THESIS
2023
1 online resource (xi, 101 pages) : illustrations (chiefly color)
Abstract
The solutions to certain partial differential equations may exhibit large variations in
small regions in the physical domain. When numerical methods are applied to solve these
equations, overly fine meshes are often necessary to accurately resolve the solution behaviors,
resulting in significant increases in computational expenses. In such cases, mesh
refinement methods can be employed to generate nonuniform meshes, where the mesh
resolutions are higher only in the areas with significant solution variations. This can help
reduce computational costs while maintaining the accuracy of numerical solutions. The
adaptive moving mesh method (also called r-refinement) is a mesh refinement method
that seeks to find a coordinate transformation between the computational and the physical
domain bas...[
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The solutions to certain partial differential equations may exhibit large variations in
small regions in the physical domain. When numerical methods are applied to solve these
equations, overly fine meshes are often necessary to accurately resolve the solution behaviors,
resulting in significant increases in computational expenses. In such cases, mesh
refinement methods can be employed to generate nonuniform meshes, where the mesh
resolutions are higher only in the areas with significant solution variations. This can help
reduce computational costs while maintaining the accuracy of numerical solutions. The
adaptive moving mesh method (also called r-refinement) is a mesh refinement method
that seeks to find a coordinate transformation between the computational and the physical
domain based on specific criteria. In this thesis, we explore the theories and techniques
of the adaptive moving mesh method and propose two applications. First, we develop a
moving mesh finite element method for the minimum compliance problem in topology optimization.
Our numerical experiments demonstrate that using a coarser mesh alongside
the moving mesh technique can yield desirable output configurations while also enhancing
computational efficiency. Second, we propose a moving mesh method for the simulation
of the finite-time blowup solution of the Landau-Lifshitz-Gilbert equation. With iterative
remeshing, we are able to simulate the blowup solution with the magnitude of the maximum
gradient up to 10
4, and that of the minimum mesh size being 10
−5. We study the
self-similar patterns and the blowup rates of the solutions and also verify the numerical
results by comparing them to established analytical results in a recent study.
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