This dissertation presents a multilevel algorithm to solve constant and variable coeffcient elliptic boundary value problems on adaptively refined structured meshes in 2D and 3D. Cacheaware algorithms for optimizing the operations to exploit the cache memory subsystem areshown. Keywords: Multigrid, Cache Aware, Adaptive Mesh Refinement, Partial Differential Equations, Numerical Solution.

Transient multi-phase flow problems in porous media are ubiquitous in engineering and environmental systems and processes; examples include heat exchangers, reservoir simulation, environmental remediation, magma flow in the earth crust and water management in porous electrodes of PEM fuel cells. This thesis focuses on the development of accurate and computationally efficient numerical models to simulate such flows. The research challenges addressed in this work fall in two areas. For a numerical standpoint, conventional numerical methods including Newton-Raphson linearization and a simple upwind scheme do not always provide the required computational efficiency or sufficiently accurate resolution of the flow field. From a modelling perspective, closure schemes required in volume-averaged formulations, such as the generalized Leverett J function for capillary pressure, are specific to certain media (e.g. lithologic media) and are not valid for fibrous porous media, which are of central interest in fuel cells. This thesis presents a set of algorithms that are integrated efficiently to achieve computations that are more than two orders of magnitude faster compared to traditional techniques. The method uses an adaptive operator splitting method based on an a posteriori criterion to separate the flow from the transport equations which eliminates unnecessary and costly solution of the implicit pressure-velocity term at every time step; adaptive meshing to reduce the size of the discretized problem; efficient block preconditioned solver techniques for fast solution of the discrete equations; and a recently developed artificial diffusion strategy to stabilize the numerical solution of the transport equation. The significant improvements in accuracy and efficiency of the approach is demosntrated using numerical experiments in 2D and 3D. The method is also extended to advection-dominated problems to specifically investigate two-phase flow in heterogeneous porous media involving capillary transport. Both hydrophilic and hydrophobic media are considered, and insights relevant to fuel cell electrodes are discussed.

Accurate numerical solution of time-dependent, high-dimensional partial differential equations (PDEs) usually requires efficient numerical techniques and massive-scale parallel computing. In this thesis, we implement and evaluate discretization schemes suited for PDEs of higher dimensionality, focusing on high order of accuracy and low computational cost. Spatial discretization is particularly challenging in higher dimensions. The memory requirements for uniform grids quickly grow out of reach even on large-scale parallel computers. We utilize high-order discretization schemes and implement adaptive mesh refinement on structured hyperrectangular domains in order to reduce the required number of grid points and computational work. We allow for anisotropic (non-uniform) refinement by recursive bisection and show how to construct, manage and load balance such grids efficiently. In our numerical examples, we use finite difference schemes to discretize the PDEs. In the adaptive case we show how a stable discretization can be constructed using SBP-SAT operators. However, our adaptive mesh framework is general and other methods of discretization are viable. For integration in time, we implement exponential integrators based on the Lanczos/Arnoldi iterative schemes for eigenvalue approximations. Using adaptive time stepping and a truncated Magnus expansion, we attain high levels of accuracy in the solution at low computational cost. We further investigate alternative implementations of the Lanczos algorithm with reduced communication costs. As an example application problem, we have considered the time-dependent Schrödinger equation (TDSE). We present solvers and results for the solution of the TDSE on equidistant as well as adaptively refined Cartesian grids. / eSSENCE

