Almost Block Diagonal Linear Systems: Sequential and Parallel Solution Techniques, and Applications

Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in numerical methods for two-point boundary value problems for ordinary differential equations and in related partial differential equation problems. The stable, efficient sequential solution of ABDs has received much attention over the last fifteen years and the parallel solution more recently. We survey the fields of application with emphasis on how ABDs and bordered ABDs (BABDs) arise. We outline most known direct solution techniques, both sequential and parallel, and discuss the comparative efficiency of the parallel methods. Finally, we examine parallel iterative methods for solving BABD systems. Copyright (C) 2000 John Wiley & Sons, Ltd

Accelerating the Performance of a Novel Meshless Method Based on Collocation With Radial Basis Functions By Employing a Graphical Processing Unit as a Parallel Coprocessor

In recent times, a variety of industries, applications and numerical methods including the meshless method have enjoyed a great deal of success by utilizing the graphical processing unit (GPU) as a parallel coprocessor. These benefits often include performance improvement over the previous implementations. Furthermore, applications running on graphics processors enjoy superior performance per dollar and performance per watt than implementations built exclusively on traditional central processing technologies. The GPU was originally designed for graphics acceleration but the modern GPU, known as the General Purpose Graphical Processing Unit (GPGPU) can be used for scientific and engineering calculations. The GPGPU consists of massively parallel array of integer and floating point processors. There are typically hundreds of processors per graphics card with dedicated high-speed memory. This work describes an application written by the author, titled GaussianRBF to show the implementation and results of a novel meshless method that in-cooperates the collocation of the Gaussian radial basis function by utilizing the GPU as a parallel co-processor. Key phases of the proposed meshless method have been executed on the GPU using the NVIDIA CUDA software development kit. Especially, the matrix fill and solution phases have been carried out on the GPU, along with some post processing. This approach resulted in a decreased processing time compared to similar algorithm implemented on the CPU while maintaining the same accuracy

Hp-spectral Methods for Structural Mechanics and Fluid Dynamics Problems

We consider the usage of higher order spectral element methods for the solution of problems in structures and fluid mechanics areas. In structures applications we study different beam theories, with mixed and displacement based formulations, consider the analysis of plates subject to external loadings, and large deformation analysis of beams with continuum based formulations. Higher order methods alleviate the problems of locking that have plagued finite element method applications to structures, and also provide for spectral accuracy of the solutions. For applications in computational fluid dynamics areas we consider the driven cavity problem with least squares based finite element methods. In the context of higher order methods, efficient techniques need to be devised for the solution of the resulting algebraic systems of equations and we explore the usage of element by element bi-orthogonal conjugate gradient solvers for solving problems effectively along with domain decomposition algorithms for fluid problems. In the context of least squares finite element methods we also explore the usage of Multigrid techniques to obtain faster convergence of the the solutions for the problems of interest. Applications of the traditional Lagrange based finite element methods with the Penalty finite element method are presented for modelling porous media flow problems. Finally, we explore applications to some CFD problems namely, the flow past a cylinder and forward facing step

Hp-spectral Methods for Structural Mechanics and Fluid Dynamics Problems

We consider the usage of higher order spectral element methods for the solution of problems in structures and fluid mechanics areas. In structures applications we study different beam theories, with mixed and displacement based formulations, consider the analysis of plates subject to external loadings, and large deformation analysis of beams with continuum based formulations. Higher order methods alleviate the problems of locking that have plagued finite element method applications to structures, and also provide for spectral accuracy of the solutions. For applications in computational fluid dynamics areas we consider the driven cavity problem with least squares based finite element methods. In the context of higher order methods, efficient techniques need to be devised for the solution of the resulting algebraic systems of equations and we explore the usage of element by element bi-orthogonal conjugate gradient solvers for solving problems effectively along with domain decomposition algorithms for fluid problems. In the context of least squares finite element methods we also explore the usage of Multigrid techniques to obtain faster convergence of the the solutions for the problems of interest. Applications of the traditional Lagrange based finite element methods with the Penalty finite element method are presented for modelling porous media flow problems. Finally, we explore applications to some CFD problems namely, the flow past a cylinder and forward facing step

High Resolution Numerical Methods for Coupled Non-linear Multi-physics Simulations with Applications in Reactor Analysis

