Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “numerical differentiation”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,045 records · Page 58
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Stability analysis of a one-dimensional multiphysics model of a molten salt fast reactor

Reactor designs with a fast neutron spectrum have recently moved into the research focus for possible Generation IV reactor concepts - a representative of which is the molten salt fast reactor (MSFR). In order to assess the safety and reliability of this reactor concept, it is necessary to obtain knowledge of the system behaviour. An important part of this research concerns the dynamical stability of the MSFR. Research regarding the MSFR stability so far has either been using linearized models for stability criteria or only been examining certain transients at fixed parameter values. This work delivers a comprehensive stability analysis for a non-linearized MSFR model, and a wide range of values for all relevant parameters. For this purpose, a one-dimensional MSFR model was set up, taking neutron kinetics and thermal hydraulics in the form of a system of differential equations into account. The stability of this model was investigated by means of the numerical tool MATCONT, which was used to monitor the evolution of a so-called fixed-point solution, here referring to the steady state at operating conditions of the system, while system parameters got varied. The MATCONT analyses showed no loss of stability for any of the considered parameter variations and no solutions that might co-exist in parallel with the steady-state fixed-point solution. Therefore these results indicate a stable fixed point to which all solution transients converge, making the considered MSFR model insusceptible to deviations from the equilibrium. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
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 OSTI2024

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

ReactionMechanismSimulator.jl: A modern approach to chemical kinetic mechanism simulation and analysis

Abstract We present ReactionMechanismSimulator.jl (RMS), a modern differentiable software for the simulation and analysis of chemical kinetic mechanisms, including multiphase systems. RMS has already been applied to problems in combustion, pyrolysis, polymers, pharmaceuticals, catalysis, and electrocatalysis. RMS is written in Julia, making it easy to develop and allowing it to take advantage of Julia's extensive numerical computing ecosystem. In addition to its extensive library of optimized analytic Jacobians, RMS can generate and use Jacobians computed using automatic differentiation and symbolically generated analytic Jacobians. RMS is demonstrated to be faster than Cantera and Chemkin in several benchmarks. RMS also implements an extensive set of features for analyzing chemical mechanisms, including a library of easy‐to‐call plotting functions, molecular structure resolved flux diagram generation, crash analysis, traditional sensitivity analysis, transitory sensitivity analysis, and an automatic mechanism analysis toolkit. RMS implements efficient adjoint and parallel forward sensitivity analyses. We also demonstrate the ease of adding new features to RMS.

Johnson, Matthew S.↗
NASA NTRS1998

Flow Applications of the Least Squares Finite Element Method

The main thrust of the effort has been towards the development, analysis and implementation of the least-squares finite element method (LSFEM) for fluid dynamics and electromagnetics applications. In the past year, there were four major accomplishments: 1) special treatments in computational fluid dynamics and computational electromagnetics, such as upwinding, numerical dissipation, staggered grid, non-equal order elements, operator splitting and preconditioning, edge elements, and vector potential are unnecessary; 2) the analysis of the LSFEM for most partial differential equations can be based on the bounded inverse theorem; 3) the finite difference and finite volume algorithms solve only two Maxwell equations and ignore the divergence equations; and 4) the first numerical simulation of three-dimensional Marangoni-Benard convection was performed using the LSFEM.

Jiang, Bo-Nan↗
NASA NTRS1986

Development of a numerical procedure to map a general 3-d body onto a near-circle

Conformal mapping is a classical technique utilized for solving problems in aerodynamics and hydrodynamics. Conformal mapping is utilized in the construction of grids around airfoils, engine inlets and other aircraft configurations. These shapes are transformed onto a near-circle image for which the equations of fluid motion are discretized on the mapped plane and solved numerically by utilizing the appropriate techniques. In comparison to other grid-generation techniques such as algerbraic or differential type, conformal mapping offers an analytical and accurate form even if the grid deformation is large. One of the most appealing features is that the grid can be constrained to remain orthogonal to the body after the transformation. Hence, the grid is suitable for analyzing the supersonic flow past a blunt object. The associated shock as a coordinate surface adjusts its position in the course of computation until convergence is reached. The present work applied conformal mapping to 3-D bodies with no axis of symmetry such as the Aerobraking Flight Experiment (AFE) vehicle, transforming the AFE shape onto a near-circle image. A numerical procedure and code are used to generate grids around the AFE body.

Hommel, M. J.↗
NASA NTRS1989

Numerical studies of convective heat transfer in an inclined semi-annular enclosure

Natural convection heat transfer in a two-dimensional differentially heated semiannular enclosure is studied. The enclosure is isothermally heated and cooled at the inner and outer walls, respectively. A commercial software based on the SIMPLER algorithm was used to simulate the velocity and temperature profiles. Various parameters that affect the momentum and heat transfer processes were examined. These parameters include the Rayleigh number, Prandtl number, radius ratio, and the angle of inclination. A flow regime extending from conduction-dominated to convection-dominated flow was examined. The computed results of heat transfer are presented as a function of flow parameter and geometric factors. It is found that the heat transfer rate attains a minimum when the enclosure is tilted about +50 deg with respect to the gravitational direction.

