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Results for “Functional expansion”

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

Working with Bezier Curves as bases for Functional Expansion Tallies

Functional expansion tallies (FETs) are a powerful tool for getting more information per history from Monte Carlo simulations, but in the past they have been constrained to orthogonal bases. Bezier curves, from computer aided design (CAD), could be well suited for FET due to their ability to assume many arbitrary shapes, but are non-orthogonal. Recent development has made non-orthongal FET possible. The convergence of B ´ezier curve FETs in both polynomial order and number of samples is explored. These bases are well suited for representing normal distributions, and opens the door to possible other CAD derived FET bases.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Working with Bézier Curves as bases for Functional Expansion Tallies

Functional expansion tallies (FETs) are powerful tools for getting more information per history from Monte Carlo simulations, but in the past they have been constrained to orthogonal bases. Bézier curves are used widely in computer aided design (CAD) geometry kernels and could be well-suited for FETs due to their ability to assume many arbitrary shapes, but they use nonorthogonal bases. Recent developments in 2021 have made nonorthogonal FETs possible. The convergence of Bézier curve FETs in both polynomial order and number of samples is explored in this work. It is shown that these bases are well-suited for representing normal distributions and this opens the door to the possibility of other CAD-derived FET bases.

97 - MATHEMATICS AND COMPUTING↗
DOE OSTI2022

3D spherical functional expansion tallies in Serpent 2 Monte Carlo code

This work extends the application of functional expansion tallies to 3D spherical geometries. The 3D Zernike polynomials are set as an orthonormal polynomials basis for the functional reconstruction. The study describes the construction of the complete set of polynomials, a natural expansion of the spherical harmonics polynomials where 3D Zernike moments can be evaluated as a linear combination of the geometrical moments. The 3D Zernike polynomials formulation and the computational approach implemented in Serpent 2 are presented and tested through the Godiva model from the ICSBEP criticality benchmark test cases. The implementation results are in agreement with a reference solution described in a fine-resolution mesh, enhancing also the performance and memory demand. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2025

Functional Expansion Tallies Using MCNP6 PTRAC Files [Slides]

Accuracy is vital for many radiation transport applications. Monte Carlo (MC) codes are the most accurate, but they are expensive for fine meshes. Functional Expansion Tallies (FETs) yield smoother, lower-variance results with far fewer degrees of freedom.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1984

Discrete orthogonal function expansions for non-uniform grids using the fast Fourier transform

A technique for applying discrete Fourier series to infinite domains is presented. The technique uses mappings designed to minimize truncation error and can be applied to solve mixed initial boundary value problems among others. The method is alias-free and yields consistent differentiation and integration operators. The mapping-induced truncation error is explicitly expressible and small in nearly all cases of interest. The method is illustrated for three problems involving convection, diffusion, and vortex interaction.

Cain, A. B.↗
NASA NTRS1992

Functional expansion representations of artificial neural networks

In the past few years, significant interest has developed in using artificial neural networks to model and control nonlinear dynamical systems. While there exists many proposed schemes for accomplishing this and a wealth of supporting empirical results, most approaches to date tend to be ad hoc in nature and rely mainly on heuristic justifications. The purpose of this project was to further develop some analytical tools for representing nonlinear discrete-time input-output systems, which when applied to neural networks would give insight on architecture selection, pruning strategies, and learning algorithms. A long term goal is to determine in what sense, if any, a neural network can be used as a universal approximator for nonliner input-output maps with memory (i.e., realized by a dynamical system). This property is well known for the case of static or memoryless input-output maps. The general architecture under consideration in this project was a single-input, single-output recurrent feedforward network.

Gray, W. Steven↗
DOE OSTI2024

A Characteristics Approach to the Finite Element Method

Herein, we present a new method for solving the linear Boltzmann transport equation. Two commonly used and well-understood methods for solving partial differential equations are the method of characteristics (MOC) and the finite element method (FEM). We propose a new method that combines the fundamental concept of the FEM with the analytic solution from the MOC to obtain coefficients for the FEM basis function expansion. Traditionally, coefficients for the FEM basis function expansion are obtained via matrix inversion. Instead, we solve for the coefficients with the MOC and represent the underlying fields with the basis function expansion using these coefficients. We provide a convergence study for our method with results from two sets of FEM basis functions: Gauss-Legendre and Gauss-Lobatto sets. We also compare two different variations of our method categorized as short characteristics and intermediate characteristics.

42 ENGINEERING↗
NASA NTRS1977

A note of expansions of functions of velocity in the two-body problem

Fourier expansions based on both the true anomaly and the mean anomaly are obtained for the functions of velocity in the two-body problem; the series of coefficients is written from classical formulae involving associated Legendre polynomials, Gegenbauer polynomials, or Bessel functions. The Fourier expansions are compared with the expansions in powers of eccentricity developed by Broucke (1974) through use of computerized Poisson series manipulation.

