A Deficiency Problem of the Least Squares Finite Element Method for Solving Radiative Transfer in Strongly Inhomogeneous Media

The accuracy and stability of the least squares finite element method (LSFEM) and the Galerkin finite element method (GFEM) for solving radiative transfer in homogeneous and inhomogeneous media are studied theoretically via a frequency domain technique. The theoretical result confirms the traditional understanding of the superior stability of the LSFEM as compared to the GFEM. However, it is demonstrated numerically and proved theoretically that the LSFEM will suffer a deficiency problem for solving radiative transfer in media with strong inhomogeneity. This deficiency problem of the LSFEM will cause a severe accuracy degradation, which compromises too much of the performance of the LSFEM and makes it not a good choice to solve radiative transfer in strongly inhomogeneous media. It is also theoretically proved that the LSFEM is equivalent to a second order form of radiative transfer equation discretized by the central difference scheme

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

Isogeometric Analysis for Electromagnetism

The combination of numerical analysis with the scanning technology has been seeing increased use in many research areas. There is an emerging need for high-fidelity geometric modeling and meshing for practical applications. The Isogeometric Analysis (IGA) is a comprehensive computational framework, which integrates geometric modeling and meshing with analysis. Different from other existing numerical methods, the IGA can generate analysis ready models without loss of geometrical accuracy. In IGA, the continuity and the quality of a solution can be conveniently controlled and refined. These features enable IGA to integrate modeling, analysis, and design in a unified framework, the root idea of IGA. The IGA for electromagmetics is studied here for steady and transient electromagnetics as well as electromagnetic scattering. The solution procedure and the associated Matlab codes are developed to simulate the electromagnetic radiation on a biological tissues. The scattered and the total electrical fields are computed over the complex geometry of a brain section with realistic material properties. A perfectly matched layer (PML) is developed to model the far field boundary condition. The IGA platform developed here offers a reliable simulation within an accurate representation of the geometry. The results of this research can be used both in evaluating the potential health and safety risks of electromagnetic radiations and in optimizing the design of radiating devices used in non-invasive diagnostics and therapies

Path-tracing Monte Carlo Library for 3D Radiative Transfer in Highly Resolved Cloudy Atmospheres

Interactions between clouds and radiation are at the root of many difficulties in numerically predicting future weather and climate and in retrieving the state of the atmosphere from remote sensing observations. The large range of issues related to these interactions, and in particular to three-dimensional interactions, motivated the development of accurate radiative tools able to compute all types of radiative metrics, from monochromatic, local and directional observables, to integrated energetic quantities. In the continuity of this community effort, we propose here an open-source library for general use in Monte Carlo algorithms. This library is devoted to the acceleration of path-tracing in complex data, typically high-resolution large-domain grounds and clouds. The main algorithmic advances embedded in the library are those related to the construction and traversal of hierarchical grids accelerating the tracing of paths through heterogeneous fields in null-collision (maximum cross-section) algorithms. We show that with these hierarchical grids, the computing time is only weakly sensitivive to the refinement of the volumetric data. The library is tested with a rendering algorithm that produces synthetic images of cloud radiances. Two other examples are given as illustrations, that are respectively used to analyse the transmission of solar radiation under a cloud together with its sensitivity to an optical parameter, and to assess a parametrization of 3D radiative effects of clouds.Comment: Submitted to JAMES, revised and submitted again (this is v2

Analysis of radiative heat transfer in an absorbing-emitting-scattering medium using fluent

Thermal radiation is transferred by electromagnetic waves or photons. Thermal radiation is a very complex phenomenon and although the governing equations are known but they are difficult to solve. The analysis of radiative heat transfer in presence of participating medium is very difficult because radiative intensity as function of position, direction, wavelength, temperature and time. In some cases, these dependencies are not straightforward.The transient term of the radiative transfer equation (RTE) can be neglected i.e. steady-state RTE assumption does not lead to significant errors, since the temporal variations of the observables are in the range of 10-9 to 10-15. The problem has been solved using the FLUENT software to analyze the radiative heat transfer problems, based on finite volume method (FVM). The available computational fluid dynamics software package FLUENT is applied to various test problems using DISCRETE ORDINATE MODEL (DOM MODEL) and the results obtained have been compared with other published results. The test problems include Isothermal Absorbing-Emitting medium in square enclosure, purely scattering medium in square and rectangular enclosure, three-dimensional heat generation in cuboid enclosure, collimated incidence in square enclosure, non-isothermal gray participating medium between two parallel surfaces and also anisotropically scattering gray planar medium in square enclosure. The medium is homogeneous and diffuse gray surface is used.The user defined function (UDF) has been written in C programming language for temperature profile, linear anisotropic phase function and Rayleigh phase function. The effect of absorption coefficient, scattering coefficient, scattering albedo, emissivity, anisotropic factor, grid size, angular discretization and angle of incidence on heat flux and temperature in the medium has been studied. The effect of Rayleigh phase function on heat flux has also studied

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells