Peridynamics (PD), a reformulation of the Classical Continuum Mechanics (CCM), is a new and promising meshless and nonlocal computational method in solid mechanics. To permit discontinuities, the PD integro-differential equation contains spatial integrals and time derivatives. PD can be considered as the continuum version of molecular dynamics. This feature of PD makes it a good candidate for multi-scale analysis of materials. Concurrently, the topology optimization has also been rapidly growing in view of the need to design lightweight and high performance structures. Therefore, this thesis presents the potential for a peridynamics-based topology optimization approach. To avoid the gradient calculations, a heuristic topology optimization method is employed. The minimization of the PD strain energy density is set as the objective function. The structure is optimized based on a modified solid isotropic material with a penalization approach and a projection scheme is utilized to obtain distinct results. Several test cases have been studied to analyze the suitability of the proposed method in topology optimization. / Graduate

Around the world, the energy over consumption issue has been one of the key socio-economic and political challenges, which has drastically worsened over the last few years. Over the years engineers and environmentalists have proposed several approaches to improve energy efficiency. One is to reduce energy demand by improving consumption habits and a second approach is to introduce the use of a "greener" concept by using biomaterials in a diverse and more efficient manner in engineering construction to create energy efficient environments. This work will investigate the effects of using "green" stabilized earth materials to provide and enhance thermal regulation for indoor environments. This effects can be compared to what skin does to regulate body temperature in humans, animals, and plants. On this effort the thermal behavior of several biomaterials will be analyzed using a computational tool in order to test the mechanical properties of biomaterials and also several geometry configurations to minimize the energy needed for heating and cooling an environment. In this research a localized radial basis function (LRBF) meshless method, developed by the Computational Mechanics Lab (CML) at the University of Central Florida, has been implemented to test several wall geometrical configuration using known biomaterials such as clay. The advantage of using the LRBF meshless method in this particular research is based in the accuracy of the numerical method and also because it decreases computation time regardless of model complexity geometry without the need of mesh generation. This research includes a complete description of the LRBF meshless method, as well as a quantification of cooling methods that have been used by past civilizations and recent construction standards but have not been validated on scientific basis. Results are presented which will demonstrate the effectiveness of using integrated sheets of biomaterials in engineering construction to increase energy efficiency in indoor environments.

Meshless methods are a relatively new class of schemes for the numerical solution of partial differential equations. Their special characteristic is that they do not require a mesh but only need a distribution of points in the computational domain. The approximation at any point of spatial derivatives appearing in the partial differential equations is performed using a local cloud of points called the "connectivity" (or stencil). A point distribution can be more easily generated than a grid since we have less constraints to satisfy. The present work uses two meshless methods; an existing scheme called Least Squares Kinetic Upwind Method (LSKUM) and a new scheme called Kinetic Meshless Method (KMM). LSKUM is a "kinetic" scheme which uses a "least squares" approximation} for discretizing the derivatives occurring in the partial differential equations. The first part of the thesis is concerned with some theoretical properties and application of LSKUM to 3-D point distributions. Using previously established results we show that first order LSKUM in 1-D is positivity preserving under a CFL-like condition. The 3-D LSKUM is applied to point distributions obtained from FAME mesh. FAME, which stands for Feature Associated Mesh Embedding, is a composite overlapping grid system developed at QinetiQ (formerly DERA), UK, for store separation problems. The FAME mesh has a cell-based data structure and this is first converted to a node-based data structure which leads to a point distribution. For each point in this distribution we find a set of nearby nodes which forms the connectivity. The connectivity at each point (which is also the "full stencil" for that point) is split along each of the three coordinate directions so that we need six split (or half or one-sided) stencils at each point. The split stencils are used in LSKUM to calculate the split-flux derivatives arising in kinetic schemes which gives the upwind character to LSKUM. The "quality" of each of these stencils affects the accuracy and stability of the numerical scheme. In this work we focus on developing some numerical criteria to quantify the quality of a stencil for meshless methods like LSKUM. The first test is based on singular value decomposition of the over-determined problem and the singular values are used to measure the ill-conditioning (generally caused by a flat stencil). If any of the split stencils are found to be ill-conditioned then we use the full stencil for calculating the corresponding split flux derivative. A second test that is used is based on an accuracy measurement. The idea of this test is that a "good" stencil must give accurate estimates of derivatives and vice versa. If the error in the computed derivatives is above some specified tolerance the stencil is classified as unacceptable. In this case we either enhance the stencil (to remove disc-type degenerate structure) or switch to full stencil. It is found that the full stencil almost always behaves well in terms of both the tests. The use of these two tests and the associated modifications of defective stencils in an automatic manner allows the solver to converge without any blow up. The results obtained for a 3-D configuration compare favorably with wind tunnel measurements and the framework developed here provides a rational basis for approaching the connectivity selection problem. The second part of the thesis deals with a new scheme called Kinetic Meshless Method (KMM) which was developed as a consequence of the experience obtained with LSKUM and FAME mesh. As mentioned before the full stencil is generally better behaved than the split stencils. Hence the new scheme is constructed so that it does not require split stencils but operates on a full stencil (which is like a centered stencil). In order to obtain an upwind bias we introduce mid-point states (between a point and its neighbour) and the least squares fitting is performed using these mid-point states. The mid-point states are defined in an upwind-biased manner at the kinetic/Boltzmann level and moment-method strategy leads to an upwind scheme at the Euler level. On a standard 4-point Cartesian stencil this scheme reduces to finite volume method with KFVS fluxes. We can also show the rotational invariance of the scheme which is an important property of the governing equations themselves. The KMM is extended to higher order accuracy using a reconstruction procedure similar to finite volume schemes even though we do not have (or need) any cells in the present case. Numerical studies on a model 2-D problem show second order accuracy. Some theoretical and practical advantages of using a kinetic formulation for deriving the scheme are recognized. Several 2-D inviscid flows are solved which also demonstrate many important characteristics. The subsonic test cases show that the scheme produces less numerical entropy compared to LSKUM, and is also better in preserving the symmetry of the flow. The test cases involving discontinuous flows show that the new scheme is capable of resolving shocks very sharply especially with adaptation. The robustness of the scheme is also very good as shown in the supersonic test cases.

