MC 5479
Speaker
Tian
Qiao
Department
of
Applied
Mathematics,
University
of
Waterloo
Title
Design of Tubular Network Systems Using Circle Packing and Discrete Optimization
Abstract
In
this
thesis,
we
describe
the
design
of
a
tubular
network
systems
that
must
occupy,
as
best
as
possible,
regions
that
demonstrate
some
kind
of
longitudinal
symmetry.
In
order
to
simplify
the
problem,
the
region
of
the
container
is
discretized
into
a
sequence
of
prism
blocks
.
The
problem
is
decomposed
into
two
parts:
1.
Pack
tubes
in
these
blocks,
2.
Connect
these
packed
tubes
at
the
ends
of
each
block.
In the first part, since each block is prismatic, the problem of packing tubes is equivalent to the packing of circles in the cross-sectional area of each block. In this case, we assume that the cross-sectional area of each block is a polygon. We investigate a series of algorithms to pack circles, including a rather naive approach as well as the GGL [2] circle packing algorithm. Then we modify the GGL algorithm to pack circles in regions that are more complicated. Based on the GGL, we will also invent new algorithm that provides more satisfactory packing results.
In the second part, we connect the packed tubes from Part one to form a complete network system. First we consider the simplest case -- constructing a tubular system in a container with no variations, i.e., a single block. We solve this problem in terms of the travelling salesman problem (TSP) which is a classical problem in discrete optimization. For containers with varying cross-sections, we connect tubes at end of each block independently instead of constructing a complete system. This problem can be reduced to a perfect matching (PM) problem at each end. We apply similar integer programming algorithms to both perfect matching problem and TSP. However, the design of complete tubular network system in a container exhibiting longitudinal symmetry remains an open problem for future work.