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. http://resolver.caltech.edu/CaltechTHESIS:05312015-134909133
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)|
|Defense Date:||8 May 2015|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Gemma Mason|
|Deposited On:||02 Jun 2015 23:19|
|Last Modified:||12 Apr 2016 17:37|
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