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All sourcesNASA NTRSDOE OSTI
DOE OSTI2014

Predictive Engineering Tools for Injection-Molded Long-Carbon-Fiber Thermoplastic Composites - FY 2014 Fourth Quarterly Report

During the last quarter of FY 2014, the following technical progress has been made toward project milestones: 1) Autodesk, Inc. (Autodesk) has implemented a new fiber length distribution (FLD) model based on an unbreakable length assumption with Reduced Order Modeling (ROM) by the Proper Orthogonal Decomposition (POD) approach in the mid-plane, dual-domain and 3D solvers. 2) Autodesk improved the ASMI 3D solver for fiber orientation prediction using the anisotropic rotary diffusion (ARD) – reduced strain closure (RSC) model. 3) Autodesk received consultant services from Prof. C.L. Tucker at the University of Illinois on numerical simulation of fiber orientation and fiber length. 4) PlastiComp, Inc. (PlastiComp) suggested to Purdue University a procedure for fiber separation using an inert-gas atmosphere in the burn-off furnace. 5) Purdue University (Purdue) hosted a face-to-face project review meeting at Purdue University on August 6-7, 2014. 6) Purdue conducted fiber orientation measurements for 3 PlastiComp plaques: fast-fill 30wt% LCF/PP edged-gated, slow-fill 50wt% LCF/PP edge-gated, and slow-fill 50wt% LCF/PP center-gated plaques, and delivered the orientation data for these plaques at the selected locations (named A, B, and C) to PNNL. 7) PNNL conducted ASMI mid-plane analyses for the above PlastiComp plaques and compared the predicted fiber orientations with the measured data provided by Purdue at Locations A, B, and C on these plaques. 8) PNNL planned the project review meeting (August 6-7, 2014) with Purdue. 9) PNNL performed ASMI analyses for the Toyota complex parts with and without ribs, having different wall thicknesses, and using the PlastiComp 50wt% LCF/PP, 50wt% LCF/PA66, 30wt% LCF/PP, and 30wt% LCF/PA66 materials to provide guidance for tool design and modifications needed for molding these parts. 10) Magna Exteriors and Interiors Corp. (Magna) molded plaques from the 50% LCF/PP and 50% LCF/PA66 materials received from Plasticomp in order to extract machine purgings (purge materials) from Magna’s 200-Ton Injection Molding machine targeted to mold the complex part. 11) Toyota and Magna discussed with PNNL tool modification for molding the complex part.

36 MATERIALS SCIENCE↗
DOE OSTI2025

A predictive analytical model of electrical transport in multi-principal-element alloys

A predictive analytical model is presented for the electrical conductivity of multi-principal-element alloys (MPEAs), including those containing aluminum, transition metals, and refractory metals. Given that the lattice parameter of the Wigner-Seitz cell of an MPEA is similarly variable to a bulk metallic glass, it is postulated that electron scattering can be approximated by a series of two-level systems. Here, the resulting reduced-order model enabled an accurate determination of electrical resistivity and electron thermal conductivity based on the scattering of electrons in a two-level system across a Bloch-potential-based virtual crystal approximation. Model results are compared to experimental four-point probe electrical resistivity measurements between 300 K and 700 K for Al 0.3 CoCrCuFeNi, CoCrFeMnNi, (CoCrFeMnNi) 0.98 W 0.02 , (CoCrFeMnNi) 0.95 W 0.05 , and Nb 4 Ta 4 V 3 Ti, for model validation.

Analytical model↗
DOE OSTI2024

Physics-Informed Active Learning With Simultaneous Weak-Form Latent Space Dynamics Identification

