Coupling, Conservation, and Performance in Numerical Simulations
- Author(s): Ding, Ounan
- Advisor(s): Schroeder, Craig
- et al.
This thesis considers three aspects of the numerical simulations, which are
coupling, conservation, and performance. We conduct a project and address
one challenge from each of these aspects.
We propose a novel penalty force to enforce contacts with accurate Coulomb
friction. The force is compatible with fully-implicit time integration and the
use of optimization-based integration. In addition to processing collisions
between deformable objects, the force can be used to couple rigid bodies to
deformable objects or the material point method. The force naturally leads to
stable stacking without drift over time, even when solvers are not run to
convergence. The force leads to an asymmetrical system, and we provide a
practical solution for handling these.
Next we present a new technique for transferring momentum and velocity between
particles and MAC grids based on the Affine-Particle-In-Cell (APIC) framework
previously developed for co-located
grids. We extend the original APIC paper and show that
the proposed transfers preserve linear and angular momentum and also satisfy
all of the original APIC properties.
Early indications in the original APIC paper suggested that APIC might be
suitable for simulating high Reynolds fluids due to favorable retention of
vortices, but these properties were not studied further. We use two
dimensional Fourier analysis to investigate dissipation in the limit $\dt=0$.
We investigate dissipation and vortex retention numerically to quantify the
effectiveness of APIC compared with other transfer algorithms.
Finally we present an efficient solver for problems typically seen in
Microfluidic ``lab on a chip'' devices are small devices that operate on small
length scales on small volumes of fluid. Designs for microfluidic chips are
generally composed of standardized and often repeated components connected by
long, thin, straight fluid channels. We propose a novel discretization
algorithm for simulating the Stokes equations on geometry with these features,
which produces sparse linear systems with many repeated matrix blocks. The
discretization is formally third order accurate for velocity and second order
accurate for pressure in the $L^\infty$ norm. We also propose a novel linear
system solver based on cyclic reduction, reordered sparse Gaussian elimination,
and operation caching that is designed to efficiently solve systems with
repeated matrix blocks.