A Parametric Study of the Accuracy of Several Radiative Transport Solution Methods for a Set of 2-D Benchmark Problems

2013 ◽  
Author(s):  
John Tencer ◽  
John R. Howell
Keyword(s):  
Radiation Transport ◽  
Benchmark Problems ◽  
Discrete Ordinates ◽  
Test Problems ◽  
Governing Equations ◽  
Element Discretization ◽  
Discontinuous Boundary ◽  
All Solutions ◽  
Solution Methods ◽  
Ray Effects

Several of the most popular deterministic radiation transport methods for heat transfer are compared. Relative solution error is compared as a function of optical thickness and scattering albedo. The test problems are chosen to represent a range of problem types. Problems with discontinuous boundary conditions are included to evaluate ‘ray effects’ for discrete ordinates solutions. A brief derivation and a statement of the governing equations for each method is included so that the details of the precise method used is clear. All solutions are generated using finite element discretization. Where applicable, any stabilization used is included in the statement of the governing equations.

scholarly journals A Comparison of Angular Discretization Techniques for the Radiative Transport Equation

2015 ◽  
Author(s):  
John Tencer
Keyword(s):  
Spherical Harmonics ◽  
Radiation Transport ◽  
Radiative Flux ◽  
Benchmark Problems ◽  
Discrete Ordinates ◽  
Flux Divergence ◽  
Discontinuous Boundary ◽  
All Solutions ◽  
Mesh Resolution ◽  
Ray Effects

Two of the most popular deterministic radiation transport methods for treating the angular dependence of the radiative intensity for heat transfer: the discrete ordinates and simplified spherical harmonics approximations are compared. A problem with discontinuous boundary conditions is included to evaluate ray effects for discrete ordinates solutions. Mesh resolution studies are included to ensure adequate convergence and evaluate the effects of the contribution of false scattering. All solutions are generated using finite element spatial discretization. Where applicable, any stabilization used is included in the description of the approximation method or the statement of the governing equations. A previous paper by the author presented results for a set of 2D benchmark problems for the discrete ordinates method using the PN-TN quadrature of orders 4, 6, and 8 as well as the P1, M1, and SP3 approximations. This paper expands that work to include the Lathrop-Carlson level symmetric quadrature of order up to 20 as well as the Lebedev quadrature of order up to 76 and simplified spherical harmonics of odd orders from 1 to 15. Two 3D benchmark problems are considered here. The first is a canonical problem of a cube with a single hot wall. This case is used primarily to demonstrate the potentially unintuitive interaction between mesh resolution, quadrature order, and solution error. The second case is meant to be representative of a pool fire. The temperature and absorption coefficient distributions are defined analytically. In both cases, the relative error in the radiative flux or the radiative flux divergence within a volume is considered as the quantity of interest as these are the terms that enter into the energy equation. The spectral dependence of the optical properties and the intensity is neglected.

Verification for Ray Effects Elimination Module of Radiation Shielding Code ARES by Kobayashi Benchmarks

2014 ◽  
Author(s):  
Mengteng Chen ◽  
Bin Zhang ◽  
Yixue Chen
Keyword(s):  
Radiation Transport ◽  
Source Region ◽  
Anisotropic Scattering ◽  
Relative Difference ◽  
Radiation Shielding ◽  
Benchmark Problems ◽  
Discrete Ordinates ◽  
Ray Effects ◽  
Good Agreement

ARES is a multi-group of anisotropic scattering transport shielding code based on discrete ordinates method. The 3D radiation transport benchmark problems proposed by Kobayashi were calculated by ARES with sub-module ARES_RayEffect which using first collision method for ray effects mitigation. ARES_RayEffect calculates uncollided flux and first collision source moments for ARES. The uncollided flux is obtained by a ray tracing calculation between a source point and a target mesh center. In addition, ARES_RayEffect has a modifying factor function to improve the quality of uncollided flux calculation. For verification, the results of MCNP code are used as reference solution and the results of TORT with FNSUNCL3 are compared. ARES_RayEffect introduced the modifying factor to reduce the relative difference of meshes near the source region. For example, at the position (15,15,15) in Problem 1 case i, the relative difference of the result of ARES with ARES_RayEffect is −2.34%, while relative difference of the result of TORT with FNSUNCL3 is −11.92%. The calculated total neutron fluxes show good agreement with the MCNP solutions. For the pure absorber cases, the maximum differences are less than 3%. For the half scattering cases, the maximum differences are less than 11%. Numerical results demonstrate that ray effects can be effectively mitigated.

Application of Modified Discrete Ordinates Method With the Concept of Blocked-Off Region Method to Radiative Heat Transfer Problems in Irregular Geometries

2010 ◽  
Author(s):  
H. Amiri ◽  
S. H. Mansouri ◽  
A. Safavinejad
Keyword(s):  
Heat Transfer ◽  
Radiative Heat Transfer ◽  
Radiative Heat ◽  
Curvilinear Coordinates ◽  
Discrete Ordinates ◽  
Test Problems ◽  
Discrete Ordinates Method ◽  
Irregular Geometries ◽  
Transfer Problems ◽  
Ray Effects

