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Results for “DIFFERENTIAL ALGEBRA”

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 OSTI2025

SDA: a symbolic differential algebra package in C++

Truncated Power Series Algebra (TPSA), or Differential Algebra (DA), is a well-established tool in accelerator physics, commonly used for generating high-order maps of dynamic systems, as well as in symplectic tracking, normal form analysis, verified integration, optimization, and fast multipole methods. This package is the first to perform symbolic DA computations, enabling traceability of initial condition contributions and runtime reduction for repeated DA calculations, potentially expanding DA’s applications.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Rythmos: Solution and Analysis Package for Differential-Algebraic and Ordinary-Differential Equations

Time integration is a central component for most transient simulations. It coordinates many of the major parts of a simulation together, e.g., a residual calculation with a transient solver, solution with the output, various operator-split physics, and forward and adjoint solutions for inversion. Even though there is this variety in these transient simulations, there is still a common set of algorithms and procedures to progress transient solutions for ordinary-differential equations (ODEs) and differential-alegbraic equations (DAEs). Rythmos is a collection of these algorithms that can be used for the solution of transient simulations. It provides common time-integration methods, such as Backward and Forward Euler, Explicit and Implicit Runge-Kutta, and Backward-Difference Formulas. It can also provide sensitivities, and adjoint components for transient simulations. Rythmos is a package within Trilinos, and requires some other packages (e.g., Teuchos and Thrya) to provide basic time-integration capabilities. It also can be coupled with several other Trilinos packages to provide additional capabilities (e.g., AztecOO and Belos for linear solutions, and NOX for non-linear solutions). The documentation is broken down into three parts: Theory Manual, User's Manual, and Developer's Guide. The Theory Manual contains the basic theory of the time integrators, the nomenclature and mathematical structure utilized within Rythmos, and verification results demonstrating that the designed order of accuracy is achieved. The User's Manual provides information on how to use the Rythmos, description of input parameters through Teuchos Parameter Lists, and description of convergence test examples. The Developer's Guide is a high-level discussion of the design and structure of Rythmos to provide information to developers for the continued development of capabilities. Details of individual components can be found in the Doxygen webpages.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Regularization and index reduction for linear differential–algebraic systems

In this paper, a necessary and sufficient condition for the existence of a proportional semistate feedback is established such that the closed loop system is regular and of index at most two. The condition is characterized by a rank condition that involves only the original system coefficient matrices. Also, a new rank condition ensuring that a given linear time-invariant descriptor system is regular and of index at most some specific value is also derived. The developed theory is illustrated through physical and numerical examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

On a Reduction of the Order in a Differential-Algebraic System

The conditions of solvability and the structure of a generalized Green operator of the Cauchy problem for a linear differential-algebraic system are found. The sufficient conditions of reducibility of a differential-algebraic equation to a sequence of systems joining differential and algebraic equations are constructed. An original classification and a single scheme of construction of the solutions of differentialalgebraic equations are proposed.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

scikit-SUNDAE ((SUN)DIALS Differential Algebraic Equations) [SWR-24-137]

Scikit-SUNDAE provides Python bindings to SUNDIALS integrators. The implicit differential algebraic (IDA) solver and C-based variable-coefficient ordinary differential equations (CVODE) solver are both included. The name SUNDAE combines (SUN)DIALS and DAE, which stands for differential algebraic equations. Solvers specific to DAE problems are not frequently available in Python. An ordinary differential equation (ODE) solver is also included for completeness. ODEs can be categorized as a subset of DAEs (i.e., DAEs with no algebraic constraints). https://pypi.org/project/scikit-sundae

Randall, Corey↗
DOE OSTI2020

Battery models, systems, and methods using robust fail-safe iteration free approach for solving differential algebraic equations

Battery models using robust fail-safe iteration free approach for solving Differential Algebraic Equations, and associated systems and methods are disclosed. In one embodiment, a method includes generating a model of the rechargeable battery; determining one or more initial conditions for one or more algebraic variables of the model using a solver; holding differential variables of the model static by a switch function while determining the one or more initial conditions; applying the initial conditions to the model by the switch function; and determining one or more parameters for the rechargeable battery by solving the algebraic and differential equations.

25 ENERGY STORAGE↗
DOE OSTI2021

Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations

In this paper we present a non-intrusive data-driven approach for model reduction of nonlinear systems. The approach considers the particular case of nonlinear partial differential equations (PDEs) that form systems of partial differential–algebraic equations (PDAEs) when lifted to polynomial form. Such systems arise, for example, when the governing equations include Arrhenius reaction terms (e.g., in reacting flow models) and thermodynamic terms (e.g., the Helmholtz free energy terms in a phase-field solidification model). Using the known structured form of the lifted algebraic equations, the approach computes the reduced operators for the algebraic equations explicitly, using straightforward linear algebra operations on the basis matrices. The reduced operators for the differential equations are inferred from lifted snapshot data using operator inference, which solves a linear least squares regression problem. The approach is illustrated for the nonlinear model of solidification of a pure material. The lifting transformations reformulate the solidification PDEs as a system of PDAEs that have cubic structure. The operators of the lifted system for this solidification example have affine dependence on key process parameters, permitting us to learn a parametric reduced model with operator inference. Numerical experiments show the effectiveness of the resulting reduced models in capturing key aspects of the solidification dynamics.

