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At least 37 records · Page 2

Asymptotically Compatible Reproducing Kernel Collocation and Meshfree Integration for Nonlocal Diffusion

Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The numerical scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for a linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme, which is stable. In addition, assembling the stiffness matrix of the nonlocal problem requires costly computational resources because high-order Gaussian quadrature is necessary to evaluate the integral. We thus provide a remedy to the problem by introducing a quasi-discrete nonlocal diffusion operator for which no numerical quadrature is further needed after applying the RK collocation scheme. The quasi-discrete nonlocal diffusion operator combined with RK collocation is shown to be convergent to the correct local diffusion problem by taking the limits of nonlocal interaction and spatial resolution simultaneously. The theoretical results are then validated with numerical experiments. We additionally illustrate a connection between the proposed technique and an existing optimization based approach based on generalized moving least squares.

97 MATHEMATICS AND COMPUTING↗

Assessment of the Griffin Reactor Multiphysics Application Using the Empire Micro Reactor Design Concept

In late 2019, INL and ANL agreed to jointly develop the reactor physics code named Griffin based on the integration of the two code suites, MAMMOTH/Rattlesnake (INL) and MC2 - 3/PROTEUS (ANL). Griffin is being developed based on the MOOSE framework and MOOSE quality assurance procedures. This decision was made to be able to allow DOE-NE to efficiently invest funding to this area and to provide effective and timely support for existing and potential users; the latter includes industry and government organizations who are developing various types of advanced reactors in the near and long term. Since MAMMOTH/Rattlesnake has been developed based on the MOOSE framework, the INL/ANL Griffin development team agreed to build Griffin beginning with a merger of MAMMOTH and Rattlesnake into a single code and moving forward by implementing capabilities from the PROTEUS suite into Griffin. Moving forward, both ANL and INL efforts are equally invested in the Griffin project, with management support, to provide an advanced reactor multiphysics tool to assist in reactor design, optimization, and safety analysis. Much work remains in moving Griffin forward to migrate PROTEUS capabilities and to optimize performance to meet user needs. The main objective of this work is to assess the current status of Griffin capabilities in terms of performance and accuracy, to determine priorities for PROTEUS migration, and to identify capabilities and features to improve for supporting the code integration effort. For this assessment, the Empire micro reactor problem that was developed in the ARPA-E MEITNER program was selected as an advance reactor concept of interest to the technical community. The Empire reactor problem was expanded from its original incomplete specification to be a small heat-pipe-cooled micro reactor core with ~113 cm radius and 70 cm in height, composed of 18 fuel assemblies, 12 control drums, and beryllium radial and axial reflectors. In the current model, using 5 cm axial reflectors specified in the original Empire assembly model, more than 10% of neutrons leak axially and through the empty center safety hole, as well as through heat pipe channels in fuel assembly elements that extend through the top reflector region. Several calculation models of the core were defined for systematic assessment, including 2-D and 3-D fuel assemblies and whole cores with cylindrical boundaries. Cross sections were generated using Serpent 2, and meshes were produced using the Argonne mesh tool or the INL neutronics meshing tools combined with CUBIT. Cross sections and meshes were converted to the ISOXML and Exodus formats, respectively, so that Griffin and PROTEUS could use consistent data for solving the reactor problems. With the prepared cross sections and meshes, PROTEUS was run first to ensure that all input data were correctly generated and input options in terms of angle, mesh, and energy group were accurately determined. Comparisons against Serpent 2 solutions were made in terms of eigenvalue and pin power. The same calculations and comparisons were then conducted using Griffin. For the fuel assembly and whole core problems, the PROTEUS eigenvalues agreed well with reference Serpent 2 solutions within 100 and 30 pcm, respectively, and pin power differences relative to Serpent 2 were overall less than 2.2% and RMS 0.8% for the whole core models. This indicated that all input data were properly prepared. Using the same data, Griffin was run selecting the SAAF-CFEM SN solver with Legendre-Gaussian quadrature and NDA and DSA for acceleration. It was found that the SAAF-CFEM solver of Griffin required finer meshes to achieve eigenvalue and pin power solutions in good agreement with Serpent 2, consequently requiring more memory requirement and longer computation time. On the other hand, the SPH-Diffusion 2-D core calculations performed using Griffin were able to recover the exact eigenvalue from the reference Serpent 2 solutions, resulting in a pin-power distribution with an RMS of 0.6% and maximum absolute difference of less than 1.4%. The runtimes for SPH-Diffusion for the 2-D core were less than 3 minutes on 40 cores. During this evolution of this evaluation, many updates were made in Griffin by the Griffin development team of INL (focusing on software updates) and ANL (reviewing and supporting software updates) to complete this assessment. Observations from the code assessment are presented in the conclusion section of this report, followed by a discussion of recommendations for future work.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

Generalized Airy functions for use in one-dimensional quantum mechanical problems

The solution of the one dimensional, time independent, Schroedinger equation in which the energy minus the potential varies as the nth power of the distance is obtained from proper linear combinations of Bessel functions. The linear combinations called generalized Airy functions, reduce to the usual Airy functions Ai(x) and Bi(x) when n equals 1 and have the same type of simple asymptotic behavior. Expressions for the generalized Airy functions which can be evaluated by the method of generalized Gaussian quadrature are obtained.