The parametric greedy latent space dynamics identification (gLaSDI) framework has demonstrated promising potential for accurate and efficient modeling of high-dimensional nonlinear physical systems. However, it remains challenging to handle noisy data. Here, to enhance robustness against noise, we incorporate the weak-form estimation of nonlinear dynamics (WENDy) into gLaSDI. In the proposed weak-form gLaSDI (WgLaSDI) framework, an autoencoder and WENDy are trained simultaneously to discover intrinsic nonlinear latent-space dynamics of high-dimensional data. Compared with the standard sparse identification of nonlinear dynamics (SINDy) employed in gLaSDI, WENDy enables variance reduction and robust latent space discovery, therefore leading to more accurate and efficient reduced-order modeling. Furthermore, the greedy physics-informed active learning in WgLaSDI enables adaptive sampling of optimal training data on the fly for enhanced modeling accuracy. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including viscous and inviscid Burgers' equations, time-dependent radial advection, and the Vlasov equation for plasma physics. With data that contains 5%–10% Gaussian white noise, WgLaSDI outperforms gLaSDI by orders of magnitude, achieving 1%–7% relative errors. Compared with the high-fidelity models, WgLaSDI achieves 121 to 1779x speed-up.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1985

Model Reduction for Control System Design

An approach and a technique for effectively obtaining reduced order mathematical models of a given large order model for the purposes of synthesis, analysis and implementation of control systems is developed. This approach involves the use of an error criterion which is the H-infinity norm of a frequency weighted error between the full and reduced order models. The weightings are chosen to take into account the purpose for which the reduced order model is intended. A previously unknown error bound in the H-infinity norm for reduced order models obtained from internally balanced realizations was obtained. This motivated further development of the balancing technique to include the frequency dependent weightings. This resulted in the frequency weighted balanced realization and a new model reduction technique. Two approaches to designing reduced order controllers were developed. The first involves reducing the order of a high order controller with an appropriate weighting. The second involves linear quadratic Gaussian synthesis based on a reduced order model obtained with an appropriate weighting.

Enns, D. F.↗
DOE OSTI2021

From Machine Learning to Nuclear Digital Twins

This presentation provides multiple case studies of Machine Learning, Uncertainty Quantification, Reduced Order Modeling, to achieve digital transformation and provide the components for nuclear digital twinning. The presentation is for an international virtual event titled: Consultancy Meeting on Applications of AI and Pattern Recognition Techniques for Uncertainty Quantification in Nuclear Power Modelling and Simulation held Oct 21-22, 2021.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2018

Geologic CO 2 sequestration monitoring design: A machine learning and uncertainty quantification based approach

Monitoring is a crucial aspect of geologic carbon dioxide (CO 2 ) sequestration risk management. Effective monitoring is critical to ensure CO 2 is safely and permanently stored throughout the life-cycle of a geologic CO 2 sequestration project. Effective monitoring involves deciding: (i) where is the optimal location to place the monitoring well(s), and (ii) what type of data (pressure, temperature, CO 2 saturation, etc.) should be measured taking into consideration the uncertainties at geologic sequestration sites. We have developed a filtering-based data assimilation procedure to design effective monitoring approaches. To reduce the computational cost of the filtering-based data assimilation process, a machine-learning algorithm: Multivariate Adaptive Regression Splines is used to derive computationally effcient reduced order models from results of full-physics numerical simulations of CO 2 injection in saline aquifer and subsequent multi-phase fluid flow. We use example scenarios of CO 2 leakage through legacy wellbore and demonstrate a monitoring strategy can be selected with the aim of reducing uncertainty in metrics related to CO 2 leakage. We demonstrate the proposed framework with two synthetic examples: a simple validation case and a more complicated case including multiple monitoring wells. The examples demonstrate that the proposed approach can be effective in developing monitoring approaches that take into consideration uncertainties.

58 GEOSCIENCES↗
DOE OSTI2023

Impacts of Different Operation Conditions and Geological Formation Characteristics on CO2 Sequestration in Citronelle Dome, Alabama

