SEARCH · Engineering Papers

Results for “differential equations”

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.

SourceResearch connectionFrom yearThrough year

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

At least 1,099 records · Page 61

All sources NASA NTRS DOE OSTI

NASA NTRS1997

Multi-Body Analysis of a Tiltrotor Configuration

The paper describes the aeroelastic analysis of a tiltrotor configuration. The 1/5 scale wind tunnel semispan model of the V-22 tiltrotor aircraft is considered. The analysis is performed by means of a multi-body code, based on an original formulation. The differential equilibrium problem is stated in terms of first order differential equations. The equilibrium equations of every rigid body are written, together with the definitions of the momenta. The bodies are connected by kinematic constraints, applied in form of Lagrangian multipliers. Deformable components are mainly modelled by means of beam elements, based on an original finite volume formulation. Multi-disciplinar problems can be solved by adding user-defined differential equations. In the presented analysis the equations related to the control of the swash-plate of the model are considered. Advantages of a multi-body aeroelastic code over existing comprehensive rotorcraft codes include the exact modelling of the kinematics of the hub, the detailed modelling of the flexibility of critical hub components, and the possibility to simulate steady flight conditions as well as wind-up and maneuvers. The simulations described in the paper include: 1) the analysis of the aeroelastic stability, with particular regard to the proprotor/pylon instability that is peculiar to tiltrotors, 2) the determination of the dynamic behavior of the system and of the loads due to typical maneuvers, with particular regard to the conversion from helicopter to airplane mode, and 3) the stress evaluation in critical components, such as the pitch links and the conversion downstop spring.

Ghiringhelli, G. L.↗

DOE OSTI2020

A model for radiative heating of a high-Z pusher

Several approaches to Inertial Confinement Fusion (ICF), including double-shell, pushered-single-shell, and the Revolver designs, have fuel surrounded by pushers made from high-Z materials. An advantage of these designs is that radiation emitted by the hot fuel will be absorbed and re-radiated into the fuel to reduce cooling. This process is referred to as radiation-trapping, and it lowers the fuel temperature required for ignition. To elucidate the physics of radiation trapping, a simplified model comprising a set of coupled differential equations has been developed to model the arbitrary time-dependent wall temperature from the Hammer and Rosen solution [J. H. Hammer and M. D. Rosen, Phys. Plasmas 10, 1829 (2003)] to a Marshak wave. The derivation of the model and a set of analytical power-law solutions will be presented. A discussion on numerical implementation of the differential equations into a model for burn in ICF capsules is also included.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1972

Analytical treatment of gas flows through multilayer insulation, project 1

A theoretical investigation of gas flow inside a multilayer insulation system was made for the case of the broadside pumping process. A set of simultaneous first-order differential equations for the temperature and pressure of the gas molecules through the perforations on the insulation layers. A modified Runge-Kutta method was used for numerical experiment. The numerical stability problem was also investigated. It was shown that when the relaxation time is less than the time period over which the gas properties change appreciably, the set of differential equations can be replaced by a set of algebraic equations for solution. Numerical examples were given and comparisons with experimental data were made.

Lin, J. T.↗

NASA NTRS1977

Varieties of operator manipulation

A change in MACSYMA syntax is proposed to accommodate the operator manipulators necessary to implement direct and indirect methods for the solution of differential equations, calculus of finite differences, and the fractional calculus, as well as their modern counterparts. To illustrate the benefits and convenience of this syntax extension, an example is given to show how MACSYMA's pattern-matching capability can be used to implement a particular set of operator identities which can then be used to obtain exact solutions to nonlinear differential equations.

Doohovskoy, A.↗

DOE OSTI2025

Quantum Algorithm for Linear Non-unitary Dynamics with Near-Optimal Dependence on All Parameters

We introduce a family of identities that express general linear non-unitary evolution operators as a linear combination of unitary evolution operators, each solving a Hamiltonian simulation problem. This formulation can exponentially enhance the accuracy of the recently introduced linear combination of Hamiltonian simulation (LCHS) method [An, Liu, and Lin, Physical Review Letters, 2023]. For the first time, this approach enables quantum algorithms to solve linear differential equations with both optimal state preparation cost and near-optimal scaling in matrix queries on all parameters.

