Postgraduate research opportunities Verifiable and efficient simulation for structural-health digital twins

The correctness and validity of simulation models for structural health is under scrutiny in this project.

Digital twins for structural-health monitoring (SHM) heavily rely on physics simulation to inform their prediction. The issue is that simulation models are often deterministic and cannot capture the complexity of a stochastic physical world. Simulation models can be made stochastic by means of uncertainty propagation methods, but this is often computationally prohibitive. A digital twin must be able to update nearly in real time, whilst stochastic models typically need several hours if not days to complete a single simulation. How can we make these simulation models more realistic whilst complying with the efficiency constraints of a digital twin? 

Moreover, to achieve high fidelity, simulation models are often very complex and their correctness is often questioned. Are there enough measurements, data and information to make these simulation models so detailed and complex? What can be done to improve the robustness and correctness of these models? Is the construction of these models just based on idealised information conceived at the design stage or is it based on actual data?

These issues motivate the investigation of alternative ways to solve ordinary and partial differential equations that go beyond the well-established paradigm of finite element analysis. We propose the use of Lagrange-Taylor or simply Taylor models to address the issues of correctness and validity of simulation models. Taylor models are mathematical representations used in computer-aided formal verification to model and analyse the behaviour of nonlinear and hybrid systems. 

A Taylor model is a combination of a Taylor series expansion and an interval Lagrange reminder. The Taylor series represents the local behaviour of a nonlinear function around a nominal point, and the interval Lagrange reminder represents the uncertainty in the inputs. The resulting Taylor model provides a rigorous representation of the behaviour of the nonlinear system in a region around the nominal point, considering the uncertainty in the inputs. The advantage of Taylor models is that they allow for efficient and precise analysis of nonlinear systems, even in the presence of uncertainty, by providing guaranteed bounds on the behaviour of the system. 

Taylor models are still an active area of research, as their scalability, implementation and computational complexity is still not very well understood. The accuracy of Taylor models is measured with the size of the interval that bounds the uncertainty that propagates from the inputs to the output response of interest, like displacement, stresses, strains, etc. The accuracy of Taylor models is known to degrade as the number of variables in the nonlinear system increases, as well as when the system exhibits high nonlinear behaviour. 

Several approaches could be explored to address the scalability concerns, including:

• Use of reduced-order models to represent the nonlinear behaviour of the system.
• The combination of Taylor models with finite element analysis.
• The adaptive refinement based on monitoring data. 

In SHM, Taylor models can be used to represent the behaviour of a structure and its components, taking into account the uncertainty in the loads, structural parameters, and environmental conditions. By modelling the structure as a set of Taylor models, engineers can analyse its behaviour and predict its response under various conditions, including those that may not have been encountered in the past.

On the other hand, a digital twin is a virtual representation of a physical system or process that can be used to simulate, monitor, and analyse its behaviour. In SHM, a digital twin can be used to represent a structure and its components, and to incorporate data from various sensors and monitoring systems. This allows engineers to better understand the behaviour of the structure under various loads and environmental conditions, and to make informed decisions about its maintenance and repair.

By combining Taylor models and digital twins, engineers can improve the accuracy and efficiency of the SHM process, and obtain a more complete and up-to-date understanding of the structure's behaviour and condition. Overall, the combination of Taylor models and digital twins provides a powerful tool for improving the validity and efficiency of SHM, and for ensuring the safety and longevity of structures.