A Finite Element Method for Angular Discretization of the Radiation Transport Equation on Spherical Geodesic Grids

Discrete ordinate ($S_N$) and filtered spherical harmonics ($FP_N$) based schemes have been proven to be robust and accurate in solving the Boltzmann transport equation but they have their own strengths and weaknesses in different physical scenarios. We present a new method based on a finite element approach in angle that combines the strengths of both methods and mitigates their disadvantages. The angular variables are specified on a spherical geodesic grid with functions on the sphere being represented using a finite element basis. A positivity-preserving limiting strategy is employed to prevent non-physical values from appearing in the solutions. The resulting method is then compared with both $S_N$ and $FP_N$ schemes using four test problems and is found to perform well when one of the other methods fail.Comment: 24 pages, 13 figure

A New Spherical Harmonics Scheme for Multi-Dimensional Radiation Transport I: Static Matter Configurations

Recent work by McClarren & Hauck [29] suggests that the filtered spherical harmonics method represents an efficient, robust, and accurate method for radiation transport, at least in the two-dimensional (2D) case. We extend their work to the three-dimensional (3D) case and find that all of the advantages of the filtering approach identified in 2D are present also in the 3D case. We reformulate the filter operation in a way that is independent of the timestep and of the spatial discretization. We also explore different second- and fourth-order filters and find that the second-order ones yield significantly better results. Overall, our findings suggest that the filtered spherical harmonics approach represents a very promising method for 3D radiation transport calculations.Comment: 29 pages, 13 figures. Version matching the one in Journal of Computational Physic

Mixed-Cell Methods for Diffusion Problems in Multiphase Systems.

We simulate diffusion in multimaterial systems with a cell-centered Eulerian mesh in two dimensions. A system with immiscible fluids contains sharp interfaces. An Eulerian mesh is fixed in space and does not move with the material. Therefore, cells with an interface contain multiple fluids; these are known as mixed cells. The treatment of mixed cells can vary in computational cost and accuracy. In some cases, the primary source of inaccuracy can be attributed to approximations made in modeling the mixed cells. This thesis focuses on the treatment of mixed cells based on the diffusion approximation of the transport equation. We introduce five subgrid, mixed-cell models. Two models have a single temperature for each cell, while the other three allow a separate temperature for each phase. The single-temperature models are implemented using the Support-Operators Method, which is derived herein. The first single-temperature model utilizes an effective tensor diffusivity that distinguishes diffusion tangent and normal to the interface. The second single-temperature model specifies a unique diffusivity in each corner of a mixed cell, which is effectively a mesh refinement of the mixed cell. The three multi-temperature models have increasingly accurate levels of approximation of the flux: (i) flux is calculated between cell-centers for each phase, (ii) flux is calculated between the centroid of each phase, and (iii) flux normal to an interface is calculated between centroids of each phase. The physical interpretations of these models are: (i) each phase occupies the entire cell, (ii) oblique flux is continuous, (iii) only normal flux is continuous. The standard approximation, using the harmonic mean of the diffusivities present in a mixed cell as an effective diffusivity, is also tested for comparison. We also derive two time-dependent analytical solutions for diffusion in a two-phase system, in both one and two dimensions. With the standard model as a reference point, the accuracy of the new models is quantified, and the convergence rates of the error are determined between pairs of spatial resolutions for the two problems with analytical solutions. Simulations of multiphysics and multimaterial phenomenon may benefit from increased mixed-cell fidelity achieved in this dissertation.PHDApplied PhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/107150/1/leftynm_1.pd

Anisotropic Adaptivity and Subgrid Scale Modelling for the Solution of the Neutron Transport Equation with an Emphasis on Shielding Applications

This thesis demonstrates advanced new discretisation and adaptive meshing technologies that improve the accuracy and stability of using finite element discretisations applied to the Boltzmann transport equation (BTE). This equation describes the advective transport of neutral particles such as neutrons and photons within a domain. The BTE is difficult to solve, due to its large phase space (three dimensions of space, two of angle and one each of energy and time) and the presence of non-physical oscillations in many situations. This work explores the use of a finite element method that combines the advantages of the two schemes: the discontinuous and continuous Galerkin methods. The new discretisation uses multiscale (subgrid) finite elements that work locally within each element in the finite element mesh in addition to a global, continuous, formulation. The use of higher order functions that describe the variation of the angular flux over each element is also explored using these subgrid finite element schemes. In addition to the spatial discretisation, methods have also been developed to optimise the finite element mesh in order to reduce resulting errors in the solution over the domain, or locally in situations where there is a goal of specific interest (such as a dose in a detector region). The chapters of this thesis have been structured to be submitted individually for journal publication, and are arranged as follows. Chapter 1 introduces the reader to motivation behind the research contained within this thesis. Chapter 2 introduces the forms of the BTE that are used within this thesis. Chapter 3 provides the methods that are used, together with examples, of the validation and verification of the software that was developed as a result of this work, the transport code RADIANT. Chapter 4 introduces the inner element subgrid scale finite element discretisation of the BTE that forms the basis of the discretisations within RADIANT and explores its convergence and computational times on a set of benchmark problems. Chapter 5 develops the error metrics that are used to optimise the mesh in order to reduce the discretisation error within a finite element mesh using anisotropic adaptivity that can use elongated elements that accurately resolves computational demanding regions, such as in the presence of shocks. The work of this chapter is then extended in Chapter 6 that forms error metrics for goal based adaptivity to minimise the error in a detector response. Finally, conclusions from this thesis and suggestions for future work that may be explored are discussed in Chapter 7.Open Acces

FUN3D Manual: 12.6

This manual describes the installation and execution of FUN3D version 12.6, including optional dependent packages. FUN3D is a suite of computational fluid dynamics simulation and design tools that uses mixed-element unstructured grids in a large number of formats, including structured multiblock and overset grid systems. A discretely-exact adjoint solver enables efficient gradient-based design and grid adaptation to reduce estimated discretization error. FUN3D is available with and without a reacting, real-gas capability. This generic gas option is available only for those persons that qualify for its beta release status

FUN3D Manual: 12.5

This manual describes the installation and execution of FUN3D version 12.5, including optional dependent packages. FUN3D is a suite of computational uid dynamics simulation and design tools that uses mixed-element unstructured grids in a large number of formats, including structured multiblock and overset grid systems. A discretely-exact adjoint solver enables ecient gradient-based design and grid adaptation to reduce estimated discretization error. FUN3D is available with and without a reacting, real-gas capability. This generic gas option is available only for those persons that qualify for its beta release status

FUN3D Manual: 12.9

This manual describes the installation and execution of FUN3D version 12.9, including optional dependent packages. FUN3D is a suite of computational fluid dynamics simulation and design tools that uses mixed-element unstructured grids in a large number of formats, including structured multiblock and overset grid systems. A discretely-exact adjoint solver enables efficient gradient-based design and grid adaptation to reduce estimated discretization error. FUN3D is available with and without a reacting, real-gas capability. This generic gas option is available only for those persons that qualify for its beta release status