In this work, a numerical approach based on meshless methods is proposed to obtain eigenmodes of Laplace-Beltrami operator on manifolds, and its performance is compared against existing alternative methods. Radial Basis Function (RBF)-based methods allow one to obtain interpolation and differentiation matrices easily by using scattered data points. We derive expressions for such matrices for the Laplace-Beltrami operator via so-called Reilly’s formulas and use them to solve the respective eigenvalue problem. Numerical studies of proposed methods are performed in order to demonstrate convergence on simple examples of one-dimensional curves and two-dimensional surfaces.

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.

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Projection methods for the incompressible Navier-Stokes equations (NSE) are efficient, but introduce numerical boundary layers and have limited temporal accuracy due to their fractional step nature. The Pressure Poisson Equation (PPE) reformulations represent a class of methods that replace the incompressibility constraint by a Poisson equation for the pressure, with a suitable choice of the boundary condition so that the incompressibility is maintained. PPE reformulations of the NSE have important advantages: the pressure is no longer implicitly coupled to the velocity, thus can be directly recovered by solving a Poisson equation, and no numerical boundary layers are generated; arbitrary order time-stepping schemes can be used to achieve high order accuracy in time. In this thesis, we focus on numerical approaches of the PPE reformulations, in particular, the Shirokoff-Rosales (SR) PPE reformulation. Interestingly, the electric boundary conditions, i.e., the tangential and divergence boundary conditions, provided for the velocity in the SR PPE reformulation render classical nodal finite elements non-convergent. We propose two alternative methodologies, mixed finite element methods and meshfree finite differences, and demonstrate that these approaches allow for arbitrary order of accuracy both in space and in time. / Mathematics

The Element Free Galerkin (EFG) method, which is based on the Moving Least Squares (MLS) approximation, is developed here for vibration, buckling and static analysis of homogenous and FGM plate with curvilinear stiffeners. Numerical results for different stiffeners configurations and boundary conditions are presented. All results are verified using the commercial finite element software ANSYS® and other available results in literature. In addition, the vibration analysis of plates with curvilinear stiffeners is carried out using Ritz method. A 24 by 28 in. curvilinear stiffened panel was machined from 2219-T851 aluminum for experimental validation of the Ritz and meshfree methods of vibration mode shape predictions. Results were obtained for this panel mounted vertically to a steel clamping bracket using acoustic excitation and a laser vibrometer. Experimental results appear to correlate well with the meshfree and Ritz method results. In reality, many engineering structures are subjected to random pressure loads in nature and cannot be assumed to be deterministic. Typical engineering structures include buildings and towers, offshore structures, vehicles and ships, are subjected to random pressure. The vibrations induced from gust loads, engine noise, and other auxiliary electrical system can also produce noise inside aircraft. Consequently, all flight vehicles operate in random vibration environment. These random loads can be modeled by using their statistical properties. The dynamical responses of the structures which are subjected to random excitations are very complicated. To investigate their dynamic responses under random loads, the meshfree method is developed for random vibration analysis of curvilinearly-stiffened plates. Since extensive efforts have been devoted to study the buckling and vibration analysis of stiffened panel to maximize their natural frequencies and critical buckling loads, these structures are subjected to in-plane loading while the vibration analysis is considered. In these cases the natural frequencies calculated by neglecting the in-plane compression are usually over predicted. In order to have more accurate results it might be necessary to take into account the effects of in-plane load since it can change the natural frequency of plate considerably. To provide a better view of the free vibration behavior of the plate with curvilinear stiffeners subjected to axial/biaxial or shear stresses several numerical examples are studied. The FEM analysis of curvilinearly stiffened plate is quite computationally expensive, and the meshfree method seems to be a proper substitution to reduce the CPU time. However it will still require many simulations. Because of the number of simulations may be required in the solution of an engineering optimization problem, many researchers have tried to find approaches and techniques in optimization which can reduce the number of function evaluations. In these problems, surrogate models for analysis and optimization can be very efficient. The basic idea in surrogate model is to reduce computational cost and giving a better understanding of the influence of the design variables on the different objectives and constrains. To use the advantage of both meshfree method and surrogate model in reducing CPU time, the meshfree method is used to generate the sample points and combination of Kriging (a surrogate model) and Genetic Algorithms is used for design of curvilinearly stiffened plate. The meshfree and kriging results and CPU time were compared with those obtained using EBF3PanelOpt. / Ph. D.

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.

Avec l'avancé des méthodes numériques, de nouvelles méthodes dites « sans maillage » sont apparues pour remédier à certaines limitations de la méthode des éléments finis. Ces méthodes ont la particularité de n’employer aucun maillage prédéfini : elles utilisent un ensemble de nœuds dispersés dans le domaine considéré et sur ses frontières. L’objectif de cette étude est de montrer l’intérêt de l’application des méthodes sans maillage basées sur les fonctions de base radiale pour la simulation des procédés de mise en forme en général et de poinçonnage rapide en particulier. Une attention particulière sera portée sur la méthode RPIM qui offre l’avantage de proposer une interpolation nodale. La démarche proposée dans ce document consiste: à rappeler succinctement les principes essentiels des méthodes sans maillage en précisant leurs avantages par rapport aux méthodes classiques, à présenter et à mettre en œuvre la technique numérique de la méthode sans maillage RPIM avec un calibrage des paramètres caractéristiques, et enfin, à traiter des exemples numériques de procédés de mise en forme avec amorçage et propagation de fissure qui confirmeront ces avantages. / In recent years, new methods named Meshfree methods have been developed to surmount limitations of the finite element method. The main characteristic of these methods is to not employ any pre-defined mesh: they use a set of nodes scattered within the problem domain as well as sets of nodes scattered on the boundaries of the domain. A particular attention is paid to the RPIM method, which proposes a nodal interpolation. The followed steps are: a presentation of « Meshfree methods » and their advantages compared to the traditional methods, an introduction to the meshfree RPIM method with a calibration of its associated parameters, and finally, a discussion on results obtained with the RPIM in forming processes exhibiting initiation and propagation of a crack showing the interest of the approach.

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.