The discrete ordinates method (DOM) for the solution of radiative heat transfer problems have received significant attention and development owing to their good compromise between accuracy, flexibility and moderate computational requirement. However, the DOM suffers from the ray effects related to the discretization of the angular distribution of the radiation intensity. The modified discrete ordinate method (MDOM) proved to significantly mitigate ray effects originated from discontinuities or abrupt changes of the wall temperature. This article presents blocked-off region treatment of irregular geometries using a modified discrete ordinates method in Cartesian coordinates. The Cartesian based 2D algorithm can be used to solve radiative heat transfer in irregular geometries by dividing the region into active and inactive regions. It is easier and convenient way of handling 2D irregular geometries than to write an algorithm in curvilinear coordinates. It is capable of handling participating (absorbing, emitting and isotropic or anisotropic scattering) or non participating gray media enclosed by gray diffuse walls. Both radiative and non-radiative equilibrium situations are considered. The walls of the enclosures can have either heat flux or temperature boundary conditions. Cases with curved and obstacle and radiation shield are considered. Some test problems are considered and results are validated with the available results in the literature. Results are found to be accurate for all kinds of situations.

scholarly journals DAPHNE-3D: A NEW TRANSPORT SOLVER FOR UNSTRUCTURED TETRAHEDRAL MESHES

2021 ◽  
Vol 247 ◽  
pp. 03004
Author(s):  
Evangelia Diamantopoulou ◽  
Daniele Sciannandrone
Keyword(s):  
Radiation Transport ◽  
Benchmark Problems ◽  
Discrete Ordinates ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Tetrahedral Meshes ◽  
Discontinuous Galerkin Finite Element ◽  
Unstructured Tetrahedral Meshes ◽  
The One ◽  
Good Agreement

A new Discrete Ordinates transport solver for unstructured tetrahedral meshes is presented. The solver uses the Discontinuous Galërkin Finite Element Method with linear or quadratic expansion of the flux within each cell. The solution of the one-group problem is obtained with non-preconditioned fixed-point or GMRES iterations. Precision and performances of the solver are evaluated on the 3D Radiation Transport Benchmark Problems proposed by Kobayashi, showing very good agreement with the reference and good computing times in serial execution.

An Angular Finite Element Method for the Solution of the Even-Parity Equation

2009 ◽  
Author(s):  
R. Becker ◽  
R. Koch ◽  
M. F. Modest ◽  
H.-J. Bauer
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Basis Functions ◽  
New Method ◽  
Discrete Ordinates ◽  
Test Case ◽  
Element Approximation ◽  
Promising Alternative ◽  
Element Discretization ◽  
Spatial Coordinates

The present article introduces a new method to solve the radiative transfer equation (RTE). First, a finite element discretization of the solid angle dependence is derived, wherein the coefficients of the finite element approximation are functions of the spatial coordinates. The angular basis functions are defined according to finite element principles on subdivisions of the octahedron. In a second step, these spatially dependent coefficients are discretized by spatial finite elements. This approach is very attractive, since it provides a concise derivation for approximations of the angular dependence with an arbitrary number of angular nodes. In addition, the usage of high-order angular basis functions is straightforward. In the current paper the governing equations are first derived independently of the actual angular approximation. Then, the design principles for the angular mesh are discussed and the parameterization of the piecewise angular basis functions is derived. In the following, the method is applied to two-dimensional test cases which are commonly used for the validation of approximation methods of the RTE. The results reveal that the proposed method is a promising alternative to the well-established practices like the Discrete Ordinates Method (DOM) and provides highly accurate approximations. A test case known to exhibit the ray effect in the DOM verifies the ability of the new method to avoid ray effects.

Intensity-Dependent Dispersion in Nonlinear Phononic Layered Systems

2011 ◽  
Author(s):  
Kevin L. Manktelow ◽  
Michael J. Leamy ◽  
Massimo Ruzzene
Keyword(s):  
Discrete Systems ◽  
Dispersion Analysis ◽  
Constitutive Relationship ◽  
Analysis Tool ◽  
Governing Equations ◽  
Cubic Nonlinearity ◽  
Layered Systems ◽  
Element Discretization ◽  
Adequate Representation ◽  
Intensity Wave

Phononic crystals are typically considered to operate in regimes where a linear constitutive relationship provides an adequate representation. For high intensity wave propagation, however, weak nonlinearities can affect performance. For example, a cubic nonlinearity gives rise to frequency shifting and thus a shift in band gap location. In the study of nonlinear optics, a cubic term has been treated using a quasi-linear constitutive relationship with intensity dependent properties. This technique is explored herein for generating nonlinear dispersion relationships for the elastic case. In addition, a perturbation method developed previously for discrete systems, used in conjunction with a finite element discretization, is proposed as an alternative dispersion analysis tool. Simulations of the fully nonlinear governing equations are provided as validation of the predicted dispersion curves.

A Fuzzy Simulated Evolution Algorithm for Hard Problems

2014 ◽  
pp. 892-912
Author(s):  
Michael Mutingi
Keyword(s):  
Set Theory ◽  
Large Scale ◽  
Cell Formation ◽  
Computational Experiments ◽  
Benchmark Problems ◽  
Problem Complexity ◽  
Simulated Evolution ◽  
Solution Methods ◽  
Hard Problems ◽  
Evolution Algorithm

As problem complexity continues to increase in industry, developing efficient solution methods for solving hard problems, such as heterogeneous vehicle routing and integrated cell formation problems, is imperative. The focus of this chapter is to develop from the classical simulated evolution algorithm, a Fuzzy Simulated Evolution Algorithm (FSEA) that incorporates the concepts of fuzzy set theory, evolution, and constructive perturbation. The aim is to improve the search efficiency of the algorithm by enhancing the major phases of the algorithm through initialization, evaluation, selection, and reconstruction. Illustrative examples are provided to demonstrate the candidate application areas and to show the strength of the algorithm. Computational experiments are conducted based on benchmark problems in the literature. Results from the computational experiments demonstrate the strength of the algorithm. It is anticipated that the application of the FSEA metaheuristic can be extended to other hard large scale problems.

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