For many years, terrorism has threatened life, property and business. Targets are largely in urban areas where there is a greater density of life and economic value. Governments, insurers and engineers have sought to mitigate these threats through understanding the effects of urban bombings, increasing the resilience of buildings and improving estimates of financial loss for insurance purposes. This has led to a desire for an improved approach to the prediction of blast propagation in urban cityscapes. Urban geometry has a significant impact on blast wave propagation. Presently, only computational fluid dynamics (CFD) methods adequately simulate these effects. However, for large-scale urban domains, these methods are both challenging to use and are computationally expensive. Adaptive mesh refinement (AMR) methods alleviate the problem, but are difficult to use for the non-expert and require significant tuning. We aim to make CFD urban blast simulation a primary choice for governments, insurers and engineers through improvements to AMR and by studying the performance of CFD in relation to other methods used by the industry. We present a new AMR flagging approach based on a second derivative error norm for compressive shocks (ENCS). This is compared with existing methods and is shown to lead to a reduction in overall refinement without affecting solution quality. Significant improvements to feature tracking over long distances are demonstrated, making the method easier to tune and less obtuse to non-experts. In the chapter that follows, we consider blast damage in urban areas. We begin with a validation and a numerical study, investigating the effects of simple street geometry on blast resultants. We then investigate the sensitivity of their distribution to the location of the charge. We find that moving the charge by a small distance can lead to a significant change in peak overpressures and creates a highly localised damage field due to interactions between the blast wave and the geometry. We then extend the investigation to the prediction of insured losses following a large-scale bombing in London. A CFD loss model is presented and compared with simpler approaches that do not account for urban geometry. We find that the simpler models lead to significant over-predictions of loss, equivalent to several hundred million pounds for the scenario considered. We use these findings to argue for increased uptake of CFD methods by the insurance industry. In the final chapter, we investigate the influence of urban geometry on the propagation of blast waves. An earlier study on the confinement effects of narrow streets is repeated at a converged resolution and we corroborate the findings. We repeat the study, this time introducing a variable porosity into the building facade. We observe that the effect of this porosity is as significant as the confinement effect, and we recommend to engineers that they consider porosity effects in certain cases. We conclude the study by investigating how alterations to building window layout can improve the protective effects of a facade. Maintaining the window surface area constant, we consider a range of layouts and observe how some result in significant reductions to blast strength inside the building.

Adaptive finite element methods for the solution of partial differential equations require effective methods of mesh refinement and coarsening, fast multilevel solvers for the systems of FE equations need a hierarchical structure of the grid. In the paper a technology is presented for the application of irregular hierarchical triangular meshes arising from refinement by only dividing elements into four congruent triangles. The paper describes the necessary data structures and data structure management, the principles and algorithms of refining and coarsening the mesh, and also a specific assembly technique for the FE equations system. Aspects of the parallel implementation on MIMD computers with a message passing communication are included.

Cette thèse, intitulée NSIBM : un solveur parallèle de Navier-Stokes avec raffinement automatique basé sur la méthode des frontières immergées, a été effectuée au sein du laboratoire iCube, département de mécanique, à Strasbourg, dans le quartier de l'Orangerie, sous la direction du professeur Yannick Hoarau. L'essentiel du travail effectué consiste en le développement d'un programme capable de résoudre numériquement l'équation de Navier-Stokes qui régit des fluides en mouvement. Une attention particulière a été portée à la production de maillages conformes aux géométries proposées et à leur génération. Les moyens mis en œuvre ici pour gérer l'éternel problème de la finesse du maillage opposée au trop grand nombre de cellules sont multiples : le raffinement, la parallélisation et les frontières immergées. Dans un premier temps, j'ai conçu un générateur de maillage en deux et trois dimensions en y intégrant la possibilité de diviser des cellules, et cela de manière automatique, par des critères géométriques, numériques ou physiques. Il permet également de supprimer des cellules, de manière à ne pas mailler le vide ou les parties solides de la géométrie.Dans un deuxième temps, j'ai rendu ce code parallèle en lui donnant la capacité d'utiliser plusieurs processeurs, afin de calculer plus vite et donc d'utiliser davantage de mailles. Cette étape fait appel à deux technologies : Metis, qui partage équitablement les mailles sur le nombre choisi de processeurs et OpenMPI, qui est l'interface de communication entre ces processeurs. Enfin, la méthode des frontières immergées a été introduite au code pour gérer les bords non verticaux ou horizontaux dans un maillage cartésien, c'est-à-dire formé de rectangles ou de pavés droits. Elle consiste à donner un caractère hybride à une cellule traversée par une frontière par l'introduction d'un terme numérique de forçage simulant la présence de la paroi.Ce travail de développement a ensuite été mis à l'épreuve et validé dans une série de cas tests en deux comme en trois dimensions. Des exemples de maillages complexes générés facilement sont donnés. / This thesis, entitled NSIBM: a parallel Navier-Stokes solver with automatic mesh refinement based on immersed boundary method, has been conducted within the iCube laboratory dedicated to mechanics and located in Strasbourg. It has been supervised by Professor Yannick Hoarau. This work mainly deals with coding a program able to solve the Navier-Stokes equations that governs moving fluids, in a numerical way. Particular attention was paid to the production of meshes that suit given geometries and their generation.The means used here to handle the eternal problem of the fineness of the mesh opposed to too many cells are several~:refinement, parallelization and the immersed boundary method.Initially, I designed a two and three-dimensional mesh generator that includes the possibility of dividing cells,in an automatic way, by geometrical, numerical or physical criteria. It also allows to remove cells, where there is no point keeping it. Secondly, I parallelized the program by giving him the ability to use multiple processors to calculate faster and therefore use bigger meshes.This step uses two available libraries~: \textit{Metis}, which gives a optimal mesh partition, and \textit{openMPI}, which deals with communication between nodes. Finally, the immersed boundary method has been implemented to handle non-vertical or non-horizontal edges in a cartesian grid. Its principle is to confer a hybrid status to a cell which is crossed by an edge by adding a numerical force term simulating the presence of the boundary. This development work was then tested and validated in a serie of test cases in two and three dimensions. Examples of complex meshes easily generated are given.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第22861号 / 情博第740号 / 新制||情||127(附属図書館) / 京都大学大学院情報学研究科先端数理科学専攻 / (主査)教授 青柳 富誌生, 教授 磯 祐介, 教授 田口 智清 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

