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In this thesis, a parallel solver was developed for the modelling of blood flow through a number of patient-specific geometries. A locally conservative Galerkin (LCG) spatial discretisation was applied along with an artificial compressibility and characteristic based split (CBS) scheme to solve the 3D incompressible Navier-Stokes equations. The Spalart-Allmaras one equation turbulence model was also optionally employed. The solver was constructed using FORTRAN and the Message Passing Interface (MPI). Parallel testing demonstrated linear or better than linear speedup on hybrid patient-specific meshes. These meshes were unstructured with structured boundary layers. From the parallel testing it is clear that the significance of inter-processor communication is negligible in a three dimensional case. Preliminary tests on a short patient-specific carotid geometry demonstrated the need for ten or more boundary layer meshes in order to sufficiently resolve the peak wall shear stress (WSS) along with the peak time-averaged WSS. A time sensitivity study was also undertaken along with the assessment of the order of the real time step term. Three backward difference formulae (BDF) were tested and no significant difference between them was detected. Significant speedup was possible as the order of time discretisation increased however, making the choice of BDF important in producing a timely solution. Followed by the preliminary investigation, four more carotid geometries were investigated in detail. A total of six haemodynamic wall parameters have been brought together to analyse the regions of possible atherogenesis within each carotid. The investigations revealed that geometry plays an overriding influence on the wall parameter distribution. Each carotid artery displayed high time-averaged WSS at the apex, although the value increased significantly with a proximal stenosis. Two out of four meshes contained a region of low time-averaged WSS distal to the flow divider and within the largest connecting artery (internal or external carotid artery), indicating a potential region of atherosclerosis plaque formation. The remaining two meshes already had a stenosis in the corresponding region. This is in excellent agreement with other established works. From the investigations, it is apparent that a classification system of stenosis severity may be possible with potential application as a clinical diagnosis aid. Finally, the flow within a thoracic aortic aneurysm was investigated in order to assess the influence of a proximal folded neck. The folded neck had a significant effect on the wall shear stress, increasing by up to 250% over an artificially smoothed neck. High wall shear stresses may be linked to aneurysm rupture. Being proximal to the aneurysm, this indicated that local geometry should be taken into account when assessing the rupture potential of an aneurysm.

Incompressible flows -- flows in which variations in the density of a fluid are negligible -- arise in a wide variety of applications, from hydraulics to aerodynamics. The incompressible Navier-Stokes equations which govern such flows are also of fundamental physical and mathematical interest. They are believed to hold the key to understanding turbulent phenomena; precise conditions for the existence and uniqueness of solutions remain unknown -- and establishing such conditions is the subject of one of the Clay Mathematics Institute's Millennium Prize Problems. Typical solutions of incompressible flow problems involve both fine- and large-scale phenomena, so that a uniform finite element mesh of sufficient granularity will at best be wasteful of computational resources, and at worst be infeasible because of resource limitations. Thus adaptive mesh refinements are required. In industry, the adaptivity schemes used are ad hoc, requiring a domain expert to predict features of the solution. A badly chosen mesh may cause the code to take considerably longer to converge, or fail to converge altogether. Typically, the Navier-Stokes solve will be just one component in an optimization loop, which means that any failure requiring human intervention is costly. Therefore, I pursue technological foundations for a solver of the incompressible Navier-Stokes equations that provides robust adaptivity starting with a coarse mesh. By robust, I mean both that the solver always converges to a solution in predictable time, and that the adaptive scheme is independent of the problem -- no special expertise is required for adaptivity. The cornerstone of my approach is the discontinuous Petrov-Galerkin (DPG) finite element methodology developed by Leszek Demkowicz and Jay Gopalakrishnan. For a large class of problems, DPG can be shown to converge at optimal rates. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements. Several approximations to Navier-Stokes are of interest, and I study each of these in turn, culminating in the study of the steady 2D incompressible Navier-Stokes equations. The Stokes equations can be obtained by neglecting the convective term; these are accurate for "creeping" viscous flows. The Oseen equations replace the convective term, which is nonlinear, with a linear approximation. The steady-state incompressible Navier-Stokes equations approximate the transient equations by neglecting time variations. Crucial to this work is Camellia, a toolbox I developed for solving DPG problems which uses the Trilinos numerical libraries. Camellia supports 2D meshes of triangles and quads of variable polynomial order, allows simple specification of variational forms, supports h- and p-refinements, and distributes the computation of the stiffness matrix, among other features. The central contribution of this dissertation is design and development of mathematical techniques and software, based on the DPG method, for solving the 2D incompressible Navier-Stokes equations in the laminar regime (Reynolds numbers up to about 1000). Along the way, I investigate approximations to these equations -- the Stokes equations and the Oseen equations -- followed by the steady-state Navier-Stokes equations. / text

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Shallow water equations (SWEs) are a set of hyperbolic partial differential equations that describe the flow below a pressure surface in a fluid. They are widely applicable in the domain of fluid dynamics. To meet the needs of engineers working on the area of fluid dynamics, a method known as spectral/hp element method has been developed which is a scheme that can be used with complicated geometry. The use of discontinuous Galerkin (DG) discretisation permits discontinuity of the numerical solution to exist at inter-element surfaces. In the DG method, the solution within each element is not reconstructed by looking to neighbouring elements, thus the transfer information between elements will be ensured through the numerical fluxes. As a consequence, the accuracy of the method depends largely on the definition of the numerical fluxes. There are many different type of numerical fluxes computed from Riemann solvers. Four of them will be applied here respectively for comparison through a 2D Rossby wave test case. / AFRIKAANSE OPSOMMING: Vlakwatervergelykings (SWEs) is ’n stel hiperboliese parsiële differensiaalvergelykings wat die vloei onder ’n oppervlak wat druk op ’n vloeistof uitoefen beskryf. Hulle het wye toepassing op die gebied van vloeidinamika. Om aan die behoeftes van ingenieurs wat werk op die gebied van vloeidinamika te voldoen is ’n metode bekend as die spektraal /hp element metode ontwikkel. Hierdie metode kan gebruik word selfs wanneer die probleem ingewikkelde grenskondisies het. Die Diskontinue Galerkin (DG) diskretisering wat gebruik word laat diskontinuïteit van die numeriese oplossing toe om te bestaan by tussenelement oppervlakke. In die DG metode word die oplossing binne elke element nie gerekonstrueer deur te kyk na die naburige elemente nie. Dus word die oordrag van informasie tussen elemente verseker deur die numeriese stroomterme. Die akkuraatheid van hierdie metode hang dus grootliks af van die definisie van die numeriese stroomterme. Daar is baie verskillende tipe numeriese strometerme wat bereken kan word uit Riemann oplossers. Vier van hulle sal hier gebruik en vergelyk word op ’n 2D Rossby golf toets geval.

Dissertation (PhD)--University of Stellenbosch, 2002. / ENGLISH ABSTRACT: This document presents a Galerkin FE formulation for the full-wave, frequency domain, electromagnetic analysis of three dimensional structures relevant to microwave engineering, together with the investigation of two techniques to enhance the formulation's computational efficiency. The first technique considered is the fast multi pole method (FMM) and the second technique is adaptive refinement of the discretization, based on a posteriori error estimation. Thus, the motivation for the work presented in this document is to increase the computational efficiency of the FE formulation considered. The FE formulation considered is widely used within the microwave engineering, finite element community. Tetrahedral, rectilinear, curl-conforming, mixed- and full order, hierarchical vector elements are used. The formulation is extended to incorporate a cavity backed aperture employing the appropriate half-space Green function within a BI boundary condition, which represents a specific member of a large class of hybrid FE-BI formulations. The formulation is also extended to model coaxial ports via a Neumann boundary condition, using a priori knowledge of the dominant modal