Major concerns of carbon dioxide (CO2) sequestration in subsurface formations are knowledge of the well injectivity and gas storage capacity of the formation, the CO2 pressure and saturation plume extensions during and after injection, and the risks associated with CO2 leakage and fault reactivation. Saline reservoirs are considered as one of the target formations for CO2 sequestration through structural, residual, dissolution, and mineral trapping mechanisms. The boundary condition of the saline reservoir dictates the pressure and saturation plume extension of the injected supercritical CO2 that could expand over large distances. This can lead to sources of risk, e.g., leakage and/or fault reactivation due to presence of wells, thief zones, and geological discontinuities. Therefore, there is a critical need to develop a model that describes how risk-related performance metrics (i.e., the CO2 saturation plume size, the pressure differential plume area, and the pressure differential at specific locations) vary as a function of the size of injection, time following injection, injection operations, and geologic environment. In this study, a systematic reservoir modeling studies of anthropogenic CO2 sequestration in Citronelle dome, Alabama, was performed where all relevant scenarios and conditions to address the questions of the saturation and pressure plume size in the area of review (AoR) and post-injection site care (PISC) are considered. The objective for this study was firstly to systematically simulate CO2 sequestration, i.e., saturation dynamics, and pressure behavior over a range of operational and geological conditions and to derive conclusions about the factors influencing saturation and pressure plume size, post-injection behavior, and the risk associated with them, by developing third-generation reduced order models (ROMs) for reservoir behavior. Finally, to assess the uncertainty associated with our studies, Latin Hypercube Sampling (LHS) together with an experimental design technique, i.e., Plackett–Burman design, was used. Application of Pareto charts and respond surfaces enabled us to determine the most important parameters impacting saturation and pressure plume sizes and to quantify the auto- and cross-correlation among different parameters in both history-matched and upscaled models.

58 GEOSCIENCES↗
DOE OSTI2026

Ensemble Kalman filter for data assimilation coupled with low-resolution computations techniques applied in fluid dynamics

This paper presents an innovative Reduced-order model (ROM) for merging experimental and simulation data using data assimilation (DA) to estimate the "True" state of a fluid dynamics system, leading to more accurate predictions. Our methodology introduces a novel approach by implementing the ensemble Kalman filter (EnKF) within a reduced-dimensional framework, grounded in a robust theoretical foundation and applied to fluid dynamics. To address the substantial computational demands of DA, the proposed ROM employs low-resolution (LR) techniques to drastically reduce computational costs. This innovative approach involves downsampling datasets for DA computations, followed by an advanced reconstruction technique based on low-cost singular value decomposition (lcSVD). The lcSVD method, a key innovation in this paper, has never been applied to DA before and offers a highly efficient way to enhance resolution with minimal computational resources. Our results demonstrate significant reductions in both computation time and RAM usage through these LR techniques without compromising the accuracy of the estimations. For instance, in a turbulent test case, for a data compression rate of 15.9, the LR approach can achieve a speed-up of 13.7 and a RAM compression of 90.9% while maintaining a low relative root mean square error (RRMSE) of 2.6%, compared to 0.8% in the high-resolution (HR) reference. Furthermore, we highlight the effectiveness of the EnKF in estimating and predicting the state of fluid flow systems based on limited observations and given low-fidelity numerical data. This paper highlights the potential of the proposed DA method in fluid dynamics applications, particularly for improving computational efficiency in CFD and related fields. Its ability to balance accuracy with low computational and memory costs makes it especially suitable for large-scale and real-time applications, such as environmental monitoring or engineering design. This method will be incorporated into ModelFLOWs-app.

Data Assimilation↗
DOE OSTI2020

Fast Computational Algorithms for Partial Differential Equations and Uncertainty Quantifications

This project concerned the construction, testing and analysis of computational algorithms for solving parameterized and stochastic partial differential equations. The study and understanding of equations of this type is of fundamental importance in numerous engineering and scientific applications. Examples include simulation of plasma dynamics in models of electric propulsion and nuclear fusion, simulation of multiphase flows, such as the flow of water, gas and oil in reservoirs, and structural analysis of the dependence of structures on materials. Parametrization is used in such settings when properties of the models such as viscosity of fluids or electric resistivity of materials are not precisely understood and instead are treated as random variables. The resulting solutions are themselves random, and having such solutions will enable engineers to use probabilistic methods to assess the likelihood of events, for example, whether a pollutant in a liquid will exceed a limit, and to use such analyses to develop ways to ensure positive outcomes. Construction of accurate (high resolution) computational solutions is expensive, requiring significant computer time and computational resources, and there is need to reduce computational cost to make simulation useful and effective. The aim of the project was to construct algorithms to efficiently compute surrogate solutions to parameterized problems to allow for efficient and accurate simulation. The technical approach used focused on two related strategies, based on rank-reduction methods and reduced-order models. These methods construct surrogate solutions of parameter-dependent models by projection or interpolation into low-dimensional approximation spaces. Cost savings are achieved if the low-dimensional spaces can be identified and constructed efficiently and if the resulting low-dimensional algebraic systems can be solved cheaply. Accomplishments include: Theoretical and empirical demonstration of the effectiveness of fast multigrid solution strategies for computing low-rank representations of parameter-dependent solutions to discrete partial differential equations, including the first proof establishing so-called textbook convergence properties for low-rank methods. Development of efficient solution algorithms for solving nonlinear parameter-dependent partial differential equations used in models of fluid dynamics. Developent of efficient algorithms for low-rank representation of solutions of time-dependent simulations of fluid dynamics using multi-dimensional tensor representations of solutions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Forward sensitivity analysis and mode dependent control for closure modeling of Galerkin systems

