Abstract

The finite element method (FEM) is one of the great triumphs of modern day applied mathematics, numerical analysis and software and computing technology. Every area of the sciences and engineering has been positively impacted by the ability to model and study complex physical and natural systems described by systems of partial differential equations (PDE) via the FEM .

In parallel the recent developments in sensor, measurement, and signalling technologies enables the phenomenological study of systems as diverse as protein signalling in the cell, to turbulent combustion in jet engines, to plastic deformation in bridges.

The connection between sensor data and FEM is currently restricted to data assimilation for solving inverse problems or the calibration of PDE based models. This however places unwarranted faith in the fidelity of the underlying mathematical description of the actual system under study. If one concedes that there is ‘missing physics’ or mis-specification between generative reality and the mathematical abstraction defining the FEM then a framework to systematically characterise and propagate this uncertainty in FEM is required.

This talk will present a formal statistical construction of the FEM which systematically blends both mathematical description with observational data and provides both small and large scale examples from 3D printed structures to working rail bridges currently operated by the United Kingdom Network Rail.