Simulation of acoustic waves in a fluid inside a nuclear tank

[Hanna-che]

Overview:

My project is part of a collaboration between the Master CSMI and Avnir Energy, a company specializing in the development of non-destructive monitoring tools for sensitive industrial structures, particularly in the nuclear sector.
Their overall aim is to detect possible anomalies in structures such as nuclear tank. To achieve this, they generate waves in the fluid or structure (by striking it), capture the signals generated using sensors, and use time signal reversal to analyze and identify the origin and nature of any defects.

Project objectives:

My general objective is to model and simulate acoustic waves in a fluid at rest, contained in a tank. This involves two objectives: The first is to simulate acoustic waves in a fluid contained in a simplified geometry (2D, 3D), and the second is to couple the fluid with the structure, by modeling the fluid-structure interaction.

Method:

This acoustic simulation in a fluid at rest is based on the wave equation. It is defined by : $$\frac{\partial^2 u}{\partial t^2} - c^2 \Delta u = f \quad \text{in } \Omega \times [0, T]$$, with u the pressure field (in Pa), c the celerity of sound in the fluid (in m/s) and f the source term.
The implementation was carried out using Feel++'s CFPDE toolbox. As this toolbox does not handle second derivatives in time, it is necessary to reformulate the problem in coupled equations. Homogeneous Neumann edge conditions are also introduced, and the fluid is initially assumed to be at rest. The complete system is rewritten as :

Simulations

To model a localized impact, I used a source term with the shape of a Ricker wavelet. Initially, I carried out tests with no concrete physical meaning, and then performed my tests in water. The celerity of sound in water is c = 1480 m/s.
The following 2D visualizations were obtained using Paraview with the “Warp by Scalar” option, obtained at time steps $t = 10^{-5}$ (first time step), $5\times10^{-4}$, $7\times10^{-4}$ and $10^{-3}$, with a mesh size h = 0.01 :

The following 3D visualizations were obtained at time steps $t = 3.4\times10^{-4}$, $7\times10^{-4}$ and $10^{-3}$, and seconds at time steps $t = 4\times10^{-4}$, $7\times10^{-4}$ and $10^{-3}$, using Paraview's “Contour” option, with a mesh size of h = 0.1.

Convergence studies :

To validate my numerical model, I carried out several types of test.
First, with an analytical function, built by ourselves. The $L^2$ and $H^1$ errors were calculated automatically by Feel++. For convergence in space, I found the theoretical orders (2 in $L^2$ and 1 in $H^1$).
For convergence in time, the results did not show the expected slope (2 because I use an implicit BDF scheme of order 2). Hypotheses about the dominant error in space and the domain of convergence already reached were put forward and tested, but no satisfactory results were obtained.
As a third test, I carried out a mesh convergence study. I compared my numerical solutions with a very fine numerical reference solution at certain points. The differences were very small (down to $10^{-12}$), making convergence difficult to visualize, especially in 3D.

Conclusion and Next Steps:

In conclusion, these various tests have enabled me to highlight limitations linked to the increasing complexity of the problem, particularly in 3D. Indeed, adding a dimension greatly increases the cost in memory and computation time, requiring a coarser mesh. What's more, current solvers (LU, GMRES) are limited for long, fine simulations. The FieldSplit preconditioner will be necessary for my convergence studies.

The next step in this work is fluid-structure coupling. This means coupling the wave equation with the elasticity equation. It will also be necessary to take into account the deformation of the fluid domain, via the ALE method. This will enable us to model real-life physical situations, in line with Avnir Energy's needs. This study forms the core of my future internship.