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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Microstructure-Sensitive Uncertainty Quantification for Crystal Plasticity Finite Element Constitutive Models Using Stochastic Collocation Methods

Uncertainty quantification (UQ) plays a major role in verification and validation for computational engineering models and simulations, and establishes trust in the predictive capability of computational models. In the materials science and engineering context, where the process-structure-property-performance linkage is well known to be the only road mapping from manufacturing to engineering performance, numerous integrated computational materials engineering (ICME) models have been developed across a wide spectrum of length-scales and time-scales to relieve the burden of resource-intensive experiments. Within the structure-property linkage, crystal plasticity finite element method (CPFEM) models have been widely used since they are one of a few ICME toolboxes that allows numerical predictions, providing the bridge from microstructure to materials properties and performances. Several constitutive models have been proposed in the last few decades to capture the mechanics and plasticity behavior of materials. While some UQ studies have been performed, the robustness and uncertainty of these constitutive models have not been rigorously established. In this work, we apply a stochastic collocation (SC) method, which is mathematically rigorous and has been widely used in the field of UQ, to quantify the uncertainty of three most commonly used constitutive models in CPFEM, namely phenomenological models (with and without twinning), and dislocation-density-based constitutive models, for three different types of crystal structures, namely face-centered cubic (fcc) copper (Cu), body-centered cubic (bcc) tungsten (W), and hexagonal close packing (hcp) magnesium (Mg). Our numerical results not only quantify the uncertainty of these constitutive models in stress-strain curve, but also analyze the global sensitivity of the underlying constitutive parameters with respect to the initial yield behavior, which may be helpful for robust constitutive model calibration works in the future.

36 MATERIALS SCIENCE↗
DOE OSTI2016

Analysis of Time Dependent Data and PRA

In the past decades, several numerical simulation codes have been employed to simulate accident dynamics (e.g., RELAP5-3D, MELCOR, MAAP). In order to evaluate the impact of uncertainties into accident dynamics, several stochastic methodologies have been coupled with these codes. These stochastic methods range from classical Monte-Carlo and Latin Hypercube sampling to stochastic polynomial methods. Similar approaches have been introduced into the risk and safety community where stochastic methods (such as RAVEN, ADAPT, MCDET, ADS) have been coupled with safety analysis codes in order to evaluate the safety impact of timing and sequencing of events. These approaches are usually called Dynamic PRA or simulation-based PRA methods. These uncertainties and safety methods usually generate a large number of simulation runs (database storage may be on the order of gigabytes or higher). The scope of this paper is to present a broad overview of methods and algorithms that can be used to analyze and extract information from large data sets containing time dependent data. In this context, 'extracting information' means constructing input-output correlations, finding commonalities, and identifying outliers. Some of the algorithms presented here have been developed or are under development within the RAVEN statistical framework. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2022

Block triangular preconditioning for stochastic Galerkin method

Here, in this paper, we develop a new block triangular preconditioner for solving partial differential equations with random coefficients. We prove spectral bounds for the preconditioned system. Several numerical examples are provided to demonstrate the efficiency of this preconditioner, especially for stochastic problems with large variance.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Computationally-efficient stochastic cluster dynamics method for modeling damage accumulation in irradiated materials

An improved version of a recently developed stochastic cluster dynamics (SCD) method (Marian and Bulatov, 2012) [6] is introduced as an alternative to rate theory (RT) methods for solving coupled ordinary differential equation (ODE) systems for irradiation damage simulations. SCD circumvents by design the curse of dimensionality of the variable space that renders traditional ODE-based RT approaches inefficient when handling complex defect population comprised of multiple (more than two) defect species. Several improvements introduced here enable efficient and accurate simulations of irradiated materials up to realistic (high) damage doses characteristic of next-generation nuclear systems. The first improvement is a procedure for efficiently updating the defect reaction-network and event selection in the context of a dynamically expanding reaction-network. Next is a novel implementation of the τ-leaping method that speeds up SCD simulations by advancing the state of the reaction network in large time increments when appropriate. Lastly, a volume rescaling procedure is introduced to control the computational complexity of the expanding reaction-network through occasional reductions of the defect population while maintaining accurate statistics. The enhanced SCD method is then applied to model defect cluster accumulation in iron thin films subjected to triple ion-beam (Fe{sup 3+}, He{sup +} and H{sup +}) irradiations, for which standard RT or spatially-resolved kinetic Monte Carlo simulations are prohibitively expensive.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2013

