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In meshfree methods, partial differential equations are solved on an unstructured cloud of points distributed throughout the computational domain. In collocated meshfree methods, the differential operators are directly approximated at each grid point based on a local cloud of neighboring points. The set of neighboring nodes used to construct the local approximation is determined using a variable search radius. The variable search radius establishes an implicit nodal connectivity and hence a mesh is not required. As a result, meshfree methods have the potential flexibility to handle problem sets where the computational grid may undergo large deformations as well as where the grid may need to undergo adaptive refinement. In this work we develop the sharp interface formulation of the immersed boundary method for collocated meshfree approximations. We use the framework to implement three meshfree methods: General Finite Differences (GFD), Smoothed Particle Hydrodynamics (SPH), and Moving Least Squares (MLS). We evaluate the numerical accuracy and convergence rate of these methods by solving the 2D Poisson equation. We demonstrate that GFD is computationally more efficient than MLS and show that its accuracy is superior to a popular corrected form of SPH and comparable to MLS. We then use GFD to solve several canonic steady state fluid flow problems on meshfree grids generated using uniform and variable radii Poisson disk algorithm.

Large displacement and large strain problems of mechanical systems can be solved mainly by four methods. These are Floating Frame of Reference, Incremental Finite Element, Large Rotation Vector and Absolute Nodal Coordinate Formulations (ANCF). Due to exact rigid body representation, simple mass matrix structure and non-incremental formulation, ANCF is more convenient in analyzing flexible multibody systems. However, it is limited to problems with regular boundaries, currently. The aim of the thesis is to improve the current ANCF in order to handle various problems with irregular boundaries. For this purpose, firstly meshfree ANCF has been developed to analyze flexible multibody systems. Verification of the developed meshfree formulation has been performed for beam type structures and accurate results have been obtained. Then, &ldquo / ANCF with Virtual Element Mapping Method&rdquo / has been proposed to overcome the boundary problems of the current formulations. The proposed method has been implemented to plane stress, plane strain, plate/shell and 3D solid finite elements. Verification of the proposed method has been performed by using the patch test problems available in the literature. Besides, it has been verified by various flexible multibody problems with large deformations. Additionally, shape function polynomials for thin plate assumption have been derived. It is observed that developed formulations and methods can be useful not only for flexible multibody systems but also for structural mechanics problems subjected to large deformations and/or rotations. The proposed methods and formulations are more efficient than the current formulations in the literature due to extended shape limits of finite elements.

Thesis (MEng)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: In this thesis we set out to design an algorithm for solving the all-pairs fixed-radius nearest neighbours search problem for a massively parallel heterogeneous system. The all-pairs search problem is stated as follows: Given a set of N points in d-dimensional space, find all pairs of points within a horizon distance of one another. This search is required by any nonlocal or meshless numerical modelling method to construct the neighbour list of each mesh point in the problem domain. Therefore, this work is applicable to a wide variety of fields, ranging from molecular dynamics to pattern recognition and geographical information systems. Here we focus on nonlocal solid mechanics methods. The basic method of solving the all-pairs search is to calculate, for each mesh point, the distance to each other mesh point and compare with the horizon value to determine if the points are neighbours. This can be a very computationally intensive procedure, especially if the neighbourhood needs to be updated at every time step to account for changes in material configuration. The problem also becomes more complex if the analysis is done in parallel. Furthermore, GPU computing has become very popular in the last decade. Most of the fastest supercomputers in the world today employ GPU processors as accelerators to CPU processors. It is also believed that the next-generation exascale supercomputers will be heterogeneous. Therefore the focus is on how to develop a neighbour searching algorithm that will take advantage of next-generation hardware. In this thesis we propose a CPU - multi GPU algorithm, which is an extension of the fixed-grid method, for the fixed-radius nearest neighbours search on massively parallel systems. / AFRIKAANSE OPSOMMING: In hierdie tesis het ons die ontwerp van ’n algoritme vir die oplossing van die alle-pare vaste-radius naaste bure soektog probleem vir groot skaal parallele heterogene stelsels aangepak. Die alle-pare soektog probleem is as volg gestel: Gegewe ’n stel van N punte in d-dimensionele ruimte, vind al die pare van punte wat binne ’n horison afstand van mekaar af is. Die soektog word deur enige nie-lokale of roosterlose numeriese metode benodig om die bure-lys van alle rooster-punte in die probleem te kry. Daarom is hierdie werk van toepassing op ’n wye verskeidenheid van velde, wat wissel van molekulêre dinamika tot patroon herkenning en geografiese inligtingstelsels. Hier is ons fokus op nie-lokale soliede meganika metodes. Die basiese metode vir die oplossing van die alle-pare soektog is om vir elke rooster-punt, die afstand na elke ander rooster-punt te bereken en te vergelyk met die horison lente, om dus so te bepaal of die punte bure is. Dit kan ’n baie berekenings intensiewe proses wees, veral as die probleem by elke stap opgedateer moet word om die veranderinge in die materiaal konfigurasie daar te stel. Die probleem word ook baie meer kompleks as die analise in parallel gedoen word. Verder het GVE’s (Grafiese verwerkings eenhede) baie gewild geword in die afgelope dekade. Die meeste van die vinnigste superrekenaars in die wêreld vandag gebruik GVE’s as versnellers te same met SVE’s (Sentrale verwerkings eenhede). Dit is ook van mening dat die volgende generasie exa-skaal superrekenaars GVE’s sal implementeer. Daarom is die fokus op hoe om ’n bure-lys soektog algoritme te ontwikkel wat gebruik sal maak van die volgende generasie hardeware. In hierdie tesis stel ons ’n SVE - veelvoudige GVE algoritme voor, wat ’n verlenging van die vaste-rooster metode is, vir die vaste-radius naaste bure soektog op groot skaal parallele stelsels.

