The method of lines solution of discrete ordinates method for radiative heat transfer in cylindrical enclosures

A radiation code based on method of lines solution of discrete ordinates method for radiative heat transfer in axisymmetric cylindrical enclosures containing absorbing-emitting medium was developed and tested for predictive accuracy by applying it to (i) test problems with black and grey walls (ii) a gas turbine combustor simulator enclosing a non-homogeneous absorbing-emitting medium and benchmarking its steady-state predictions against exact solutions and measurements. Comparisons show that it provides accurate solutions for radiative heat fluxes and can be used with confidence in conjunction with CFD codes based on the same approach.     

Simultaneous approximation for the Bézier variant of Baskakov-Beta operators

In the present paper, we study the rate of convergence in simultaneous approximation for the Bézier variant of the Baskakov-Beta operators by using the decomposition technique of functions of bounded variation.     

Numerical method for calculation of the incompressible flow in general curvilinear co‐ordinates with double staggered grid

A solution methodology has been developed for incompressible flow in general curvilinear co‐ordinates. Two staggered grids are used to discretize the physical domain. The first grid is a MAC quadrilateral mesh with pressure arranged at the centre and the Cartesian velocity components located at the middle of the sides of the mesh. The second grid is so displaced that its corners correspond to the centre of the first grid. In the second grid the pressure is placed at the corner of the first grid. The discretized mass and momentum conservation equations are derived on a control volume. The two pressure grid functions are coupled explicitly through the boundary conditions and implicitly through the velocity of the field. The introduction of these two grid functions avoids an averaging of pressure and velocity components when calculating terms that are generated in general curvilinear co‐ordinates. The SIMPLE calculation procedure is extended to the present curvilinear co‐ordinates with double grids. Application of the methodology is illustrated by calculation of well‐known external and internal problems: viscous flow over a circular cylinder, with Reynolds numbers ranging from 10 to 40, and lid‐driven flow in a cavity with inclined walls are examined. The numerical results are in close agreement with experimental results and other numerical data. Copyright © 2003 John Wiley & Sons, Ltd.     

Evaluation of quadrature schemes for the discrete ordinates method

After reviewing the presently available quadrature schemes for the discrete ordinates method, the accuracy of different schemes is analyzed and evaluated. It is shown from a comprehensive error analysis that the moment conditions have to satisfied not only for the principal coordinates directions, as it is mostly carried out, but for any arbitrary test direction. Among the schemes with approximately 50 discrete ordinates the DCT-020-2468 quadrature was found to give the best accuracy. The highest accuracy among all schemes is achieved by the LC-11 quadrature which requires 96 discrete ordinates. This scheme is rarely used up to date and deserves more attention for high accuracy predictions.     

三角B\''{e}zier曲面和四边B\''{e}zier曲面之间的相互转化

B\'{e}zier曲面有两种不同的形式:三角B\'{e}zier曲面和四边B\'{e}zier曲面,它们有着不同的基底和不同的几何拓扑结构, 但是它们也有很多共同的性质,因此三角B\'{e}zier曲面和四边B\'{e}zier曲面之间的相互转化就成为CAGD 里一个重要研究课题.在本文中, 我们用函数复合的方法实现两者之间的相互转化.被复合的两个函数, 一个用Polar形式表示,另一个用常见的Bernstein基形式表示.     

空间参数曲线的双目标能量极小化方法及其应用

能量极小化方法已广泛用于平面曲线的构造,而在空间曲线构造方面的应用尚少。首先介绍了空间参数曲线的弯曲能和扭曲能,然后提出了一种以弯曲能和扭曲能同时极小为目标的空间参数曲线构造方法,最后以空间三次Bézier曲线为例,探讨了该方法在曲线的构造、延拓、平滑等问题中的应用。所提出的方法更符合空间参数曲线既需考虑弯曲又需考虑扭曲的特点。     

GPU accelerated simulations of 3D deterministic particle transport using discrete ordinates method