Eaves, J. O.↗

Unsteady, combined radiation and conduction in an absorbing, scattering, and emitting medium.

The transient cooldown of a gray, absorbing, isotropic scattering, emitting, and conducting medium bounded by gray, diffusely emitting and reflecting parallel plates is considered. Numerical solutions are obtained for the initial boundary-value problem with a discontinuous decrease in temperature at one boundary. The quasi-steady equation of radiative transfer is solved using Gaussian quadrature and a matrix eigenvector technique together with explicit numerical solution of the unsteady energy equation. Temperature and energy flux distributions are presented for variations of optical thickness, boundary emissivity, albedo, and conduction-radiation parameter.

Weston, K. C.↗

Note on the modified two-stream approximation of Sagan and Pollack.

The modified two-stream approximation is derived from the radiative transfer equation using a two-point gaussian quadrature and neglecting terms of order l equal to or greater than 3 in the Legendre expansion of the phase function. The results are compared with the Eddington approximation and with exact results for the special cases of perfect absorption, perfect scattering, and a semi-infinite layer with isotropic scattering.

Lyzenga, D. R.↗

Averaged initial Cartesian coordinates for long lifetime satellite studies

A set of initial Cartesian coordinates, which are free of ambiguities and resonance singularities, is developed to study satellite mission requirements and dispersions over long lifetimes. The method outlined herein possesses two distinct advantages over most other averaging procedures. First, the averaging is carried out numerically using Gaussian quadratures, thus avoiding tedious expansions and the resulting resonances for critical inclinations, etc. Secondly, by using the initial rectangular Cartesian coordinates, conventional, existing acceleration perturbation routines can be absorbed into the program without further modifications, thus making the method easily adaptable to the addition of new perturbation effects. The averaged nonlinear differential equations are integrated by means of a Runge Kutta method. A typical step size of several orbits permits rapid integration of long lifetime orbits in a short computing time.

Pines, S.↗

Sensitivity of inelastic response to numerical integration of strain energy

The exact solution to the quasi-static, inelastic response of a cantilever beam of rectangular cross section subjected to a bending moment at the tip is obtained. The material of the beam is assumed to be linearly elastic-linearly strain-hardening. This solution is then compared with three different numerical solutions of the same problem obtained by minimizing the total potential energy using Gaussian quadratures of two different orders and a Newton-Cotes scheme for integrating the strain energy of deformation. Significant differences between the exact dissipative strain energy and its numerical counterpart are emphasized. The consequence of this on the nonlinear transient responses of a beam with solid cross section and that of a thin-walled beam on elastic supports under impulsive loads are examined.

Kamat, M. P.↗

Recent developments in the Green's function method

A recent computational development on the Green's function method (the method used in the computer program SOUSSA: Steady, Oscillatory and Unsteady Subsonic and Supersonic Aerodynamics) is presented. A scheme consisting of combined numerical (Gaussian quadrature) and analytical procedures for the evaluation of the source and doublet integrals used in the program is presented. This combination results in 80 to 90% reduction in computer time.

Tseng, K.↗

Molecular collisions in a laser field - Effect of the laser linewidth

The effect of the laser linewidth on molecular collision processes taking place in a laser field is studied, using an approximation scheme that replaces the actual frequency distribution of the field by a finite number of frequencies and weights. The choice of the frequencies and weights is conveniently accomplished by the method of Gaussian quadrature. Close-coupling calculations are performed on model systems, and the results indicate that the neglect of the laser linewidth may be justified in most collision processes in a laser field.

Lee, H. W.↗

Thermal microwave emission from a three-layer random medium with three-dimensional variations

The radiative transfer theory has been used to study the thermal microwave emission from a three-layer random medium. The random medium is characterized by correlation functions that are Gaussian in the horizontal direction and exponential in the vertical direction. Gaussian quadrature methods are used to solve for the brightness temperatures. The spectral and angular dependence of the brightness temperature are illustrated numerically. In particular we compare the case of a two-layer random medium whose brightness temperature decreases with frequency with the case of a three-layer random medium whose brightness temperature increases with frequency.

Tsang, L.↗

Radiative transfer theory for passive microwave remote sensing of a two-layer random medium with cylindrical structures

A model of the vegetation layer as a two-layer random medium with a small correlation length l sub rho in the horizontal direction, and a large correlation length l sub z in the vertical direction, is presented for fields with cylindrical structures. As l sub z approaches infinity, closed form solutions are derived for the brightness temperatures; the kernels in the scattering terms of the radiative transfer equations result in delta functions showing that forward scattering is dominant over all other directions. The results are compared with the Gaussian quadrature method data for numerical solution of the radiative transfer equations.

