On the geometry of Population-Based Structural Health Monitoring: Part I — sheaves on graphs

Population-Based Structural Health Monitoring (PBSHM) has recently emerged as a means of using data from multiple structures to enhance diagnostic capability in any given individual structure. As part of a sequence of papers on the foundations of PBSHM, a geometric picture was proposed for the methodology. Although this picture provided some insight into PBSHM generally, it could not provide a rigorous basis for study as it was based on a fibre-bundle viewpoint that did not exactly match the ‘geometry’ of PBSHM. Assuming that a rigorous framework could provide valuable guidance into designing workable PBSHM systems, the objective of this paper (and a companion paper) is to provide a rigorous geometrical picture. The current paper achieves this by eschewing the fibre bundle as the main geometrical object of interest and replacing it with a cellular sheaf. It is shown how a population of structures, represented in abstract form, can be encapsulated in a cellular sheaf, together with the SHM feature data for the structures. Furthermore, a geometrical operation – a variant of local PCA alignment – is introduced which actually implements (linear) transfer learning between structures and naturally provides a generalisation of some previously-proposed algorithms. The methodology is validated on a population of simulated structures in the context of novelty detection. Further analysis of a multi-class SHM problem exposes some issues for PCA-alignment and statistic-alignment algorithms generally where rotations are included. As this paper possibly represents the first use of sheaves in engineering dynamics, substantial background material is included here and other aspects of sheaf technology are discussed.