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Eulerian Geometric Discretizations of Manifolds and Dynamics


Mullen, Patrick Gary (2012) Eulerian Geometric Discretizations of Manifolds and Dynamics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/5GSC-W341.


This thesis explores new methods for geometric, structure-preserving Eulerian discretizations of dynamics, including Lie advection and incompressible fluids, and the manifolds in which these dynamics occur. The result is a novel method for discrete Lie advection of differential forms, a new family of structure-preserving fluid integrators, and a new set of energies for optimizing meshes appropriate for some discrete geometric operators. First, high-resolution nite volume methods are leveraged to introduce a new method for discretizing the Lie advection of discrete differential forms, along with the related contraction operator, on regular grids. Through its geometric approach, the method exactly preserves properties such as the closedness of Lie advected closed forms. This results in an extension of nite volume techniques applicable to forms of arbitrary degree. After this, attention is turned to simplicial meshes, where new meshing techniques are developed to give formal error bounds on the discrete diagonal Hodge star, an important operator for geometric computations. Utilizing weighted Delaunay triangulations, both the primal mesh and its dual are optimized simultaneously over the entire space of orthogonal primal/dual pairs. Improved accuracy of the solution of Poisson equations is demonstrated as a practical application, as well as an increase in percentage of well-centered elements. Finally, a new structure-preserving method for the incompressible Navier-Stokes equations on simplicial meshes is developed, offering in the inviscid case the exact conservation of either the discrete energy or symplectic form. This leads to capturing the correct energy decay when viscosity is added, resulting in dissipation independent of grid and time resolution.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Eulerian, Geometric Integrators, Fluids, Lie Advection, Meshing
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Computer Science
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Desbrun, Mathieu
Thesis Committee:
  • Desbrun, Mathieu (chair)
  • Schroeder, Peter
  • Barr, Alan H.
  • Alliez, Pierre
Defense Date:8 September 2011
Record Number:CaltechTHESIS:09092011-162631888
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6662
Deposited By: Patrick Mullen
Deposited On:19 Oct 2011 18:57
Last Modified:03 Oct 2019 23:53

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