Parallel Algorithms for Radiation Transport on Unstructured Grids

A method and results of numerical modeling of radiative heating of the back surface of the MSRO (Mars Sample Return Orbiter) space vehicle of the European Space Agency are presented. To determine radiation heat fluxes, the method of discrete ordinates on unstructured tetrahedral grids is used. The radiative model is based on the radiation-transfer equation in a multigroup approximation. Numerical calculation has been performed for the most thermally stressed point of the assumed trajectory of the entry of an MSRO-type space vehicle into the Mars atmosphere. A comparison with the discrete ordinates method on structured grids is made. Good agreement between the results of calculations on structured and unstructured grids is demonstrated. The level of radiation heat fluxes to the back surface of the MSRO space vehicle is predicted.

Ray effects through two- and three-dimensional isotropically scattering, participating media are investigated using four ordinates methods: discrete ordinates, using a classical quadrature scheme and a refined azimuthal discretization around the azimuthal angle endpoints; and finite volume, with and without refined azimuthal discretization around the azimuthal angle endpoints. The discrete ordinates method with refined azimuthal angular discretization consists of a coupling of Gaussian quadrature for polar discretization and control azimuthal angle discretization splitting into a quadruple range. For both the discrete ordinates method with refined azimuthal angular discretization and the finite volume method with refined azimuthal angular discretization, the azimuthal direction is discretized such that azimuthal grid points are close to the azimuthal angle endpoints. Radiative heat transfer through isotropically scattering, participating and optically thin media within enclosures considering uniform and strip diffuse boundary conditions is considered. Excellent agreement between the discrete ordinates method/ finite volume method with refined azimuthal angular discretization and the benchmark literature results is achieved for high-order azimuthal discretization, whereas finer high-order azimuthal discretization is required for the finite volume method. Ray effects are eliminated for combinations of higher-order azimuthal discretization and refined azimuthal discretization around the azimuthal angle endpoints. It has also been found that, in order to eliminate ray effects, the discrete ordinates and finite volume methods’ quadrature sets’ order must be increased as the medium’s aspect ratio decreases or when the size of the diffuse strip of incident radiation on one of the medium’s surfaces is decreased.

Measurements of the directional and spatial distribution of the radiation intensity have been made for an irradiated India ink suspension, and the results have been used to determine radiation fluxes associated with the forward and backward directions. The inherent radiative properties of the ink have also been measured and used with appropriate theories (the discrete ordinate and three-flux methods) to predict the radiation field within the suspension. For the prescribed laboratory conditions, the radiation is concentrated within a cone of half angle θ ≈ 0.85 rad throughout the suspension, and for this region good agreement is obtained between the intensity measurements and predictions based on the discrete ordinate method. Good agreement is also obtained between radiation fluxes measured for the forward direction and predictions based on both the discrete ordinate and three-flux methods. Agreement between the predicted and measured intensities for θ ≳ 0.85 rad, as well as agreement between predicted and measured radiation fluxes for the backward direction, is less satisfactory. Differences are attributed to uncertainties in the radiation intensity and phase function measurements.

Towards the Parallelisation of Pressure Correction Method on Unstructured Grids

The stationary monochromatic radiative transfer equation (RTE) is a partial differential transport equation stated on a five-dimensional phase space, the Cartesian product of physical and angular domain. We solve the RTE with a Galerkin FEM in physical space and collocation in angle, corresponding to a discrete ordinates method (DOM). To reduce the complexity of the problem and to avoid the “curse of dimension”, we adapt the sparse grid combination technique to the solution space of the RTE and show that we obtain a sparse DOM which uses essentially only as many degrees of freedom as required for a purely spatial transport problem. For smooth solutions, the convergence rates deteriorate only by a logarithmic factor. We compare the sparse DOM to the standard full DOM and a sparse tensor product approach developed earlier with Galerkin FEM in physical space and a spectral method in angle. Numerical experiments confirm our findings.

Light going through a participating medium like smoke can be scattered or absorbed by every point in the medium. To accurately render such a medium we must compute the radiance resulting at every point inside the medium because of these physical effects, which have been modeled by the radiative transfer equation. Computing the radiance at any point inside a participating medium amounts to numerically solving this radiative transport equation. Discrete Ordinate Method (DOM) is a widely used solution method. DOM is computationally intensive. Fattal [Fat09] proposed Light Propagation Maps (LPM) to expedite DOM computation. In this paper we propose a streaming based parallelization of LPM to run on SIMD graphics hardware. Our method is fast and scalable. We report more than 20x speed improvement by using our method as compared to Fattal’s original method. Using our approach we are able to render 64x64x64 dynamic volumes with multiple scattering of light at interactive speed on complex lighting, and are able to render volumes of any size independent of the GPU memory capability.

Performance evaluation of four radiative transfer methods in solving multi-dimensional radiation and/or conduction heat transfer problems