Wang, L.-W.↗
NASA NTRS1993

A critical analysis of the accuracy of several numerical techniques for combustion kinetic rate equations

A detailed analysis of the accuracy of several techniques recently developed for integrating stiff ordinary differential equations is presented. The techniques include two general-purpose codes EPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREK1D, and GCKP4 developed specifically to solve chemical kinetic rate equations. The accuracy study is made by application of these codes to two practical combustion kinetics problems. Both problems describe adiabatic, homogeneous, gas-phase chemical reactions at constant pressure, and include all three combustion regimes: induction, heat release, and equilibration. To illustrate the error variation in the different combustion regimes the species are divided into three types (reactants, intermediates, and products), and error versus time plots are presented for each species type and the temperature. These plots show that CHEMEQ is the most accurate code during induction and early heat release. During late heat release and equilibration, however, the other codes are more accurate. A single global quantity, a mean integrated root-mean-square error, that measures the average error incurred in solving the complete problem is used to compare the accuracy of the codes. Among the codes examined, LSODE is the most accurate for solving chemical kinetics problems. It is also the most efficient code, in the sense that it requires the least computational work to attain a specified accuracy level. An important finding is that use of the algebraic enthalpy conservation equation to compute the temperature can be more accurate and efficient than integrating the temperature differential equation.

Radhadrishnan, Krishnan↗
NASA NTRS1978

An extension of MacCormack's method for flows with higher-order equations and in different configurations

The numerical scheme for the computation of a shock discontinuity developed by MacCormack has been extended to solve a number of differential equations, including cases explicitly containing higher-order derivatives: (1) Korteweg-de Vries equation with a term of third-order derivative, (2) a system of nonlinear equations governing nonsteady one-dimensional plasma flow in cylindrical coordinate, (3) equations of solar wind. Comparisons with previous results are made, if available, to illustrate the advantages of the present method. The question of convergence of the numerical calculation is discussed.

Ying, S. J.↗
DOE OSTI2019

The Bloch equation for spin dynamics in electron storage rings: Computational and theoretical aspects

In this paper, we describe our work on spin polarization in high-energy electron storage rings which we base on the Full Bloch equation (FBE) for the polarization density and which aims towards the [Formula: see text] option of the proposed Future Circular Collider (FCC-ee) and the proposed Circular Electron Positron Collider (CEPC). The FBE takes into account non spin-flip and spin-flip effects due to synchrotron radiation including the spin-diffusion effects and the Sokolov–Ternov effect with its Baier–Katkov generalization as well as the kinetic-polarization effect. This mathematical model is an alternative to the standard mathematical model based on the Derbenev–Kondratenko formulas. For our numerical and analytical studies of the FBE, we develop an approximation to the latter to obtain an effective FBE. This is accomplished by finding a third mathematical model based on a system of stochastic differential equations (SDEs) underlying the FBE and by approximating that system via the method of averaging from perturbative ODE theory. We also give an overview of our algorithm for numerically integrating the effective FBE. This discretizes the phase space using spectral methods and discretizes time via the additive Runge–Kutta (ARK) method which is a high-order semi-implicit method. We also discuss the relevance of the third mathematical model for spin tracking.

Physics↗
DOE OSTI2022

Validation and parameterization of a novel physics-constrained neural dynamics model applied to turbulent fluid flow

We report, in fluid physics, data-driven models to enhance or accelerate time to solution are becoming increasingly popular for many application domains, such as alternatives to turbulence closures, system surrogates, or for new physics discovery. In the context of reduced order models of high-dimensional time-dependent fluid systems, machine learning methods grant the benefit of automated learning from data, but the burden of a model lies on its reduced-order representation of both the fluid state and physical dynamics. In this work, we build a physics-constrained, data-driven reduced order model for Navier–Stokes equations to approximate spatiotemporal fluid dynamics in the canonical case of isotropic turbulence in a triply periodic box. The model design choices mimic numerical and physical constraints by, for example, implicitly enforcing the incompressibility constraint and utilizing continuous neural ordinary differential equations for tracking the evolution of the governing differential equation. We demonstrate this technique on a three-dimensional, moderate Reynolds number turbulent fluid flow. In assessing the statistical quality and characteristics of the machine-learned model through rigorous diagnostic tests, we find that our model is capable of reconstructing the dynamics of the flow over large integral timescales, favoring accuracy at the larger length scales. More significantly, comprehensive diagnostics suggest that physically interpretable model parameters, corresponding to the representations of the fluid state and dynamics, have attributable and quantifiable impact on the quality of the model predictions and computational complexity.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

HomPINNs: Homotopy physics-informed neural networks for learning multiple solutions of nonlinear elliptic differential equations

