Accelerator architectures are used to accelerate the simulation of nonlinear hyperbolic PDEs. Three different architectures, a multicore CPU using threading, IBM’s Cell Processor, and Nvidia’s Tesla GPUs are investigated. Speed-ups of between 40-75× relative to a single CPU core in single precision are obtained using the Cell processor and the GPU. The three implementations are extended to parallel computing clusters by making use of the Message Passing Interface (MPI). The resulting hybrid-parallel code is investigated for performance and scalability on both a GPU and Cell computing cluster.

Accelerator architectures are used to accelerate the simulation of nonlinear hyperbolic PDEs. Three different architectures, a multicore CPU using threading, IBM’s Cell Processor, and Nvidia’s Tesla GPUs are investigated. Speed-ups of between 40-75× relative to a single CPU core in single precision are obtained using the Cell processor and the GPU. The three implementations are extended to parallel computing clusters by making use of the Message Passing Interface (MPI). The resulting hybrid-parallel code is investigated for performance and scalability on both a GPU and Cell computing cluster.

This paper discusses how a boundary-based surface fitting approach can be utilised to smoothly reconstruct a given human face where the scan data corresponding to the face is provided. In particular, the paper discusses how a solution to the Biharmonic equation can be used to set up the corresponding boundary value problem. We show how a compact explicit solution method can be utilised for efficiently solving the chosen Biharmonic equation. Thus, given the raw scan data of a 3D face, we extract a series of profile curves from the data which can then be utilised as boundary conditions to solve the Biharmonic equation. The resulting solution provides us a continuous single surface patch describing the original face.

The discussion in this paper focuses on how boundary based smooth shape design can be carried out. For this we treat surface generation as a mathematical boundary-value problem. In particular, we utilize elliptic Partial Differential Equations (PDEs) of arbitrary order. Using the methodology outlined here a designer can therefore generate the geometry of shapes satisfying an arbitrary set of boundary conditions. The boundary conditions for the chosen PDE can be specified as curves in 3-space defining the profile geometry of the shape. We show how a compact analytic solution for the chosen arbitrary order PDE can be formulated enabling complex shapes to be designed and manipulated in real time. This solution scheme, although analytic, satisfies exactly, even in the case of general boundary conditions, where the resulting surface has a closed form representation allowing real time shape manipulation. In order to enable users to appreciate the powerful shape design and manipulation capability of the method, we present a set of practical examples.

This thesis centers on the development and analysis of numerical multiscale methods for multiscale problems arising in steady heat conduction, heat transfer and wave propagation in heterogeneous media. In a multiscale problem several scales interact with each other to form a system which has variations over a wide range of scales. A direct numerical simulation of such problems requires resolving the small scales over a computational domain, typically much larger than the microscopic scales. This demands a tremendous computational cost. We develop and analyse multiscale methods based on the heterogeneous multiscale methods (HMM) framework, which captures the macroscopic variations in the solution at a cost much lower than traditional numerical recipes. HMM assumes that there is a macro and a micro model which describes the problem. The micro model is accurate but computationally expensive to solve. The macro model is inexpensive but incomplete as it lacks certain parameter values. These are upscaled by solving the micro model locally in small parts of the domain. The accuracy of the method is then linked to how accurately this upscaling procedure captures the right macroscopic effects. In this thesis we analyse the upscaling error of existing multiscale methods and also propose a micro model which significantly reduces the upscaling error invarious settings. In papers I and IV we give an analysis of a finite difference HMM (FD-HMM) for approximating the effective solutions of multiscale wave equations over long time scales. In particular, we consider time scales T^ε = O(ε−k ), k =1, 2, where ε represents the size of the microstructures in the medium. In this setting, waves exhibit non-trivial behaviour which do not appear over short time scales. We use new analytical tools to prove that the FD-HMM accurately captures the long time effects. We first, in Paper I, consider T^ε =O(ε−2 ) and analyze the accuracy of FD-HMM in a one-dimensional periodicsetting. The core analytical ideas are quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic wave equations.The analysis naturally reveals the role of consistency in HMM for high order approximation of effective quantities over long time scales. Next, in paperIV, we consider T^ε = O(ε−1 ) and use the tools in a multi-dimensional settingto analyze the accuracy of the FD-HMM in locally-periodic media where fast and slow variations are allowed at the same time. Moreover, in papers II and III we propose new multiscale methods which substantially improve the upscaling error in multiscale elliptic, parabolic and hyperbolic partial differential equations. In paper II we first propose a FD-HMM for solving elliptic homogenization problems. The strategy is to use the wave equation as the micro model even if the macro problem is of elliptic type. Next in paper III, we use this idea in a finite element HMM setting and generalize the approach to parabolic and hyperbolic problems. In a spatially fully discrete a priori error analysis we prove that the upscaling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media. / <p>QC 20150216</p> / Multiscale methods for wave propagation