Dans le présent travail, nous mettons en œuvre un développement numérique d’une méthode de discrétisation mixte MEF/Meshless pour la résolution de problème mécanique fortement non-linéaire. Une attention particulière est attribuée à la construction des fonctions de forme par approximation diffuse. Dans le but de traiter des problèmes de la mécanique des solides en transformations finies, nous développons une méthode numérique dite « mixte » unissant à la fois la méthode numérique Meshless afin de discrétiser les zones à fort gradient de déformation, et la méthode des Eléments Finis (MEF) pour les zones les moins sollicitées. Nous veillons donc à assurer le couplage entre ces deux méthodes de discrétisation à travers la mise en œuvre de techniques spécifiques, notamment pour la continuité du domaine et sa consistance en premier lieu, mais aussi pour le choix de la méthode d’intégration numérique appropriée. Nous testons par la suite la fiabilité et la performance du modèle par la mise en place d’une étude comparative avec un modèle MEF standard conçu pour les mêmes conditions numériques, afin de tester la performance des techniques numériques attribuées au modèle « mixte ». Nous développons, par la suite, un modèle de comportement à travers une description des transformations finies. Nous adoptons ainsi une discrétisation spatiale en éléments «assumed strain», et une discrétisation temporelle adaptées. Pour valider notre modèle « mixte » retenu, nous réalisons une étude comparative avec des exemples simples de calcul non linéaire / In this work, we implement a development of mixed discretization MEF-Meshless for solving strongly nonlinear mechanical problem. Particular attention is given to the construction of the shape functions by diffuse approximation. In order to deal with problems of solid mechanics with large strain, we develop a so-called “mixed” numerical method combining both Meshless to discretize areas of high deformation gradient, and Finite Element Method (MEF) for non-concerned areas. We ensure coupling between both discretization methods through the implementation of specific techniques, including the continuity and consistency of the field and the choice of the appropriate method of numerical integration. We test later the reliability and performance of the model by the introduction of a comparative study with a standard FEM model designed for the same numerical conditions to evaluate numerical techniques attributed to our “mixed” model. We develop, thereafter, a model of behavior through a large strain description. We adopt spatial discretization elements “assumed strain” and a suitable time discretization. To validate our “mixed” model, we perform a comparative study of nonlinear simple calculation examples

Bezsíťové metody reprezentují alternativu ke standardním diskretizačním technikám, které pro svůj chod vyžadují síť. V posledních desetiletích bylo vynaloženo mnoho úsilí k ověření konkurenceschopnosti bezsíťových metod v různých inženýrských odvětvích. Diplomová práce je zaměřena na aplikaci vhodné bezsíťové metody ve výpočetní aeroakustice. Stěžejní část této práce se zabývá úlohami šíření zvuku, které lze modelovat pomocí linearizovaných Eulerových rovnic. Obecně lze tyto rovnice zařadit mezi lineární hyperbolické systémy. Pro úlohy aeroakustiky se jako vhodná bezsíťová metoda jeví Finite point method (FPM), která byla úspěšně použita pro řešení úloh dynamiky tekutin. Odvozením této metody a návrhy k dosažení vysoké přesnosti se věnuje další část práce. Úlohy šíření zvuku se známým řešením jsou testovány vlastním softwarem vyvinutým speciálně pro tyto účely.

L’objet de cette thèse a été de développer une méthode numérique multi-approche et multiéchelle appliquée à la simulation d’écoulements diphasiques de fluides non miscibles, incompressibles et isothermiques et plus particulièrement à l’atomisation primaire. Cette méthode repose sur une approche couplée entre un maillage local raffiné et un maillage global plus large. Le couplage est explicite avec raffinement en temps, c’est-à-dire que chaque domaine évolue selon son propre pas de temps. Afin de prendre en compte les différentes échelles en temps et en espace dans le processus d’atomisation, cette méthode numérique couple deux méthodes numériques diphasiques différentes : une méthode de capture de l’interface dans le domaine local raffiné près de l’injecteur et une méthode de sous-maille dans le domaine global grossier et la région du spray dispersé. Le code développé et parallélisé dans le logiciel OpenFOAMR s’avère capable de réduire de manière significative le temps de calcul d’une simulation aux grandes échelles de l’atomisation dans un injecteur coaxial, tout en prédisant de manière fiable les données expérimentales. / The purpose of this work has been to develop a multi-approach and multi-scale numerical method applied to the simulation of two-phase flows involving non miscible, incompressible and isothermal fluids, and more specifically primary atomization. This method is based on a coupled approach between a refined local mesh and a coarser global mesh. The coupling is explicit with refinement in time, i.e. each domain evolves following its own time-step. In order to account for the different scales in space and time of the atomization process, this numerical method couples two different two-phase numerical methods: an interface capturing method in the refined local domain near the injector and a sub-grid method in the coarser global domain in the dispersed spray region. The code has been developed and parallelized in the OpenFOAMR software. It is able to reduce significantly the computational cost of a large eddy simulation of a coaxial atomization, while predicting with accuracy the experimental data.

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