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Results for “stiff ordinary 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.

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At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Order conditions for nonlinearly partitioned Runge-Kutta methods

Recently, a new class of nonlinearly partitioned Runge–Kutta (NPRK) methods was proposed for nonlinearly partitioned systems of autonomous ordinary differential equations y' = F(y, y). The target class of problems are those in which different scales, stiffnesses, or physics are coupled in a nonlinear way, wherein the desired partition cannot be written in a classical additive or component-wise fashion. Here we use a rooted-tree analysis to derive full-order conditions for NPRKM methods, where M denotes the number of nonlinear partitions. Due to the nonlinear coupling and thereby the mixed product differentials, it turns out that the standard node-colored rooted tree analysis used in analyzing ODE integrators does not naturally apply. Instead we develop a new edge-colored rooted-tree framework to address the nonlinear coupling. The resulting order conditions are enumerated, are provided directly for up to fourth order with M = 2 and third order with M = 3, and are related to existing order conditions of additive and partitioned RK methods. We conclude with an example that shows how the nonlinear order conditions can be used to obtain an embedded estimate of the state-dependent nonlinear coupling strength in a dynamical system.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1986

Computational methods for the identification of spatially varying stiffness and damping in beams

A numerical approximation scheme for the estimation of functional parameters in Euler-Bernoulli models for the transverse vibration of flexible beams with tip bodies is developed. The method permits the identification of spatially varying flexural stiffness and Voigt-Kelvin viscoelastic damping coefficients which appear in the hybrid system of ordinary and partial differential equations and boundary conditions describing the dynamics of such structures. An inverse problem is formulated as a least squares fit to data subject to constraints in the form of a vector system of abstract first order evolution equations. Spline-based finite element approximations are used to finite dimensionalize the problem. Theoretical convergence results are given and numerical studies carried out on both conventional (serial) and vector computers are discussed.

Banks, H. T.↗
NASA NTRS1985

New integration techniques for chemical kinetic rate equations. 2: Accuracy comparison

A comparison of the accuracy of several techniques recently developed for solving stiff differential equations is presented. The techniques examined include two general purpose codes EEPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREKID, and GCKP84 developed specifically to solve chemical kinetic rate equations. The accuracy comparisons are made by applying these solution procedures 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. The comparisons show that LSODE is 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 an iterative solution of the algebraic enthalpy conservation equation for the temperature can be more accurate and efficient than computing the temperature by integrating its time derivative.

Radhakrishnan, K.↗
NASA NTRS1986

New integration techniques for chemical kinetic rate equations. II - Accuracy comparison

A comparison of the accuracy of several techniques recently developed for solving stiff differential equations is presented. The techniques examined include two general purpose codes EEPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREKID, and GCKP84 developed specifically to solve chemical kinetic rate equations. The accuracy comparisons are made by applying these solution procedures 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. The comparisons show that LSODE is 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 an iterative solution of the algebraic enthalpy conservation equation for the temperature can be more accurate and efficient than computing the temperature by integrating its time derivative.

Radhakrishnan, K.↗
NASA NTRS1988

The ATOMFT integrator - Using Taylor series to solve ordinary differential equations

This paper discusses the application of ATOMFT, an integration package based on Taylor series solution with a sophisticated user interface. ATOMFT has the capabilities to allow the implementation of user defined functions and the solution of stiff and algebraic equations. Detailed examples, including the solutions to several astrodynamics problems, are presented. Comparisons with its predecessor ATOMCC and other modern integrators indicate that ATOMFT is a fast, accurate, and easy method to use to solve many differential equation problems.

Berryman, Kenneth W.↗
NASA NTRS1985

Three-dimensional baroclinic instability of a Hadley cell for small Richardson number

A three-dimensional, linear stability analysis of a baroclinic flow for Richardson number, Ri, of order unity is presented. The model considered is a thin horizontal, rotating fluid layer which is subjected to horizontal and vertical temperature gradients. The basic state is a Hadley cell which is a solution of the complete set of governing, nonlinear equations and contains both Ekman and thermal boundary layers adjacent to the rigid boundaries; it is given in a closed form. The stability analysis is also based on the complete set of equations; and perturbation possessing zonal, meridional, and vertical structures were considered. Numerical methods were developed for the stability problem which results in a stiff, eighth-order, ordinary differential eigenvalue problem. The previous work on three-dimensional baroclinic instability for small Ri was extended to a more realistic model involving the Prandtl number, sigma, and the Ekman number, E, and to finite growth rates and a wider range of the zonal wavenumber.

