Many academic areas of the sciences, which might seem contrived at their outset,
often find relevance in practical fields like engineering. Prime examples include
quantum mechanics in semiconductor production, number theory’s importance in
cybersecurity, and adjustments for general relativity in global positioning systems.
This thesis looks up the abstraction hierarchy, focussing on the concepts of algebraic
topology and, to a lesser degree, differential geometry, to address challenges
in structural health monitoring (SHM) and nonlinear dynamics.
The first component of this thesis explores obtaining series solutions to nonlinear differential
equations by considering generating series expansions, a framework rooted
in differential geometry. Here, novel optimisations are presented for computationally
determined series solutions, which output impulse responses of nonlinear systems.
The process is applied to a benchmark study, yielding unseen solution depths, providing
greater insights into the behaviour of nonlinear systems.
Much of the data-based SHM literature is dedicated to machine learning methods;
overlooking informative shape features within data. The cornerstone of this thesis
introduces and applies Topological Data Analysis (TDA), harnessing the data’s
shape, such as holes and voids, for SHM decision-making. Topological arguments are
shown to enhance SHM insights regarding damage detection and environmental and
operational variation removal, as well as augmenting established machine-learning
approaches. A multi-faceted understanding of data is crucial for SHM since information
is limited, and decisions carry dire safety and economic consequences.
A key challenge in SHM is discerning damage effects from benign environmental
fluctuations. This thesis addresses this problem by imbuing 1D time series with a
topology, where different trends give unique shapes, and inferences are made via
topological reasoning. A significant portion of this thesis evaluates the Z24 Bridge,
showcasing TDA’s capability to identify damage amidst dominant external factors.
This thesis shows that abstract mathematical concepts yield beneficial outcomes by
providing unique insights; underscoring the potential of TDA and generating series
in SHM and nonlinear dynamics. By leveraging such novel analyses, one might
identify data subtleties, indicative of structural issues, which might otherwise be
overlooked with machine learning and traditional SHM means.