fields. Results are presented in support of the formulation and its extensions, including novel results on the coupling between microstrip patch antennas on a perforated substrate. The FMM is investigated first, with the purpose of optimizing the non-local BI component of the cavity FE-BI formulation, in light of its coupling with the differential equation based, sparse FEM. The FMM results in a partly sparse factorization of the BI contribution to the system matrix. Error control schemes for the FMM are thoroughly reviewed and an additional, novel scheme is empirically devised. The second technique investigated, which is more directly related to the FEM and larger in scope, is the use of a posteriori error estimation, in order to optimize the FE discretization through adaptive refinement. A overview of available a posteriori error estimation techniques in the general FE literature is given as well as a survey of available techniques that are specifically tailored to Maxwell's equations. Two known approaches within the applied mathematics literature are adapted to the FE formulation at hand, resulting in two novel, residual based error estimation procedures for this FE formulation - one explicit in nature and the other implicit. The two error estimators are then used to drive a single p adaptive analysis cycle of the FE formulation, experimentally demonstrating their effectiveness. A quasi-static condition is introduced and successfully used to enhance the adaptive algorithm's effectiveness, independently of the error estimation procedure employed. The novel error estimation schemes and adaptive results represent the main research contributions of this study. / AFRIKAANSE OPSOMMING: Hierdie dokument beskryf 'n Galerkin eindige element (EE) formulering vir die volgolf, frekwensiegebied, elektromagnetiese analise van driedimensionele strukture relevant vir mikrogolfingenieurwese, saam met die ondersoek van twee tegnieke om die numeriese effektiwiteit van die formulering te verbeter. Die eerste tegniek wat ondersoek word, is die vinnige multipooi metode (VMM) en die tweede is die aanpasbare verfyning van die EE diskretisering, gebaseer op a posteriori foutberaming. Dus, die motivering vir hierdie werk is om die numeriese effektiwiteit van die genoemde EE formulering te verbeter. Die bogenoemde EE formulering word algemeen gebruik deur die mikrogolfingenieurswese, eindige element-gemeenskap. Tetrahedriese, reglynige, curl-ondersteunende, hierargiese vektorelemente van gemengde- en volledige ordes word gebruik. Die formulering word uitgebrei om holtes in 'n oneindige grondvlak te kan hanteer, deur gebruik te maak van die toepaslike Green funksie binne 'n grensintegraal (GI) grensvoorwaarde, wat 'n spesifieke lid is van 'n groot klas, hibriede, EE-GI formulerings. Die formulering word ook uitgebrei om koaksiale poorte to modelleer via 'n Neumann grensvoorwaarde, deur die gebruik van a priori kennis van die koaksiale, dominante modus-velde. Resultate word gelewer om die formulering, saam met die uitbreidings daarvan, te ondersteun, insluitende oorspronklike resultate in verband met die koppeling tussen mikrostrook plakantennes op 'n geperforeerde substraat. Die VMM word eerste ondersoek, met die doelom die nie-lokale, GI komponent van die EEGI formulering vir holtes te optimeer, weens die koppeling daarvan met die yl, differensiaalvergelyking- gebaseerde, eindige element-metode. Die VMM lei tot 'n gedeeltelik-yl faktorisering van die GI bydrae tot die algehele matriksvergelyking. Skemas om die VMM fout te beheer word deeglik ondersoek en 'n addisionele, oorspronklike skema word empiries ontwikkel. Die tweede tegniek wat ondersoek word, wat meer direk verband hou met die eindige elementmetode, en van groter omvang is, is die gebruik van a posteriori foutberaming om die EE diskretisasie te optimeer deur middel van aanpasbare verfyning. 'n Oorsig van beskikbare, a posteriori foutberamingstegnieke in die algemene EE literatuur word gegee, asook 'n opname van beskikbare tegnieke wat spesifiek gerig is op Maxwell se vergelykings. Twee bekende benaderings binne die toegepaste wiskunde-literatuur word aangepas by die bogenoemde EE formulering, wat lei tot twee oorspronklike residu-gebaseerde foutberamingstegnieke vir hierdie formulering - een van 'n eksplisiete aard en die ander implisiet. Die twee foutberamingstegnieke word gebruik om 'n enkel, p-aanpasbare analisesiklus aan te dryf, wat die effektiwiteit van die foutberamingstegnieke eksperimenteel demonstreer. 'n Kwasi-statiese vereiste word beskryf en suksesvol gebruik om die aanpasbare algoritme se effektiwiteit te verhoog, onafhanklik van die foutberamingstegniek wat gebruik word. Die oorspronklike foutberamingstegnieke en aanpasbare algoritme-resultate verteenwoordig die hoof navorsingsbydraes van hierdie studie.

