The implementation of a general-purpose tetrahedral mesh phantom geometry library for the Monte Carlo radiation transport code EGSnrc is described. Recently, tetrahedral mesh geometries have been proposed as standard reference phantoms to advance the state of the art over rectilinear voxel phantoms. Prior to this work, EGSnrc already supported voxelized geometries, but not tetrahedral meshes. Other major radiation transport codes such as MCNP6, Geant4, and PHITS, are also capable of simulating the interaction of ionizing radiation with tetrahedral mesh phantoms. Tetrahedral mesh phantoms have a number of advantages over voxel phantoms including improved modelling fidelity and locally varying element resolution. In addition, CAD geometries can be converted into meshes, which can then be directly used in simulations. In this work, an EGSnrc tetrahedral mesh geometry library called EGS_Mesh is implemented. The implementation uses fast computational geometry algorithms from the literature and is accelerated using an octree spatial partitioning scheme. For a preliminary verification, results obtained using EGS_Mesh are compared to classical EGSnrc geometries and theoretical results (including a Fano test) and found to match within 0.1%. To demonstrate the capability of EGS_Mesh to simulate transport in complex mesh phantoms from the literature, results using the ICRP 145 reference human phantoms are compared to published results obtained using Geant4. The comparison has found agreement mostly within 5% of the Geant4 results, but with some differences up to 10%.

This thesis considers the application of the type II third order WENO finite volume reconstruction for unstructured tetrahedral meshes proposed by Zhang and Shu in (CCP, 2009) and the third order multirate Runge-Kutta time-stepping to the solution of Maxwell's equations. The dependance of accuracy of the third order WENO scheme on the small parameter in the definition of non-linear weights is studied in detail for one-dimensional uniform meshes and numerical results confirming the theoretical analysis are presented for the linear advection equation. This analysis is found to be crucial in the design of the efficient three-dimensional WENO scheme, full details of which are presented. Several multirate Runge-Kutta (MRK) schemes which advance the solution with local time-steps assigned to different multirate groups are studied. Analysis of accuracy of three different MRK approaches for linear problems based on classic order-conditions is presented. The most flexible and efficient multirate schemes based on works by Tang and Warnecke (JCM, 2006) and Liu, Li and Hu (JCP, 2010) are implemented in three-dimensional finite volume time-domain (FVTD) method. The main characteristics of chosen MRK schemes are flexibility in defining the time-step ratios between multirate groups and consistency of the scheme. Various approaches to partition the three-dimensional computational domain into multirate groups to maximize the achievable speedup are discussed. Numerical experiments with three-dimensional electromagnetic problems are presented to validate the performance of the proposed FVTD method. Three-dimensional results agree with theoretical and numerical accuracy analysis performed for the one-dimensional case. The proposed implementation of multirate schemes demonstrates greater speedup than previously reported in literature. / February 2016

Some boundary value problems yield anisotropic solutions, e.g. solutions with boundary layers. If such problems are to be solved with the finite element method (FEM), anisotropically refined meshes can be advantageous. In order to construct these meshes or to control the error one aims at reliable error estimators. For isotropic meshes such estimators are known but they fail when applied to anisotropic meshes. Rectangular (or cuboidal) anisotropic meshes were already investigated. In this paper an error estimator is presented for tetrahedral or triangular meshes which offer a much greater geometrical flexibility.

Many architecure, medical and engineering applications need a spacial support for various numerical computations (i.e. FEM simulations). Tetrahedral meshes are one of perspective spatial representations for them. In this thesis, several possibilities of effective tetrahedral mesh representation for its generating and processing are described. A computer library for the mesh processing is proposed which can be characterized by memory  efficient imposition of the mesh while preserving the ability to apply topological and geometric algorithms effectively. The library is implemented in C++ language using templates. Time and space complexity of typical mesh operations is compared with CGAL library and according to measurements the proposed library has lower memory requirements than CGAL.

No description available.

Developments are reported of unstructured-mesh methods for simulating stratified, turbulent and shear flows. The numerical model employs nonoscillatory forward in-time integrators for anelastic and incompressible flow PDEs, built on Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) and a preconditioned conjugate residual elliptic solver. Finite-volume spatial discretisation adopts an edge-based data structure. Tetrahedral-based and hybrid-based median-dual options for unstructured meshes are developed, enabling flexible spatial resolution. Viscous laminar and detached eddy simulation (DES) flow solvers are developed based on the edge-based NFT MPDATA scheme. The built-in implicit large eddy simulation (ILES) capability of the NFT scheme is also employed and extended to fully unstructured tetrahedral and hybrid meshes. Challenging atmospheric and engineering problems are solved numerically to validate the model and to demonstrate its applications. The numerical problems include simulations of stratified, turbulent and shear flows past obstacles involving complex gravity-wave phenomena in the lee, critical-level laminar-turbulence transitioning and various vortex structures in the wake. Qualitative flow patterns and quantitative data analysis are both presented in the current study.