Since the introduction of modern computers, numerical methods for atmospheric simulations have routinely been applied for weather prediction, and in the last fifty years, there has been a steady improvement in the accuracy of forecasts. Accurate numerical models of the atmosphere are also becoming more important as researchers rely on global climate simulations to assess and understand the impact of global warming. The choice of grid in a numerical model is an important design decision and no obvious optimal choice exists for computations in spherical geometry. Despite this disadvantage, grid-based methods are found in all current circulation models. A different approach to the issue of discretizing the surface of the sphere is given by meshless methods, of which radial basis function (RBF) methods are becoming prevalent. In this thesis, RBF methods for simulation of atmospheric flows are explored. Several techniques are introduced to increase the efficiency of the methods. By utilizing a novel algorithm for adaptively placing the node points, accuracy is shown to improve by over one order of magnitude for two relevant test problems. The computational cost can also be reduced by using a local finite difference-like RBF scheme. However, this requires a stabilization mechanism for the hyperbolic problems of interest here. A hyper-viscosity scheme is introduced to address this issue. Another stability issue arising from the ill-conditioning of the RBF basis for almost-flat basis functions is also discussed in the thesis, and two algorithms are proposed for dealing with this stability problem. The algorithms are specifically tailored for the task of creating finite difference weights using RBFs and are expected to overcome the issue of stationary error in local RBF collocation.

Doctor of Philosophy / Department of Mechanical and Nuclear Engineering / Prakash Krishnaswami / Xiao J. Xin / Over the past two decades meshless methods have attracted much attention owing to their advantages in adaptivity, higher degree of solution field continuity, and capability to handle moving boundary and changing geometry. In this work, a meshless integral method based on the regularized boundary integral equation has been developed and applied to two-dimensional linear elasticity and elastoplasticity with small or large deformation. The development of the meshless integral method and its application to two-dimensional linear elasticity is described first. The governing integral equation is obtained from the weak form of elasticity over a local sub-domain, and the moving least-squares approximation is employed for meshless function approximation. This formulation incorporates: a subtraction method for singularity removal in the boundary integral equation, a special numerical integration for the calculation of integrals with weak singularity which further improves accuracy, a collocation method for the imposition of essential boundary conditions, and a method for incorporation of natural boundary conditions in the system governing equation. Next, elastoplastic material behavior with small deformation is introduced into the meshless integral method. The constitutive law is rate-independent flow theory based on von Mises yielding criterion with isotropic hardening. The method is then extended to large deformation plasticity based on Green-Naghdi’s theory using updated Lagrangian description. The Green-Lagrange strain is decomposed into the elastic and plastic part, and the elastoplastic constitutive law is employed that relates the Green-Lagrange strain to the second Piola-Kirchhoff stress. Finally, a pre- and post-processor for the meshless method using node- and pixel-based approach is presented. Numerical results from the meshless integral method agree well with available analytical solutions or finite element results, and the comparisons demonstrate that the meshless integral method is accurate and robust. This research lays the foundation for modeling and simulation of metal cutting processes.

