63 research outputs found

    On a convergent DSA preconditioned source iteration for a DGFEM method for radiative transfer

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    We consider the numerical approximation of the radiative transfer equation using discontinuous angular and continuous spatial approximations for the even parts of the solution. The even-parity equations are solved using a diffusion synthetic accelerated source iteration. We provide a convergence analysis for the infinite-dimensional iteration as well as for its discretized counterpart. The diffusion correction is computed by a subspace correction, which leads to a convergence behavior that is robust with respect to the discretization. The proven theoretical contraction rate deteriorates for scattering dominated problems. We show numerically that the preconditioned iteration is in practice robust in the diffusion limit. Moreover, computations for the lattice problem indicate that the presented discretization does not suffer from the ray effect. The theoretical methodology is presented for plane-parallel geometries with isotropic scattering, but the approach and proofs generalize to multi-dimensional problems and more general scattering operators verbatim

    Analysis of Iterative Methods for the Linear Boltzmann Transport Equation

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    In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the discretisation parameters. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport.Comment: 27 pages, 8 figure

    Fast Microwave Tomography Algorithm for Breast Cancer Imaging

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    Microwave tomography has shown promise for breast cancer imaging. The microwaves are harmless to body tissues, which makes microwave tomography a safe adjuvant screening to mammography. Although many clinical studies have shown the effectiveness of regular screening for the detection of breast cancer, the anatomy of the breast and its critical tissues challenge the identification and diagnosis of tumors in this region. Detection of tumors in the breast is more challenging in heterogeneously dense and extremely dense breasts, and microwave tomography has the potential to be effective in such cases. The sensitivity of microwaves to various breast tissues and the comfort and safety of the screening method have made microwave tomography an attractive imaging technique. Despite the need for an alternative screening technique, microwave tomography has not yet been introduced as a screening modality in regular health care, and is still subject to research. The main obstacles are imperfect hardware systems and inefficient imaging algorithms. The immense computational costs for the image reconstruction algorithm present a crucial challenge. 2D imaging algorithms are proposed to reduce the amount of hardware resources required and the imaging time. Although 2D microwave tomography algorithms are computationally less expensive, few imaging groups have been successful in integrating the acquired 3D data into the 2D tomography algorithms for clinical applications. The microwave tomography algorithms include two main computation problems: the forward problem and the inverse problem. The first part of this thesis focuses on a new fast forward solver, the 2D discrete dipole approximation (DDA), which is formulated and modeled. The effect of frequency, sampling number, target size, and contrast on the accuracy of the solver are studied. Additionally, the 2D DDA time efficiency and computation time as a single forward solver are investigated.\ua0 The second part of this thesis focuses on the inverse problem. This portion of the algorithm is based on a log-magnitude and phase transformation optimization problem and is formulated as the Gauss-Newton iterative algorithm. The synthetic data from a finite-element-based solver (COMSOL Multiphysics) and the experimental data acquired from the breast imaging system at Chalmers University of Technology are used to evaluate the DDA-based image reconstruction algorithm. The investigations of modeling and computational complexity show that the 2D DDA is a fast and accurate forward solver that can be embedded in tomography algorithms to produce images in seconds. The successful development and implementation in this thesis of 2D tomographic breast imaging with acceptable accuracy and high computational cost efficiency has provided significant savings in time and in-use memory and is a dramatic improvement over previous implementations

    NUMERICAL INVESTIGATION AND PARALLEL COMPUTING FOR THERMAL TRANSPORT MECHANISM DURING NANOMACHINING

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    Nano-scale machining, or Nanomachining is a hybrid process in which the total thermal energy necessary to remove atoms from a work-piece surface is applied from external sources. In the current study, the total thermal energy necessary to remove atoms from a work-piece surface is applied from two sources: (1) localized energy from a laser beam focused to a micron-scale spot to preheat the work-piece, and (2) a high-precision electron-beam emitted from the tips of carbon nano-tubes to remove material via evaporation/sublimation. Macro-to-nano scale heat transfer models are discussed for understanding their capability to capture and its application to predict the transient heat transfer mechanism required for nano-machining. In this case, thermal transport mechanism during nano-scale machining involves both phonons (lattice vibrations) and electrons; it is modeled using a parabolic two-step (PTS) model, which accounts for the time lag between these energy carriers. A numerical algorithm is developed for the solution of the PTS model based on explicit and implicit finite-difference methods. Since numerical solution for simulation of nanomachining involves high computational cost in terms of wall clock time consumed, performance comparison over a wide range of numerical techniques has been done to devise an efficient numerical solution procedure. Gauss-Seidel (GS), successive over relaxation (SOR), conjugate gradient (CG), d -form Douglas-Gunn time splitting, and other methods have been used to compare the computational cost involved in these methods. Use of the Douglas-Gunn time splitting in the solution of 3D time-dependent heat transport equations appears to be optimal especially as problem size (number of spatial grid points and/or required number of time steps) becomes large. Parallel computing is implemented to further reduce the wall clock time required for the complete simulation of nanomachining process. Domain decomposition with inter-processor communication using Message Passing Interface (MPI) libraries is adapted for parallel computing. Performance tuning has been implemented for efficient parallelization by overlapping communication with computation. Numerical solution for laser source and electron-beam source with different Gaussian distribution are presented. Performance of the parallel code is tested on four distinct computer cluster architecture. Results obtained for laser source agree well with available experimental data in the literature. The results for electron-beam source are self-consistent; nevertheless, they need to be validated experimentally

