are obtained from a multilevel physical model in which duty cycle-dependent deterministic loads and related probabilistic variations are accounted for.

Newell, J. F.↗

NASA NTRS1989

Probabilistic modeling for simulation of aerodynamic uncertainties in propulsion systems

The numerical simulation of the probabilistic aerothermodynamic response of propulsion system components to randomness in their environment was explored. The reusable rocket engine turbopumps were selected as an example because of the severe cryogenic environment in which they operate. The thermal and combustion instabilities, coupled with the engine thrust requirements from start up to shut down, lead to randomness in the flow variables and uncertainties in the aerodynamic loading. The probabilistic modeling of the turbopumps aerodynamic response was accomplished using the panel method coupled with fast probability integration methods. The aerodynamic response in the form of probabilistic rotor blades and splitter loading were predicted and the results presented for specified flow coefficient and rotor preswirl variance. Possible future applications of the aerothermodynamic probabilistic modeling in engine transient simulation, condition monitoring and engine life prediction are briefly discussed.

Hamed, Awatef↗

NASA NTRS1988

Probabilistic structural analysis methods development for SSME

The development of probabilistic structural analysis methods is a major part of the SSME Structural Durability Program and consists of three program elements: composite load spectra, probabilistic finite element structural analysis, and probabilistic structural analysis applications. Recent progress includes: (1) the effects of the uncertainties of several factors on the HPFP blade temperature pressure and torque, (2) the evaluation of the cumulative distribution function of structural response variables based on assumed uncertainties on primitive structural variables, and (3) evaluation of the failure probability. Collectively, the results obtained demonstrate that the structural durability of critical SSME components can be probabilistically evaluated.

Chamis, C. C.↗

NASA NTRS1990

Probabilistic analysis of a materially nonlinear structure

A probabilistic finite element program is used to perform probabilistic analysis of a materially nonlinear structure. The program used in this study is NESSUS (Numerical Evaluation of Stochastic Structure Under Stress), under development at Southwest Research Institute. The cumulative distribution function (CDF) of the radial stress of a thick-walled cylinder under internal pressure is computed and compared with the analytical solution. In addition, sensitivity factors showing the relative importance of the input random variables are calculated. Significant plasticity is present in this problem and has a pronounced effect on the probabilistic results. The random input variables are the material yield stress and internal pressure with Weibull and normal distributions, respectively. The results verify the ability of NESSUS to compute the CDF and sensitivity factors of a materially nonlinear structure. In addition, the ability of the Advanced Mean Value (AMV) procedure to assess the probabilistic behavior of structures which exhibit a highly nonlinear response is shown. Thus, the AMV procedure can be applied with confidence to other structures which exhibit nonlinear behavior.

Millwater, H. R.↗

NASA NTRS1990

Probabilistic modeling for simulation of aerodynamic uncertainties in propulsion systems

The numerical simulation of the probabilistic aerothermodynamic response of propulsion system components to randomness in their environment was explored. The reusable rocket engine turbopumps were selected as an example because of the severe cryogenic environment in which they operate. The thermal and combustion instabilities, coupled with the engine thrust requirements from start up to shut down, lead to randomness in the flow variables and uncertainties in the aerosynamic loading. The probabilistic modeling of the turbopumps' aerodynamic response was accomplished using the panel method coupled with fast probability integration methods. The aerodynamic response in the form of probabilistic rotor blades and splitter loading were predicted and the results presented for specified flow coefficient and rotor preswirl variance. Possible future applications of the aerothermodynamic probabilistic modeling in engine transient simulation, condition monitoring, and engine life prediction are briefly discussed.

Hamed, A.↗

NASA NTRS1991

Probabilistic structural analysis of adaptive/smart/intelligent space structures

A three-bay, space, cantilever truss is probabilistically evaluated for adaptive/smart/intelligent behavior. For each behavior, the scatter (ranges) in buckling loads, vibration frequencies, and member axial forces are probabilistically determined. Sensitivities associated with uncertainties in the structure, material and load variables that describe the truss are determined for different probabilities. The relative magnitude for these sensitivities are used to identify significant truss variables that control/classify its behavior to respond as an adaptive/smart/intelligent structure. Results show that the probabilistic buckling loads and vibration frequencies increase for each truss classification, with a substantial increase for intelligent trusses. Similarly, the probabilistic member axial forces reduce for adaptive and intelligent trusses and increase for smart trusses.

Pai, Shantaram S.↗