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Geometric, Variational Discretization of Continuum Theories


Gawlik, Evan S. (2010) Geometric, Variational Discretization of Continuum Theories. Senior thesis (Minor), California Institute of Technology. doi:10.7907/HW8J-FQ68.


This study derives geometric, variational discretizations of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincaré systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes.

Item Type:Thesis (Senior thesis (Minor))
Subject Keywords:Structure-preserving integrators, geometric mechanics, variational integrators, magnetohydrodynamics, nematic liquid crystals, microstretch continua
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Minor Option:Control and Dynamical Systems
Awards:Axline Merit Scholars, 2007-2010. Fredrick J. Zeigler Memorial Award, 2008. Henry Ford II Scholar Award, 2009. George W. Housner Award, 2010. Don Shepard Award, 2010.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Desbrun, Mathieu (co-advisor)
  • Marsden, Jerrold E. (co-advisor)
Thesis Committee:
  • None, None
Defense Date:11 June 2010
Record Number:CaltechTHESIS:06252010-144116673
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:5961
Deposited By: Evan Gawlik
Deposited On:28 Jun 2010 21:05
Last Modified:08 Nov 2019 18:12

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