This study is concerned with the development and implementation of a preconditioner for a set of hyperbolic partial differential equations resulting from a new 5-moment closure for the prediction of gas flows both in and out of local equilibrium. This new 5-moment closure offers a robust and efficient system of first-order hyperbolic partial differential equations that has proven to provide an accurate treatment of one-dimensional gases, both in and for significant departures from local thermodynamic equilibrium. However, numerical computations using this model have proven to be difficult as a result of a singularity in the closing flux of the system. This also causes infinitely large wavespeeds in the system. The main goal of this work is to mitigate these numerical issues. Since the solution of a hyperbolic system is characterized by the waves of the system, one could suggest to scale these wavespeeds to remove the arbitrarily large speeds without altering the solution of the system. To accomplish this, this work starts with a detailed study of the behaviour of the system’s wavespeeds, given by the eigenvalues of the flux Jacobian of the system. Since, it is not possible to solve for these eigenvalues explicitly, it is suggested to approximate them by interpolation between the few states at which these waves can be solved for explicitly. With an estimate for the wavespeeds, the nature of the singularity in the system can be analyzed mathematically. The results of this mathematical analysis are used to develop a preconditioner matrix to remove the singularity from the model. To implement the proposed preconditioned model numerically, a centred-difference scheme with artificial dissipation is proposed. A dual-time-stepping strategy is developed and implemented with implicit Euler time marching for both physical and pseudo time iteration. This dual-time treatment allows the preconditioned system to remain applicable to time-accurate problems and is found to greatly increase the robustness of the solution of the steady-state problems. Solutions to several canonical problems for both continuum and non-equilibrium flow are computed and comparisons are made to classical models.

The aim of this paper is to show how the spine of a PDE surface can be generated and how it can be used to efficiently parameterise a PDE surface. For the purpose of the work presented here an approximate analytic solution form for the chosen PDE is utilised. It is shown that the spine of the PDE surface is then computed as a by-product of this analytic solution. Furthermore, it is shown that a parameterisation can be introduced on the spine enabling intuitive manipulation of PDE surfaces.

We propose the use of Partial Differential Equations (PDEs) for shape modelling within visual cyberworlds. PDEs, especially those that are elliptic in nature, enable surface modelling to be defined as boundary-value problems. Here we show how the PDE based on the Biharmonic equation subject to suitable boundary conditions can be used for shape modelling within visual cyberworlds. We discuss an analytic solution formulation for the Biharmonic equation which allows us to define a function based geometry whereby the resulting geometry can be visualised efficiently at arbitrary levels of shape resolutions. In particular, we discuss how function based PDE surfaces can be readily integrated within VRML and X3D environments

No / Manipulation of PDE surfaces using a set of interactively defined parameters is considered. The PDE method treats surface design as a boundary-value problem and ensures that surfaces can be defined using an appropriately chosen set of boundary conditions and design parameters. Here we show how the data input to the system, from a user interface such as the mouse of a computer terminal, can be efficiently used to define a set of parameters with which to manipulate the surface interactively in real time.

No / This paper presents a surface generation tool designed for the construction of aircraft geometry. The software generates complex geometries which can be crafted or modified by the user in real time. The surface generation is based on partial differential equations (PDEs). The PDE method can produce different configurations of aircraft shapes interactively. Each surface is generated by a number of curves representing the character lines of a given part of the aircraft shape that can be manipulated in real time. Different surfaces then blend to create the full shape of the airplane. An important function of the proposed tool is its ability to change the aircraft shape through the adjustments of parameters associated with the initial curves. The user can apply linear transformations to the curves generating the airplane through simple input from the computer keyboard and the mouse. The updated curves can then be used to generate the surface leading to different configurations of a given airplane shape. The work presents detailed descriptions on the PDE method, parametric design and manipulation of aircrafts along with graphical demonstrations of its abilities and a series of examples to illustrate the capacity of the methodology implemented.