Mathematics applied to seeking out structural defects

Improving nondestructive testing through guided wave imaging

“Based on six years of experience working within the aeronautics sector with the Direction Générale de l’Armement (the French Government Defence procurement and technology agency), I know that, when it comes to aircraft maintenance, nondestructive testing presents a significant gap with the methods proposed in the applied mathematics literature”. These are the words of Laurent Bourgeois, professor at ENSTA Paris and member of the project team Poems (Propagation des Ondes: Étude Mathématique et Simulation - Wave Propagation: Mathematical Study and Simulation), a joint undertaking involving the Inria Saclay Centre, ENSTA Paris and the CNRS. Nondestructive testing involves identifying and locating defects in structures without damaging them. In the industrial sphere, this type of testing generally involves the use of ultrasound waves, which are passed through the material. The signal that comes back can then be used to evaluate any obstacles the waves may have encountered, thereby identifying any defects in the structure.

One method is applied to slender structures known as “waveguides” such as pipes, thin plates and rails, in which waves are guided by the geometry. Low-frequency waves are in general used in this case. “This makes it possible to image large areas in one go, but the resolution is poorer than with high-frequency waves”, explains Arnaud Recoquillay, nondestructive testing researcher at CEA-LIST. “Traditional guided wave engineering methods are unable to analyse high-frequency signals, which are a great deal more complex.” Laurent Bourgeois proposed to use applied mathematics to harness a wider frequency range, enabling more precise imaging.

Following the waveguide

Assisted by his colleagues from Poems, a project team renowned for its expertise in spectral theory and diffraction problems more generally, he turned to the linear sampling method, a mathematical tool developed in the 1990s. “This involves dividing the area you are looking to inspect into a sampling grid and determining, for each point, whether or not it belongs to the defect, using mathematical criteria based on measurements”, explains Laurent Bourgeois.

A PhD coordinated alongside CEA-LIST, which Arnaud Recoquillay defended in early 2018, explored the shift from theory to practice: fitting sensors along a guide, with the capacity to send and receive the ultrasound waves needed for nondestructive testing. A proof of principle was established, but only for “closed” guides, i.e. guides which are homogeneous and out in the open.

And so Inria and CEA-LIST joined forces for a second PhD, which was funded by the CEA and which has just been completed.  This had a new objective: to adapt this sampling method to a guide partially submerged in an environment, whether this involved steel rods in concrete or pipes in water. “There are two main problems with this”, explains Laurent Bourgeois. “The first is that we only have access to part of the structure, preventing us from fitting sensors all the way along it. The second problem is that part of the guide is ‘open’, meaning it is in contact with another environment, where some of the waves will be lost.” 

From the simplest to the most complex

There were a number of stages to this PhD, which was assigned to Jean-François Fritsch, then a student at ENSTA Paris. The researchers began by working on a simplified version of the problem, considering a structure governed by anti-plane elasticity (meaning it behaves like a fluid from the point of view of the waves). Using equations which are well-known in such cases, they were able to provide mathematical proof for the method and to move onto the following stage: studying the junction between two guides, whether two pipes or two plates welded together.

“Successfully applying the sampling method to this type of junction was a good start, enabling us to then implement it on a partially submerged guide”, explains Arnaud Recoquillay. “The section where the guide enters a new environment marks a sort of junction.” The final step involved simulating this specific case.  “The sampling method, in a situation where you have a fluid interacting with a solid, is highly technical, as the elasticity of a structure adds a lot of complexity from a mathematical point of view”, says Laurent Bourgeois. “It took a lot of patience and determination on the part of the PhD student just to write the equations.” 

But the results were conclusive: the sampling method can be used to identify defects in a partially submerged guide, even if precision is slightly poorer than with a closed guide. “It was an ambitious PhD, exploring the direct problem of wave propagation and the inverse problem of imaging at the same time”, explains Arnaud Recoquillay.

Journal articles, a patent and a range of applications

From an academic point of view, research linked to the PhD has already led to three journal articles, with more expected to follow. A patent has also been filed for the method and the experimental protocol for identifying defects on guide junctions. CEA-LIST will now carry out the experimental validation of the method developed through this PhD.  Once they have a proof of concept, adapting the method to industrial constraints will enable it to be transfered to nondestructive testing professionals.

Given that it potentially concerns all slender structures, this will have a wide range of applications, from pipes in nuclear power plants and railway or metro tracks to steel embedded in concrete for anchoring bridges, underwater cables and buried pipes. What’s more, given that the method works on thin plates, it could also be used on aircraft fuselages. These are all fields where the new method will help to speed up the process of seeking out defects, while enhancing precision.

Inria and CEA-LIST, meanwhile, will continue to work together to explore the subject. “One of my main areas of focus is structural health monitoring, which involves identifying defects in structures using permanently fitted sensors”, explains Arnaud Recoquillay. “Both the type and the number of sensors are much more limited, which means our sampling method will need to be further optimised. This is cutting-edge mathematical research, which is in the expertise of Poems, and so the collaboration takes on its full meaning.” After yet another success, it surely won't be long before more joint research subjects come along.

Find out more

• Multi-sensor and multi-modal technology: optimising data acquisition, CEA-LIST, 14/4/2022.
• EP19: a closer look at nondestructive testing in aeronautics, Safran, 18/6/2021.