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Results for “engineering, methods, limits, mathematics”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2011

Remarks on Some Engineering Methods and Their Limits

The author makes the point that engineering practice and the mathematical modelling applied to engineering problems are different and that mathematical solutions that exist for engineering problems are incomprehensible to engineers. Two practical examples are presented to illustrate the point: parameter fitting error estimates and oscillator design. Some insights are offered to help translate between the languages of mathematics and engineering.

42 ENGINEERING↗
DOE OSTI2017

Verification Suite for the Application of the Limiting Surface Density Method to Arrays of 9975 Packages

Criticality Safety Engineering at the Savannah River Site’s K-Area Complex seeks to reduce the computational burden of analysis for the K-Area Complex. By applying an adapted Limiting Surface Density hand calculation method, the need for lengthy detailed Monte Carlo models of the storage arrays may be reduced or eliminated. After promising initial results using models of idealized but heterogeneous packages, application of the method to large arrays of highly heterogeneous 9975 shipping packages is attempted. A SCALE KENO-VI model of the 9975 package, with true dimensions and radial and axial heterogeneity is developed. A suite of array arrangements representing both arrays with equal number of packages per side (package-cubic) and arrays with unequal number of packages per side (non-cubic) are evaluated in SCALE. The cubic arrangements are used to derive the 9975-specific material and geometry constants for the LSD method. These constants are then used to implement the LSD method and compare the LSD- predicted critical mass per package to SCALE-predicted critical mass per package. Sensitivities to certain material and geometric nuances are evaluated for inclusion in the verification suite. Finally, the ability of the method to predict neutron multiplication changes induced by fissile mass changes is examined as a segue for evaluating credible abnormal conditions with the LSD method.

42 ENGINEERING↗
DOE OSTI2024

TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems

Deep learning (DL), in particular deep neural networks, by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties—such as stability, conservation, and positivity—and accuracy are required. DL methods in their original forms are often not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates for missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, hence promoting better generalization. This paper develops a model-constrained DL approach and its variant TNet—a Tikhonov neural network—which are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solutions with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for one-dimensional (1D) deconvolution, 5 for an inverse 2D heat conductivity problem, 100 for inverse initial conditions for a time-dependent 2D Burgers’s equation, and 50 for inverse initial conditions for 2D Navier–Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. Furthermore, this is possible owing to the model-constrained term, replications, and randomization.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

On the benefit of using both hand calculation methods and computer experiments algorithms to solve complex nuclear criticality safety problems - 14299

A companion paper shows the 'anomalies', and even the safety issues, that may be raised by the very usual problem of making safe a fissile mass inside a controlled geometry. These anomalies mainly come from the solving of such problems which belong to the mathematical family of 'robust inversion' including two inversion parameters (the fissile mass and the geometry) and one nuisance parameter (the moderation ratio of the fissile material). Since a 'brute force' method of resolution requires a lot of calculations and since a 'smart' method of resolution requires a lot of engineer time, it is tempting to solve these problems by a 'custom made' approach rather than to achieve a 'whole' resolution. The present paper shows how one can use a hand calculation method (derivation of the one-speed diffusion theory) to perform the whole resolution of such kind of problems with a minimum calculations and human efforts. The powerful capability of predicting the 'anomalies' highlighted in the companion paper is shown. Then the limitations of such a method are discussed: limitation in the geometry of the problem and bias in absolute values prediction. Finally, new algorithms of computer experiments design, dedicated to solve robust inversion problems and based on Kriging surrogates, are tested to solve these problems with limiting calculations and human efforts. The performance of these new algorithms may supplement the one-speed diffusion theory limitations. (authors)

42 ENGINEERING↗
DOE OSTI2024

Diffusion Limit–Preserving Lumped DFEMs on AMR Meshes

Here, we present sweep-compatible, novel upwinding recipes for the bilinear discontinuous (BLD) finite element method (FEM) that allows lumped BLD to be used on adaptive mesh refinement (AMR) meshes for thick transport applications without adding additional degrees of freedom at hanging nodes that exist on refinement boundaries. We analyze the properties of the upwinding and lumping that are needed for BLD to get the thick diffusion limit on such meshes, present results demonstrating locking with the wrong recipe, and present results showing error convergence and robustness properties for two diffusive problems on a variety of AMR meshes.