Diffusive-Time-of-Flight (DTOF), representing the travel time of pressure front propagation, has found many applications in unconventional reservoir performance analysis. The computation of DTOF typically involves upwind finite difference of the Eikonal equation and solution using the fast marching method (FMM). However, application of the finite difference based FMM to irregular grid systems remains a challenge. In this paper, we present a novel and robust method for solving the Eikonal equation using finite volume discretization and the FMM.The finite volume form of the Eikonal equation is derived by differential manipulation, volume integration and the divergence theorem. Using product rule, the differential term is first converted to the divergence form. Then volume integrals that contain divergence terms are converted to surface integrals using the divergence theorem. Consequently, the spatial coordinates are replaced by cell volumes and transmissibilities which are universal for both structured and unstructured grids in finite volume simulators. When applied with the upstream scheme, the finite volume form evolves into a set of quadratic equations, and fast marching method is implemented to solve these equations.The implementation is first validated with analytical solutions using isotropic and anisotropic models with homogeneous reservoir properties. Consistent DTOF distributions are obtained between the proposed approach and the analytical solutions. Next, the implementation is applied to unconventional reservoirs with hydraulic and natural fractures. Our approach relies on cell volumes and connections (transmissibilities) rather than the grid geometry, and thus can be easily applied to complex grid systems. For illustrative purposes, we present applications of the proposed method to embedded discrete fracture models (EDFM), dual-porosity dual-permeability models, and unstructured PEBI grids with heterogeneous reservoir properties. Visualization of the DTOF provides flow diagnostics such as evolution of the drainage volume of the wells and well interactions.The novelty of the proposed approach is its broad applicability to arbitrary grid systems and ease of implementation in commercial reservoir simulators. This makes the approach well-suited for field applications with complex grid geometry and complex well architecture.

Chapter 17 - The Method of Discrete Ordinates (SN-Approximation)

Entropy generation through radiative transfer in participating media: analysis and numerical computation

Radiative transfer models have been widely used to interpret the radar backscatter from forested areas. Most of these models are based on an iterative solution of the radiative transfer equation, usually solved up to first or second order, thus taking into account single and double scattering. Although this method leads to results agreeing well with copolarized backscatter measurements, it produces less accurate estimates for horizontal-vertical (HV) polarization. This paper presents a radiative transfer backscatter model that accounts for multiple scattering by using the discrete ordinate and eigenvalue method applied to a layered medium. Using parameters derived from an architectural tree model, calculations at C- and L-band are compared with HV data acquired for a maritime pine forest in the southwest of France during the Spaceborne Imaging Radar-C missions. Good agreement is found at C-band for all values of forest biomass, and reasonable agreement at L-band for high biomass, when the soil backscatter plays a minor role. For low biomass, the L-band modeling is inadequate because of difficulties in estimating the soil backscatter. Comparison with calculations from a first-order radiative transfer model shows that multiple scattering is significant, especially at C-band.

The discrete ordinates method (DOM) or its variant, the finite angle method (FAM), is a popular solution method for the radiative transfer equation (RTE). Accurate solution to the RTE using the DOM or FAM requires many solid angles (directions) in multidimensional geometry. In combustion gases, where the absorption coefficient oscillates wildly and the RTE must be solved repeatedly, this method becomes computationally intractable. Here, the FAM is hybridized with the P<sub>1</sub> approximation, which is efficient since it requires solution to a single partial differential equation as opposed to a set of directional RTEs in the FAM. The P<sub>1</sub> approximation is accurate when the intensity field is fairly isotropic, as evidenced in optically thick media. Hence, the hybridization employs the FAM for optically thin and intermediate spectral intervals and employs the P<sub>1</sub> approximation for optically thick spectral intervals. The objective of the present study is to determine optimal parameters for hybridization that can provide the best compromise between accuracy and efficiency. Using a narrowband-based box model for carbon dioxide and water vapor, the nongray radiative transfer equation is solved in media with nonuniform properties enclosed in multidimensional enclosures. Two different approaches-cutoff and filter optical thickness-are investigated for hybridizing. Several problems, both two- and three-dimensional, and with and without coupling to other modes of heat transfer are considered. The filter approach was found to be the best choice for prediction of the radiative source and temperature (in the case of a coupled mode), while the cutoff approach was found to be the best for prediction of wall radiative heat fluxes.

A theoretical study of radiation between two concentric spheres using a modified discrete ordinates method associated with Legendre transform

The objective of the Cranked-Arrow Wing Aerodynamics Project International (CAWAPI) was to allow a comprehensive validation of Computational Fluid Dynamics methods against the CAWAP flight database. A major part of this work involved the generation of high-quality computational grids. Prior to the grid generation an IGES file containing the air-tight geometry of the F-16XL aircraft was generated by a cooperation of the CAWAPI partners. Based on this geometry description both structured and unstructured grids have been generated. The baseline structured (multi-block) grid (and a family of derived grids) has been generated by the National Aerospace Laboratory NLR. Although the algorithms used by NLR had become available just before CAWAPI and thus only a limited experience with their application to such a complex configuration had been gained, a grid of good quality was generated well within four weeks. This time compared favourably with that required to produce the unstructured grids in CAWAPI. The baseline all-tetrahedral and hybrid unstructured grids has been generated at NASA Langley Research Center and the USAFA, respectively. To provide more geometrical resolution, trimmed unstructured grids have been generated at EADS-MAS, the UTSimCenter, Boeing Phantom Works and KTH/FOI. All grids generated within the framework of CAWAPI will be discussed in the article. Both results obtained on the structured grids and the unstructured grids showed a significant improvement in agreement with flight test data in comparison with those obtained on the structured multi-block grid used during CAWAP.

Over the last year, a considerable portion of FEFLO, a general-purpose legacy CFD code operating on unstructured grids, was ported to run on GPUs. Like so many other legacy codes, FEFLO is an adaptive, edge-based nite element code for the solution of compressible and incompressible ow, which was primarily written in Fortran 77 and has previously been ported to vector, shared memory parallel and distributed memory parallel machines. Due to the large size of FEFLO and the likelihood of human error in porting, as well as the desire for continued development within a single codebase, a specialized Python script, based on FParser (Peterson, 2009), was written to perform automated translation from the OpenMP-parallelized loops to GPU kernels implemented in CUDA, along with GPU memory management, while integrating with the existing framework for distributed memory parallelism via MPI. The present paper describes extensions of the script and algorithmic techniques that enable the ecient running on GPUs of the modules that are not straightforward to port. In particular, we consider LU-SGS algorithms, linelet preconditioning and particle-mesh algorithms.