Applied Dynamical Systems↗

NASA NTRS1975

DYNGEN: A program for calculating steady-state and transient performance of turbojet and turbofan engines

The DYNGEN, a digital computer program for analyzing the steady state and transient performance of turbojet and turbofan engines, is described. The DYNGEN is based on earlier computer codes (SMOTE, GENENG, and GENENG 2) which are capable of calculating the steady state performance of turbojet and turbofan engines at design and off-design operating conditions. The DYNGEN has the combined capabilities of GENENG and GENENG 2 for calculating steady state performance; to these the further capability for calculating transient performance was added. The DYNGEN can be used to analyze one- and two-spool turbojet engines or two- and three-spool turbofan engines without modification to the basic program. A modified Euler method is used by DYNGEN to solve the differential equations which model the dynamics of the engine. This new method frees the programmer from having to minimize the number of equations which require iterative solution. As a result, some of the approximations normally used in transient engine simulations can be eliminated. This tends to produce better agreement when answers are compared with those from purely steady state simulations. The modified Euler method also permits the user to specify large time steps (about 0.10 sec) to be used in the solution of the differential equations. This saves computer execution time when long transients are run. Examples of the use of the program are included, and program results are compared with those from an existing hybrid-computer simulation of a two-spool turbofan.

Sellers, J. F.↗

DOE OSTI2026

Data and scripts associated with a manuscript modeling microbial regulation of priming effects

This data package is associated with the publication “Modeling Microbial Regulatory Feedback in Organic Matter Decomposition Identifies Copiotrophic Traits as Key Drivers of Positive Priming” published as a preprint on BioRXiv by Ahamed et al. (2026); https://doi.org/10.1101/2024.08.11.607483. The package contains MATLAB scripts and saved simulation outputs used to implement a cybernetic model of microbial regulation during complex organic matter (OM) decomposition governing priming effects. It includes models of (i) single microbial functional groups (copiotrophic or oligotrophic degraders) and (ii) binary consortia composed of degraders and non-degraders with contrasting or common growth traits. Simulation results were generated using Monte Carlo analyses, with randomized key model parameters across a range of environmental mixing fractions of complex and labile OM. The dataset was created to provide a transparent and reusable computational framework for systematically exploring how microbial growth traits, metabolic regulation, and community composition influence OM decomposition dynamics and priming effects. For details on how to navigate data packages generated by this project, see https://data.ess-dive.lbl.gov/portals/PNNLRiverCorridorSFA/About. In addition to a readme, this data package also includes a file-level metadata (FLMD) file that describes each file and a data dictionary (DD) that describes the variable definitions. This package includes: (1) annotated MATLAB code implementing the system of ordinary differential equations and cybernetic control laws; (2) saved output files containing data (e.g., biomass, substrates, enzyme levels, priming metrics); and (3) scripts for processing saved outputs and regenerating figures. Specifically, the data package contains three main MATLAB scripts: runPrimingModel.m, runPlotData.m, and runPlotSuppFigS1.m, along with this readme and supporting documentation. Users should begin with runPrimingModel.m, which contains the annotated code implementing the system of ordinary differential equations and cybernetic control laws. This script runs the Monte Carlo simulations of microbial OM decomposition and allows users to modify microbial trait definitions, adjust parameter distributions, or define new community configurations. Simulation outputs are automatically saved as .mat files in the folder named SavedData, which stores all pre-generated results included in this package. The second script, runPlotData.m, reads files from the SavedData folder and processes them to regenerate the figures presented in the manuscript. The third script, runPlotSuppFigS1.m, specifically generates Figure S1 in the Supplementary Material of the manuscript. The package also includes the aforementioned files in non-proprietary .txt format. If users intend to use them, they should first save the files in their respective .m or .mat formats prior to execution in MATLAB.

Biomass concentration↗

NASA NTRS1990

A fully Sinc-Galerkin method for Euler-Bernoulli beam models

A fully Sinc-Galerkin method in both space and time is presented for fourth-order time-dependent partial differential equations with fixed and cantilever boundary conditions. The Sinc discretizations for the second-order temporal problem and the fourth-order spatial problems are presented. Alternate formulations for variable parameter fourth-order problems are given which prove to be especially useful when applying the forward techniques to parameter recovery problems. The discrete system which corresponds to the time-dependent partial differential equations of interest are then formulated. Computational issues are discussed and a robust and efficient algorithm for solving the resulting matrix system is outlined. Numerical results which highlight the method are given for problems with both analytic and singular solutions as well as fixed and cantilever boundary conditions.