The modeling of nuclear reactors involves the solution of a multi-physics problem with widely varying time and length scales. This translates mathematically to solving a system of coupled, non-linear, and stiff partial differential equations (PDEs). Multi-physics applications possess the added complexity that most of the solution fields participate in various physics components, potentially yielding spatial and/or temporal coupling errors. This dissertation deals with the verification aspects associated with such a multi-physics code, i.e., the substantiation that the mathematical description of the multi-physics equations are solved correctly (both in time and space). Conventional paradigms used in reactor analysis problems employed to couple various physics components are often non-iterative and can be inconsistent in their treatment of the non-linear terms. This leads to the usage of smaller time steps to maintain stability and accuracy requirements, thereby increasing the overall computational time for simulation. The inconsistencies of these weakly coupled solution methods can be overcome using tighter coupling strategies and yield a better approximation to the coupled non-linear operator, by resolving the dominant spatial and temporal scales involved in the multi-physics simulation. A multi-physics framework, KARMA (K(c)ode for Analysis of Reactor and other Multi-physics Applications), is presented. KARMA uses tight coupling strategies for various physical models based on a Matrix-free Nonlinear-Krylov (MFNK) framework in order to attain high-order spatio-temporal accuracy for all solution fields in amenable wall clock times, for various test problems. The framework also utilizes traditional loosely coupled methods as lower-order solvers, which serve as efficient preconditioners for the tightly coupled solution. Since the software platform employs both lower and higher-order coupling strategies, it can easily be used to test and evaluate different coupling strategies and numerical methods and to compare their efficiency for problems of interest. Multi-physics code verification efforts pertaining to reactor applications are described and associated numerical results obtained using the developed multi-physics framework are provided. The versatility of numerical methods used here for coupled problems and feasibility of general non-linear solvers with appropriate physics-based preconditioners in the KARMA framework offer significantly efficient techniques to solve multi-physics problems in reactor analysis

Variationsformulierungen und funktionale Approximationsalgorithmen in der stochastischen Plastizität von Materialien

A class of abstract stochastic variational inequalities of the second kind described by uncertain parameters is considered within the framework of infinitesimal and large displacement elastoplasticity theory. Particularly the focus is set on the rate-independent evolutionary problem with general hardening whose material characteristics are assumed to have positively-definite distributions. By exhibiting the structure of the evolutionary equations in a convex setting the mathematical formulation is carried over to the computationally more suitable mixed variational description for which the existence and uniqueness of the solution is studied. Time discretised as usual with backward Euler, the inequality is reduced to a minimisation problem for a convex functional on discrete tensor product subspaces whose unique minimiser is obtained via a stochastic closest point projection algorithm based on "white noise analysis". To this end a description in the language of non-dissipative and dissipative operators is used, both employing the stochastic Galerkin method in its fully intrusive or non-intrusive variant. The former method represents the direct, purely algebraic way of computing the response in each iteration of Newton-like methods. As the solution is given in a form of polynomial chaos expansion, i.e. an explicit functional relationship between the independent random variables, the subsequent evaluations of its functionals (the mean, variance, or probabilities of exceedence) are shown to be very cheap, but with limited accuracy. Due to this reason, the intrusive method is contrasted to the less efficient but more accurate non-intrusive variant which evaluates the residuum in each iteration via high-dimensional integration rules based on random or deterministic sampling - Monte Carlo and related techniques. In addition to these, the problem is also solved with the help of the stochastic collocation method via sparse grid techniques. Finally, the methods are validated on a series of test examples in plain strain conditions whose reference solution is computed via direct integration methods.Im Rahmen der Elastoplastizitätstheorie infinitesimaler und starker Verschiebungen wird eine Klasse von abstrakten, stochastischen Variationsungleichungen betrachtet, welche durch unsichere Parameter beschrieben werden. Im Speziellen wird das raten-unabhängige Evolutionsproblem mit allgemeiner Verfestigung betrachtet, dessen Materialeigenschaften-Verteilung als durch die Maximum-Entropie Methode gegeben angenommen wird. Durch die Darstellung der Struktur der Evolutionsgleichungen in einem konvexen Rahmen wird die Existenz und Eindeutigkeit der Lösung betrachtet und die mathematische Formulierung in eine berechnungstechnisch besser passende gemischt-variationale Beschreibung überführt. Innerhalb eines Euler-rückwarts Zeitschrittes reduziert sich die Ungleichung auf ein Minimierungsproblem für ein konvexes Energiefunktional auf diskreten Tensorproduktunterräumen, dessen eindeutige Lösung mithilfe eines stochastischen nächstgelegenen-Punkt-Projektionsalgorithmus basierend auf der "white noise Analyse" bestimmt wird. Hierzu wird eine Beschreibung basierend auf nicht-dissipativen und dissipativen Operatoren benutzt und die sogenannte intrusive stochastische Galerkinmethode in den Berechnungsprozess eingeführt. Diese Methode stellt einen direkten algebraischen Weg zur Berechnung der Lösung in jeder Iteration von Newton-ähnlichen Verfahren dar. Da die Lösung in der Form einer polynomiellen Chaos-Entwicklung gegeben ist, also einer expliziten Beschreibung des funktionalen Zusammenhangs der unabhängigen Zufallsvariablen, sind die nachfolgenden Auswertungen von Funktionalen dieser Lösung (Mittelwert, Varianz, Überschreitungswahrscheinlichkeit) berechnungstechnisch sehr günstig. Zusätzlich wird die Methode mit der nicht-intrusiven Variante verglichen, einem pseudo-Galerkin Verfahren, welches das Residuum in jeder Iteration mit Methoden zur hochdimensionalen Integration basierend auf zufälligen oder deterministischen Abtastverfahren auswertet. Abschließend wird die Methode mit einer Reihe von Testbeispielen mit einfachen Spannungsbedingungen validiert, deren Referenzlösungen über direkte Integrationsverfahren berechnet werden

A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss–Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established