Chuang, S. L.↗

Radiative transfer theory for passive microwave remote sensing of random media

The radiative transfer theory is applied to a random medium for the case of passive remote sensing. A general numerical solution with Gaussian quadrature method is studied. For a medium with laminar structure or a medium with cylindrical structure, closed form solutions are obtained by solving the radiative transfer equations of a two-layer random medium. Numerical results for the general case and the two limiting cases are compared.

Chuang, S. L.↗

A mathematical characterization of vegetation effect on microwave remote sensing from the Earth

In passive microwave remote sensing of the earth, a theoretical model that utilizes the radiative transfer equations was developed to account for the volume scattering effects of the vegetation canopy. Vegetation canopies such as alfalfa, sorghum, and corn are simulated by a layer of ellipsoidal scatterers and cylindrical structures. The ellipsoidal scatterers represent the leaves of vegetation and are randomly positioned and oriented. The orientation of ellipsoids is characterized by a probability density function of Eulerian angles of rotation. The cylindrical structures represent the stalks of vegetation and their radii are assumed to be much smaller than their lengths. The underlying soil is represented by a half-space medium with a homogeneous permittivity and uniform temperature profile. The radiative transfer quations are solved by a numerical method using a Gaussian quadrature formula to compute both the vertical and horizontal polarized brightness temperature as a function of observation angle. The theory was applied to the interpretation of experimental data obtained from sorghum covered fields near College Station, Texas.

Choe, Y.↗

The aerodynamics of propellers and rotors using an acoustic formulation in the time domain

The aerodynamics of propellers and rotors is especially complicated because of the highly three-dimensional and compressible nature of the flow field. However, in linearized theory the problem is governed by the wave equation, and a numerically-efficient integral formulation can be derived. This reduces the problem from one in space to one over a surface. Many such formulations exist in the aeroacoustics literature, but these become singular integral equations if one naively tries to use them to predict surface pressures, i.e., for aerodynamics. The present paper illustrates how one must interpret these equations in order to obtain nonambiguous results. After the regularized form of the integral equation is derived, a method for solving it numerically is described. This preliminary computer code uses Legendre-Gaussian quadrature to solve the equation. Numerical results are compared to experimental results for ellipsoids, wings, and rotors, including effects due to lift. Compressibility and the farfield boundary conditions are satisfied automatically using this method.

Long, L. N.↗

Passive and active remote sensing of atmospheric precipitation

Both passive and active remote sensing of atmospheric precipitation are studied with the vector radiative transfer equations by making use of the Mie scattering phase functions and incorporating the raindrop-size distributions. For passive remote sensing, the Gaussian quadrature method is employed to solve for the brightness temperatures. For active remote sensing, an iterative approach carrying out to the second order in albedo is used to calculate for the bistatic coefficients, the backscattering cross sections/unit volume, and the interchannel cross talks. The calculated results are plotted as a function of rainfall rates and compared to various available experimental data. The theoretical model is easily applied to the remote sensing of aerosol particles, smoke, fog, and haze at infrared and visible frequencies.

Jin, Y. Q.↗

A method for computing the leading-edge suction in a higher-order panel method

Experimental data show that the phenomenon of a separation induced leading edge vortex is influenced by the wing thickness and the shape of the leading edge. Both thickness and leading edge shape (rounded rather than point) delay the formation of a vortex. Existing computer programs used to predict the effect of a leading edge vortex do not include a procedure for determining whether or not a vortex actually exists. Studies under NASA Contract NAS1-15678 have shown that the vortex development can be predicted by using the relationship between the leading edge suction coefficient and the parabolic nose drag. The linear theory FLEXSTAB was used to calculate the leading edge suction coefficient. This report describes the development of a method for calculating leading edge suction using the capabilities of the higher order panel methods (exact boundary conditions). For a two dimensional case, numerical methods were developed using the double strength and downwash distribution along the chord. A Gaussian quadrature formula that directly incorporates the logarithmic singularity in the downwash distribution, at all panel edges, was found to be the best method.

Ehlers, F. E.↗

A 4-node Bilinear Isoparametric Element in Rockwell NASTRAN

The development and evaluation of the Rockwell NASTRAN four node quadrilateral (QUAD4) element is presented. The element derivation utilizes bilinear isoparametric techniques both for membrane and bending characteristics. The QUAD4 element coordinate system, membrane properties, lumped mass matrix, and treatment of warping are based upon the COSMIC/NASTRAN QDMEMI element while the bending characteristics are based on a paper by T. J. R. Hughes. The effects of warping on the bending stiffness, consistent mass, and geometric stiffness are based upon a paper by R. H. MacNeal. Numerical integration is accomplished by Gaussian quadrature on a 2 x 2 grid. Practical user support features include variable element thickness, thermal analysis and layered composite material definitions.

Liao, C.↗