Applications of level set theory to heat transfer problems and automatic mesh generation.

Abstract

The goal of the research is to extend level set theory to moving boundary Stefan problems and automatic quadrilateral mesh generation for both plane and three-dimensional geometry by combining it with the finite element formulation. The Stefan problem is one of the classical free or moving boundary problems and is necessarily a nonlinear problem because the Stefan condition is imposed on unknown moving boundaries. The method developed in this dissertation is a combination of the front tracking method and the fixed domain method. The fixed entire domain is used without depending on an enthalpy or freezing index formulation because the moving front is tracked using a high order function by level set theory. The elliptic automatic mesh generation algorithm is developed to generate unstructured surface quadrilateral meshes. In order to generate surface meshes, the quadrilateral meshes in a planar domain are constructed first. These meshes are generated by a looping algorithm, which is one of the domain decomposition methods. Node visibility is one of constraints used to choose an optimal splitting line subdividing the geometry. The concepts of level set theory are utilized to determine node visibility using the characteristics of the level set function. The quadrilateral meshes on the parent domain are mapped to the general curved surface by an elliptic algorithm after implementing the surface reconstruction process to overcome the limitations of the conventional elliptic algorithm. For surface reconstruction, the moving least squares method is implemented so that the necessary information from the surface function can be obtained. The method developed in this research is extended to generate local replacement meshes for specially shaped attachments in an automobile body structure in order to replace manual operations required to prepare a finite element model.