published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

The Boltzmann Transport Equation (BTE) has been the keystone of the kinetic theory, which is at the center of Statistical Mechanics bridging the gap between the atomic structures and the continuum-like behaviors. The existence of solutions has been a great mathematical challenge and still remains elusive. As a grazing limit of the Boltzmann operator, the Fokker-Planck-Landau (FPL) operator is of primary importance for collisional plasmas. We have worked on the following three different projects regarding the most important kinetic models, the BTE and the FPL Equations. (1). A Discontinuous Galerkin Solver for Nonlinear BTE. We propose a deterministic numerical solver based on Discontinuous Galerkin (DG) methods, which has been rarely studied. As the key part, the weak form of the collision operator is approximated within subspaces of piecewise polynomials. To save the tremendous computational cost with increasing order of polynomials and number of mesh nodes, as well as to resolve loss of conservations due to domain truncations, the following combined procedures are applied. First, the collision operator is projected onto a subspace of basis polynomials up to first order. Then, at every time step, a conservation routine is employed to enforce the preservation of desired moments (mass, momentum and/or energy), with only linear complexity. The asymptotic error analysis shows the validity and guarantees the accuracy of these two procedures. We applied the property of ``shifting symmetries" in the weight matrix, which consists in finding a minimal set of basis matrices that can exactly reconstruct the complete family of collision weight matrix. This procedure, together with showing the sparsity of the weight matrix, reduces the computation and storage of the collision matrix from O(N3) down to O(N^2). (2). Spectral Gap for Linearized Boltzmann Operator. Spectral gaps provide information on the relaxation to equilibrium. This is a pioneer field currently unexplored form the computational viewpoint. This work, for the first time, provides numerical evidence on the existence of spectral gaps and corresponding approximate values. The linearized Boltzmann operator is projected onto a Discontinuous Galerkin mesh, resulting in a ``collision matrix". The original spectral gap problem is then approximated by a constrained minimization problem, with objective function the Rayleigh quotient of the "collision matrix" and with constraints the conservation laws. A conservation correction then applies. We also study the convergence of the approximate Rayleigh quotient to the real spectral gap. (3). A Conservative Scheme for Approximating Collisional Plasmas. We have developed a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equations coupled with Poisson equations. The original problem is splitted into two subproblems: collisonless Vlasov problem and collisonal homogeneous Fokker-Planck-Landau problem. They are handled with different numerical schemes. The former is approximated using Runge-Kutta Discontinuous Galerkin (RKDG) scheme with a piecewise polynomial basis subspace covering all collision invariants; while the latter is solved by a conservative spectral method. To link the two different computing grids, a special conservation routine is also developed. All the projects are implemented with hybrid MPI and OpenMP. Numerical results and applications are provided. / text

My main research topic is about developing new domain decomposition algorithms for the solution of systems of partial differential equations. This was mainly applied to fluid dynamics problems (as compressible Euler or Stokes equations) and electromagnetics (time-harmonic and time-domain first order system of Maxwell's equations). Since the solution of large linear systems is strongly related to the application of a discretization method, I was also interested in developing and analyzing the application of high order methods (such as Discontinuos Galerkin methods) to Maxwell's equations (sometimes in conjuction with time-discretization schemes in the case of time-domain problems). As an active member of NACHOS pro ject (besides my main afiliation as an assistant professor at University of Nice), I had the opportunity to develop certain directions in my research, by interacting with permanent et non-permanent members (Post-doctoral researchers) or participating to supervision of PhD Students. This is strongly refflected in a part of my scientific contributions so far. This memoir is composed of three parts: the first is about the application of Schwarz methods to fluid dynamics problems; the second about the high order methods for the Maxwell's equations and the last about the domain decomposition algorithms for wave propagation problems.

