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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Nyström type exponential integrators for strongly magnetized charged particle dynamics

Solving for charged particle motion in electromagnetic fields (i.e. the particle pushing problem) is a computationally intensive component of particle-in-cell (PIC) methods for plasma physics simulations. This task is especially challenging when the plasma is strongly magnetized due numerical stiffness arising from the wide range of time scales between highly oscillatory gyromotion and long term macroscopic behavior. A promising approach to solve these problems is by a class of methods known as exponential integrators that can solve linear problems exactly and are A-stable. This work extends the standard exponential integration framework to derive Nyström-type exponential integrators that integrates the Newtonian equations of motion as a second-order differential equation directly. In particular, we derive second-order and third-order Nyström-type exponential integrators for strongly magnetized particle pushing problems. Numerical experiments show that the Nyström-type exponential integrators exhibit significant improvement in computation speed over the standard exponential integrators.

general physics↗
NASA NTRS1991

Exponential integration algorithms applied to viscoplasticity

Four, linear, exponential, integration algorithms (two implicit, one explicit, and one predictor/corrector) are applied to a viscoplastic model to assess their capabilities. Viscoplasticity comprises a system of coupled, nonlinear, stiff, first order, ordinary differential equations which are a challenge to integrate by any means. Two of the algorithms (the predictor/corrector and one of the implicits) give outstanding results, even for very large time steps.

Freed, Alan D.↗
NASA NTRS1991

Mutual impedance of parallel and perpendicular coplanar surface monopoles

One dimensional integral formulas are derived for mutual impedance of arbitrary size, coplanar, parallel, and perpendicular surface monopoles. The integrals in formulas are expressed as exponential integrals where possible. The mutual impedance expression for parallel monopoles is a summation of exponential integrals and one-dimensional integrals. For perpendicular monopoles, the mutual impedance is in closed form, containing exponential integrals only. The final expressions are in a form suitable for numerical computation. Since the expressions contain at most one-dimensional integrals, they can be utilized to reduce the matrix filling time in the moment method formulations, especially when inhomogeneous sectioning is preferred. Additionally, they can be used in rectangular surface patch modeling of conducting surfaces with edges which are at an angle to the surface patches, providing the angle is small. To this end, the expressions were utilized in the moment method analysis of linearly tapered slot antennas. Very good accuracy was obtained with a reduction in computer time.

Koksal, Adnan↗
NASA NTRS1977

A new algorithm for the integration of exponential and logarithmic functions

An algorithm for symbolic integration of functions built up from the rational functions by repeatedly applying either the exponential or logarithm functions is discussed. This algorithm does not require polynomial factorization nor partial fraction decomposition and requires solutions of linear systems with only a small number of unknowns. It is proven that if this algorithm is applied to rational functions over the integers, a computing time bound for the algorithm can be obtained which is a polynomial in a bound on the integer length of the coefficients, and in the degrees of the numerator and denominator of the rational function involved.

Rothstein, M.↗
NASA NTRS1994

Efficient and Accurate Explicit Integration Algorithms with Application to Viscoplastic Models

Several explicit integration algorithms with self-adative time integration strategies are developed and investigated for efficiency and accuracy. These algorithms involve the Runge-Kutta second order, the lower Runge-Kutta method of orders one and two, and the exponential integration method. The algorithms are applied to viscoplastic models put forth by Freed and Verrilli and Bodner and Partom for thermal/mechanical loadings (including tensile, relaxation, and cyclic loadings). The large amount of computations performed showed that, for comparable accuracy, the efficiency of an integration algorithm depends significantly on the type of application (loading). However, in general, for the aforementioned loadings and viscoplastic models, the exponential integration algorithm with the proposed self-adaptive time integration strategy worked more (or comparably) efficiently and accurately than the other integration algorithms. Using this strategy for integrating viscoplastic models may lead to considerable savings in computer time (better efficiency) without adversely affecting the accuracy of the results. This conclusion should encourage the utilization of viscoplastic models in the stress analysis and design of structural components.

Arya, Vinod K.↗
NASA NTRS1991

Analytical solution for boundary heat fluxes from a radiating rectangular medium

Reference is made to the work of Shah (1979) which demonstrated the possibility of partially integrating the radiative equations analytically to obtain an 'exact' solution. Shah's solution was given as a double integration of the modified Bessel function of order zero. Here, it is shown that the 'exact' solution for a rectangular region radiating to cold black walls can be conveniently derived, and expressed in simple form, by using an integral function, Sn, analogous to the exponential integral function appearing in plane-layer solutions.

