Numerical simulations are presented for three dimensional viscous incompressible free shear flows. The numerical method is based on solving the vorticity equation using Vortex-In-Cell method. In this method, the vorticity field is discretized into a finite set of Lagrangian elements (particles) and the computational domain is covered by Eulerian mesh. Velocity field is computed on the mesh by solving Poisson equation. The solution proceeds in time by advecting the particles with the flow. Second order Adam-Bashford method is used for time integration. Exchange of information between Lagrangian particles and Eulerian grid is carried out using the M’4 interpolation scheme. The classical inviscid scheme is enhanced to account for stretching and viscous effects. For that matter, two schemes are used. The first one used periodic remeshing of the vortex particles along with fourth order finite difference approximation for the partial derivatives of the stretching and viscous terms. In the second scheme, derivatives are approximated by least squares polynomial. The novelty of this work is signified by using the moving least squares technique within the framework of the Vortex-in-Cell method and implementing it to a three dimensional temporal mixing layer. Comparisons of the mean flow and velocity statistics are made with experimental studies. The results confirm the validity of the present schemes. Both schemes also demonstrate capability to qualitatively capture significant flow scales, and allow gaining physical insight as to the development of instabilities and the formation of three dimensional vortex structures. The two schemes show acceptable low numerical diffusion as well.

Numerical simulations are presented for three dimensional viscous incompressible free shear flows. The numerical method is based on solving the vorticity equation using Vortex-In-Cell method. In this method, the vorticity field is discretized into a finite set of Lagrangian elements (particles) and the computational domain is covered by Eulerian mesh. Velocity field is computed on the mesh by solving Poisson equation. The solution proceeds in time by advecting the particles with the flow. Second order Adam-Bashford method is used for time integration. Exchange of information between Lagrangian particles and Eulerian grid is carried out using the M’4 interpolation scheme. The classical inviscid scheme is enhanced to account for stretching and viscous effects. For that matter, two schemes are used. The first one used periodic remeshing of the vortex particles along with fourth order finite difference approximation for the partial derivatives of the stretching and viscous terms. In the second scheme, derivatives are approximated by least squares polynomial. The novelty of this work is signified by using the moving least squares technique within the framework of the Vortex-in-Cell method and implementing it to a three dimensional temporal mixing layer. Comparisons of the mean flow and velocity statistics are made with experimental studies. The results confirm the validity of the present schemes. Both schemes also demonstrate capability to qualitatively capture significant flow scales, and allow gaining physical insight as to the development of instabilities and the formation of three dimensional vortex structures. The two schemes show acceptable low numerical diffusion as well.

In this thesis, I analyze the weight functions used in moving least squares (MLS) methods to construct implicit surfaces that interpolate or approximate polygon soup. I found that one previous method that presented an analytic solution to the integrated moving least squares method has issues with degeneracies because they changed the weight functions to decrease too slowly. Inspired by their method, I derived a bound for the choice of weight function for implicit moving least squares (IMLS) methods to avoid these degeneracies in two-dimensions and in three-dimensions. Based on this bound, I give a theoretical proof of the correctness of the moving least squares interpolation and approximation scheme with weight function used in Shen et al. when used on closed polyhedrons. Further, previous IMLS implicit surface reconstruction algorithms that ll holes and gaps create surfaces with obvious bulges due to an intrinsic property of MLS. I propose a generalized IMLS method using a Gaussian distribution function to re-weight each polygon, making nearer polygons dominate and reducing the bulges on holes and gaps.

Numerical simulations are presented for three dimensional viscous incompressible free shear flows. The numerical method is based on solving the vorticity equation using Vortex-In-Cell method. In this method, the vorticity field is discretized into a finite set of Lagrangian elements (particles) and the computational domain is covered by Eulerian mesh. Velocity field is computed on the mesh by solving Poisson equation. The solution proceeds in time by advecting the particles with the flow. Second order Adam-Bashford method is used for time integration. Exchange of information between Lagrangian particles and Eulerian grid is carried out using the M’4 interpolation scheme. The classical inviscid scheme is enhanced to account for stretching and viscous effects. For that matter, two schemes are used. The first one used periodic remeshing of the vortex particles along with fourth order finite difference approximation for the partial derivatives of the stretching and viscous terms. In the second scheme, derivatives are approximated by least squares polynomial. The novelty of this work is signified by using the moving least squares technique within the framework of the Vortex-in-Cell method and implementing it to a three dimensional temporal mixing layer. Comparisons of the mean flow and velocity statistics are made with experimental studies. The results confirm the validity of the present schemes. Both schemes also demonstrate capability to qualitatively capture significant flow scales, and allow gaining physical insight as to the development of instabilities and the formation of three dimensional vortex structures. The two schemes show acceptable low numerical diffusion as well.