Optimizing the implementation of the target motion sampling temperature treatment technique - How fast can it get?

This article discusses the optimization of the target motion sampling (TMS) temperature treatment method, previously implemented in the Monte Carlo reactor physics code Serpent 2. The TMS method was introduced in [1] and first practical results were presented at the PHYSOR 2012 conference [2]. The method is a stochastic method for taking the effect of thermal motion into account on-the-fly in a Monte Carlo neutron transport calculation. It is based on sampling the target velocities at collision sites and then utilizing the 0 K cross sections at target-at-rest frame for reaction sampling. The fact that the total cross section becomes a distributed quantity is handled using rejection sampling techniques. The original implementation of the TMS requires 2.0 times more CPU time in a PWR pin-cell case than a conventional Monte Carlo calculation relying on pre-broadened effective cross sections. In a HTGR case examined in this paper the overhead factor is as high as 3.6. By first changing from a multi-group to a continuous-energy implementation and then fine-tuning a parameter affecting the conservativity of the majorant cross section, it is possible to decrease the overhead factors to 1.4 and 2.3, respectively. Preliminary calculations are also made using a new and yet incomplete optimization method in which the temperature of the basis cross section is increased above 0 K. It seems that with the new approach it may be possible to decrease the factors even as low as 1.06 and 1.33, respectively, but its functionality has not yet been proven. Therefore, these performance measures should be considered preliminary. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2018

A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem

We study a multigrid method for solving large linear systems of equations with tensor product structure here. Such systems are obtained from stochastic finite element discretization of stochastic partial differential equations such as the steady-state diffusion problem with random coefficients. When the variance in the problem is not too large, the solution can be well approximated by a low-rank object. In the proposed multigrid algorithm, the matrix iterates are truncated to low rank to reduce memory requirements and computational effort. The technique is proved convergent with an analytic error bound. Numerical experiments show its effectiveness in solving the Galerkin systems compared to the original multigrid solver, especially when the number of degrees of freedom associated with the spatial discretization is large.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Approximate method for stochastic chemical kinetics with two-time scales by chemical Langevin equations

The frequently used reduction technique is based on the chemical master equation for stochastic chemical kinetics with two-time scales, which yields the modified stochastic simulation algorithm (SSA). For the chemical reaction processes involving a large number of molecular species and reactions, the collection of slow reactions may still include a large number of molecular species and reactions. Consequently, the SSA is still computationally expensive. Because the chemical Langevin equations (CLEs) can effectively work for a large number of molecular species and reactions, this paper develops a reduction method based on the CLE by the stochastic averaging principle developed in the work of Khasminskii and Yin [SIAM J. Appl. Math. 56, 1766–1793 (1996); ibid. 56, 1794–1819 (1996)] to average out the fast-reacting variables. This reduction method leads to a limit averaging system, which is an approximation of the slow reactions. Because in the stochastic chemical kinetics, the CLE is seen as the approximation of the SSA, the limit averaging system can be treated as the approximation of the slow reactions. As an application, we examine the reduction of computation complexity for the gene regulatory networks with two-time scales driven by intrinsic noise. For linear and nonlinear protein production functions, the simulations show that the sample average (expectation) of the limit averaging system is close to that of the slow-reaction process based on the SSA. It demonstrates that the limit averaging system is an efficient approximation of the slow-reaction process in the sense of the weak convergence.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2012

