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This thesis presents the development of GPU accelerated solvers for use in simulation of the primary atomization phenomenon. By using the open source continuum mechanics library, OpenFOAM, as a basis along with the NVidia CUDA API linear system solvers have been developed so that the multiphase solver runs in part on GPUs. This aims to reduce the enormous computational cost associated with modelling primary atomization. The modelling of such is vital to understanding the mechanisms that make combustion efficient. Firstly, the OpenFOAM code is benchmarked to assess both its suitability for atomization problems and to establish efficient operating parameters for comparison to GPU accelerations. This benchmarking then culminates in a comparison to an experimental test case, from the literature, dominated by surface tension, in 3D. Finally, a comparison is made with a primary atomizing liquid sheet as published in the literature. A geometric multigrid method is employed to solve the pressure Poisson equations, the first use of a geometric multigrid method in 3D GPU accelerated VOF simulation. Detailed investigations are made into the compute efficiency of the GPU accelerated solver, comparing memory bandwidth usage to hardware maximums as well as GPU idling time. In addition, the components of the multigrid method are also investigated, including the effect of residual scaling. While the GPU based multigrid method shows some improvement over the equivalent CPU implementation, the costs associated with running on GPU cause this to not be significantly greater.

We consider algorithms for preconditioning of two discontinuous Galerkin (DG) methods for second order elliptic problems, namely the symmetric interior penalty (SIPG) method and the method of Baumann and Oden. For the SIPG method we first consider two-level preconditioners using coarse spaces of either continuous piecewise polynomial functions or piecewise constant (discontinuous) functions. We show that both choices give rise to uniform, with respect to the mesh size, preconditioners. We also consider multilevel preconditioners based on the same two types of coarse spaces. In the case when continuous coarse spaces are used, we prove that a variable V-cycle multigrid algorithm is a uniform preconditioner. We present numerical experiments illustrating the behavior of the considered preconditioners when applied to various test problems in three spatial dimensions. The numerical results confirm our theoretical results and in the cases not covered by the theory show the efficiency of the proposed algorithms. Another approach for preconditioning the SIPG method that we consider is an algebraic multigrid algorithm using coarsening based on element agglomeration which is suitable for unstructured meshes. We also consider an improved version of the algorithm using a smoothed aggregation technique. We present numerical experiments using the proposed algorithms which show their efficiency as uniform preconditioners. For the method of Baumann and Oden we construct a preconditioner based on an orthogonal splitting of the discrete space into piecewise constant functions and functions with zero average over each element. We show that the preconditioner is uniformly spectrally equivalent to an appropriate symmetrization of the discrete equations when quadratic or higher order finite elements are used. In the case of linear elements we give a characterization of the kernel of the discrete system and present numerical evidence that the method has optimal convergence rates in both L2 and H1 norms. We present numerical experiments which show that the convergence of the proposed preconditioning technique is independent of the mesh size.

[pt] A solução numérica de problemas de escoamentos de fluidos com transferência de calor, envolve a solução de um conjunto de equações diferenciais parciais não lineares acopladas. O maior esforço computacional gasto na solução dessas equações, é devido a solução dos sistemas algébricos resultantes da discretização das equações de conservação. A taxa de convergência de varios métodos iterativos é sensível a natureza do problema que está sendo resolvido. Portanto, nenhum método pode ser aclamado como melhor para todos os problemas. Junto com o desenvolvimento de novos métodos iterativos, o desenvolvimento de técnicas de aceleração da convergência dos métodos iterativos conhecidos é de consideravel interesse de ponto de vista prático. O objetivo primário do presente trabalho consistiu em analisar uma classe de algoritmos para a solução de sistemas algébricos provenientes da discretização das equações de conservação de fenômenos de transporte. O segundo objetivo foi o de selecionar um método adequado e eficiente que produza um aumento da taxa de convergência. Para este propósito, selecionou-se e implementou-se um esquema de - multigrid - por correção aditiva. Esta é uma técnica recente na qual o mesmo problema diferencial é aproximado em diversas malhas cujos tamanhos de malha são geralmente múltiplos integrais. Investigou-se seu desempenho para melhorar a taxa de convergência junto com o método iterativo linha por linha TDMA, e comparou-se seu desempenho com o método de correção por blocos. / [en] The numerical solution of fluid flows problems with heat transfer requires the solution of a set of coupled non- linear partial differencial equations. The major computational effort in solving these equations is due to the solution of the algebraic systems resultant from the discretization of the conservation equations. The convergence rate of iterative methods is sensitive to the nature of the problem being solved. Therefore, no one method may be claimed to be the best for all problems. Along with the development of new iterative methods, the development of technics for accelerating the convergence of known iterative methods presents a considerable interest from a practical standpoint. The primary objective of the present work was to analise a class of algorithms for the solution of algebraic systems resulting from the discretization of transport phenomena conservation equations. The second objective was to select an adequate and efficient method which lead an increase of the convergence rate. For this purpose a multigrid additive correction scheme was selected and implemented. This is a novel technique in which the same differential problem is approximated on several grids whose mesh sizes are usually integral multiples. It was investigated its performance to improve the convergence rate in combination with the iterative line-by-line TDMA as well as its performance with the use of block correction algorithm. / [es] La solución numérica de problemas de flujo de fluidos con transferencia de calor, comprende la solución de un conjunto de ecuaciones diferenciales parciales no lineales acopladas. El mayor esfuerzo computacional en la solución de esas ecuaciones, se debe a la resolución de los sistemas algebraicos resultantes de la discretización de las ecuaciones de conservación. La tasa de convergencia de varios métodos iterativos es altamente sensible a la naturaleza del problema. Por lo tanto, ningún método puede ser considerado como el