Siegel, R.↗
NASA NTRS2017

New Mathematical Functions for Vacuum System Analysis

A new bivariate function has been found that provides solutions of integrals having the form u (sup minus eta) e (sup u) du which arise when developing predictions for the behavior of pressure within a rigid volume under high vacuum conditions in the presence of venting as well as sources characterized by power law transient decay over the range [0,1] for eta and for u greater than or equal to 0. A few properties of the new function are explored in this work. For instance the eta equals 1/2 case reproduces the Dawson function. In addition, a slight variation of the solution technique reproduces the exponential integral for eta equals 1. The technique used to generate these functions leads to an approach for solving a more general class of nonlinear ordinary differential equations, with the potential for identifying other new functions that solve other integrals.

Dawson Function↗
NASA NTRS1990

An efficient numerical integral in three-dimensional electromagnetic field computations

An improved algorithm for efficiently computing a sinusoid and an exponential integral commonly encountered in method-of-moments solutions is presented. The new algorithm has been tested for accuracy and computer execution time against both numerical integration and other existing numerical algorithms, and has outperformed them. Typical execution time comparisons on several computers are given.

Whetten, Frank L.↗
NASA NTRS1986

Analytic evaluation of two-center molecular integrals

By using the Fourier-transform technique, the explicit expressions for the one-electron - two-center overlap integrals of Slater-type atomic orbitals up to 3d are derived. The final expressions are analytic, simple, and independent of local coordinates. Furthermore, they do not contain the nonclosed-form of exponential integrals which were presented in expressions given in earlier work. It is shown that the two-electron - two-center Coulomb integrals, as well as the hybrid integrals, can simply be expressed in terms of these integrals. The numerical instability arising from the situation in which the exponents of the two orbitals are almost equal is discussed, and a solution for this problem based on a Taylor-series expansion of the integral is suggested.

Tai, H.↗
DOE OSTI2025

Classical eikonal from Magnus expansion

In a classical scattering problem, the classical eikonal is defined as the generator of the canonical transformation that maps in-states to out-states. It can be regarded as the classical limit of the log of the quantum S-matrix. In a classical analog of the Born approximation in quantum mechanics, the classical eikonal admits an expansion in oriented tree graphs, where oriented edges denote retarded/advanced worldline propagators. The Magnus expansion, which takes the log of a time-ordered exponential integral, offers an efficient method to compute the coefficients of the tree graphs to all orders. We exploit a Hopf algebra structure behind the Magnus expansion to develop a fast algorithm which can compute the tree coefficients up to the 12th order (over half a million trees) in less than an hour. In a relativistic setting, our methods can be applied to the post-Minkowskian (PM) expansion for gravitational binaries in the worldline formalism. We demonstrate the methods by computing the 3PM eikonal and find agreement with previous results based on amplitude methods. Importantly, the Magnus expansion yields a finite eikonal, while the naïve eikonal based on the time-symmetric propagator is infrared-divergent from 3PM on.

Black Holes↗
NASA NTRS1974

Mutual impedance of nonplanar-skew sinusoidal dipoles

The mutual impedance of nonplanar-skew sinusoidal dipoles is presented as a summation of several exponential integrals with complex arguments. Mathematical models are developed to show the near-zone field of the sinusoidal dipole. The mutual impedance of coupled dipoles is expressed as the sum of four monopole-mobopole impedances to simplify the analysis procedure. The subroutines for solving the parameters of the dipoles are discussed.

Richmond, J. H.↗
NASA NTRS1981

The C IV 1550 profile in type 1 Seyfert galaxies

The paper presents C IV 1550 A line profiles for the type 1 Seyfert galaxies NGC 5548, Mrk 509, NGC 7469, and MCG-2-58-22. Several line broadening mechanisms and theoretical line profiles are considered, and random motion of discrete clouds is ruled out. A spherical ensemble of discrete clouds with steady outflow or inflow produces a logarithmic profile, but does not account for the highly extended wings. A spherical ensemble with ballistic outflow produces a profile of the first exponential integral function, and fits the observed profile to the continuum level. Although C IV profiles favor the ballistic model, both Mrk 509 and NGC 7469 have significant asymmetry, and Balmer lines with a higher optical depth show higher asymmetry and redshift than all four galaxies.

Wu, C.-C.↗