In this paper, we use the newly developed method of particular solution (MPS) and one-stage method of fundamental solution (MFS-MPS) for solving partial differential equation (PDE). In the 1-D Poisson equation, we prove the solution of MFS-MPS is converge to Spectral Collocation Method using Polynomial, and show that the numerical solution similar to those of using the method of particular solution (MPS), Kansa's method, and Spectral Collocation Method using Polynomial (SCMP). In 2-D, we also test these results for the Poisson equation and find the error behaviors.

Multiquadric collocation method is highly efficient for solving partial differential equations due to its exponential error convergence rate. More amazingly, there are two ways to reduce the error: the traditional way of refining the grid, and the unexpected way of simply increasing the value of shape constant $c$ contained in the multiquadric basis function, $sqrt{r^2 + c^2}$. The latter is accomplished without increasing computational cost. It has been speculated that in a numerical solution without roundoff error, infinite accuracy can be achieved by letting $c ightarrow infty$. The ability to obtain infinitely accurate solution is limited only by the roundoff error induced instability of matrix solution with large condition number. Using the arbitrary precision computation capability of {it Mathematica}, this paper tests the above conjecture. A sharper error estimate than previously obtained is presented in this paper. A formula for a finite, optimal $c$ value that minimizes the solution error for a given grid size is obtained. Using residual errors, constants in error estimate and optimal $c$ formula can be obtained. These results are supported by numerical examples.

Many types of radial basis functions, such as multiquadrics, contain a free parameter called shape factor, which controls the flatness of RBFs. In the 1-D problems, Fornberg et al. [2] proved that with simple conditions on the increasingly flat radial basis function, the solutions converge to the Lagrange interpolating. In this report, we study and extend it to the 1-D Poisson equation RBFs direct solver, and observed that the interpolants converge to the Spectral Collocation Method using Polynomial. In 2-D, however, Fornberg et al. [2] observed that limit of interpolants fails to exist in cases of highly regular grid layouts. We also test this in the PDEs solver and found the error behavior is different from interpolating problem.

O Método dos Elementos Finitos Generalizados, MEFG, compartilha importantes características dos métodos sem malha. As funções de aproximação do MEFG, atreladas aos pontos nodais, são enriquecidas de modo análogo ao refinamento p realizado no Método das Nuvens hp. Por outro lado, por empregar uma malha de elementos para construir as funções partição da unidade, ele também pode ser entendido como uma forma não convencional do Método dos Elementos Finitos. Neste trabalho, ambas as interpretações são consideradas. Os métodos sem malha, particularmente o Método de Galerkin Livre de Elementos e o Método das Nuvens hp, são introduzidos com o propósito de estabelecer os conceitos fundamentais para a descrição do MEFG. Na seqüência, apresentam-se aplicações numéricas em análise linear e evidenciam-se características que tornam o MEFG interessante para a simulação da propagação de descontinuidades. Após discutir os modelos de dano adotados para representar o comportamento não-linear do material, são introduzidos exemplos de aplicação, inicialmente do Método das Nuvens hp e depois do MEFG, na análise de estruturas de concreto. Os resultados obtidos servem de argumento para a implementação de um procedimento p-adaptativo, particularmente com o MEFG. Propõe-se, então a adaptação do Método dos Resíduos em Elementos Equilibrados à formulação do MEFG. Com vistas ao seu emprego em problemas não-lineares, algumas modificações são introduzidas à formulação do estimador. Mostra-se que a medida obtida para representar o erro, apesar de fundamentada em diversas hipóteses nem sempre possíveis de serem satisfeitas, ainda assim viabiliza a análise não-linear p-adaptativa. Ao final, são enumeradas propostas para a aplicação do MEFG em problemas caracterizados pela propagação de defeitos / The Generalized Finite Element Method, GFEM, shares several features with the so called meshless methods. The approximation functions used in the GFEM are associated with nodal points like in meshless methods. In addition, the enrichment of the approximation spaces can be done in the same fashion as in the meshless hp-Cloud method. On the other hand, the partition of unity used in the GFEM is provided by Lagrangian finite element shape functions. Therefore, this method can also be understood as a variation of the Finite Element Method. Indeed, both interpretations of the GFEM are valid and give unique insights into the method. The meshless character of the GFEM justified the investigation of meshless methods in this work. Among them, the Element Free Galerkin Method and the hp-Cloud Method are described aiming to introduce key concepts of the GFEM formulation. Following that, several linear problems are solved using these three methods. Such linear analysis demonstrates several features of the GFEM and its suitability to simulate propagating discontinuities. Next, damage models employed to model the nonlinear behavior of concrete structures are discussed and numerical analysis using the hp-Cloud Method and the GFEM are presented. The results motivate the implementation of a p-adaptive procedure tailored to the GFEM. The technique adopted is the Equilibrated Element Residual Method. The estimator is modified to take into account nonlinear peculiarities of the problems considered. The hypotheses assumed in the definition of the error measure are sometimes violated. Nonetheless, it is shown that the proposed error indicator is effective for the class of p-adaptive nonlinear analysis investigated. Finally, several suggestions are enumerated considering future applications of the GFEM, specially for the simulation of damage and crack propagation

