We are interested in numerical methods for the optimization constrained second order ordinary differential equations arising in biofilm modelling. This class of problems is challenging for several reasons. One of the reasons is that the underlying solution has a steep slope, making it difficult to resolve. We propose a new numerical method with techniques such as domain decomposition and asynchronous iterations for solving certain types of ordinary differential equations more efficiently. In fact, for our class of problems after applying the techniques of domain decomposition with overlap we are able to solve the ordinary differential equations with a steep slope on a larger domain than previously possible. After applying asynchronous iteration techniques, we are able to solve the problem with less time.~We provide theoretical conditions for the convergence of each of the techniques. The other reason is that the second order ordinary differential equations are coupled with an optimization problem, which can be viewed as the constraints. We propose a numerical method for solving the coupled problem and show that it converges under certain conditions. An application of the proposed methods on biofilm modeling is discussed. The numerical method proposed is adopted to solve the biofilm problem, and we are able to solve the problem with larger thickness of the biofilm than possible before as is shown in the numerical experiments. / Mathematics

The electrical activity of the heart is a well studied process. Mathematical modeling and computer simulations are used to study the cardiac electrical activity: several mathematical models exist, among them the microscopic model, which is based on the explicit representation of individual cells. The cardiac tissue is viewed as two separate domains: the intra-cellular and extra-cellular domains, Ωᵢ and Ωₑ, respectively, separated by cellular membranes Γ. The microscopic model consists of a set of Poisson equations, one for each sub-domain, Ωᵢ and Ωₑ, coupled on interfaces Γ with nonlinear transmission conditions involving a system of ODEs. The unusual transmission conditions on Γ make the model challenging to solve numerically. In this thesis, we first focus on the dimensional analysis of the microscopic model. We then reformulate the problem on the interface Γ using a Steklov-Poincaré operator. We discretize the model in space using finite element methods. We prove the existence of a semi-discrete solution using a reformulation of the model as an ODE system on the interface Γ. We derive stability and error estimates for the finite element method. Afterwards, we consider five numerical schemes including the Godunov splitting method, two implicit methods, (Backward Euler (BE) and second order Backward Differentiation Formula (BDF2)), and two semi-implicit methods (Forward Backward Euler (FBE), and second order Semi-implicit Backward Differentiation Formula (SBDF2)). A convergence analysis of the implicit and semi-implicit methods is performed and the results are compared with manufactured solutions that we have proposed. Numerical results are presented to compare the stability, accuracy and efficiency of the methods. CPU times needed to solve the problem over a single cell using FBE, SBDF2 and Godunov splitting methods are reported. The results show that FBE and Godunov splitting methods achieve better numerical accuracy and efficiency than implicit and SBDF2 schemes, for a given computational time. Finally, we solve the model using FBE and Domain Decomposition Method (DDM) for two cells connected to each other by a gap junction. We investigate the influence of the space discretization and we explore the differences between a conforming and nonconforming mesh on Γ. We compare the solutions obtained with both FBE and DDM methods. The results show that both methods give the same solution. Therefore, the DDM is capable of providing an accurate solution with a minimal number of sub-domain iterations.

Computational Fluid Dynamics (CFD) applications usually involve intensive computations, which can be accelerated through using open accelerators, especially GPUs due to their common use in the scientific computing community. In addition to code acceleration, it is important to ensure that the code and algorithm are implemented numerically correctly, which is called code verification. This dissertation focuses on accelerating research CFD codes on multi-CPUs/GPUs using MPI and OpenACC, as well as the code verification for turbulence model implementation using the method of manufactured solutions and code-to-code comparisons. First, a variety of performance optimizations both agnostic and specific to applications and platforms are developed in order to 1) improve the heterogeneous CPU/GPU compute utilization; 2) improve the memory bandwidth to the main memory; 3) reduce communication overhead between the CPU host and the GPU accelerator; and 4) reduce the tedious manual tuning work for GPU scheduling. Both finite difference and finite volume CFD codes and multiple platforms with different architectures are utilized to evaluate the performance optimizations used. A maximum speedup of over 70 is achieved on 16 V100 GPUs over 16 Xeon E5-2680v4 CPUs for multi-block test cases. In addition, systematic studies of code verification are performed for a second-order accurate finite volume research CFD code. Cross-term sinusoidal manufactured solutions are applied to verify the Spalart-Allmaras and k-omega SST model implementation, both in 2D and 3D. This dissertation shows that the spatial and temporal schemes are implemented numerically correctly. / Doctor of Philosophy / Computational Fluid Dynamics (CFD) is a numerical method to solve fluid problems, which usually requires a large amount of computations. A large CFD problem can be decomposed into smaller sub-problems which are stored in discrete memory locations and accelerated by a large number of compute units. In addition to code acceleration, it is important to ensure that the code and algorithm are implemented correctly, which is called code verification. This dissertation focuses on the CFD code acceleration as well as the code verification for turbulence model implementation. In this dissertation, multiple Graphic Processing Units (GPUs) are utilized to accelerate two CFD codes, considering that the GPU has high computational power and high memory bandwidth. A variety of optimizations are developed and applied to improve the performance of CFD codes on different parallel computing systems. The program execution time can be reduced significantly especially when multiple GPUs are used. In addition, code-to-code comparisons with some NASA CFD codes and the method of manufactured solutions are utilized to verify the correctness of a research CFD code.

