DEVELOPMENT AND TESTING OF THE MODIFIED SIMPSON’S RULE FOR DISCRETE ORDINATES TRANSPORT APPLICATIONS

A new angular quadrature type termed Modified Simpson Trapezoidal (MST) is developed based on the conventional Simpson’s 1/3 rule where the angular pattern over polar levels has a trapezoid shape. An adaptive coefficient correction scheme is developed to enable our new quadrature to integrate the angular flux over subintervals separated by the interior jump irregularities. A two-dimensional test problem is employed to verify the angular discretization error in the uncollided SN scalar flux computed with our new quadrature sets, as well as conventional angular quadrature types. Numerical results show that the MST quadrature error in the point-wise scalar flux converges with second order against increasing number of discrete angles, while the error obtained with other conventional quadrature types converges slower than first order depending on the regularity of the exact point-wise uncollided angular flux. In order to reduce the number of discrete points needed, a variant of the MST quadrature, namely MSTP30, is developed by using the Quadruple Range [1] polar quadrature with fixed 30 polar angles and applying the MST quadrature to the azimuthal dependence in each polar level. The angular discretization error in the point-wise SN scalar flux obtained with MSTP30 sets converges with fourth order because the polar discretization error is sufficiently reduced that MSTP30 behaves like a one-dimensional quadrature. Furthermore, because MSTP30 computes the integral over subintervals that keep the true solution’s irregularity at the boundaries, this fourth order convergence rate is unaffected by such inevitable irregularities.

Effect of Sky Discretization for Shading Device Calculation on Building Energy Performance Simulations

The calculation of sunlit surfaces in a building has always been a relevant aspect in building energy simulation programs. Due to the high computational cost, some programs use algorithms for shading calculation for certain solar positions after discretization of hemispherical sky. The influence of the level of discretization on the estimation of incident direct radiation on building surfaces, as well as on the required computational times, are studied in this work. The direct solar energy on a window for a year, with simulation time steps of five minutes, has been simulated by using an algorithm based on Projection and Clipping Methods. A total of 6144 simulations have been carried out, varying window sizes, window orientations, typologies of shading devices, latitudes and discretization levels of the hemispherical sky. In terms of annual incident solar energy, the results show that maximum error values are about 5% for a low level of angular discretization. Errors up to 22% in hourly incident solar energy have been estimated for some of the configurations analysed. Furthermore, a great number of configurations show errors of shading factor on a window of up to 30%, which could be most relevant in studies of natural lighting. The study also shows that the improvement achieved by the most accurate discretization level implies an increase in computational cost of about 30 times.

Accelerated Solution of Discrete Ordinates Approximation to the Boltzmann Transport Equation for a Gray Absorbing–Emitting Medium Via Model Reduction

This work applies a projection-based model-reduction approach to make high-order quadrature (HOQ) computationally feasible for the discrete ordinates approximation of the radiative transfer equation (RTE) for purely absorbing applications. In contrast to traditional discrete ordinates variants, the proposed method provides easily evaluated error estimates associated with the angular discretization as well as an efficient approach for reducing this error to an arbitrary level. In particular, the proposed approach constructs a reduced basis from (high-fidelity) solutions of the radiative intensity computed at a relatively small number of ordinate directions. Then, the method computes inexpensive approximations of the radiative intensity at the (remaining) quadrature points of a high-order quadrature using a reduced-order model (ROM) constructed from this reduced basis. This strategy results in a much more accurate solution than might have been achieved using only the ordinate directions used to construct the reduced basis. One- and three-dimensional test problems highlight the efficiency of the proposed method.

Evaluation of PWR pressure vessel fast neutron fluence benchmarks from NUREG/CR-6115 with ares transport code

An accurate evaluation of PWR pressure vessel fast neutron fluence is essential to ensure pressure vessel integrity over the design lifetime. The discrete ordinates method is one of the main methods to treat such problems. In this paper, evaluations have been performed for three PWR benchmarks described in NUREG/CR-6115 using ARES transport code. The calculated results were compared to the reference values and a satisfactory agreement was obtained. In addition, the effects of SN numeric and source distribution modeling for pressure vessel fast neutron fluence calculation are investigated. Based on the fine enough grids adopted, the different spatial and angular discretization introduces derivations less than 3 %, and fix-up for negative scattering source causes no noticeable effects when calculating pressure vessel fast neutron fluence. However, the discrepancy of assembly-wise and pin-wise source modeling for peripheral assemblies reaches ~20 %, which indicates that pin-wise modeling for peripheral assemblies is essential. These results provide guidelines for pressure vessel fast neutron fluence calculation and demonstrate that the ARES transport code is capable of performing neutron transport calculations for evaluating PWR pressure vessel fast neutron fluence.

Hybrid Ballistic-Diffusive Solution of the Frequency-Dependent Phonon Boltzmann Transport Equation

The phonon Boltzmann Transport Equation (BTE) is difficult to solve on account of the directional and spectral nature of the phonon intensity, which necessitates angular and spectral discretization, and ultimately results in a large number (typically few hundreds) of four-dimensional partial differential equations. In the ballistic (large Knudsen number) regime, the phonon intensity is highly anisotropic, and therefore, angular resolution is desirable. However, in the diffusive (small Knudsen number) regime, the intensity is fairly isotropic, and hence, angular discretization is wasteful. In such scenarios, the method of spherical harmonics may be effectively used to reduce the large number of directional BTEs to a few partial differential equations. Since the Knudsen number is frequency dependent, the decision to preserve or eliminate angular discretization may be made frequency by frequency based on whether the spectral Knudsen number is large or small. In this article, a hybrid method is proposed in which for some frequency intervals (bands), full angular discretization is used, while for others, the first order spherical harmonics (P1) is invoked to reduce the number of directional BTEs. The accuracy and efficiency of the hybrid method is tested by solving several steady state and transient nanoscale heat conduction problems in two and three-dimensional geometries. Silicon is used as the candidate material. It is found that hybridization is effective in significantly improving the efficiency of solution of the BTE — sometimes by a factor of three — without significant penalty on the accuracy.