University of Bristol
Supervisor: Professor Tony Mulholland
Non-contact laser ultrasound arrays can send high frequency, broadband pulses into metallic components as they are being additively manufactured. The waves that are generated travel into the bulk of the material, are scattered by the heterogeneities they encounter, and travel back to the surface to be recorded. These time domain signals (A-scans) therefore contain knowledge of the journey the waves have taken and therefore the grain structure (the actual spatial map of the phase and orientation of the crystals) deep within the component. This knowledge can be extracted by combining this data with mathematical models of wave propagation in such materials. Typical materials of interest are Titanium alloys and the aim is to produce maps of the alpha-beta phases within a specific component as it is being built and monitor how this map changes over time as the material cools and solidifies.
This project will focus on the mathematical modelling part of this multi-disciplinary team enterprise. It will develop new mathematical models and deepen our fundamental understanding of ultrasound wave propagation in polycrystalline materials. The semi-analytical models that will emerge will then enable real-time, model-based, tomographic reconstructions using laser-based, ultrasound array, in-line sensing of additively manufactured metals.
The project will be supported by industry via Dr Andreas Schumm at EdF in the form of expert advice and access to key datasets. The project will complement experimental work that is being conducted at Strathclyde (Stratoudaki) and tomographic reconstruction work at Edinburgh (Curtis).
This project will contribute to the FIND CDT themes of Future NDE Technologies (as it models complex materials, produces volumetric imaging, aims for real time data processing, and requires digital twin modelling) and Future Manufacturing NDE (as it will focus on the inline inspection of 3D printed components).
The student would not be expected to have prior knowledge of the mathematical modelling techniques to be used in this project and full support and training will be given throughout. The project would suit a student that enjoys “pen and paper” mathematical derivations and is seeking a career in industry or academia that involves the modelling associated with NDT.
- Build a mathematical model of high frequency, ultrasound wave propagation in heterogeneous materials. The material is locally anisotropic and the grain orientations will be modelled using ideas from stochastic (Markovian) processes. Monochromatic shear waves propagating in two-dimensional media will be modelled at first and a stochastic differential equation framework will be developed. Explicit expressions for the moments of the probability distributions of the power transmission and reflection coefficients will then be derived. The dependency of the moments of these coefficients on the material microstructure will then be established. (LOW RISK)
- Extend the model to a broadband impulse and derive semi-analytical expressions for the moments of the pulse-echo response. The stochastic differential equations will be described by a propagator matrix which in turn will lead to a series of Ricatti equations for each of the moments of the pulse-echo response. Their solution will be used to examine (analytically) the dependency of the moments of the pulse-echo response (for example, analytical expressions for the attenuation spectrum) on the material microstructure. Other wave modes, input waves, and inspection scenarios will be investigated. (MEDIUM/HIGH RISK).
- A finite element simulation of ultrasound wave propagation in heterogeneous, polycrystalline materials (EBSD images will be imported in to a suitable solver) will be developed in order to test the analytical predictions developed above. (LOW RISK).