Antar, B. N.↗
NASA NTRS1993

Wavelet multiresolution analyses adapted for the fast solution of boundary value ordinary differential equations

We present ideas on how to use wavelets in the solution of boundary value ordinary differential equations. Rather than using classical wavelets, we adapt their construction so that they become (bi)orthogonal with respect to the inner product defined by the operator. The stiffness matrix in a Galerkin method then becomes diagonal and can thus be trivially inverted. We show how one can construct an O(N) algorithm for various constant and variable coefficient operators.

Jawerth, Bjoern↗
NASA NTRS1982

Integrating matrix formulations for vibrations of rotating beams including the effects of concentrated masses

By expressing partial differential equations of motion in matrix notation, utilizing the integrating matrix as a spatial operator, and applying the boundary conditions, the resulting ordinary differential equations can be cast into standard eigenvalue form upon assumption of the usual time dependence. As originally developed, the technique was limited to beams having continuous mass and stiffness properties along their lengths. Integrating matrix methods are extended to treat the differential equations governing the flap, lag, or axial vibrations of rotating beams having concentrated masses. Inclusion of concentrated masses is shown to lead to the same kind of standard eigenvalue problem as before, but with slightly modified matrices.

Lakin, W. D.↗
NASA NTRS2018

Atmospheric Chemistry Modeling Using a Regression Forest Model

Atmospheric chemistry is central to many environmental issues such as air pollution, climate change, and stratospheric ozone loss. Chemistry Transport Models (CTM) are a central tool for understanding these issues, whether for research or for forecasting. These models split the atmosphere in a large number of grid-boxes and consider the emission of compounds into these boxes and their subsequent transport, deposition, and chemical processing. The chemistry is represented through a series of simultaneous ordinary differential equations, one for each compound. Given the difference in life-times between the chemical compounds (milli-seconds for O1D to years for CH4) these equations are numerically stiff and solving them consists of a significant fraction of the computational burden of a CTM. We have investigated a machine learning approach to solving the differential equations instead of solving them numerically. From an annual simulation of the GEOS-Chem model we have produced a training dataset consisting of the concentration of compounds before and after the differential equations are solved, together with some key physical parameters for every grid-box and time-step. From this dataset we have trained a machine learning algorithm (regression forest) to be able to predict the concentration of the compounds after the integration step based on the concentrations and physical state at the beginning of the time step. We have then included this algorithm back into the GEOS-Chem model, bypassing the need to integrate the chemistry. This machine learning approach shows many of the characteristics of the full simulation and has the potential to be substantially faster. There are a wide range of application for such an approach - generating boundary conditions, for use in air quality forecasts, chemical data assimilation systems, centennial scale climate simulations etc. We discuss our approches' speed and accuracy, and highlight some potential future directions for improving this approach.

Keller, Christoph A.↗
NASA NTRS2019

Atmospheric Chemistry Modeling Using Machine Learning

Atmospheric chemistry models are a central tool to study the impact of chemical constituents on the environment, vegetation and human health. These models split the atmosphere in a large number of grid-boxes and consider the emission of compounds into these boxes and their subsequent transport, deposition, and chemical processing. The chemistry is represented through a series of simultaneous ordinary differential equations, one for each compound. Given the difference in life-times between the chemical compounds (milli-seconds for O1D to years for CH4) these equations are numerically stiff and solving them consists of a significant fraction of the computational burden of a chemistry model. We have investigated a machine learning approach to emulate the chemistry instead of solving the differential equations numerically. From a one-month simulation of the GEOS-Chem model we have produced a training dataset consisting of the concentration of compounds before and after the differential equations are solved, together with some key physical parameters for every grid-box and time-step. From this dataset we have trained a machine learning algorithm (regression forest) to be able to predict the concentration of the compounds after the integration step based on the concentrations and physical state at the beginning of the time step. We have then included this algorithm back into the GEOS-Chem model, bypassing the need to integrate the chemistry. This machine learning approach shows many of the characteristics of the full simulation and has the potential to be substantially faster. There are a wide range of application for such an approach - generating boundary conditions, for use in air quality forecasts, chemical data assimilation systems, etc. We discuss speed and accuracy of our approach, and highlight some potential future directions for improving it.

