Thesis (M.S.)--Oregon State University, 2008. / Printout. Includes bibliographical references (leaf 24). Also available on the World Wide Web.

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Numerous approaches have been proposed for solving partial differential equations; all these methods have their own advantages and disadvantages depending on the problems being treated. In recent years there has been much development of particle methods for mechanical problems. Among these are the Smoothed Particle Hydrodynamics (SPH), Reproducing Kernel Particle Method (RKPM), Element Free Galerkin (EFG) and Moving Least Squares (MLS) methods. This development is motivated by the extension of their applications to mechanical and engineering problems. Since numerical experiments are one of the basic tools used in computational mechanics, in physics, in biology etc, a robust spatial discretization would be a significant contribution towards solutions of a number of problems. Even a well-defined stable and convergent formulation of a continuous model does not guarantee a perfect numerical solution to the problem under investigation. Particle methods especially SPH and RKPM have advantages over meshed methods for problems, in which large distortions and high discontinuities occur, such as high velocity impact, fragmentation, hydrodynamic ram. These methods are also convenient for open problems. Recently, SPH and its family have grown into a successful simulation tools and the extension of these methods to initial boundary value problems requires further research in numerical fields. In this thesis, several problem areas of the SPH formulation were examined. Firstly, a new approach based on ‘Hamilton’s variational principle’ is used to derive the equations of motion in the SPH form. Secondly, the application of a complex Von Neumann analysis to SPH method reveals the existence of a number of physical mechanisms accountable for the stability of the method. Finally, the notion of the amplification matrix is used to detect how numerical errors propagate permits the identification of the mechanisms responsible for the delimitation of the domain of numerical stability. By doing so, we were able to erect a link between the physics and the numerics that govern the SPH formulation.

Thesis (Ph. D.)--University of Iowa, 2007. / Supervisor: Shaoping Xiao.. Includes bibliographical references (leaves 174-182).

Thesis (M.S. in Modeling, Virtual Environment, and Simulation (MOVES))--Naval Postgraduate School, March 2007. / Thesis Advisor(s): Christian Darken. "March 2007." Includes bibliographical references (p. 115). Also available in print.

Numerous approaches have been proposed for solving partial differential equations; all these methods have their own advantages and disadvantages depending on the problems being treated. In recent years there has been much development of particle methods for mechanical problems. Among these are the Smoothed Particle Hydrodynamics (SPH), Reproducing Kernel Particle Method (RKPM), Element Free Galerkin (EFG) and Moving Least Squares (MLS) methods. This development is motivated by the extension of their applications to mechanical and engineering problems. Since numerical experiments are one of the basic tools used in computational mechanics, in physics, in biology etc, a robust spatial discretization would be a significant contribution towards solutions of a number of problems. Even a well-defined stable and convergent formulation of a continuous model does not guarantee a perfect numerical solution to the problem under investigation. Particle methods especially SPH and RKPM have advantages over meshed methods for problems, in which large distortions and high discontinuities occur, such as high velocity impact, fragmentation, hydrodynamic ram. These methods are also convenient for open problems. Recently, SPH and its family have grown into a successful simulation tools and the extension of these methods to initial boundary value problems requires further research in numerical fields. In this thesis, several problem areas of the SPH formulation were examined. Firstly, a new approach based on ‘Hamilton’s variational principle’ is used to derive the equations of motion in the SPH form. Secondly, the application of a complex Von Neumann analysis to SPH method reveals the existence of a number of physical mechanisms accountable for the stability of the method. Finally, the notion of the amplification matrix is used to detect how numerical errors propagate permits the identification of the mechanisms responsible for the delimitation of the domain of numerical stability. By doing so, we were able to erect a link between the physics and the numerics that govern the SPH formulation.

Une part importante de l’activité en calcul scientifique et analyse numérique est consacrée aux problèmes de transport d’une quantité par un champ donné (ou lui-même calculé numériquement). Les questions de conservations étant essentielles dans ce domaine, on formule en général le problème de façon eulérienne sous la forme d’un bilan au niveau de chaque cellule élémentaire du maillage, et l’on gère l’évolution en suivant les valeurs moyennes dans ces cellules au cours du temps. Une autre approche consiste à suivre les caractéristiques du champ et à transporter les valeurs ponctuelles le long de ces caractéristiques. Cette approche est délicate à mettre en oeuvre, n’assure pas en général une parfaite conservation de la matière transportée, mais peut permettre dans certaines situations de transporter des quantités non régulières avec une grande précision, et sur des temps très longs (sans conditions restrictives sur le pas de temps comme dans le cas des méthodes eulériennes). Les travaux de thèse présentés ici partent de l’idée suivante : dans le cadre des méthodes utilisant un suivi de caractéristiques, transporter une quantité supplémentaire géométrique apportant plus d’informations sur le problème (on peut penser à un tenseur des contraintes dans le contexte de la mécanique des fluides, une métrique sous-jacente lors de l’adaptation de maillage, etc. ). Un premier pan du travail est la formulation théorique d’une méthode de transport de telles quantités. Elle repose sur le principe suivant : utiliser la différentielle du champ de transport pour calculer la différentielle du flot, nous donnant une information sur la déformation locale du domaine nous permettant de modifier nos quantités géométriques. Cette une approche a été explorée dans dans le contexte des méthodes particulaires plus particulièrement dans le domaine de la physique des plasmas. Ces premiers travaux amènent à travailler sur des densités paramétrées par un couple point/tenseur, comme les gaussiennes par exemple, qui sont un contexte d’applications assez naturelles de la méthode. En effet, on peut par la formulation établie transporter le point et le tenseur. La question qui se pose alors et qui constitue le second axe de notre travail est celle du choix d’une distance sur des espaces de densités, permettant par exemple d’étudier l’erreur commise entre la densité transportée et son approximation en fonction de la « concentration » au voisinage du point. On verra que les distances Lp montrent des limites par rapport au phénomène que nous souhaitons étudier. Cette étude repose principalement sur deux outils, les distances de Wasserstein, tirées de la théorie du transport optimal, et la distance de Fisher, au carrefour des statistiques et de la géométrie différentielle. / In applied mathematics, question of moving quantities by vector is an important question : fluid mechanics, kinetic theory… Using particle methods, we're going to move an additional quantity giving more information on the problem. First part of the work is the theorical formulation for this kind of transport. It's going to use the differential in space of the vector field to compute the differential of the flow. An immediate and natural application is density who are parametrized by and point and a tensor, like gaussians. We're going to move such densities by moving point and tensor. Natural question is now the accuracy of such approximation. It's second part of our work , which discuss of distance to estimate such type of densities.

Thesis (M.S.)--University of Alabama at Birmingham, 2009. / Description based on contents viewed June 2, 2009; title from PDF t.p. Includes bibliographical references (p. 71-76).

Thesis (Ph.D.)--University of Western Australia, 2007.

No description available.