42 ENGINEERING↗
DOE OSTI2015

The application of ICASPA to critical and safe mass calculations - 14586

ICASPA, an Improved Critical and Safe Parameters Algorithm, calculates the size of a fissile body for a given value of k-effective. ICASPA uses an iterative predictor/corrector algorithm in which the MONK9A code is used to derive a precise k-effective value for a given candidate body size; auxiliary equations are then used to adjust the body size. The process is repeated until MONK9A outputs the chosen target value of k-effective. EDF Energy operates eight nuclear power stations comprising one Pressurized Water Reactor (PWR) and seven Advanced Gas-cooled Reactor (AGR) stations. At these sites, the term 'fuel route' is used to encompass all of the ex-reactor arrangements for the handling and storage of fuel. The criticality safety cases for these fuel routes are based on the use of geometrically safe arrangements for the required quantities of fuel. The allowed quantities and arrangements of fuel are safely sub-critical, even under the contingency of accidental moderator ingress. The fuel route safety cases consider all credible potential fault conditions, including events such as dropped fuel. Any such events might change the geometry of the fuel elements, but it is difficult to calculate exact k-effective values for realistic models of dropped fuel. This difficulty is overcome by the use of an alternative approach in which limiting safe masses are calculated for bounding (worst-case) rearrangements of fuel rods and moderators, using geometries similar to those covered by criticality handbook data. Where required, the resulting safe masses are used to set safety case limits and conditions. For example, some fire extinguishants, such as water and ABC dry powder, are also effective moderators. Hence, in some fuel route areas, administrative criticality controls are used to either prohibit or limit the permitted quantities of these materials. This paper describes the work that has been carried out to adopt ICASPA as the approved calculation method for these safe mass calculations. The work has been carried out within the company quality assurance requirements for nuclear safety related software. It has involved the development of a configuration controlled version of ICASPA and the local verification and validation of the resulting code. The benefits from the adoption of ICASPA are also presented. These include accuracy improvements and, because the method uses the MONK9A code for k-effective calculations, the ability to use more sophisticated models for limiting case geometries. (authors)

42 ENGINEERING↗
NASA NTRS2010

Using MathCad to Evaluate Exact Integral Formulations of Spacecraft Orbital Heats for Primitive Surfaces at Any Orientation

With the advent of high speed computing Monte Carlo ray tracing techniques has become the preferred method for evaluating spacecraft orbital heats. Monte Carlo has its greatest advantage where there are many interacting surfaces. However Monte Carlo programs are specialized programs that suffer from some inaccuracy, long calculation times and high purchase cost. A general orbital heating integral is presented here that is accurate, fast and runs on MathCad, a generally available engineering mathematics program. The integral is easy to read, understand and alter. The integral can be applied to unshaded primitive surfaces at any orientation. The method is limited to direct heating calculations. This integral formulation can be used for quick orbit evaluations and spot checking Monte Carlo results.

Pinckney, John↗
DOE OSTI2012

Los Alamos National Laboratory Computational Physics Summer Student Workshop (Final Report)

The numerical solution of highly compressible, multi-material flows is an ongoing research area. These types of flows can be solved with a Lagrangian type mesh which moves with the material in a simulation to allow precise material interface tracking. Currently, researchers at Los Alamos National Laboratory and elsewhere are investigating cell-centered Lagrangian algorithms with the aim of producing methods that have second-order accuracy, preserve symmetry, and do not generate spurious vorticity. The new cell-centered algorithms solve a Riemann-like problem at the vertex of a cell. Professor Phil Roe at the University of Michigan has proposed a new structure for Lagrangian hydrodynamic algorithms that does not rely on the solution of the Riemann problem. The new approach utilizes Flux Corrected Transport (FCT) and it implements a form of vorticity control. The first step in the development of this method has been to construct an algorithm that solves the acoustic equations on an Eulerian mesh. The algorithm, which builds on the work of Morton and Roe, calculates fluxes at cell vertices, attains second-order accuracy using FCT, and has the special property of preserving vorticity. Results are presented that confirm the second order accuracy of the scheme and the vorticity preserving properties. The results are compared to the output produced by a MUSCL-Hancock algorithm. Some discussion of limiting methods for the FCT algorithm is also given.