Keller, Christoph A.↗
NASA NTRS2019

Atmospheric Chemistry Modeling and Air Quality Forecasting Using Machine Learning

Atmospheric chemistry models are a central tool to study the impact of chemical constituents on the environment, vegetation and human health. These models split the atmosphere in a large number of grid-boxes and consider the emission of compounds into these boxes and their subsequent transport, deposition, and chemical processing. The chemistry is represented through a series of simultaneous ordinary differential equations, one for each compound. Given the difference in life-times between the chemical compounds (milli-seconds for O1D to years for CH4) these equations are numerically stiff and solving them consists of a significant fraction of the computational burden of a chemistry model.We have investigated a machine learning approach to emulate the chemistry instead of solving the differential equations numerically. From a one-month simulation of the GEOS-Chem model we have produced a training dataset consisting of the concentration of compounds before and after the differential equations are solved, together with some key physical parameters for every grid-box and time-step. From this dataset we have trained a machine learning algorithm (regression forest) to be able to predict the concentration of the compounds after the integration step based on the concentrations and physical state at the beginning of the time step. We have then included this algorithm back into the GEOS-Chem model, bypassing the need to integrate the chemistry.This machine learning approach shows many of the characteristics of the full simulation and has the potential to be substantially faster. There are a wide range of application for such an approach - generating boundary conditions, for use in air quality forecasts, chemical data assimilation systems, etc. We discuss speed and accuracy of our approach, and highlight some potential future directions for improving it.

Keller, Christoph A.↗
DOE OSTI2024

Multiphysics Time-Integration for Turbulent Combustion at the Exascale

Turbulent reacting flow systems are often modeled with coupled time-dependent partial differential equations (PDEs). Solving such equations can easily tax the world's largest supercomputers. One pragmatic strategy for attacking such problems is to split the PDEs into components that can more easily be solved in isolation. This generic operator-splitting strategy leads to a set of ordinary differential equations (ODEs) that need to be solved as part of an "outer-loop" time-stepping approach. In many combustion applications, the ODEs to be solved can be very stiff, exhibiting timescales that span many orders of magnitude. The SUNDIALS library provides a plethora of robust time integration algorithms for solving these ODEs on exascale-capable computing hardware, yet for many complex applications (such multicomponent fuels or emissions predictions), the chemical models remain too complex to solve using reasonable resources. The Quasi-Steady State Approximation (QSSA) can be an effective tool for reducing the size and stiffness of the simulations. In this talk, I will discuss the use of the SUDIALS library of ODE solvers together with automatic code generation tools to solve complex turbulent reacting flow problems using QSSA models.

chemistry↗
NASA NTRS1986

The identification of a distributed parameter model for a flexible structure

A computational method is developed for the estimation of parameters in a distributed model for a flexible structure. The structure we consider (part of the RPL experiment) consists of a cantilevered beam with a thruster and linear accelerometer at the free end. The thruster is fed by a pressurized hose whose horizontal motion effects the transverse vibration of the beam. The Euler-Bernoulli theory is used to model the vibration of the beam and treat the hose-thruster assembly as a lumped or point mass-dashpot-spring system at the tip. Using measurements of linear acceleration at the tip, it is estimated that the parameters (mass, stiffness, damping) and a Voight-Kelvin viscoelastic structural damping parameter for the beam using a least squares fit to the data. Spline based approximations to the hybrid (coupled ordinary and partial differential equations) system are considered; theoretical convergence results and numerical studies with both simulation and actual experimental data obtained from the structure are presented and discussed.

Banks, H. T.↗
NASA NTRS2019

Machine Learning Application to Atmospheric Chemistry Modeling

Atmospheric chemistry models are a central tool to study the impact of chemical constituents on the environment, vegetation and human health. These models split the atmosphere in a large number of grid-boxes and consider the emission of compounds into these boxes and their subsequent transport, deposition, and chemical processing. The chemistry is represented through a series of simultaneous ordinary differential equations, one for each compound. Given the difference in life-times between the chemical compounds (milli-seconds for O (sup 1) D (Deuterium) to years for CH4) these equations are numerically stiff and solving them consists of a significant fraction of the computational burden of a chemistry model. We have investigated a machine learning approach to emulate the chemistry instead of solving the differential equations numerically. From a one-month simulation of the GEOS-Chem model we have produced a training dataset consisting of the concentration of compounds before and after the differential equations are solved, together with some key physical parameters for every grid-box and time-step. From this dataset we have trained a machine learning algorithm (regression forest) to be able to predict the concentration of the compounds after the integration step based on the concentrations and physical state at the beginning of the time step. We have then included this algorithm back into the GEOS-Chem model, bypassing the need to integrate the chemistry. This machine learning approach shows many of the characteristics of the full simulation and has the potential to be substantially faster. There are a wide range of application for such an approach - generating boundary conditions, for use in air quality forecasts, chemical data assimilation systems, etc. We discuss speed and accuracy of our approach, and highlight some potential future directions for improving it.

