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Master's Dissertation
DOI
10.11606/D.76.2004.tde-30032005-111208
Document
Author
Full name
Lia Munhoz Benati Napolitano
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2004
Supervisor
Committee
Bernardes, Esmerindo de Sousa (President)
Batista Neto, João do Espírito Santo
Hernandes, Antonio Carlos
Title in Portuguese
"Implementação numérica do método Level Set para propagação de curvas e superfícies"
Keywords in Portuguese
Choques
Diferenças Finitas
Equações de Hamilton-Jacobi
Leis de Conservação Hiperbólica
Método Level Set
Abstract in Portuguese
Nesta dissertação de Mestrado será apresentada uma poderosa técnica numérica, conhecida como método Level Set, capaz de simular e analisar movimentos de curvas em diferentes cenários físicos. Tal método - formulado por Osher e Sethian [1] - está sedimentado na seguinte idéia: representar uma determinada curva (ou superfície) Γ como a curva de nível zero (zero level set) de uma função Φ de maior dimensão (denominada função Level Set). A equação diferencial do tipo Hamilton-Jacobi que descreve a evolução da função Level Set é discretizada através da utilização de acurados esquemas hiperbólicos e, como resultado de tal acurácia, obtém-se uma formulação numérica capaz de tratar eficazmente mudanças topológicas e/ou descontinuidades que, eventualmente, podem surgir no decorrer da propagação da curva (ou superfície) de nível zero. Em virtude da eficácia e versatilidade do método Level Set, esta técnica numérica está sendo amplamente aplicada à diversas áreas científicas, incluindo mecânica dos fluidos, processamento de imagens e visão computacional, crescimento de cristais, geometria computacional e ciência dos materiais. Particularmente, o propósito deste trabalho equivale ao estudo dos fundamentos do método Level Set e, por fim, visa-se aplicar tal modelo numérico à problemas existentes na área de crescimento de cristais. [1] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79:12, 1988.
Title in English
"Implementation of Level Set Method for computing curves and surfaces motion"
Keywords in English
Finite Differences
Hamilton-Jacobi Equations
Hyperbolic Conservation Laws
Level Set Methods
Shocks
Abstract in English
In this dissertation, we present a powerful numerical technique known as Level Set Method for computing and analyzing moving fronts in different physical settings. The method -formulated by Osher and Sethian [1] - is based on the following idea: a curve (or surface) is embedded as the zero level set of a higher-dimensional function Φ (called level set function). Then, we can link the evolution of this function Φ to the propagation of the curve itself through a time-dependent initial value problem. At any time, the curve is given by the zero level set of the time-dependent level set function Φ. The evolution of the level set function Φ is described by a Hamilton-Jacobi type partial differential equation, which can be discretised by the use of accurate methods for hyperbolic equations. As a result, the Level Set Method is able to track complex curves that can develop large spikes, sharp corners or change its topology as they evolve. Because of its versatility and efficacy, this numerical technique has found applications in a large number of areas, including fluid mechanics, image processing and computer vision, crystal growth, computational geometry and materials science. Particularly, the aim of this dissertation has been to understand the fundamentals of Level Set Method and its final goal is compute the motion of bondaries in crystal growth using this numerical model. [1] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79:12, 1988.
 
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Publishing Date
2005-04-08
 
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