NR 20140805

Modelling multiphase flow, more specifically particle-laden flow, poses multiple challenges. These difficulties are heightened when the particles are differentiated by a set of “internal” variables, such as size or temperature. Traditional treatments of such flows can be classified in two main categories, Lagrangian and Eulerian methods. The former approaches are highly accurate but can also lead to extremely expensive computations and challenges to load balancing on parallel machines. In contrast, the Eulerian models offer the promise of less expensive computations but often introduce modelling artifacts and can become more complicated and expensive when a large number of internal variables are treated. Recently, a new model was proposed to treat such situations. It extends the ten-moment Gaussian model for viscous gases to the treatment of a dilute particle phase with an arbitrary number of internal variables. In its initial application, the only internal variable chosen for the particle phase was the particle diameter. This new polydisperse Gaussian model (PGM) comprises 15 equations, has an eigensystem that can be expressed in closed form and also possesses a convex entropy. Previously, this model has been tested in one dimension. The PGM was developed with the detonation of radiological dispersal devices (RDD) as an immediate application. The detonation of RDDs poses many numerical challenges, namely the wide range of spatial and temporal scales as well as the high computational costs to accurately resolve solutions. In order to address these issues, the goal of this current project is to develop a block-based adaptive mesh refinement (AMR) implementation that can be used in conjunction with a parallel computer. Another goal of this project is to obtain the first three-dimensional results for the PGM. In this thesis, the kinetic theory of gases underlying the development of the PGM is studied. Different numerical schemes and adaptive mesh refinement methods are described. The new block-based adaptive mesh refinement algorithm is presented. Finally, results for different flow problems using the new AMR algorithm are shown, as well as the first three-dimensional results for the PGM.

The development of adaptive mesh refinement capabilities in the field of computational fluid dynamics is an essential tool for enabling the simulation of larger and more complex physical problems. While such techniques have been known for a long time, most simulations do not make use of them because of the lack of a robust implementation. In this work, we present recent progresses that have been made to develop adaptive mesh refinement features in Nek5000, a code based on the spectral element method. These developments are driven by the algorithmic challenges posed by future exascale supercomputers. First, we perform the study of the strong scaling of Nek5000 on three petascale machines in order to assess the scalability of the code and identify the current bottlenecks. It is found that strong scaling limit ranges between 5, 000 and 220, 000 degrees of freedom per core depending on the machine and the case. The need for synchronized and low latency communication for efficient computational fluid dynamics simulation is also confirmed. Additionally, we present how Hypre, a library for linear algebra, is used to develop a new and efficient code for performing the setup step required prior to the use of an algebraic multigrid solver for preconditioning the pressure equation in Nek5000. Finally, the main objective of this work is to develop new methods for estimating the error on a numerical solution of the Navier–Stokes equations via the resolution of an adjoint problem. These new estimators are compared to existing ones, which are based on the decay of the spectral coefficients. Then, the estimators are combined with newly implemented capabilities in Nek5000 for automatic grid refinement and adaptive mesh adaptation is carried out. The applications considered so far are steady and two-dimensional, namely the lid-driven cavity at Re = 7, 500 and the flow past a cylinder at Re = 40. The use of adaptive mesh refinement techniques makes mesh generation easier and it is shown that a similar accuracy as with a static mesh can be reached with a significant reduction in the number of degrees of freedom. / <p>QC 20171114</p>