Graphics Processing Unit (GPU), originally developed for real-time, high-definition 3D graphics in computer games, now provides great faculty in solving scientific applications. The basis of particle transport simulation is the time-dependent, multi-group, inhomogeneous Boltzmann transport equation. The numerical solution to the Boltzmann equation involves the discrete ordinates (S_n) method and the procedure of source iteration. In this paper, we present a GPU accelerated simulation of one energy group time-independent deterministic discrete ordinates particle transport in 3D Cartesian geometry (Sweep3D). The performance of the GPU simulations are reported with the simulations of vacuum boundary condition. The discussion of the relative advantages and disadvantages of the GPU implementation, the simulation on multi GPUs, the programming effort and code portability are also reported. The results show that the overall performance speedup of one NVIDIA Tesla M2050 GPU ranges from 2.56 compared with one Intel Xeon X5670 chip to 8.14 compared with one Intel Core Q6600 chip for no flux fixup. The simulation with flux fixup on one M2050 is 1.23 times faster than on one X5670.     

Simultaneous derivation of intensities and weighting functions in a general pseudo-spherical discrete ordinate radiative transfer treatment

The retrieval of atmospheric constituents from measurements of backscattered light requires a radiative transfer forward model that can simulate both intensities and weighting functions (partial derivatives of intensity with respect to atmospheric parameters being retrieved). The radiative transfer equation is solved in a multi-layer multiply-scattering atmosphere using the discrete ordinate method. In an earlier paper dealing with the upwelling top-of-the-atmosphere radiation field, it was shown that a full internal perturbation analysis of the plane-parallel discrete ordinate solution leads in a natural way to the simultaneous generation of analytically-derived weighting functions with respect to a wide range of atmospheric variables. In the present paper, a more direct approach is used to evaluate explicitly all partial derivatives of the intensity field. A generalization of the post-processing function is developed for the derivation of weighting functions at arbitrary optical depth and stream angles for both upwelling and downwelling directions. Further, a complete treatment is given for the pseudo-spherical approximation of the direct beam attenuation; this is an important extension to the range of viewing geometries encountered in practical radiative transfer applications. The numerical model LIDORT developed for this work is able to generate intensities and weighting functions for a wide range of retrieval scenarios, in addition to the passive remote sensing application from space. We present a number of examples in an atmosphere with O₃ absorption in the UV, for satellite (upwelling radiation) and ground-based (downwelling radiation) applications. In particular, we examine the effect of various pseudo-spherical parameterizations on backscatter intensities and weighting functions with respect to O₃ volume mixing ratio. In addition, the use of layer-integrated multiple scatter output from the model is shown to be important for satellite instruments with wide-angle off-nadir viewing geometries.     

Linearization of the interaction principle: Analytic Jacobians in the “Radiant” model

In this paper we present a new linearization of the Radiant radiative transfer model. Radiant uses discrete ordinates for solving the radiative transfer equation in a multiply-scattering anisotropic medium with solar and thermal sources, but employs the adding method (interaction principle) for the stacking of reflection and transmission matrices in a multilayer atmosphere. For the linearization, we show that the entire radiation field is analytically differentiable with respect to any surface or atmospheric parameter for which we require Jacobians (derivatives of the radiance field). Derivatives of the discrete ordinate solutions are based on existing methods developed for the LIDORT radiative transfer models. Linearization of the interaction principle is completely new and constitutes the major theme of the paper. We discuss the application of the Radiant model and its linearization in the Level 2 algorithm for the retrieval of columns of carbon dioxide as the main target of the Orbiting Carbon Observatory (OCO) mission.     

The dual bases for the Bézier-Said-Wang type generalized Ball polynomial bases and their applications

A unifying representation for the existing generalized Ball bases and the Bernstein bases are given. Then the dual bases for the Bézier-Said-Wang type generalized bases (BSWGB for short) are presented. The Marsden identity and the mutual transformation formulas between Bézier curve and Bézier-Said-Wang type generalized curve (BSWGB curve) are also given. These results are very useful for the applications of BSWGB curves and their popularization in CAGD. Numerical examples are also given to show the effectiveness of our methods.