Smith, R. C.↗

DOE OSTI2025

Bayesian learning with Gaussian processes for low-dimensional representations of time-dependent nonlinear systems

This work presents a data-driven method for learning low-dimensional time-dependent physics-based surrogate models whose predictions are endowed with uncertainty estimates. We use the operator inference approach to model reduction that poses the problem of learning low-dimensional model terms as a regression of state space data and corresponding time derivatives by minimizing the residual of reduced system equations. Standard operator inference models perform well with accurate training data that are dense in time, but producing stable and accurate models when the state data are noisy and/or sparse in time remains a challenge. Another challenge is the lack of uncertainty estimation for the predictions from the operator inference models. Our approach addresses these challenges by incorporating Gaussian process surrogates into the operator inference framework to (1) probabilistically describe uncertainties in the state predictions and (2) procure analytical time derivative estimates with quantified uncertainties. The formulation leads to a generalized least-squares regression and, ultimately, reduced-order models that are described probabilistically with a closed-form expression for the posterior distribution of the operators. The resulting probabilistic surrogate model propagates uncertainties from the observed state data to reduced-order predictions. Furthermore, we demonstrate the method is effective for constructing low-dimensional models of two nonlinear partial differential equations representing a compressible flow and a nonlinear diffusion–reaction process, as well as for estimating the parameters of a low-dimensional system of nonlinear ordinary differential equations representing compartmental models in epidemiology.

Data-driven model reduction↗

DOE OSTI2024

Sparsified time-dependent Fourier neural operators for fusion simulations

This paper presents a sparsified Fourier neural operator for coupled time-dependent partial differential equations (ST-FNO) as an efficient machine learning surrogate for fluid and particle-based fusion codes such as NIMROD (Non-Ideal Magnetohydrodynamics with Rotation - Open Discussion) and GTC (Gyrokinetic Toroidal Code). ST-FNO leverages the structures in the governing equations and utilizes neural operators to represent Green's function-like numerical operators in the corresponding numerical solvers. Once trained, ST-FNO can rapidly and accurately predict dynamics in fusion devices compared with first-principle numerical algorithms. In general, ST-FNO represents an efficient and accurate machine learning surrogate for numerical simulators for multi-variable nonlinear time-dependent partial differential equations, with the proposed architectures and loss functions. The efficacy of ST-FNO has been demonstrated using quiescent H-mode simulation data from NIMROD and kink-mode simulation data from GTC. The ST-FNO H-mode results show orders of magnitude reduction in memory and central processing unit usage in comparison with the numerical solvers in NIMROD when computing fields over a selected poloidal plane. The ST-FNO kink-mode results achieve a factor of 2 reduction in the number of parameters compared to baseline FNO models without accuracy loss.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1978

Remarks on boundary-fitted coordinate system generation

The essential part of numerical solutions of partial differential equations is the representation of gradients and integrals by, respectively, differences between points and summations over points. In order for such numerical representations to be accurate, it is necessary that these points be more closely spaced in regions of large gradients. The need for accurate representation is particularly acute near body surfaces, since the boundary conditions are generally the most influential part of a partial differential equation solution. This is especially true of viscous solutions at high Reynolds number, where very large gradients occur in the boundary layer.

Thompson, J. F.↗

DOE OSTI2021

Holomorphic modular bootstrap revisited

In this work we revisit the “holomorphic modular bootstrap”, i.e. the classification of rational conformal field theories via an analysis of the modular differential equations satisfied by their characters. By making use of the representation theory of PSL(2, $\mathbb{Z}_n$), we describe a method to classify allowed central charges and weights (c, h i ) for theories with any number of characters d. This allows us to avoid various bottlenecks encountered previously in the literature, and leads to a classification of consistent characters up to d = 5 whose modular differential equations are uniquely fixed in terms of (c, h i ). In the process, we identify the full set of constraints on the allowed values of the Wronskian index for fixed d ≤ 5.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1972

Stability of distributed parameter systems.

A theorem is derived for the stability of solutions of general linear partial differential equations. A norm of the state space in the form of multiple integrals over the spatial domain is used for the Liapunov functional. Theorems and lemmas are also given for linear time-invariant constant coefficient distributed parameter systems, a class of nonlinear distributed parameter systems and for others with a Lure-type nonlinearity. The theorem conditions are similar to those known for corresponding ordinary differential equations but with operators replacing matrices.