In a parallel vertex-centered finite element multigrid solver, segmental refinement can be used to avoid all inter-process communication on the fine grids. While domain decomposition methods generally require coupled subdomain processing for the numerical solution to a nonlinear elliptic boundary value problem, segmental refinement exploits that subdomains are almost decoupled with respect to high-frequency error components. This allows to perform multigrid with fully decoupled subdomains on the fine grids, which was proposed as a sequential low-storage algorithm by Brandt in the 1970s, and as a parallel algorithm by Brandt and Diskin in 1994. Adams published the first numerical results from a multilevel segmental refinement solver in 2014, confirming the asymptotic exactness of the scheme for a cell-centered finite volume implementation. We continue Brandt’s and Adams’ research by experimentally investigating the scheme’s accuracy with a vertex-centered finite element segmental refinement solver. We confirm that full multigrid accuracy can be preserved for a few segmental refinement levels, although we observe a different dependency on the segmental refinement parameter space. We show that various strategies for the grid transfers between the finest conventional multigrid level and the segmental refinement subdomains affect the solver accuracy. Scaling results are reported for a Cray XC30 with up to 4096 cores.

published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 1990.

Multigrid acceleration is typically used for the iterative solution of partial differential equations in physics and engineering. A typical multigrid implementation uses a base discretization method, such as finite elements or finite differences, and a set of successively coarser grids that is used for accelerating the convergence of the iterative solution on the base grid. The presented thesis extends the use of multigrid acceleration to the design optimization of a sample structural system and demonstrates it within the context of the recently introduced Cellular Automata paradigm for design optimization. Within the design context, the multigrid scheme is not only used for accelerating the analysis iterations, but is also used to help refine the design across multiple grid levels to accelerate the design convergence. A comparison of computational efficiencies achieved by different multigrid implementations, including the multigrid accelerated nested design iteration scheme, is presented. The method is described in its generic form which can be applicable not only to the Cellular Automata paradigm but also to more general finite element analysis based design schemes as well. / Master of Science

Iterative solutions of the Boltzmann transport equation are computationally intensive. Spatial multigrid methods have led to efficient iterative algorithms for solving a variety of partial differential equations; thus, it is natural to explore their application to transport equations. Manteuffel et al. conducted such an exploration in one spatial dimension, using two-cell inversions as the relaxation or smoothing operation, and reported excellent results. In this dissertation we extensively test Manteuffel??s one-dimensional method and our modified versions thereof. We demonstrate that the performance of such spatial multigrid methods can degrade significantly given strong heterogeneities. We also extend Manteuffel??s basic approach to two-dimensional problems, employing four-cell inversions for the relaxation operation. We find that for uniform homogeneous problems the two-dimensional multigrid method is not as rapidly convergent as the one-dimensional method. For strongly heterogeneous problems the performance of the two-dimensional method is much like that of the one-dimensional method, which means it can be slow to converge. We conclude that this approach to spatial multigrid produces a method that converges rapidly for many problems but not for others. That is, this spatial multigrid method is not unconditionally rapidly convergent. However, our analysis of the distribution of eigenvalues of the iteration operators indicates that this spatial multigrid method may work very well as a preconditioner within a Krylov iteration algorithm, because its eigenvalues tend to be relatively well clustered. Further exploration of this promising result appears to be a fruitful area of further research.