Keller, Christoph A.↗
NASA NTRS1988

The identification of a distributed parameter model for a flexible structure

A computational method is developed for the estimation of parameters in a distributed model for a flexible structure. The structure we consider (part of the RPL experiment) consists of a cantilevered beam with a thruster and linear accelerometer at the free end. The thruster is fed by a pressurized hose whose horizontal motion effects the transverse vibration of the beam. The Euler-Bernoulli theory is used to model the vibration of the beam and treat the hose thruster assembly as a lumped or point mass dashpot spring system at the tip. Measurements of linear acceleration at the tip are used to estimate the hose parameters (mass, stiffness, damping) and a Voigt-Kelvin viscoelastic structural damping parameter for the beam using a least squares fit to the data. Spline based approximations are considered to the hybrid (coupled ordinary and partial differential equations) systems; theoretical convergence results and numerical studies with both simulation and actual experimental data obtained from the structure are presented and discussed.

Banks, H. T.↗
DOE OSTI2023

Analysis of Neural Networks as Random Dynamical Systems

In this report we present our findings and outcomes of the NNRDS (analysis of Neural Networks as Random Dynamical Systems) project. The work is largely motivated by the analogy of a large class of neural networks (NNs) with a discretized ordinary differential equation (ODE) schemes. Namely, residual NNs, or ResNets, can be viewed as a discretization of neural ODEs (NODEs) where the NN depth plays the role of the time evolution. We employ several legacy tools from ODE theory, such as stiffness, nonlocality, autonomicity, to enable regularization of ResNets thus improving their generalization capabilities. Furthermore, armed with NN analysis tools borrowed from the ODE theory, we are able to efficiently augment NN predictions with uncertainty overcoming wellknown dimensionality challenges and adding a degree of trust towards NN predictions. Finally, we have developed a Python library QUiNN (Quantification of Uncertainties in Neural Networks) that incorporates improved-architecture ResNets, besides classical feed-forward NNs, and contains wrappers to PyTorch NN models enabling several major classes of uncertainty quantification methods for NNs. Besides synthetic problems, we demonstrate the methods on datasets from climate modeling and materials science.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Data-driven selection of stiff chemistry ODE solver in operator-splitting schemes

Most computational fluid dynamics simulations of practical combustion applications employ operator-splitting schemes, where chemistry and transport are separated and integrated with distinct numerical methods. The changes in composition due to chemistry are evaluated by solving ordinary differential equations (ODE) in each cell of the computational domain, which typically dominates the computational cost when detailed chemistry is considered. In this work, a data-driven approach for the selection of chemistry ODE solvers in operator-splitting schemes is presented. Neural networks are used to predict the ODE solvers CPU times and errors for a given thermochemical state. This allows the selection of an optimal ODE solver on a cell-by-cell, timestep-by-timestep basis. The models are trained using a wide set of thermochemical states generated through partially-stirred reactors and flames simulations. The methodology is validated by quantifying the prediction errors, the classification accuracy, and the computational speedup. The model predicts the optimal ODE solver for 70 to 95% of the validation cases and decreases the computional cost by a factor of 3 or more. The generalizability of the methodology to different chemical mechanisms and different fuels is assessed and it is shown that the model’s performance is only slightly degraded and its applicability is significantly enhanced if the inputs to the neural networks are restricted to a small set of thermochemical state variables present in most chemical mechanisms. In conclusion, the models are used in an homogeneous reactor case and a multi-dimensional CFD simulation of a diesel spray at high pressure where a speedup of more than 3 is achieved.

42 ENGINEERING↗
NASA NTRS1984

Response of large space structures with stiffness control

For large space structures, such as the 100-meter-diameter wrap-rib deployable antenna and spinning solar sail, whose out-of-plane stiffness is derived from in-plane tension, the out-of-plane motion can be actively controlled by time-varying in-plane tension. An elastic string is used to demonstrate the proposed approach, which results in a nonlinear ordinary differential equation. An approximation method is outlined from which the magnitude of time-varying tension can be determined based on the efficiency factor, the time factor or the optimal factor.

Chen, J.-C.↗