42 ENGINEERING↗
DOE OSTI2014

TH-C-BRE-01: IGART: Fast On-Line Planning for IGART

Cone beam CT IGRT provides on-line anatomical data of the patient at the treatment couch while new hybrid MRI radiotherapy systems promise to provide this data during the actual radiation delivery itself. To exploit these data, and account for tissue rotations, deformations and tumor regression during radiation therapy delivery, fast, on-line IMRT (re)planning is required.On-line IMRT re-planning needs to re-generate a similar patient specific dose distribution as done in the pre-treatment planning but then for the new state of the anatomy. For this the pre-treatment information has to be propagated to the actual treatment. A fast dose engine is required, which may be, in the case of MRI based IGRT, Monte Carlo based in order to account for the magnetic field induced dose effects. Then robust re-planning methods or class solutions should be available that account efficiently for both rigid and non-rigid anatomical changes while preserving the patient specific pre-treatment dose considerations. Preferably this is all fully automatic, but also fast interactive re-planning is an option, especially in hypo-fractionated (boost) radiotherapy. Learning Objectives: Preservation of pre-treatment prescriptions in the on-line IMRT re-planning Fast re-planning techniques and speed limitations for dose engines Pros and cons of interactive versus automatic re-planning.

60 APPLIED LIFE SCIENCES↗
NASA NTRS2000

Polar Plate Theory for Orthogonal Anisotropy

Laminated fiber-reinforced (or filamentary) composites are used today for their high strength-to-weight and stiffness-to-weight ratios. However, because of the anisotropic behavior of composites, determining the response on a macroscopic scale is challenging. This is particularly evident in the evaluation of the governing differential equations of a circular disk with the fibers of the lamina oriented with rectilinear orthogonality. This includes any situation involving a composite plate of circular geometry in which out-of-plane displacements due to load are desired, such as fastener pull through loading of a composite plate. Current analysis techniques use numerical methods with rectilinear coordinate systems to solve problems with circular geometry. These analyses over predict plate stiffness by 20% and underpredict failure by 70%. Consequently, there is a need to transform classical composite plate theory to a polar coordinate system. In order to better analyze structures with circular geometries the classical composite plate equations are transformed into the plate equations for a rectilinearly anisotropic composite in polar coordinates. A composite plate is typically a laminate of fibers in rectilinear directions. Subsequent to the lay-tip the necessary geometry is cut out of a rectangular plate. In a similar manner, the derivation of the plate equation starts with the fundamental definitions of strain, displacement and curvature and incorporates the material property angular dependence into the equilibrium equations for a differential polar element. In the transformed state, the stiffness coefficients are no longer constant, adding to the complexity of the governing differential equations. This paper discusses the new derivation and evaluation of the plate equations for a circular composite disk with orthogonal rectilinear anisotropy. The resultant new three partial differential equations, which describe the circular anisotropic plate, can be used to evaluate out-of-plane displacements for given load conditions in design of composite Structures. Without the formation of these unique plate equations, design of such structures is not a precise engineering accomplishment due to the lack of precise design tools. Such structures are over designed to compensate and costly tests need to be performed after production to validate safety. Though impossible to present the mathematical solution within the limits of this paper, the application of the newly derived plate equation in its orthotropic form for a balanced symmetric laminate compared favorably with test results and surpassed the results of standard numerical methods. Within the limitations of the problem foundation the predictive model provides a mechanism for establishing out-of-plane deflection levels for a circular composite plate. Future work in this area should include incorporating the circular composite plate equations in Finite Element Models. The new anisotropic polar plate equations call be utilized as a design tool so that the design more accurately meets the requirements, thus taking full advantage of the weight savings of composites. These accomplishments have not previously been presented to the engineering community nor ever proposed to the IAF. Should the proposed paper be accepted for presentation, attendance is assured.

Bailey, Michelle D.↗
NASA NTRS1987

An Application of Retroduction to Analyzing and Testing the Backing off of Nuts and Bolts During Dynamic Loading

The method of retroduction, adapted from the doctoral thesis of Dr. A. Croce, relies on a process of dialectic questioning that begins with the information sought, proceeds to Given items (either in the form of dimensions or limits of research). and to Known mathematical forms of analysis in design or to principles of study in research. Finally, analysis and synthesis are used to abstract the dielectic questions and to arrive at the information desired. This method is used to solve the engineering design problem of a beam and to determine why bolts and nuts vibrate apart. Both mathematical analysis and dialectic logical analysis are utilized. Results are provided of tests conducted to check the retroductive study of why and how nuts back off.