Many physical problems lead to boundary value problems for partial differential equations, which can be solved with the finite element method. In order to construct adaptive solution algorithms or to measure the error one aims at reliable a posteriori error estimators. Many such estimators are known, as well as their theoretical foundation. Some boundary value problems yield so-called anisotropic solutions (e.g. with boundary layers). Then anisotropic finite element meshes can be advantageous. However, the common error estimators for isotropic meshes fail when applied to anisotropic meshes, or they were not investigated yet. For rectangular or cuboidal anisotropic meshes a modified error estimator had already been derived. In this paper error estimators for anisotropic tetrahedral or triangular meshes are considered. Such meshes offer a greater geometrical flexibility. For the Poisson equation we introduce a residual error estimator, an estimator based on a local problem, several Zienkiewicz-Zhu estimators, and an L_2 error estimator, respectively. A corresponding mathematical theory is given.For a singularly perturbed reaction-diffusion equation a residual error estimator is derived as well. The numerical examples demonstrate that reliable and efficient error estimation is possible on anisotropic meshes. The analysis basically relies on two important tools, namely anisotropic interpolation error estimates and the so-called bubble functions. Moreover, the correspondence of an anisotropic mesh with an anisotropic solution plays a vital role. AMS(MOS): 65N30, 65N15, 35B25

The objective of this PhD research programme is to investigate the effectiveness of a finite volume method incorporating radial basis functions for simulating nonlinear transport processes. The finite volume method is the favoured numerical technique for solving the advection-diffusion equations that arise in transport simulation. The method transforms the original problem into a system of nonlinear, algebraic equations through the process of discretisation. The accuracy of this discretisation determines to a large extent the accuracy of the final solution. A new method of discretisation is presented that employs radial basis functions (rbfs) as a means of local interpolation. When combined with Gaussian quadrature integration methods, the resulting finite volume discretisation leads to accurate numerical solutions without the need for very fine meshes, and the additional overheads they entail. The resulting nonlinear, algebraic system is solved efficiently using a Jacobian-free Newton-Krylov method. By employing the new method as an extension of existing shape function-based approaches, the number of nonlinear iterations required to obtain convergence can be reduced. Furthermore, information obtained from these iterations can be used to increase the efficiency of subsequent rbf-based iterations, as well as to construct an effective parallel reconditioner to further reduce the number of nonlinear iterations required. Results are presented that demonstrate the improved accuracy offered by the new method when applied to several test problems. By successively refining the meshes, it is also possible to demonstrate the increased order of the new method, when compared to a traditional shape function basedmethod. Comparing the resources required for both methods reveals that the new approach can be many times more efficient at producing a solution of a given accuracy.

Some boundary value problems yield anisotropic solutions, e.g. solutions with boundary layers. If such problems are to be solved with the finite element method (FEM), anisotropically refined meshes can be advantageous. In order to construct these meshes or to control the error one aims at reliable error estimators. For isotropic meshes such estimators are known but they fail when applied to anisotropic meshes. Rectangular (or cuboidal) anisotropic meshes were already investigated. In this paper an error estimator is presented for tetrahedral or triangular meshes which offer a much greater geometrical flexibility.

Many physical problems lead to boundary value problems for partial differential equations, which can be solved with the finite element method. In order to construct adaptive solution algorithms or to measure the error one aims at reliable a posteriori error estimators. Many such estimators are known, as well as their theoretical foundation. Some boundary value problems yield so-called anisotropic solutions (e.g. with boundary layers). Then anisotropic finite element meshes can be advantageous. However, the common error estimators for isotropic meshes fail when applied to anisotropic meshes, or they were not investigated yet. For rectangular or cuboidal anisotropic meshes a modified error estimator had already been derived. In this paper error estimators for anisotropic tetrahedral or triangular meshes are considered. Such meshes offer a greater geometrical flexibility. For the Poisson equation we introduce a residual error estimator, an estimator based on a local problem, several Zienkiewicz-Zhu estimators, and an L_2 error estimator, respectively. A corresponding mathematical theory is given.For a singularly perturbed reaction-diffusion equation a residual error estimator is derived as well. The numerical examples demonstrate that reliable and efficient error estimation is possible on anisotropic meshes. The analysis basically relies on two important tools, namely anisotropic interpolation error estimates and the so-called bubble functions. Moreover, the correspondence of an anisotropic mesh with an anisotropic solution plays a vital role. AMS(MOS): 65N30, 65N15, 35B25