42 ENGINEERING↗
DOE OSTI2020

Optimal sensor network design for multi-scale, time-varying differential algebraic equation systems: Application to an entrained-flow gasifier refractory brick

An algorithm for optimal sensor network design for multi-scale, time-varying differential algebraic equation systems with non-separable dynamics is presented. As the process is time-varying, an integral normalized posterior error covariance of a multi-scale filter is minimized to obtain the optimal sensor locations. For reducing the computational cost, an adaptive sampling rate approach is considered for the slowly-varying variables. The algorithm is applied to a smart refractory brick with embedded sensors as part of an entrained-flow gasifier. Thermistors and interdigital capacitors are considered as candidate measurement technologies for estimating temperature and slag penetration profile along the gasifier wall. Finally, when the optimal set of sensors obtained from the algorithm is used for estimating temperature and slag penetration profiles in a multi-scale Kalman filter framework, satisfactory estimates are obtained despite high measurement noise and model mismatch.

42 ENGINEERING↗
DOE OSTI2024

Solving differential‐algebraic equations in power system dynamic analysis with quantum computing

Abstract Power system dynamics are generally modeled by high dimensional non‐linear differential‐algebraic equations (DAEs) given a large number of components forming the network. These DAEs' complexity can grow exponentially due to the increasing penetration of distributed energy resources, whereas their computation time becomes sensitive due to the increasing interconnection of the power grid with other energy systems. This paper demonstrates the use of quantum computing algorithms to solve DAEs for power system dynamic analysis. We leverage a symbolic programming framework to equivalently convert the power system's DAEs into ordinary differential equations (ODEs) using index reduction methods and then encode their data into qubits using amplitude encoding. The system non‐linearity is captured by Hamiltonian simulation with truncated Taylor expansion so that state variables can be updated by a quantum linear equation solver. Our results show that quantum computing can solve the power system's DAEs accurately with a computational complexity polynomial in the logarithm of the system dimension. We also illustrate the use of recent advanced tools in scientific machine learning for implementing complex computing concepts, that is, Taylor expansion, DAEs/ODEs transformation, and quantum computing solver with abstract representation for power engineering applications.

computational complexity↗
DOE OSTI2018

Generalized Jacobi–Galerkin method for nonlinear fractional differential algebraic equations

In this paper, we provide an approximate approach based on the Galerkin method to solve a class of nonlinear fractional differential algebraic equations. The fractional derivative operator in the Caputo sense is utilized and the generalized Jacobi functions are employed as trial functions. The existence and uniqueness theorem as well as the asymptotic behavior of the exact solution are provided. It is shown that some derivatives of the solutions typically have singularity at origin dependence on the order of the fractional derivative. The influence of the perturbed data on the exact solutions along with the convergence analysis of the proposed scheme is also established. Some illustrative examples provided to demonstrate that this novel scheme is computationally efficient and accurate.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

The Differential Algebra Based Multiple Level Fast Multipole Algorithm for 3D Space Charge Field Calculation and Photoemission Simulation

Coulomb interaction between charged particles inside a bunch is one of the most importance collective effects in beam dynamics, becoming even more significant as the energy of the particle beam is lowered to accommodate analytical and low-Z material imaging purposes such as in the time resolved Ultrafast Electron Microscope (UEM) development currently underway at Michigan State University. In addition, space charge effects are the key limiting factor in the development of ultrafast atomic resolution electron imaging and diffraction technologies and are also correlated with an irreversible growth in rms beam emittance due to fluctuating components of the nonlinear electron dynamics. In the short pulse regime used in the UEM, space charge effects also lead to virtual cathode formation in which the negative charge of the electrons emitted at earlier times, combined with the attractive surface field, hinders further emission of particles and causes a degradation of the pulse properties. Space charge and virtual cathode effects and their remediation are core issues for the development of the next generation of high-brightness UEMs. Since the analytical models are only applicable for special cases, numerical simulations, in addition to experiments, are usually necessary to accurately understand the space charge effect. In this paper we will introduce a grid-free differential algebra based multiple level fast multipole algorithm, which calculates the 3D space charge field for n charged particles in arbitrary distribution with an efficiency of O(n), and the implementation of the algorithm to a simulation code for space charge dominated photoemission processes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Evaluation of the Index and Singular Points of Linear Differential-Algebraic Equations of Higher Order

We study linear systems of ordinary differential equations of higher order with an identically singular matrix at the higher derivative of the required vector-valued function in the domain of definition. We give definitions of the index and singular points for these systems, formulate the conditions of solvability, and deduce a formula for the general solution. The algorithms for finding the index and singular points are presented.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Bounded motion design in the Earth zonal problem using differential algebra based normal form methods

Establishing long-term relative bounded motion between orbits in perturbed dynamics is a key challenge in astrodynamics to enable cluster flight with minimum propellant expenditure. Herein, we present an approach that allows for the design of long-term relative bounded motion considering a zonal gravitational model. Entire sets of orbits are obtained via high-order Taylor expansions of Poincarè return maps about reference fixed points. The high-order normal form algorithm is used to determine a change in expansion variables of the map into normal form space, in which the phase space behavior is circular and can be easily parameterized by action–angle coordinates. The action–angle representation of the normal form coordinates is then used to parameterize the original Poincarè return map and average it over a full phase space revolution by a path integral along the angle parameterization. As a result, the averaged nodal period and drift in the ascending node are obtained, for which the bounded motion conditions are straightforwardly imposed. Sets of highly accurate bounded orbits are obtained, extending over several thousand kilometers, and valid for decades.

79 ASTRONOMY AND ASTROPHYSICS↗