Kerley, James J.↗
NASA NTRS1990

Overcoming the Bellman's curse of dimensionality in large optimization problems

Decomposition of large problems into a hierarchic pyramid of subproblems was proposed in the literature as a means for optimization of engineering systems too large for all-in-one optimization. This decomposition was established heuristically. The dynamic programming (DP) method due to Bellman was augmented with an optimum sensitivity analysis that provides a mathematical basis for the above decomposition, and overcomes the curse of dimensionality that limited the original formulation of DP. Numerical examples are cited.

Sobieszczanski-Sobieski, Jaroslaw↗
NASA NTRS2024

MSFC-TVC | TB-03: Derivation of Thrust Vector Control (TVC) Actuator-Force / Gimbal-Torque Transformation Matrix

Thrust vector control (TVC) systems for rocket engine propulsion traditionally use a simple linear relationship to convert between actuator forces and torques about the engine gimbal’s center-of-rotation (COR). As shown in Equation (1), the torque about the gimbal COR is proportional to the applied actuator force and the TVC moment arm (MA)—the perpendicular distance between the TVC actuator’s line-of-action (LOA) and the engine gimbal’s COR. While this fundamental relationship remains valid and accurate in a two-dimensional (2D), one-degree-of-freedom (1-DOF) context—particularly in its non-linear formulation as described in the ER63 Technical Bulletin TB-02 (Derivation of Thrust Vector Control (TVC) Engine-Gimbal / Actuator Moment-Arm Geometry)—it becomes limited when extended to three-dimensional (3D), two-degree-of-freedom (2-DOF) analyses. In 3D, out-of-plane angular displacements can arise, causing the gimbaled engine plane-of-motion to become non-coplanar with the respective actuator plane-of-motion. Such misalignments occur due to the rod-end (RE) and/or tail-stock (TE) mounting geometry of individual TVC actuators. These geometric complexities lead to inaccuracies in calculating engine torques and corresponding actuator forces if only the traditional TVC MA relationship is employed. For higher gimbal angular displacements (e.g., greater than 5 degrees), these inaccuracies become more pronounced, necessitating a more robust mathematical framework. This document presents a comprehensive 3D (2-DOF) derivation of a transformation matrix that accurately converts between gimbaled engine torques and TVC actuator forces. By incorporating the necessary geometric and rotational considerations, this new approach corrects the limitations of the traditional TVC MA method. Subsequent sections compare the newly formulated approach to the traditional method, demonstrating its enhanced accuracy and reliability for a broad range of gimbaled engine conditions.

Thrust Vector Control↗
DOE OSTI2022

Opportunities and data requirements for data-driven prognostics and health management in liquid hydrogen storage systems

During the past decade, Prognostics and Health Management (PHM) has become an important set of tools in various areas of industry and academic reliability engineering. PHM consists of a variety of mathematical and computational methods used to support data-driven decision-making to increase the safety, availability, and reliability of complex engineering systems. In particular, PHM can provide crucial insight into reliability and safety design improvements for developing technologies where historical performance and failure data are limited. This is the case of hydrogen fueling and storage technologies. This work presents a high-level approach for designing data-driven PHM applications for bulk liquid hydrogen (LH 2 ) storage systems for hydrogen fueling stations. This paper addresses core aspects of the design, development, and implementation of data-driven PHM applications that can improve the reliability assessment of hydrogen components. The analysis focuses on the relationship between data availability and diagnostic/prognostic capabilities; potential challenges; and integration schemes for current risk mitigation measures. We identify potential condition-monitoring data sources for key components in an LH 2 storage system, including storage tanks, piping, and pumps. Further, we determine that the short-term goals for the implementation of data-driven models in PHM frameworks in hydrogen systems should focus on developing adequate data collection and analysis strategies, as well as exploring the effect on reliability, safety, and regulations for hydrogen systems.

