��������

Mathematical Modelling and Analysis of Several Diffusive Processes

Michael Joseph Brutz

Applied Mathematics,��

Date and time:��

Thursday, November 6, 2014 - 1:00pm

���dz����پ��Dz�:��

Ecot 831

�������ٰ�������:��

The underlying theme of this research is using numerical methods to
develop computationally efficient algorithms for three separate problems
driven by diffusive processes. The problems under consideration are:
contaminant dispersal through fracture networks, modelling the flow of
glacial ice, and community detection on networks.

A common feature of containment facilities for nuclear waste is to
use expansive geological formations as an added barrier to contaminant
dispersal in the event of a leak. Although these formations are generally
comprised of dense rock that is difficult to penetrate, fractures within
them provide a potential means for contaminants to rapidly transport
across the barrier. The typical width of such fractures is only on the
order of millimeters whereas the typical scale of interest for contaminant
transport is on the order of kilometers. When particle tracking methods
are used to simulate the contaminant dispersal in fracture networks, this
disparity of scales severely restricts maximum time step sizes because
features at the millimeter scale need to be resolved. Our contribution to
this problem is developing a coarse scale particle tracking method that
allows for substantially larger time steps when particles are navigating
straight fractures.

With global warming comes concerns as to how the changing temperature
will impact glacial systems and their contribution to sea level rise.
On glacial scales, ice behaves as a very slowly moving non-Newtonian
fluid, and the primary problem for numerically simulating the evolution
of ice masses comes with Glen’s flow law for the effective viscosity. The
flow law is empirically based, and its simple form has proven useful for
analytical calculations. However, its simple form also allows for the effective
viscosity to become unbounded in regions of low strain rate, and
has proven to be very problematic for numerical simulations. Our contribution
to this problem is re-examining the datasets the flow law was
originally based on to develop an alternative model that fits the data with
comparable accuracy, but without the problematic singularity.
When working with networks that represent real world systems, a
common feature of interest is to find collections of vertices that form
communities. Because the word ”community” is an ambiguous term,
our interpretation is that it is necessary to quantify what it means to
be a community at a minimum of three scales for any given problem.
These scales are at the level of: individual nodes, individual communities,
and the network as a whole. Although our work focuses on detecting
overlapping communities in the context of social networks, our primary
contribution is developing a methodology that is highly modular and can
easily be adapted to target other problem-specific notions of community.

PDF of Abstract