An Adaptive Sparse-Grid High-Order Stochastic Collocation Method for Bayesian Inference in Groundwater Reactive Transport Modeling

Although Bayesian analysis has become vital to the quantification of prediction uncertainty in groundwater modeling, its application has been hindered due to the computational cost associated with numerous model executions needed for exploring the posterior probability density function (PPDF) of model parameters. This is particularly the case when the PPDF is estimated using Markov Chain Monte Carlo (MCMC) sampling. In this study, we develop a new approach that improves computational efficiency of Bayesian inference by constructing a surrogate system based on an adaptive sparse-grid high-order stochastic collocation (aSG-hSC) method. Unlike previous works using first-order hierarchical basis, we utilize a compactly supported higher-order hierar- chical basis to construct the surrogate system, resulting in a significant reduction in the number of computational simulations required. In addition, we use hierarchical surplus as an error indi- cator to determine adaptive sparse grids. This allows local refinement in the uncertain domain and/or anisotropic detection with respect to the random model parameters, which further improves computational efficiency. Finally, we incorporate a global optimization technique and propose an iterative algorithm for building the surrogate system for the PPDF with multiple significant modes. Once the surrogate system is determined, the PPDF can be evaluated by sampling the surrogate system directly with very little computational cost. The developed method is evaluated first using a simple analytical density function with multiple modes and then using two synthetic groundwater reactive transport models. The groundwater models represent different levels of complexity; the first example involves coupled linear reactions and the second example simulates nonlinear ura- nium surface complexation. The results show that the aSG-hSC is an effective and efficient tool for Bayesian inference in groundwater modeling in comparison with conventional MCMC sim- ulations. The computational efficiency is expected to be more beneficial to more computational expensive groundwater problems.

Zhang, Guannan↗
DOE OSTI2019

Optimal Allocation of Water Resources Using a Two-Stage Stochastic Programming Method with Interval and Fuzzy Parameters

Efficient water allocation is one of the most prominent issues in water resources management. In this research, a two-stage interval-parameter stochastic fuzzy programming with type 2 membership functions was used to allocate water resources optimally to different users under uncertainty. This method can handle uncertainties expressed as probability distributions, discrete intervals, and fuzzy sets. The model considers treated wastewater as an allocable water resource in a scenario, in addition to water that is extracted from wellheads, springs, and qanats. Moreover, the loss rate of water during distribution, surplus water in the reservoir in the previous and the next period, and treated wastewater parameters have been incorporated into the model. This model was applied to a case study of water resources allocation within a multi-user and multi-reservoir context in the Zarand region of Kerman, Iran. The results indicate that reasonable solutions have been generated, in the form of interval and fuzzy information under different scenarios that could help managers provide optimal water resources allocation plans. The results also demonstrate that establishing a wastewater treatment station increases the system net benefits, surplus water in wellheads for the next period, and system reliability (level of satisfying the fuzzy goal and constraints), and decreases encountering water shortages.

58 GEOSCIENCES↗
DOE OSTI2020

Assessment of wind power scenario creation methods for stochastic power systems operations

Probabilistic scenarios of renewable energy production, such as wind, have been gaining popularity for use in stochastic variants of power systems operations scheduling problems, allowing for optimal decision-making under uncertainty. The quality of the scenarios has a direct impact on the value of the resulting decisions, but until now, methods for creating scenarios have not been compared under realistic operational conditions. Here, we compare the quality of scenario sets created using three different methods, based on a simulated re-enactment of stochastic day-ahead unit commitment and subsequent dispatch for a realistic test system. We create scenarios using a dataset of forecasted and actual wind power values, scaled to evaluate the effects of increasing wind penetration levels. We show that the choice of scenario set can significantly impact system operating cost, renewable energy use, and the ability of the system to meet demand. This result has implications for the ability of system operators to efficiently integrate renewable production into their day-ahead planning, highlighting the need for the use of performance-based assessments for scenario evaluation.