Physics-informed neural networks (PINNs) based machine learning is an emerging framework for solving nonlinear differential equations. However, due to the implicit regularity of neural network structure, PINNs can only find the flattest solution in most cases by minimizing the loss functions. In this paper, we combine PINNs with the homotopy continuation method, a classical numerical method to compute isolated roots of polynomial systems, and propose a new deep learning framework, named homotopy physics-informed neural networks (HomPINNs), for solving multiple solutions of nonlinear elliptic differential equations. The implementation of an HomPINN is a homotopy process that is composed of the training of a fully connected neural network, named the starting neural network, and training processes of several PINNs with different tracking parameters. The starting neural network is to approximate a starting function constructed by the trivial solutions, while other PINNs are to minimize the loss functions defined by boundary condition and homotopy functions, varying with different tracking parameters. These training processes are regraded as different steps of a homotopy process, and a PINN is initialized by the well-trained neural network of the previous step, while the first starting neural network is initialized using the default initialization method. Finally, several numerical examples are presented to show the efficiency of our proposed HomPINNs, including reaction-diffusion equations with a heart-shaped domain.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Implementation and (Inverse Modified) Error Analysis for Implicitly Templated ODE-Nets

We focus on learning unknown dynamics from data using ODE-nets templated on implicit numerical initial value problem solvers. First, we perform inverse modified error analysis of the ODE-nets using unrolled implicit schemes for ease of interpretation. It is shown that training an ODE-net using an unrolled implicit scheme returns a close approximation of an inverse modified differential equation (IMDE). In addition, we establish a theoretical basis for hyperparameter selection when training such ODE-nets, whereas current strategies usually treat numerical integration of ODE-nets as a black box. We thus formulate an adaptive algorithm which monitors the level of error and adapts the number of (unrolled) implicit solution iterations during the training process, so that the error of the unrolled approximation is less than the current learning loss. This helps accelerate training while maintaining accuracy. Several numerical experiments are performed to demonstrate the advantages of the proposed algorithm compared to nonadaptive unrollings and validate the theoretical analysis. Here, we also note that this approach naturally allows for incorporating partially known physical terms in the equations, giving rise to what is termed “gray box” identification.

ODE-nets↗
DOE OSTI2024

Generalization error guaranteed auto-encoder-based nonlinear model reduction for operator learning

Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data. An integral component in addressing this challenge is model reduction, which reduces both the data dimensionality and problem size. In this paper, we utilize low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet). AENet first learns the latent variables of the input data and then learns the transformation from these latent variables to corresponding output data. Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Furthermore, we establish a mathematical and statistical estimation theory that analyzes the generalization error of AENet. Finally, our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process, while also demonstrating the robustness of AENet to noise.

Auto-encoder↗
DOE OSTI2020

A Stochastic Gradient Descent Approach for Stochastic Optimal Control

In this work, we introduce a stochastic gradient descent approach to solve the stochastic optimal control problem through stochastic maximum principle. The motivation that drives our method is the gradient of the cost functional in the stochastic optimal control problem is under expectation, and numerical calculation of such an expectation requires fully computation of a system of forward backward stochastic differential equations, which is computationally expensive. By evaluating the expectation with single-sample representation as suggested by the stochastic gradient descent type optimisation, we could save computational efforts in solving FBSDEs and only focus on the optimisation task which aims to determine the optimal control process.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1975

Nonlinear seismic analysis of a reactor structure impact between core components

The seismic analysis of the FFTF-PIOTA (Fast Flux Test Facility-Postirradiation Open Test Assembly), subjected to a horizontal DBE (Design Base Earthquake) is presented. The PIOTA is the first in a set of open test assemblies to be designed for the FFTF. Employing the direct method of transient analysis, the governing differential equations describing the motion of the system are set up directly and are implicitly integrated numerically in time. A simple lumped-nass beam model of the FFTF which includes small clearances between core components is used as a "driver" for a fine mesh model of the PIOTA. The nonlinear forces due to the impact of the core components and their effect on the PIOTA are computed.

Hill, R. G.↗
NASA NTRS1987

Bifurcation properties of a stratospheric vacillation model

Nonlinear properties of the stratospheric vacillation model of Holton and Mass (1976) are studied numerically using bifurcation theory. Severe truncation and vertical differencing are used to obtain 81 nonlinear ordinary differential equations with time-dependent variables. Three branches of the steady solutions are determined using Powell's hybrid method and the pseudoarclength continuation method, and a multiplicity of stable steady-state solutions with different vertical structures are found to exist in some range of the bifurcation parameter. Periodic solutions are found which branch off from a steady solution by a Hopf bifurcation. It is suggested that the interannual variability of the stratospheric circulation in the middle and high latitudes during winter may be explained by the multiplicity of steady and periodic stable solutions.

Yoden, Shigeo↗
NASA NTRS1987

A new approach for shaping of dual-reflector antennas

The shaping of 2-D dual-reflector antenna systems to generate a prescribed distribution with uniform phase at the aperture of the second reflector is examined. This method is based on the geometrical nature of Cassegrain and Gregorian dual-reflector antennas. The method of syntheses satisfies the principles of geometrical optics which are the foundations of dual-reflector designs. Instead of setting up differential equations or heuristically designing the subreflector, a set of algebraic equations is formulated and solved numerically to obtain the desired surfaces. The caustics of the reflected rays from the subreflector can be obtained and examined. Several examples of 2-D dual-reflector shaping are shown to validate the study. Geometrical optics and physical optics are used to calculate the scattered fields from the reflectors.

Lee, Teh-Hong↗