Model reduction by projection-based approaches is often associated with losing some of the important features that contribute towards the dynamics of the retained scales. As a result, a mismatch occurs between the predicted trajectories of the original system and the truncated one. We put forth a framework to apply a continuous time control signal in the latent space of the reduced order model (ROM) to account for the effect of truncation. We set the control input using parameterized models by following energy transfer principles. Our methodology relies on observing the system behavior in the physical space and using the projection operator to restrict the feedback signal into the latent space. Then, we leverage the forward sensitivity method (FSM) to derive relationships between the feedback and the desired mode-dependent control. We test the performance of the proposed approach using two test cases, corresponding to viscous Burgers and vortex merger problems at high Reynolds number. Results show that the ROM trajectory with the applied FSM control closely matches its target values in both the data-dense and data-sparse regimes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Automated Parameterized CFD Simulations of Phase-Change Material Embedded Heat Exchangers

Thermal storage can be implemented using Phase-Change Materials (PCM), which absorb significant latent heat with a relatively small temperature change. PCM phase-change processes are transient and are driven by thermal diffusion and natural convection – the latter, especially for melting process. Modeling and simulation of PCM heat exchangers is typically computationally intensive due to the relative complex time-dependent underlying physics. This paper presents a study investigating a single straight tube with circular transverse fins in the vertical orientation, using PCM’s with 37oC nominal melting temperature, leveraging automated parametric CFD simulations. The computational domain is reduced to 2D axisymmetric for a segment of the HX comprising a region in between two fins with constant wall temperature as boundary condition. This approach allows for faster evaluations of the transient thermal characteristics for a given design space of interest. Correlations and reduced order models (ROM’s) are developed. Results indicate that the ROM’s reproduce the CFD simulations with high accuracy for designs from the training data set, however, designs from cross-validation and test sets do not always have good agreement. The limited number of designs in the training set is a possible cause for the lack of robustness of the ROM’s and thus additional training designs will be added in future work. The ROM’s solve the simulations on average 500,000 times faster than CFD.