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Computational micromechanics analysis of carbon nanotube-epoxy nanocomposites, containing aligned nanotubes, is performed using the mesh independent intrinsic extended finite element method (IXFEM). The IXFEM employs a localized intrinsic enrichment strategy to treat arbitrary discontinuities defined through the level-set method separate from the problem domain discretization, i.e. the finite element (FE) mesh. A global domain decomposition identifies local subdomains for building distinct partition of unities that appropriately suit the approximation. Specialized inherently enriched shape functions, constructed using the moving least square method, enhance the approximation space in the vicinity of discontinuity interfaces, maintaining accuracy of the solution, while standard FE shape functions are used elsewhere. Comparison of the IXFEM in solving validation problems with strong and weak discontinuities against a standard finite element method (FEM) and analytic solutions validates the enriched intrinsic bases, and shows anticipated trends in the error convergence rates. Applying the IXFEM to model composite materials, through a representative volume element (RVE), the filler agents are defined as individual weak bimaterial interfaces. Though a series of RVE studies, calculating the effective elastic material properties of carbon nanotube-epoxy nanocomposite systems, the benefits in substituting the conventional mesh dependent FEM with the mesh independent IXFEM when completing micromechanics analysis, investigating effects of high filler count or an evolving microstructure, are demonstrated. / Master of Science

No / This paper focuses on the metamodel-based collaborative optimization (CO). The objective is to improve the computational efficiency of CO in order to handle multidisciplinary design optimization problems utilising high fidelity models. To address these issues, two levels of metamodel building techniques are proposed: metamodels in the disciplinary optimization are based on multi-fidelity modelling (the interaction of low and high fidelity models) and for the system level optimization a combination of a global metamodel based on the moving least squares method and trust region strategy is introduced. The proposed method is demonstrated on a continuous fiber-reinforced composite beam test problem. Results show that methods introduced in this paper provide an effective way of improving computational efficiency of CO based on high fidelity simulation models.

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.