08 HYDROGEN↗
NASA NTRS1996

Uncertainty and Intelligence in Computational Stochastic Mechanics

Classical structural reliability assessment techniques are based on precise and crisp (sharp) definitions of failure and non-failure (survival) of a structure in meeting a set of strength, function and serviceability criteria. These definitions are provided in the form of performance functions and limit state equations. Thus, the criteria provide a dichotomous definition of what real physical situations represent, in the form of abrupt change from structural survival to failure. However, based on observing the failure and survival of real structures according to the serviceability and strength criteria, the transition from a survival state to a failure state and from serviceability criteria to strength criteria are continuous and gradual rather than crisp and abrupt. That is, an entire spectrum of damage or failure levels (grades) is observed during the transition to total collapse. In the process, serviceability criteria are gradually violated with monotonically increasing level of violation, and progressively lead into the strength criteria violation. Classical structural reliability methods correctly and adequately include the ambiguity sources of uncertainty (physical randomness, statistical and modeling uncertainty) by varying amounts. However, they are unable to adequately incorporate the presence of a damage spectrum, and do not consider in their mathematical framework any sources of uncertainty of the vagueness type. Vagueness can be attributed to sources of fuzziness, unclearness, indistinctiveness, sharplessness and grayness; whereas ambiguity can be attributed to nonspecificity, one-to-many relations, variety, generality, diversity and divergence. Using the nomenclature of structural reliability, vagueness and ambiguity can be accounted for in the form of realistic delineation of structural damage based on subjective judgment of engineers. For situations that require decisions under uncertainty with cost/benefit objectives, the risk of failure should depend on the underlying level of damage and the uncertainties associated with its definition. A mathematical model for structural reliability assessment that includes both ambiguity and vagueness types of uncertainty was suggested to result in the likelihood of failure over a damage spectrum. The resulting structural reliability estimates properly represent the continuous transition from serviceability to strength limit states over the ultimate time exposure of the structure. In this section, a structural reliability assessment method based on a fuzzy definition of failure is suggested to meet these practical needs. A failure definition can be developed to indicate the relationship between failure level and structural response. In this fuzzy model, a subjective index is introduced to represent all levels of damage (or failure). This index can be interpreted as either a measure of failure level or a measure of a degree of belief in the occurrence of some performance condition (e.g., failure). The index allows expressing the transition state between complete survival and complete failure for some structural response based on subjective evaluation and judgment.

Ayyub, Bilal M.↗
DOE OSTI2023

Uncertainty quantification in scientific machine learning: Methods, metrics, and comparisons

Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. This is because in addition to aleatoric uncertainty associated with noisy data, there is also uncertainty due to limited data, but also due to NN hyperparameters, overparametrization, optimization and sampling errors as well as model misspecification. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this work, we present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. Further, to demonstrate the applicability and reliability of our framework, we present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions. In the Appendix, we include a comprehensive description of all the UQ methods employed. Further, to help facilitate the deployment of UQ in Scientific Machine Learning research and practice, we present and develop in [1] an open-source Python library (github.com/Crunch-UQ4MI/neuraluq), termed NeuralUQ, that is accompanied by an educational tutorial and additional computational experiments.

11 physics-informed neural networks↗
DOE OSTI2023

Projection pursuit adaptation on polynomial chaos expansions

Here, the present work addresses the issue of accurate stochastic approximations in high-dimensional parametric space using tools from uncertainty quantification (UQ). The basis adaptation method and its accelerated algorithm in polynomial chaos expansions (PCE) were recently proposed to construct low-dimensional approximations adapted to specific quantities of interest (QoI). The present paper addresses one difficulty with these adaptations, namely their reliance on quadrature point sampling, which limits the reusability of potentially expensive samples. Projection pursuit (PP) is a statistical tool to find the “interesting” projections in high-dimensional data and thus bypass the curse-of-dimensionality. In the present work, we combine the fundamental ideas of basis adaptation and projection pursuit regression (PPR) to propose a novel method to simultaneously learn the optimal low-dimensional spaces and PCE representation from given data. While this projection pursuit adaptation (PPA) can be entirely data-driven, the constructed approximation exhibits mean-square convergence to the solution of an underlying governing equation and thus captures the supports and probability distributions associated with the physics constraints. The proposed approach is demonstrated on a borehole problem and a structural dynamics problem, demonstrating the versatility of the method and its ability to discover low-dimensional manifolds with high accuracy with limited data. In addition, the method can learn surrogate models for different quantities of interest while reusing the same data set.

97 MATHEMATICS AND COMPUTING↗