25 ENERGY STORAGE↗
DOE OSTI2016

SURROGATE MODEL DEVELOPMENT AND VALIDATION FOR RELIABILITY ANALYSIS OF REACTOR PRESSURE VESSELS

In nuclear light water reactors (LWRs), the reactor coolant, core and shroud are contained within a massive, thick walled steel vessel known as a reactor pressure vessel (RPV). Given the tremendous size of these structures, RPVs typically contain a large population of pre-existing flaws introduced in the manufacturing process. After many years of operation, irradiation-induced embrittlement makes these vessels increasingly susceptible to fracture initiation at the locations of the pre-existing flaws. Because of the uncertainty in the loading conditions, flaw characteristics and material properties, probabilistic methods are widely accepted and used in assessing RPV integrity. The Fracture Analysis of Vessels – Oak Ridge (FAVOR) computer program developed by researchers at Oak Ridge National Laboratory is widely used for this purpose. This program can be used in order to perform deterministic and probabilistic risk-informed analyses of the structural integrity of an RPV subjected to a range of thermal-hydraulic events. FAVOR uses a one-dimensional representation of the global response of the RPV, which is appropriate for the beltline region, which experiences the most embrittlement, and employs an influence coefficient technique to rapidly compute stress intensity factors for axis-aligned surface-breaking flaws. The Grizzly code is currently under development at Idaho National Laboratory (INL) to be used as a general multiphysics simulation tool to study a variety of degradation mechanisms in nuclear power plant components. The first application of Grizzly has been to study fracture in embrittled RPVs. Grizzly can be used to model the thermo-mechanical response of an RPV under transient conditions observed in a pressurized thermal shock (PTS) scenario. The global response of the vessel provides boundary conditions for local 3D models of the material in the vicinity of a flaw. Fracture domain integrals are computed to obtain stress intensity factors, which can in turn be used to assess whether a fracture would initiate at a pre-existing flaw. To use Grizzly for probabilistic analysis, it is necessary to have a way to rapidly evaluate stress intensity factors. To accomplish this goal, a reduced order model (ROM) has been developed to efficiently represent the behavior of a detailed 3D Grizzly model used to calculate fracture parameters. This approach uses the stress intensity factor influence coefficient method that has been used with great success in FAVOR. Instead of interpolating between tabulated solutions, as FAVOR does, the ROM approach uses a response surface methodology to compute fracture solutions based on a sampled set of results used to train the ROM. The main advantages of this approach are that the process of generating the training data can be fully automated, and the procedure can be readily used to consider more general flaw configurations. This paper demonstrates the procedure used to generate a ROM to rapidly compute stress intensity factors for axis-aligned flaws. The results from this procedure are in good agreement with those produced using the traditional influence coefficient interpolation procedure, which gives confidence in this method. This paves the way for applying this procedure for more general flaw configurations.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

$\text{GPLaSDI}$: Gaussian Process-based interpretable Latent Space Dynamics Identification through deep autoencoder

Numerically solving partial differential equations (PDEs) can be challenging and computationally expensive. This has led to the development of reduced-order models (ROMs) that are accurate but faster than full order models (FOMs). Recently, machine learning advances have enabled the creation of non-linear projection methods, such as Latent Space Dynamics Identification (LaSDI). LaSDI maps full-order PDE solutions to a latent space using autoencoders and learns the system of ODEs governing the latent space dynamics. By interpolating and solving the ODE system in the reduced latent space, fast and accurate ROM predictions can be made by feeding the predicted latent space dynamics into the decoder. In this paper, we introduce GPLaSDI, a novel LaSDI-based framework that relies on Gaussian process (GP) for latent space ODE interpolations. Using GPs offers two significant advantages. First, it enables the quantification of uncertainty over the ROM predictions. Second, leveraging this prediction uncertainty allows for efficient adaptive training through a greedy selection of additional training data points. This approach does not require prior knowledge of the underlying PDEs. Consequently, GPLaSDI is inherently non-intrusive and can be applied to problems without a known PDE or its residual. Here we demonstrate the effectiveness of our approach on the Burgers equation, Vlasov equation for plasma physics, and a rising thermal bubble problem. Our proposed method achieves between 200 and 100,000 times speed-up, with up to 7% relative error.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Multifidelity importance sampling

Estimating statistics of model outputs with the Monte Carlo method usually requires a large number of model evaluations. This leads to long runtimes if the model is expensive to evaluate. Importance sampling is one method that can lead to a reduction in the number of model evaluations. Importance sampling uses a biasing distribution to sample the model more efficiently, but generating such a biasing distribution can be difficult and usually also requires model evaluations. A different strategy to speed up Monte Carlo sampling is to replace the computationally expensive high-fidelity model with a computationally cheap surrogate model; however, because the surrogate model outputs are only approximations of the high-fidelity model outputs, the estimate obtained using a surrogate model is in general biased with respect to the estimate obtained using the high-fidelity model. We introduce a multifidelity importance sampling (MFIS) method, which combines evaluations of both the high-fidelity and a surrogate model. It uses a surrogate model to facilitate the construction of the biasing distribution, but relies on a small number of evaluations of the high-fidelity model to derive an unbiased estimate of the statistics of interest. We prove that the MFIS estimate is unbiased even in the absence of accuracy guarantees on the surrogate model itself. The MFIS method can be used with any type of surrogate model, such as projection-based reduced-order models and data-fit models. Moreover, the MFIS method is applicable to black-box models, i.e., where only inputs and the corresponding outputs of the high-fidelity and the surrogate model are available but not the details of the models themselves. We demonstrate on nonlinear and time-dependent problems that our MFIS method achieves speedups of up to several orders of magnitude compared to Monte Carlo with importance sampling that uses the high-fidelity model only.