17 WIND ENERGY↗
DOE OSTI2024

Uncertainty Quantification and Sensitivity Analysis of Non-Nuclear Advanced Controls Testbed Reactor Mockup

The research presented in this report describes our progress in applying stochastic methods and uncertainty quantification, parametric study, and variance-based sensitivity analysis (also known as Sobol sensitivity analysis) to a full-core model of a nuclear thermal propulsion (NTP) system simulated with Griffin, with the goal of developing a reduced order (surrogate) model which can be rapidly sampled while perturbing multiple input parameters. In this NTP system, reactivity and power feedback affect the rotation of control drums, which are controlled by a hybrid proportional, integral and derivative (PID) controller, actuated by the power demand and reactivity feedback from the numerical model. This model uses reactor kinetic feedback (mean generation time and $\beta$ from a transient Griffin simulation executed with the improved quasi-static method to provide the kinetic parameters) as inputs to functions which control the CD rotation angle. Using a number of stochastic method approaches, we developed a dual purpose training-surrogate model of the NTP system using polynomial regression. The trained model can be rapidly sampled while simultaneously perturbing various input parameters of the model, such as coefficients on the PID control, or temperature (directly affect the neutron cross section). The surrogate model delivers accurate results orders-of-magnitude faster (minutes, not days) than the base model. Once the base model has been trained, distributions of the uncertain parameters can be changed at will to investigate the effects of perturbing multiple inputs and their effect on the output. For example, coefficients used in the PID control system may vary due to some physical interference, or there may be uncertainty in the temperature of the neutron cross sections in various regions of the reactor. A distribution can be placed on these parameters and operational boundaries can be determined. The goal of this work is to support development of an advanced control system to operate CDs in a functioning NTP system.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2014

TU-C-17A-08: Improving IMRT Planning and Reducing Inter-Planner Variability Using the Stochastic Frontier Method: Validation Based On Clinical and Simulated Data

Purpose: Intensity modulated radiation therapy always requires compromises between PTV coverage and organs at risk (OAR) sparing. We previously developed metrics that correlate doses to OAR to specific patients’ morphology using stochastic frontier analysis (SFA). Here, we aim to examine the validity of this approach using a large set of realistically simulated dosimetric and geometric data. Methods: SFA describes a set of treatment plans as an asymmetric distribution with respect to a frontier defining optimal plans. Eighty head and neck IMRT plans were used to establish a metric predicting the mean dose to parotids as a function of simple geometric parameters. A database of 140 parotids was used as a basis distribution to simulate physically plausible data of geometry and dose. Distributions comprising between 20 and 5000 were simulated and the SFA was applied to obtain new frontiers, which were compared to the original frontier. Results: It was possible to simulate distributions consistent with the original dataset. Below 160 organs, the SFA could not always describe distributions as asymmetric: a few cases showed a Gaussian or half-Gaussian distribution. In order to converge to a stable solution, the number of organs in a distribution must ideally be above 100, but in many cases stable parameters could be achieved with as low as 60 samples of organ data. Mean RMS value of the error of new frontiers was significantly reduced when additional organs are used. Conclusion: The number of organs in a distribution showed to have an impact on the effectiveness of the model. It is always possible to obtain a frontier, but if the number of organs in the distribution is small (< 160), it may not represent de lowest dose achievable. These results will be used to determine number of cases necessary to adapt the model to other organs.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

An adaptive tau-leaping method for stochastic simulations of reaction-diffusion systems

Stochastic modelling is critical for studying many biochemical processes in a cell, in particular when some reacting species have low population numbers. For many such cellular processes the spatial distribution of the molecular species plays a key role. The evolution of spatially heterogeneous biochemical systems with some species in low amounts is accurately described by the mesoscopic model of the Reaction-Diffusion Master Equation. The Inhomogeneous Stochastic Simulation Algorithm provides an exact strategy to numerically solve this model, but it is computationally very expensive on realistic applications. We propose a novel adaptive time-stepping scheme for the tau-leaping method for approximating the solution of the Reaction-Diffusion Master Equation. This technique combines effective strategies for variable time-stepping with path preservation to reduce the computational cost, while maintaining the desired accuracy. The numerical tests on various examples arising in applications show the improved efficiency achieved by the new adaptive method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Home energy management under realistic and uncertain conditions: A comparison of heuristic, deterministic, and stochastic control methods

