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Full and Model-Reduced Structure-Preserving Simulation of Incompressible Fluids


Mason, Gemma Ellen (2015) Full and Model-Reduced Structure-Preserving Simulation of Incompressible Fluids. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9KK98QG.


This thesis outlines the construction of several types of structured integrators for incompressible fluids. We first present a vorticity integrator, which is the Hamiltonian counterpart of the existing Lagrangian-based fluid integrator. We next present a model-reduced variational Eulerian integrator for incompressible fluids, which combines the efficiency gains of dimension reduction, the qualitative robustness to coarse spatial and temporal resolutions of geometric integrators, and the simplicity of homogenized boundary conditions on regular grids to deal with arbitrarily-shaped domains with sub-grid accuracy.

Both these numerical methods involve approximating the Lie group of volume-preserving diffeomorphisms by a finite-dimensional Lie-group and then restricting the resulting variational principle by means of a non-holonomic constraint. Advantages and limitations of this discretization method will be outlined. It will be seen that these derivation techniques are unable to yield symplectic integrators, but that energy conservation is easily obtained, as is a discretized version of Kelvin's circulation theorem.

Finally, we outline the basis of a spectral discrete exterior calculus, which may be a useful element in producing structured numerical methods for fluids in the future.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:numerical analysis, geometric integration, computational fluid dynamics
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Desbrun, Mathieu
Thesis Committee:
  • Hou, Thomas Y. (chair)
  • Owhadi, Houman
  • Schroeder, Peter
  • Desbrun, Mathieu
Defense Date:8 May 2015
Record Number:CaltechTHESIS:05312015-134909133
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8948
Deposited By: Gemma Mason
Deposited On:02 Jun 2015 23:19
Last Modified:04 Oct 2019 00:08

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