Cette thèse consiste à explorer une nouvelle approche pour la simulation d'objets flexibles par la mécanique des milieux continus, dans le cadre d'applications graphiques interactives telles que le jeu vidéo ou l'entraînement aux gestes chirurgicaux. Elle s'inscrit en continuité d'un stage de M2-R sur ce même sujet. Il est important de pouvoir régler simplement un compromis entre précision et temps de calcul suivant la nature de l'application. Les approches actuelles de simulation utilisent principalement la méthode des éléments finis. Celle-ci repose sur un maillage volumique des objets qu'il est souvent difficile d'adapter dynamiquement aux besoins de l'application. La nouveauté introduite par cette thèse est d'utiliser des repères déformables comme primitives cinématiques, avec des champs de déplacements inspirés des méthodes de 'skinning' utilisées en informatique graphique. Le but est d'éviter ainsi les difficultés liées au maillage volumique, ainsi que de faciliter le raffinement et la simplification adaptatives par simple ajout ou suppression de repère déformable là où c'est souhaitable. Ce travail est financé par le projet européen 'Passport for Virtual Surgery', dont le but est de créer automatiquement des modèles physiques pour l'entraînement aux gestes de chirurgie hépatique, à partir de données médicales et anatomiques personnalisées. Dans ce contexte, Guillaume, en collaboration avec d'autres membres du projet, mettra en place les outils nécessaires pour construire la scène physique à partir d'images médicales segmentées et de connaissances anatomiques génériques. Le foie sera dans un premier temps représenté par des modèles physiques précédemment développés à EVASION et étendus aux opérations de découpe. Par la suite, il y appliquera son nouveau modèle mécanique basé sur des repères déformables. The aim of this thesis is to develop a new approach for the simulation of flexible objects based on the continous middle method, related with interactive graphics applications such as video games or training in surgery. It is a continuity of the M2 research internship on the same topic. It is important to simply settle a compromise between accuracy and time computing according to the application. Current simulation approaches mainly use the finite element method, which is based on a volumetric mesh of the simulated objects. It is often difficult to dynamically adapt the needs to the application. The novelty of this thesis is to use deformable reference frames as kinematic primitives, with displacement fields based on 'skinning' methods used in computer graphics. The aim is to avoid the difficulties associated with volumetric mesh, and make the refinement and the adaptive simplification easier by adding or deleting deformable reference frames if necessary. This work is funded by the European project 'Passport for Virtual Surgery', which aims to automatically create models for physical training in gestures of liver surgery, from medical and anatomical custom data. In this context, Guillaume, in collaboration with other members of the project, will develop the tools necessary to build the physical scene from segmented medical images and generic anatomical knowledge. The liver will initially be represented by physical models previously developed in the EVASION team and then extended to cutting operations. Thereafter, Guillaume will apply his new mechanical model based on deformable reference frames. / Physically based deformable models have become ubiquitous in computer graphics. It allow to synthetize real behaviors, based on the physical laws from continuum mechanics. In this thesis, we focus on interactive simulations such as to video games or surgical simulators. The majority of the existing works focused up to here on the animation of objects made of homogeneous materials. Nevertheless, plenty of real objects, for instance like the biological structures, consist of multiple imbricated materials. Their decomposition in homogeneous zones requires a high-resolution spatial discretization to solve the variations of the material properties, which requires prohibitive computation time. In this context, we present new real time simulation techniques for deformable objects which can be cut. First of all, we present a real time method for cutting deformable objects in which, contrary to the previous methods, the object deforms on the cutting tool contact and cuts occur only when the pressure reaches a certain level. The independence of the physical, collision and visual models makes the topological changes easier. The GPU computing and local modifications enable fast execution. Then, a dynamic meshless method is described, which uses reference frames as control nodes instead of using points, with a displacement field formulation similar to skinning. It allows to easily tune the weights and benefits from the rigor of physical methods as the finite elements. The introduction of integration points, reducing the samples number by a least squares approximation, speeds up the spatial integrations. Other pre-computations are proposed in order to speed up the simulation time. Finally, new anisotropic shape functions are defined to encode the variations of material properties thanks to the introduction of the compliance distance. These complex shape functions uncouple the material resolution of the displacement functions ones. It allow an extremely sparse nodes sampling. The use of the compliance distance allows an automatic nodes distribution with regard to the material properties.