We report home energy management systems (HEMS) have been shown to reduce energy bills and to provide grid services including peak demand reduction and demand flexibility. However, uncertainty in residential energy systems is a significant issue and can reduce the benefits of a HEMS to the homeowner or grid operator. Sources of uncertainty include weather forecasts, predictions of energy-related occupant activities (e.g., hot water draws), and parameter estimation for the building envelope and energy-consuming equipment. This paper tackles the problem of uncertainty by developing a framework that simulates HEMS in uncertain conditions and evaluates the performance of multiple control strategies. A linear, reduced-order residential building model for model predictive control applications is derived and compared to a full-order model. Stochastic model predictive control is shown to perform better than deterministic and heuristic methods when considering realistic forecasts with uncertainty. The framework can evaluate the performance of HEMS in real-world applications, which can help de-risk HEMS deployment.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2025

The tensor-train stochastic finite volume method for uncertainty quantification

The stochastic finite volume method offers an efficient one-pass approach for assessing uncertainty in hyperbolic conservation laws. Still, it struggles with the curse of dimensionality when dealing with multiple stochastic variables. Here, we introduce the stochastic finite volume method within the tensor-train framework to counteract this limitation. This integration, however, comes with its own set of difficulties, mainly due to the propensity for shock formation in hyperbolic systems. To overcome these issues, we have developed a tensor-train-adapted stochastic finite volume method that employs a global WENO reconstruction, making it suitable for such complex systems. This approach represents the first step in designing tensor-train techniques for hyperbolic systems and conservation laws involving shocks.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1993

A Discrete Probability Function Method for the Equation of Radiative Transfer

A discrete probability function (DPF) method for the equation of radiative transfer is derived. The DPF is defined as the integral of the probability density function (PDF) over a discrete interval. The derivation allows the evaluation of the PDF of intensities leaving desired radiation paths including turbulence-radiation interactions without the use of computer intensive stochastic methods. The DPF method has a distinct advantage over conventional PDF methods since the creation of a partial differential equation from the equation of transfer is avoided. Further, convergence of all moments of intensity is guaranteed at the basic level of simulation unlike the stochastic method where the number of realizations for convergence of higher order moments increases rapidly. The DPF method is described for a representative path with approximately integral-length scale-sized spatial discretization. The results show good agreement with measurements in a propylene/air flame except for the effects of intermittency resulting from highly correlated realizations. The method can be extended to the treatment of spatial correlations as described in the Appendix. However, information regarding spatial correlations in turbulent flames is needed prior to the execution of this extension.

Sivathanu, Y. R.↗
DOE OSTI2026

Identifying stochastic dynamics via finite expression methods

Modeling stochastic differential equations (SDEs) is crucial for understanding complex dynamical systems in various scientific fields. Recent methods often employ neural network-based models, which typically represent SDEs through a combination of deterministic and stochastic terms. However, these models usually lack interpretability and have difficulty in generalizing beyond their training domain. Here, this paper introduces the Finite Expression Method (FEX), a symbolic learning approach designed to derive interpretable mathematical representations of the deterministic component of SDEs. For the stochastic component, we integrate FEX with advanced generative modeling techniques to provide a comprehensive representation of SDEs. The numerical experiments on linear, nonlinear, and multidimensional SDEs demonstrate that FEX generalizes well beyond the training domain and delivers more accurate long-term predictions compared to neural network-based methods. The symbolic expressions identified by FEX not only improve prediction accuracy but also offer valuable scientific insights into the underlying dynamics of the systems.

Complex dynamical systems↗