This thesis uses Proper Orthogonal Decomposition to model parametric and coupled systems. First, Proper Orthogonal Decomposition and its properties are introduced as well as how to numerically compute the decomposition. Next, a test case was used to show how well POD can be used to simulate and control a system. Finally, techniques for modeling a parametric system over a given range and a coupled system split into subdomains were explored, as well as numerical results. / Master of Science

Three-dimensional (3-D) system integration is believed to be a promising technology and has gained tremendous momentum in the semiconductor industry recently. The Silicon interposer is the key enabler for the 3-D systems, and is expected to have high input/output counts, fine wiring lines and many TSVs. Modeling and design of the silicon interposer can be challenging and is becoming a critical task. This dissertation mainly focuses on developing an efficient modeling approach for silicon interposers in 3-D systems. The developed numerical methods can be classified as several categories. 1. The investigation of the coupling effects in large TSV arrays in silicon interposers. The importance of coupling between TSVs for low resistivity silicon substrates is quantified both in frequency and time domains. This has been compared with high resistivity silicon substrates. 2. The development of an electromagnetic modeling approach for non-uniform TSVs. To model the complex TSV structures, an approach for modeling conical TSVs is proposed first. Later a hybrid modeling method which combines the conical TSV modeling method and cylindrical modeling method is proposed to model the non-uniform TSV structures. 3. The development of a hybrid modeling approach for power delivery networks (PDN) with through-silicon vias (TSVs). The proposed approach extends multi-layer finite difference method (M-FDM) to include TSVs by extracting their parasitic behavior using an integral equation based solver. 4. The development of an efficient approach for modeling signal paths with TSVs in silicon interposers. The proposed method utilizes the 3-D finite-difference frequency-domain (FDFD) method to model the redistribution layer (RDL) transmission lines. A new formulation on incorporating multiport networks into the 3-D FDFD formulation is presented to include the parasitic effects of TSV arrays in the system matrix. 5. The development of a 3-D FDFD non-conformal domain decomposition method. The proposed method allows modeling individual domains independently using the FDFD method with non-matching meshing grids at interfaces. This non-conformal domain decomposition method is applied to model interconnections in silicon interposer.