Park, K. E.↗

NASA NTRS1976

The development of the Poincare-similar elements with eccentric anomaly as the independent variable

A new set of element differential equations for the perturbed two-body motion is derived. The elements are canonical and are similar to the classical canonical Poincare elements, which have time as the independent variable. The phase space is extended by introducing the total energy and time as canonically conjugated variables. The new independent variable is, to within an additive constant, the eccentric anomaly. These elements are compared to the Kustaanheimo-Stiefel (KS) element differential equations, which also have the eccentric anomaly as the independent variable. For several numerical examples, the accuracy and stability of the new set are equal to those of the KS solution. This comparable accuracy result can probably be attributed to the fact that both sets have the same time element and very similar energy elements. The new set has only 8 elements, compared to 10 elements for the KS set. Both sets are free from singularities due to vanishing eccentricity and inclination.

Bond, V. R.↗

NASA NTRS1973

Analysis of gas flow through a multilayer insulation system.

A theoretical investigation of gas flow inside a multilayer insulation system has been made for the case of the broadside pumping process. A set of simultaneous first-order differential equations for the temperature and pressure of the gas mixture was obtained by considering the diffusion mechanism of the gas molecules through the perforations on the insulation layers. A modified Runge-Kutta method was used for numerical experiment. The numerical stability problem was investigated. It has been shown that when the relaxation time is small compared with the time period over which the gas properties change appreciably, the set of differential equations can be replaced by a set of algebraic equations for solution. Numerical examples were given and comparison with experimental data was made.

Lin, J. T.↗

NASA NTRS1973

Thermal engineering research

Shih, C. C.↗

DOE OSTI2023

Encoding nonlinear and unsteady aerodynamics of limit cycle oscillations using nonlinear sparse Bayesian learning

This article investigates the applicability of a recently proposed, nonlinear sparse Bayesian learning (NSBL) algorithm to identify and estimate the complex aerodynamics of limit cycle oscillations. NSBL provides a semi-analytical framework for determining the data-optimal sparse model nested within a (potentially) over-parameterized model. This is particularly relevant to nonlinear dynamical systems where modelling approaches involve the use of physics-based and data-driven components. In such cases, the data-driven components, where analytical descriptions of the physical processes are not readily available, are often prone to overfitting, meaning that the empirical aspects of these models will often involve the calibration of an unnecessarily large number of parameters. While an overparameterized model may fit the observed data well, such models may be inadequate for making predictions in regimes that are different from those wherein the data were recorded. In view of this, it is desirable to not only calibrate the model parameters, but also identify the optimal compromise between data fit and model complexity. In this article, we exhibit the optimal model discovery for an aeroelastic system wherein the structural dynamics are well-known and described by a differential equation model, coupled with a semi-empirical aerodynamic model for laminar separation flutter, resulting in low-amplitude limit cycle oscillations (LCO). To illustrate the performance of the algorithm, in this article, we use synthetic data and demonstrate the ability of the algorithm to correctly rediscover the optimal model and model parameters, given a known data-generating model. The synthetic data are generated from a forward simulation of a known differential equation model with parameters selected so as to mimic the dynamics observed in wind-tunnel experiments. Subsequently, we demonstrate the performance of the algorithm for model selection using noisy LCO data from wind tunnel experiments. As there is no ground truth available for the experimental data case, we provide a comparison between NSBL and Bayesian model selection to validate the results, and demonstrate the use of NSBL as an efficient alternative to traditional methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1975

Application of Hamilton's law of varying action

The law of varying action enunciated by Hamilton in 1834-1835 permits the direct analytical solution of the problems of mechanics, both stationary and nonstationary, without consideration of force equilibrium and the theory of differential equations associated therewith. It has not been possible to obtain direct analytical solutions to nonstationary systems through the use of energy theory, which has been limited for 140 years to the principle of least action and to Hamilton's principle. It is shown here that Hamilton's law permits the direct analytical solution to nonstationary, initial value systems in the mechanics of solids without any knowledge or use of the theory of differential equations. Solutions are demonstrated for nonconservative, nonstationary particle motion, both linear and nonlinear.

Bailey, C. D.↗