We study two finite element methods for solving time-harmonic electromagnetic and acoustic problems: the discontinuous Petrov-Galerkin (DPG) method and the hybrid discontinuous Galerkin (HDG) method. The DPG method for the Helmholtz equation is studied using a test space normed by a modified graph norm. The modification scales one of the terms in the graph norm by an arbitrary positive scaling parameter. We find that, as the parameter approaches zero, better results are obtained, under some circumstances. A dispersion analysis on the multiple interacting stencils that form the DPG method shows that the discrete wavenumbers of the method are complex, explaining the numerically observed artificial dissipation in the computed wave approximations. Since the DPG method is a nonstandard least-squares Galerkin method, its performance is compared with a standard least-squares method having a similar stencil. We study the HDG method for complex wavenumber cases and show how the HDG stabilization parameter must be chosen in relation to the wavenumber. We show that the commonly chosen HDG stabilization parameter values can give rise to singular systems for some complex wavenumbers. However, this failure is remedied if the real part of the stabilization parameter has the opposite sign of the imaginary part of the wavenumber. For real wavenumbers, results from a dispersion analysis for the Helmholtz case are presented. An asymptotic expansion of the dispersion relation, as the number of mesh elements per wave increase, reveal values of the stabilization parameter that asymptotically minimize the HDG wavenumber errors. Finally, a dispersion analysis of the mixed hybrid Raviart-Thomas method shows that its wavenumber errors are an order smaller than those of the HDG method. We conclude by presenting some contributions to the development of software tools for using the DPG method and their application to a terahertz photonic structure. We attempt to simulate field enhancements recently observed in a novel arrangement of annular nanogaps.

O Método dos Elementos Finitos é a forma mais difundida de análise estrutural numérica, com aplicações nas mais diversas teorias estruturais. Contudo, no estudo das cascas e alguns outros usos, suas deficiências impulsionaram a pesquisa em outros métodos de resolução de Equações Diferenciais Parciais. O presente trabalho utiliza uma dessas alternativas, o Método de Galerkin Livre de Elementos (Element-Free Galerkin) para estudar as cascas. Inicia com a observação da aproximação usada no método, os Moving Least Squares e os Multiple-Fixed Least Squares. A seguir, estabelece uma formulação que combina a teoria de placas moderadamente espessas de Reissner-Mindlin à teoria da Elasticidade Plana e se utiliza da aproximação estudada para analisar placas e chapas deste tipo. Depois, expõe uma teoria geometricamente exata de cascas inicialmente curvas onde as curvaturas iniciais são impostas como deformações livres de tensão a partir de uma configuração de referência plana. Tal teoria exclui a necessidade de coordenadas curvilíneas e consequentemente da utilização de objetos como os símbolos de Cristoffel, já que todas as integrações e imposições são feitas na configuração plana de referência, em um sistema ortonormal de coordenadas. A imposição das condições essenciais de contorno é feita por forma fraca, resultando em um funcional híbrido de deslocamentos que permite a maleabilidade necessária ao uso dos Moving Least Squares. Esse trabalho se propõe a particularizar tal teoria para o caso de pequenos deslocamentos e deformações (linearidade geométrica), mantendo a consistência das definições de tensões e deformações generalizadas enquanto permite uma imposição da forma fraca resultante, depois de discretizada, por um sistema linear de equações. Por fim, exemplos numéricos são usados para discutir sua eficácia e exatidão. / The Finite Element Method is the most spread numerical analysis tool, applied to a wide range of structural theories. However, for the study of shells and other problems, some of its deficiencies have stimulated research in other methods for solving the derived Partial Differential Equations. The present work uses one of those alternatives, the Element Free Galerkin Method, for the study of shells. It begins with the observation of the approximation used in the method, Moving Least Squares and Multiple-Fixed Least Squares. Then, it establishes a formulation that combines the Reissner-Mindlin moderately thick plate theory with plane elasticity, and uses the proponed approximation to analyze such plates and stabs. Afterwards, it demonstrates a geometrically exact shell theory that accounts for initial curvatures as a stress-free deformation from a flat reference configuration. Such theory precludes the use of curvilinear coordinates and, subsequently, the use of objects such as Cristoffel symbols, as all integrations and impositions are done in the flat reference configuration, in an orthogonal frame. The essential boundary conditions are imposed in a eak statement, rendering a hybrid displacement functional that provides the necessary conditions for the use of Moving Least Squares. This works main objective is the particularization of this theory for the small displacement and strains assumption (geometrical linearity), keeping the consistent definition of generalized stresses and strains, while allowing the imposition of the discretized weak form through a system of linear equations. Lastly, numerical simulations are carried out to assess the methods efficiency and accuracy.