Thesis (PhD) -- Stellenbosch University, 2014. / ENGLISH ABSTRACT: This work considers the efficient numerical analysis of large, aperiodic finite antenna arrays. A Method of Moments (MoM) based domain decomposition technique called the Domain Green's Function Method (DGFM) is formulated to address a wide range of array problems in a memory and runtime efficient manner. The DGFM is a perturbation approach that builds on work initially conducted by Skrivervik and Mosig for disjoint arrays on multi-layered substrates, a detailed review of which will be provided in this thesis. Novel extensions considered for the DGFM are as follows: a formulation on a higher block matrix factorisation level that allows for the treatment of a wider range of applications, and is essentially independent of the elemental basis functions used for the MoM matrix formulation of the problem. As an example of this, both conventional Rao-Wilton-Glisson elements and also hierarchical higher order basis functions were used to model large array structures. Acceleration techniques have been developed for calculating the impedance matrix for large arrays including one based on using the Adaptive Cross Approximation (ACA) algorithm. Accuracy improvements that extend the initial perturbation assumption on which the method is based have also been formulated. Finally, the DGFM is applied to array geometries in complex environments, such as that in the presence of finite ground planes, by using the Numerical Green's Function (NGF) method in the hybrid NGF-DGFM formulation. In addition to the above, the DGFM is combined with the existing domain decomposition method, viz., the Characteristic Basis Function Method (CBFM), to be used for the analysis of very large arrays consisting of sub-array tiles, such as the Low-Frequency Array (LOFAR) for radio astronomy. Finally, interesting numerical applications for the DGFM are presented, in particular their usefulness for the electromagnetic analysis of large, aperiodic sparse arrays. For this part, the accuracy improvements of the DGFM are used to calculate quantities such as embedded element patterns, which is a major extension from its original formulation. The DGFM has been integrated as part of an efficient array analysis tool in the commercial computational electromagnetics software package, FEKO. / AFRIKAANSE OPSOMMING: In hierdie werkstuk word die doeltre ende analise van eindige, aperiodiese antenna samestellings behandel. Eindige gebied benaderings wat op die Moment Metode (MoM) berus, word as vetrekpunt gebruik. `n Tegniek genaamd die Gebied Green's Funksie Metode (GGFM) word voorgestel en is geskik vir die analise van `n verskeidenheid van ontkoppelde samestellings. Die e ektiewe gebruik van rekenaargeheue en looptyd is onderliggend in die implementasie daarvan. Die GGFM is 'n perturbasie metode wat op die oorspronklike werk van Skrivervik en Mosig berus. Laasgenoemde is hoofsaaklik ontwikkel vir die analise van ontkoppelde antenna samestellings op multilaag di elektrikums. `n Deeglike oorsig van voorafgaande word in die tesis verskaf. In hierdie tesis is die bogenoemde werk op `n unieke wyse uitgebrei: `n ho er blok matriks vlak formulering is ontwikkel wat dit moontlik maak vir die analise van `n verskeidenheid strukture en wat onafhanklik is van die onderliggende basis funksies. Beide lae-vlak Rao-Wilton-Glisson (RWG) basis funksies, asook ho er orde hierargiese basis funksies word gebruik vir die modellering van groot antenna samestellings. Die oorspronklike perturbasie aanname is uitgebrei deur akkuraatheidsverbeteringe vir die tegniek voor te stel. Die Aanpasbare Kruis Benaderings (AKB) tegniek is onder andere gebruik om spoed verbeteringe vir die GGFM te bewerkstellig. Die GGFM is verder uitgebrei vir die analise van antenna samestellings in `n komplekse omgewing, bv. `n antenna samestelling bo `n eindige grondplaat. Die Numeriese Green's Funksie (NGF) metode is hiervoor ingespan en die hibriede NGF-GGFM is ontwikkel. Die GGFM is verder met die Karakteristieke Basis Funksie Metode (KBFM) gekombineer. Die analise van groot skikkings wat bestaan uit sub-skikkings, soos die wat tans by die \Low- Frequency Array (LOFAR) " vir radio astronomie in Nederland gebruik word, kan hiermee gedoen word. In die werkstuk word die GGFM ook toegepas op `n reeks interessante numeriese voorbeelde, veral die toepaslike EM analise van groot aperiodiese samestellings. Die akkuraatheidsverbeteringe vir die GGFM maak die berekening van elementpatrone vir skikkings moontlik. Die GGFM is by the sagteware pakket FEKO geintegreer.

In this thesis we study methods for solving the neutron transport equation (or linear Boltzmann equation). This is an integro-differential equation that describes the behaviour of neutrons during a nuclear fission reaction. Applications of this equation include modelling behaviour within nuclear reactors and the design of shielding around x-ray facilities in hospitals. Improvements in existing modelling techniques are an important way to address environmental and safety concerns of nuclear reactors, and also the safety of people working with or near radiation. The neutron transport equation typically has seven independent variables, however to facilitate rigorous mathematical analysis we consider the monoenergetic, steady-state equation without fission, and with isotropic interactions and isotropic source. Due to its high dimension, the equation is usually solved iteratively and we begin by considering a fundamental iterative method known as source iteration. We prove that the method converges assuming piecewise smooth material data, a result that is not present in the literature. We also improve upon known bounds on the rate of convergence assuming constant material data. We conclude by numerically verifying this new theory. We move on to consider the use of a specific, well-known diffusion equation to approximate the solution to the neutron transport equation. We provide a thorough presentation of its derivation (along with suitable boundary conditions) using an asymptotic expansion and matching procedure, a method originally presented by Habetler and Matkowsky in 1975. Next we state the method of diffusion synthetic acceleration (DSA) for which the diffusion approximation is instrumental. From there we move on to explore a new method of seeing the link between the diffusion and transport equations through the use of a block operator argument. Finally we consider domain decomposition algorithms for solving the neutron transport equation. Such methods have great potential for parallelisation and for the local application of different solution methods. A motivation for this work was to build an algorithm applying DSA only to regions of the domain where it is required. We give two very different domain decomposed source iteration algorithms, and we prove the convergence of both of these algorithms. This work provides a rigorous mathematical foundation for further development and exploration in this area. We conclude with numerical results to illustrate the new convergence theory, but also solve a physically-motivated problem using hybrid source iteration/ DSA algorithms and see significant reductions in the required computation time.

