Master's Defense | Tian Qiao, Design of Tubular Network Systems Using Circle Packing and Discrete Optimization

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.