Kinoshita, H.↗
DOE OSTI2023

The Generalized Green’s function Cluster Expansion: A Python package for simulating polarons

We present an efficient implementation of the Generalized Green’s function Cluster Expansion (GGCE), which is a new method for computing the ground-state properties and dynamics of polarons (single electrons coupled to lattice vibrations) in model electron-phonon systems. The GGCE works at arbitrary temperature and is well suited for a variety of electron-phonon couplings, including, but not limited to, site and bond Holstein and Peierls (Su-SchriefferHeeger) couplings, and couplings to multiple phonon modes with different energy scales and coupling strengths. Quick calculations can be performed efficiently on a laptop using solvers from NumPy and SciPy, or in parallel at scale using the PETSc sparse linear solver engine.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Numerically exact generalized Green's function cluster expansions for electron-phonon problems

We generalize the family of approximate momentum average methods to formulate a numerically exact, convergent hierarchy of equations whose solution provides an efficient algorithm to compute the Green's function of a particle dressed by bosons suitable in the entire parameter regime. We use this approach to extract ground-state properties and spectral functions. Our approximation-free framework, dubbed the generalized Green's function cluster expansion (GGCE), allows access to exact numerical results in the extreme adiabatic limit, where many standard methods struggle or completely fail. We showcase the performance of the method, specializing three important models of charge-boson coupling in solids and molecular complexes: the molecular Holstein model, which describes coupling between charge density and local distortions, the Peierls model, which describes modulation of charge hopping due to intersite distortions, and a more complex Holstein + Peierls system with couplings to two different phonon modes, paradigmatic of charge-lattice interactions in organic crystals. Furthermore, the GGCE serves as an efficient approach that can be systematically extended to different physical scenarios, thus providing a tool to model the frequency dependence of dressed particles in realistic settings.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS2000

Recent Trends in the Analysis of Quasioptical Systems

The recent trend in microwave instruments is the use of multiple millimeter and submillimeter wavelength bands. These systems are typically analyzed by using physical optics, Gaussian beams or ray tracing techniques. Physical optics offers high accuracy at the expense of computation time. This trade-off becomes particularly apparent in the analysis of multiple reflector antennas, such as beam waveguide antennas, where physical optics is used to compute the current on each reflector from the current on the previous reflector. At the other end of the spectrum is ray tracing approaches that ignore diffraction effects entirely. These methods are fast but sacrifice the ability to predict some effects accurately. An intermediate approach is to use an appropriate set of expansion functions to model the field between the reflectors. If the set is chosen wisely only a few coefficients need to be determined from each reflector current. The field is then computed at the next reflector through the use of the expansion functions and their coefficients rather than by using the previous reflector current. For a beam waveguide system with no enclosing tubes an excellent set of expansion functions is the Gaussian beam mode set. In many cases a preliminary design which includes the effects on diffraction may be obtained by considering only the fundamental mode and a thin lens model for the reflectors. Higher-order modes are included to model the effects of the curved reflector, which include asymmetric distortion of the beam, cross polarization, and beam truncation. This paper describes a computer code implementing higher-order Gaussian beam scattering by multiple reflector systems. There are four essential steps in the algorithm. (1) Compute the current on the first reflector using physical optics using either a feed model or by an incident set of Gaussian beam modes. (2) Find the direction of propagation for the reflected Gaussian beam-set using ray tracing. (3) Determine the waist size and location for the output beam set by examining the amplitude and phase distribution of the current on the reflector. (4) Compute the amplitudes of the individual modes in the output mode set. These steps are then repeated for each addition reflector in the chain. In each of these cases the previous Gaussian beam set provides the input field for the current calculation. Details of the four steps discussed above will be discussed. Examples will compare results from the Gaussian beam approach to pure physical optics, illustrating both its merits and limitations. Hybrid approaches capable of eliminating some of the limitations will also be discussed.

Hoppe, D. J.↗
DOE OSTI2024

Inverse aspect-ratio expanded tokamak equilibria

Following Greene et al. [Phys. Fluids 14, 671 (1971)] and Connor et al. [Phys. Plasmas 31, 577 (1988); Plasma Phys. Control. Fusion 34, 161 (1992); and Nucl. Fusion 33, 1533 (1993)], the Grad-Shafranov equation for an axisymmetric tokamak plasma equilibrium is solved via an expansion in the, supposedly small, inverse aspect-ratio of the plasma, ϵ. The displacements of equilibrium magnetic flux-surfaces due to plasma shaping are assumed to be $\mathcal{O}$(ϵ) smaller than the minor radii of the surfaces, but no other restriction is placed on the nature of the shaping. The solution of the Grad-Shafranov equation is matched to a vacuum solution that extends to infinity, and consists of an expansion in toroidal functions. The external poloidal magnetic field generated by a finite set of discrete external poloidal magnetic field-coils is calculated, and incorporated into the toroidal function expansion. In this manner, the shape of a large aspect-ratio tokamak plasma is directly related to the currents flowing in the external poloidal field-coils. Finally, a pedestal in the plasma pressure, and the associated spike in the bootstrap current, are incorporated into the model.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