Métodos de representação de superfícies a partir de pontos não-organizados se mantêm como uma das principais vertentes científicas que aquecem o estado-da-arte em Computação Gráfica e, significativamente, estão sendo reconhecidos como uma ferramenta interessante para definição de interfaces móveis no contexto de simulações numéricas de escoamento de fluidos. Não é difícil encontrar motivos para tais fatos: pelo lado da computação gráfica, por exemplo, a manipulação de conjuntos de pontos massivos com geometrias complexas e sujeitos a informações ruidosas ainda abre margem para novas metodologias. Já no âmbito da mecânica dos fluidos, onde os dados não são originados de \\emph tridimensionais, mas sim de interfaces entre fluidos imiscíveis, mecanismos de representação de superfícies a partir de pontos não-organizados podem apresentar características computacionais e propriedades geométricas que os tornem atrativos para aplicações em simulação de fenômenos físicos. O objetivo principal dessa tese de doutorado foi, portanto, o desenvolvimento de técnicas de representação de superfícies a partir de pontos não-organizados, que sejam capazes de suprir restrições de importantes trabalhos prévios. Nesse sentido, primeiramente focalizamos a elaboração de técnicas baseadas em formulações de mínimos-quadrados-móveis e de uma técnica robusta de partição da unidade implícita adaptativa em duas vias. Além de mecanismos de representação de superfícies a partir de pontos não-organizados, também propusemos um método promissor para representação de interfaces em simulação numérica de escoamento de fluidos multifásicos. Para isso, embasamo-nos numa abordagem Lagrangeana (livre-de-malhas), fundamentada no método dos mínimos-quadrados-móveis algébricos e apresentamos diversos resultados numéricos, estudos de convergências e comparações que evidenciam o potencial dessa metodologia para simulações numéricas de fenômenos físicos. Apesar de a contribuição principal deste trabalho ser o desenvolvimento de métodos para representação de superfícies a partir de pontos não-organizados, a experiência que adquirimos no desenvolvimento dessas técnicas nos conduziu à elaboração de mecanismos para representação de dados volumétricos não-organizados. Por conta disso, apresentamos dois mecanismos de representação a partir de dados volumétricos não-organizados com o intuito de serem aplicáveis a informações oriundas de malhas contendo células arbitrárias, isto é, propusemos a definição de um método de rendering unificado / Surface reconstruction from unorganized points has been one of the most promising scientific research areas in Computer Graphics. In addition, it has been used successfully for the definition of fluid interface in numerical simulation of fluid flow. There are several reasons to that fact: for instance, considering Computer Graphics, we have the handling of out-of-core data from complicated geometries and subject to noisy information that brings out opportunities for the development of new techniques. Further, considering Numerical Fluid Mechanics, where the input data does not come from tridimensional scanners, but from fluid interfaces, schemes that define the surface from unorganized points can offer geometrical and computational properties useful to numerical fluid flow simulation. The main goal of this project was the development of novel techniques for reconstructing surfaces from unorganized points with the capability to overcome the main drawbacks of important previous work. To that end, first we focused on the development of techniques based on moving-least-squares and on a robust twofold partition of unity Implicits. Added to the development of surface reconstruction from unorganized points, we proposed a novel scheme for defining fluid flow interfaces. We approach a meshless Lagrangian based on algebraic moving-least-squares surfaces. In addition, we presented several numerical results, convergence tests and comparisons, which state the power of the method to numerical simulation of physical phenomena. Although our main contributions were focused on surface reconstruction from points, we proposed methods to function reconstruction from unorganized volumetric data. Thus, we present two schemes to represent volumetric data from arbitrary meshes, i.e., a unified rendering scheme

O estudo do comportamento de fluidos é um antigo domínio das ciências da natureza. Ultimamente, fenômenos de engenharia que eram estudados empiricamente passaram a ser estudados com auxílio computacional. A Dinâmica de Fluidos Computacional (DFC) é a área da ciência da computação que estuda métodos computacionais para simulação de escoamento de fluidos, e muitas vezes é a forma mais prática, ou a única, de se observar fenômenos de interesse no escoamento. Este projeto de Mestrado procurou investigar, no âmbito da simulação de um escoamento bifásico, métodos computacionais para representar a interface entre dois fluidos imiscíveis. A separação dos fluidos por meio de uma interface é necessária para assegurar que, propriedades como viscosidade e densidade, específicas de cada fluido, sejam utilizadas corretamente para o cálculo do movimento de seus respectivos fluidos. Desenvolvemos um método lagrangeano sem a utilização de malhas com o objetivo de suprir algumas restrições de trabalhos prévios. Para representar a interface entre os dois fluidos, este método utiliza uma técnica de reconstrução de superfícies baseada em aproximações de superfícies algébricas de alta ordem. Os resultados numéricos reportados neste documento evidenciam o potencial da nossa abordagem / The study of the behaviour of fluids is an ancient field in natural sciences. Recently, engineering phenomena that were empirically studied started to be done with computacional aid. The Computational Fluid Dynamics (CFD) is the area of science that studies computational methods for computer simulation of fluid flow, and often is the most practical way, or the only, to observe phenomena of interest in flow. This Masters degree project sought to investigate, in the context of the simulation of biphasic flows, computational methods to represent the interface between two immiscible fluids. The separation of fluids by the means of an interface is required to ensure that, during the simulation, the physical properties of a fluid, like density and viscosity (specific of each fluid) are properly used in the calculus of the respective fluid motion. We developed a lagrangean method without the use of mesh with the goal of alleviating some of the previous works restrictions. To represent the interface between the two fluids, this method uses a surface reconstruction technique based on approximations of high order algebraic surfaces. The numerical results reported herein show the potential of our approach