Cette thèse présente une vision unifiée de plusieurs méthodes de décomposition de domaine : celles avec recouvrement, dites de Schwarz, et celles basées sur des compléments de Schur, dites de sous-structuration. Il est ainsi possible de changer de méthodes de manière abstraite et de construire différents préconditionneurs pour accélérer la résolution de grands systèmes linéaires creux par des méthodes itératives. On rencontre régulièrement ce type de systèmes dans des problèmes industriels ou scientifiques après discrétisation de modèles continus. Bien que de tels préconditionneurs exposent naturellement de bonnes propriétés de parallélisme sur les architectures distribuées, ils peuvent s’avérer être peu performants numériquement pour des décompositions complexes ou des problèmes physiques multi-échelles. On peut pallier ces défauts de robustesse en calculant de façon concurrente des problèmes locaux creux ou denses aux valeurs propres généralisées. D’aucuns peuvent alors identifier des modes qui perturbent la convergence des méthodes itératives sous-jacentes a priori. En utilisant ces modes, il est alors possible de définir des opérateurs de projection qui utilisent un problème dit grossier. L’utilisation de ces outils auxiliaires règle généralement les problèmes sus-cités, mais tend à diminuer les performances algorithmiques des préconditionneurs. Dans ce manuscrit, on montre en trois points quela nouvelle construction développée est performante : 1) grâce à des essais numériques à très grande échelle sur Curie—un supercalculateur européen, puis en le comparant à des solveurs de pointe 2) multi-grilles et 3) directs. / This thesis introduces a unified framework for various domain decomposition methods:those with overlap, so-called Schwarz methods, and those based on Schur complements,so-called substructuring methods. It is then possible to switch with a high-level of abstractionbetween methods and to build different preconditioners to accelerate the iterativesolution of large sparse linear systems. Such systems are frequently encountered in industrialor scientific problems after discretization of continuous models. Even though thesepreconditioners naturally exhibit good parallelism properties on distributed architectures,they can prove inadequate numerical performance for complex decompositions or multiscalephysics. This lack of robustness may be alleviated by concurrently solving sparse ordense local generalized eigenvalue problems, thus identifying modes that hinder the convergenceof the underlying iterative methods a priori. Using these modes, it is then possibleto define projection operators based on what is usually referred to as a coarse solver. Theseauxiliary tools tend to solve the aforementioned issues, but typically decrease the parallelefficiency of the preconditioners. In this dissertation, it is shown in three points thatthe newly developed construction is efficient: 1) by performing large-scale numerical experimentson Curie—a European supercomputer, and by comparing it with state of the art2) multigrid and 3) direct solvers.

In this study, a heterogeneous flow model is proposed based on a non-overlapping domain decomposition method. The model combines potential flow and incompressible viscous flow. Both flow domains contain a free surface boundary. The heterogeneous domain decomposition method is formulated following the Dirichlet-Neumann method. Both an implicit scheme and an explicit scheme are proposed. The algebraic form of the implicit scheme is of the same form of the Dirichlet--Neumann method, whereas the explicit scheme can be interpreted as the classical staggered scheme using the splitting of the Dirichlet-Neumann method. The explicit scheme is implemented based on two numerical solvers, a Boundary element method (BEM) solver for the potential flow model, and a finite element method (FEM) solver for the Navier-Stokes equations (NSE). The implementation based on the two solvers is validated using numerical examples. / Graduation date: 2013

This thesis presents a GPU accelerated implementation of a high order splitting scheme with a spectral element discretization for the incompressible Navier Stokes (INS) equations. While others have implemented this scheme on clusters of processors using the Nek5000 code, to my knowledge this thesis is the first to explore its performance on the GPU. This work implements several of the Nek5000 algorithms using OpenCL kernels that efficiently utilize the GPU memory architecture, and achieve massively parallel on chip computations. These rapid computations have the potential to significantly enhance computational fluid dynamics (CFD) simulations that arise in areas such as weather modeling or aircraft design procedures. I present convergence results for several test cases including channel, shear, Kovasznay, and lid-driven cavity flow problems, which achieve the proven convergence results.