5 research outputs found
Meshless Methods for the Neutron Transport Equation
Mesh-based methods for the numerical solution of partial differential equations (PDEs) require the division of the problem domain into non-overlapping, contiguous subdomains that conform to the problem geometry. The mesh constrains the placement and connectivity of the solution nodes over which the PDE is solved. In meshless methods, the solution nodes are independent of the problem geometry and do not require a mesh to determine connectivity. This allows the solution of PDEs on geometries that would be difficult to represent using even unstructured meshes.
The ability to represent difficult geometries and place solution nodes independent of a mesh motivates the use of meshless methods for the neutron transport equation, which often includes spatially-dependent PDE coefficients and strong localized gradients. The meshless local Petrov-Galerkin (MLPG) method is applied to the steady-state and k-eigenvalue neutron transport equations, which are discretized in energy using the multigroup approximation and in angle using the discrete ordinates approximation. The MLPG method uses weighted residuals of the transport equation to solve for basis function expansion coefficients of the neutron angular flux. Connectivity of the solution nodes is determined by the shared support domain of overlapping meshless functions, such as radial basis functions (RBFs) and moving least squares (MLS) functions.
To prevent oscillations in the neutron flux, the MLPG transport equation is stabilized by the streamline upwind Petrov-Galerkin (SUPG) method, which adds numerical diffusion to the streaming term. Global neutron conservation is enforced by using MLS basis and weight functions and appropriate SUPG parameters. The cross sections in the transport equation are approximated in accordance with global particle balance and without constraint on their spatial dependence or the location of the basis and weight functions. The equations for the strong-form meshless collocation approach are derived for comparison to the MLPG equations. Two integration schemes for the basis and weight functions in the MLPG method are presented, including a background mesh integration and a fully meshless integration approach.
The method of manufactured solutions (MMS) is used to verify the resulting MLPG method in one, two and three dimensions. Results for realistic problems, including two-dimensional pincells, a reflected ellipsoid and a three-dimensional problem with voids, are verified by comparison to Monte Carlo simulations. Finally, meshless heat transfer equations are derived using a similar MLPG approach and verified using the MMS. These heat equation are coupled to the MLPG neutron transport equations, and results for a pincell are compared to values from a commercial pressurized water reactor.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145796/1/brbass_1.pd
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Fast Radiative-Transfer-Equation-Based Image Reconstruction Algorithms for Non-Contact Diffuse Optical Tomography Systems
It is well known that the radiative transfer equation (RTE) is the most accurate deterministic light propagation model employed in diffuse optical tomography (DOT). RTE-based algorithms provide more accurate tomographic results than codes that rely on the diffusion equation (DE), which is an approximation to the RTE, in scattering dominant media. However, RTE based DOT (RTE-DOT) has limited utility in practice due to its high computational cost and lack of support for general non-contact imaging systems. In this dissertation, I developed fast reconstruction algorithms for RTE-based DOT (RTE-DOT), which consists of three independent components: an efficient linear solver for forward problems, an improved optimization solver for inverse problem, and the first light propagation model in free space that fully considers the angular dependency, which can provide a suitable measurement operator for RTE-DOT. This algorithm is validated and evaluated with numerical experiments and clinical data. According to these studies, the novel reconstruction algorithm is up to 30 times faster than traditional reconstruction techniques, while achieving comparable reconstruction accuracy