    Multilevel Schwarz methods for multigroup radiation transport problems

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    The development of advanced discretization methods for the radiation transport equation is of fundamental importance, since the numerical effort of modeling increasingly complex multidimensional problems with increasing accuracy is extremely challenging. Different expressions of this equation arise in several science fields, from nuclear fission and fusion to astrophysics, climatology and combustion. Mathematically, the radiation intensity is usually a rapidly changing function, causing a considerable loss in accuracy for many discretization methods. Depending on the coefficient ranges, the equation behaves like totally different equation types, making it very difficult to find a discretization method that is efficient in all regimes. Computationally, the huge amount of unknowns involved demands not only extremely powerful computers, but also efficient numerical methods and optimized implementations. Today, solvers covering all the coefficient ranges and still being robust in the diffusion dominated case are very scarce. In the last 20 years, Discontinous Galerkin (DG) methods have been studied for the monoenergetic problem, unsuccessfully, due to lack of stability for diffusion-dominated cases. Recently, new mathematical developments have fully explained the instability and provided a remedy by using a numerical flux depending on the scattering cross section and the mesh size. The new formulation has proven to be stable and allows the application of multigrid, matrix-free methods, reducing the memory needed for such an amount of unknowns. We use these numerical methods to address the solution of a energy dependent problem with a multigroup approach. We study the diffusion approximation to the transport problem, obtaining convergence proofs for the symmetric scattering case and advances in the nonsymmetric case, using field of values analysis. For the full transport case, we discretize by means of an asymptotic preserving, weakly penalized discontinuous Galerkin method that we solve with a multigrid preconditioned GMRES solver, using nonoverlapping Schwarz smoothers for the energy and direction dependent radiative transfer problem. To address the local thermodynamic equilibrium (LTE) constraint, we use a nonlinear additive Schwarz method to precondition the Newton solver. By solving full local radiative transfer problems for each grid cell, performed in parallel on a matrix-free implementation, we achieve a method capable to address large scale calculations arising from applications such as astrophysics, atmospheric radiation calculations and nuclear applications. To the best of our knowledge, this is the first time this preconditioner combination has been used in LTE radiation transport and in several tests we show the robustness of the approach for different mesh sizes, cross sections, energy distributions and anisotropic regimes, both in the linear and nonlinear cases

    Meshless Methods for the Neutron Transport Equation

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    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

    Discontinuous Galerkin Methods for the Linear Boltzmann Transport Equation

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    Radiation transport is an area of applied physics that is concerned with the propagation and distribution of radiative particle species such as photons and electrons within a material medium. Deterministic models of radiation transport are used in a wide range of problems including radiotherapy treatment planning, nuclear reactor design and astrophysics. The central object in many such models is the (linear) Boltzmann transport equation, a high-dimensional partial integro-differential equation describing the absorption, scattering and emission of radiation. In this thesis, we present high-order discontinuous Galerkin finite element discretisations of the time-independent linear Boltzmann transport equation in the spatial, angular and energetic domains. Efficient implementations of the angular and energetic components of the scheme are derived, and the resulting method is shown to converge with optimal convergence rates through a number of numerical examples. The assembly of the spatial scheme on general polytopic meshes is discussed in more detail, and an assembly algorithm based on employing quadrature-free integration is introduced. The quadrature-free assembly algorithm is benchmarked against a standard quadrature-based approach, and an analysis of the algorithm applied to a more general class of discontinuous Galerkin discretisations is performed. In view of developing efficient linear solvers for the system of equations resulting from our discontinuous Galerkin discretisation, we exploit the variational structure of the scheme to prove convergence results and derive a posteriori solver error estimates for a family of iterative solvers. These a posteriori solver error estimators can be used alongside standard implementations of the generalised minimal residual method to guarantee that the linear solver error between the exact and approximate finite element solutions (measured in a problem-specific norm) is below a user-specified tolerance. We discuss a family of transport-based preconditioners, and our linear solver convergence results are benchmarked through a family of numerical examples

    Discontinuous Galerkin Methods for the Linear Boltzmann Transport Equation

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    Radiation transport is an area of applied physics that is concerned with the propagation and distribution of radiative particle species such as photons and electrons within a material medium. Deterministic models of radiation transport are used in a wide range of problems including radiotherapy treatment planning, nuclear reactor design and astrophysics. The central object in many such models is the (linear) Boltzmann transport equation, a high-dimensional partial integro-differential equation describing the absorption, scattering and emission of radiation. In this thesis, we present high-order discontinuous Galerkin finite element discretisations of the time-independent linear Boltzmann transport equation in the spatial, angular and energetic domains. Efficient implementations of the angular and energetic components of the scheme are derived, and the resulting method is shown to converge with optimal convergence rates through a number of numerical examples. The assembly of the spatial scheme on general polytopic meshes is discussed in more detail, and an assembly algorithm based on employing quadrature-free integration is introduced. The quadrature-free assembly algorithm is benchmarked against a standard quadrature-based approach, and an analysis of the algorithm applied to a more general class of discontinuous Galerkin discretisations is performed. In view of developing efficient linear solvers for the system of equations resulting from our discontinuous Galerkin discretisation, we exploit the variational structure of the scheme to prove convergence results and derive a posteriori solver error estimates for a family of iterative solvers. These a posteriori solver error estimators can be used alongside standard implementations of the generalised minimal residual method to guarantee that the linear solver error between the exact and approximate finite element solutions (measured in a problem-specific norm) is below a user-specified tolerance. We discuss a family of transport-based preconditioners, and our linear solver convergence results are benchmarked through a family of numerical examples
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