Finite element modeling of ultrasonic propagation in viscoelastic media

The presence of high-contrast inclusions, thin layers, or highly deformed mesh creates challenges when using standard time-discretization algorithms for the simulation of viscoelastic wave propagation in non-destructive testing (NDT). These features push conventional algorithms to their limits in terms of the maximum time step allowed to ensure correct numerical calculation. The spectral radius of the rigidity operators defined in the «disturbed» area is largely to blame for these standard algorithms’ extremely low time steps and disproportionately high computational costs.

We developed and analyzed two numerical strategies for decoupling the overall stability of the time-discretization algorithm from the local rigidity in the «disturbed» zone.

In our first strategy, locally implicit (LI) schemes, formulated by applying an implicit scheme only for local operators, were extended to viscoelastic models. We used an energy method, independent of the operators restricted to the “disturbed” area, to prove the stability of the final algorithm. This strategy was demonstrated on the three viscoelasticity models most commonly used in NDT: Maxwell, Zener, and Kelvin-Voigt. Stability remains governed solely by the background domain–a significant improvement over the explicit reference scheme.

In our second strategy, we extended the explicit scheme stabilization procedure, coupled with a domain decomposition method, to viscoelastic models. In the «disturbed» area, the steep term is replaced by a stabilizing Chebyshev polynomial, the order of which increases the permissible time step proportionally. In practical terms, this is still an explicit method, but one that requires only local resolution at the interface. An improved, controllable stability condition is established by proven energy decay.
 

 
We then tested these strategies in realistic 2D and 3D configurations representative of ultrasonic testing in heterogeneous materials. We obtained significant improvements in time steps, number of iterations, and computational cost. The reduction in computational cost was as high as an order of magnitude in some cases, all while faithfully reproducing the attenuation and multiple reflection phenomena looked for in micro-structured materials.

 

 

Use cases, applications, technology transfer

• Advanced viscoelastic modeling to support the design of high-frequency ultrasonic tests on heterogeneous materials and complex parts (aeronautics, energy, petrochemicals, etc.).

Major project and/or partnership

• The CARNOT CIVAMONT project, which is funding a post-doctoral contract for Vinduja Vasanthan, whose research focuses on numerical methods for ultrasonic NDT. Collaboration with Sébastien Imperiale, a researcher at Inria.

Flagship publication

• « Locally implicit and stabilized explicit time schemes for transient visco-elastic wave propagation problems », V. Vasanthan, A. Imperiale, S. Imperiale, tobe published in Journal of Numerical Mathematics, https://inria.hal.science/hal-05003766/document. The paper proposes an extension of the locally implicit and explicit stabilized methods to viscoelastic wave propagation models (Maxwell, Zener, Kelvin-Voigt). 2D and 3D ultrasonic NDT test cases confirmed a significant improvement (in terms of computational cost) over the standard approaches used in the CIVA platform.