# Simulation of Fluid Dynamics Monitoring Using Ultrasonic Measurements
## 2 Numerical Models for Wave Equation  
There are several ways to calculate wave propagation. For example, the ray-based method--or ray-tracing method--is one that requires a much smaller amount of computational resources than other full-wave-based methods. We therefore applied this method in our former studies [29; 30; 31; 32]. Initially, this method considered only the deflection from [PERSON]'s law; however, nowadays, with the so-called \"fat rays\" or \"pencils\" technique, it may perform ray tracing in inhomogeneous and anisotropic media [33]. However, it is a trade-off relation between the lightness of numerical resources that the ray-based method requires and its accuracy, because the accuracy is degraded by the high-frequency or infinite-frequency approximation that the ray-based method applies [34]. The finite-difference time-domain method (FDTD) is a type of full-wave method [35]. In a finite-difference scheme, the spatial domain is discretized as grid points, which are usually evenly spaced, and all physical properties are defined at these fixed grid points or halfway between these grid points. These schemes correspond to spatially staggered schemes. This becomes a restriction on the placement of physical values and the use of symmetric differentiation operators, which require fictitious grid points located beyond the boundary. This causes difficulties for the accurate expressions of material boundaries and of boundary conditions, especially if the shape has a complex geometry. Several studies have been conducted on the FDTD methods for various media that include curved boundaries [36; 37; 38; 39], which used an interpolation method based on the distance between a discretized point and a potentially curved boundary. Unfortunately, they are only partially satisfactory in terms of accuracy and also make FDTD techniques more complex, thus partially losing one of their main advantages, which is their simplicity. The finite element time-domain method (FETD) is also a full-wave method that applies the finite element method to spatial decomposition. This method may easily define a complex geometry in the target computation domain thanks to the matured utility tools of computer-aided engineering (CAE). In addition, a curved boundary may be included when using a second-order finite element [40; 41]. Compared with the ray-based method and FDTD, FETD requires larger computation resources, which is usually limiting when calculating the propagation of a wave in a large domain and with a high-frequency source wave. The spectral-element method (SEM) [42; 43; 44] is a technique that is similar to FETD but reduces the limitation of FETD. The main difference is the type of basis function used for each method. While FETD uses Gauss points for its spatial discretization, SEM uses higher-order Gauss-Lobatto-Legendre points. SEM uses a modified formulation that leads to a perfectly diagonal mass matrix [45], and thus it is much more numerically efficient than FETD. We therefore selected SEM as the numerical method to model wave propagation in our target, i.e., liquid sodium flow. We selected a numerical code SPECFEM3D, one of the most efficiently implemented SEMs for solving wave propagation in many types of material (elastic, viscoelastic, poroelastic, etc.). SPECFEM3D has already been applied for many cases and used with several HPC environments, e.g., [46; 47; 48]. As we planned to use the frozen-flow hypothesis, i.e., the temperature and density values are temporally constant while the wave propagates in the calculation domain, we started from an acoustic wave propagation equation in a static and heterogeneous medium [49]:  
\[\
abla^{2}p-\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^{2}}=0. \tag{1}\]  
where \(\
abla\) is the Del or nabla operator, \(t\) is time, \(p\) is pressure and \(c\) is sound speed.