On topological data analysis for structural dynamics: an introduction to persistent homology
This work applies TDA to structural dynamics, offering new metrics for data scrutiny in engineering domains, but it is incremental as it adapts existing TDA methods to specific applications.
The authors introduced topological data analysis (TDA) and persistent homology to quantify data shape, applying it to nonlinear dynamics for analyzing attractors and determining optimal time-delay embeddings, and demonstrated in a structural health monitoring case study that damage alters manifold shape more than temperature effects.
Topological methods can provide a way of proposing new metrics and methods of scrutinising data, that otherwise may be overlooked. In this work, a method of quantifying the shape of data, via a topic called topological data analysis will be introduced. The main tool within topological data analysis (TDA) is persistent homology. Persistent homology is a method of quantifying the shape of data over a range of length scales. The required background and a method of computing persistent homology is briefly discussed in this work. Ideas from topological data analysis are then used for nonlinear dynamics to analyse some common attractors, by calculating their embedding dimension, and then to assess their general topologies. A method will also be proposed, that uses topological data analysis to determine the optimal delay for a time-delay embedding. TDA will also be applied to a Z24 Bridge case study in structural health monitoring, where it will be used to scrutinise different data partitions, classified by the conditions at which the data were collected. A metric, from topological data analysis, is used to compare data between the partitions. The results presented demonstrate that the presence of damage alters the manifold shape more significantly than the effects present from temperature.