42 ENGINEERING↗
DOE OSTI2019

Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations

Here, this work proposes a machine-learning framework for constructing statistical models of errors incurred by approximate solutions to parameterized systems of nonlinear equations. These approximate solutions may arise from early termination of an iterative method, a lower-fidelity model, or a projection-based reduced-order model, for example. The proposed statistical model comprises the sum of a deterministic regression-function model and a stochastic noise model. The method constructs the regression-function model by applying regression techniques from machine learning (e.g., support vector regression, artificial neural networks) to map features (i.e., error indicators such as sampled elements of the residual) to a prediction of the approximate-solution error. The method constructs the noise model as a mean-zero Gaussian random variable whose variance is computed as the sample variance of the approximate-solution error on a test set; this variance can be interpreted as the epistemic uncertainty introduced by the approximate solution. This work considers a wide range of feature-engineering methods, data-set-construction techniques, and regression techniques that aim to ensure that (1) the features are cheaply computable, (2) the noise model exhibits low variance (i.e., low epistemic uncertainty introduced), and (3) the regression model generalizes to independent test data. Finally, numerical experiments performed on several computational-mechanics problems and types of approximate solutions demonstrate the ability of the method to generate statistical models of the error that satisfy these criteria and significantly outperform more commonly adopted approaches for error modeling.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Informing Geologic CO 2 Storage Site Management Decisions under Uncertainty: Demonstration of NRAP's Integrated Assessment Model (NRAP-IAM-CS) Application

Managing risks at geologic CO 2 storage sites requires making decisions based on the quantitative predictions of long-term performance that not only take into account various subsurface hydrologic, geochemical processes and interactions but also the related uncertainties. The US DOE (United States Department of Energy) funded National Risk Assessment Partnership (NRAP) has developed a quantitative approach and a model to predict the long-term behavior of GCS sites. The approach uses an integrated assessment model (NRAP-IAM-CS) that couples reduced order models (ROMs) for various GCS components (such as storage reservoir, wells, seals, and groundwater aquifer) in a stochastic modeling framework that allows for rapid simulations of entire system behavior over thousands of years while probing uncertainties. We demonstrate application of NRAP-IAM-CS using two examples that focus on CO 2 leakage and resulting impacts. The examples are used to demonstrate how different uncertain parameters affect overall performance objectives.

03 NATURAL GAS↗
DOE OSTI2020

Closure Learning for Nonlinear Model Reduction Using Deep Residual Neural Network

Developing accurate, efficient, and robust closure models is essential in the construction of reduced order models (ROMs) for realistic nonlinear systems, which generally require drastic ROM mode truncations. We propose a deep residual neural network (ResNet) closure learning framework for ROMs of nonlinear systems. The novel ResNet-ROM framework consists of two steps: (i) In the first step, we use ROM projection to filter the given nonlinear system and construct a spatially filtered ROM. This filtered ROM is low-dimensional, but is not closed. (ii) In the second step, we use ResNet to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved ROM modes. We emphasize that in the new ResNet-ROM framework, data is used only to complement classical physical modeling (i.e., only in the closure modeling component), not to completely replace it. We also note that the new ResNet-ROM is built on general ideas of spatial filtering and deep learning and is independent of (restrictive) phenomenological arguments, e.g., of eddy viscosity type. The numerical experiments for the 1D Burgers equation show that the ResNet-ROM is significantly more accurate than the standard projection ROM. The new ResNet-ROM is also more accurate and significantly more efficient than other modern ROM closure models.

Xie, Xuping↗
NASA NTRS1981

A function space approach to smoothing with applications to model error estimation for flexible spacecraft control

A function space approach to smoothing is used to obtain a set of model error estimates inherent in a reduced-order model. By establishing knowledge of inevitable deficiencies in the truncated model, the error estimates provide a foundation for updating the model and thereby improving system performance. The function space smoothing solution leads to a specification of a method for computation of the model error estimates and development of model error analysis techniques for comparison between actual and estimated errors. The paper summarizes the model error estimation approach as well as an application arising in the area of modeling for spacecraft attitude control.

Rodriguez, G.↗