Gaussian Pulse Absorption by Perfectly Matched Layers: Pressure Acoustics, Transient

This tutorial model is a test and benchmark model for perfectly matched layers (PML) in the time domain. PMLs are widely used for problems of wave propagation in open domains. A perfectly matched layer is supposed to meet two main requirements. First, a wave must propagate without spurious reflections from the interface between the physical domain and the PML. Second, a PML must ensure long time stability of the numerical solution.

The model solves the same problem as in Gaussian Pulse in 2D Uniform Flow: Convected Wave Equation and Absorbing Layers in the absence of the background flow. The Pressure Acoustics, Transient interface solves the wave equation in a squared computational domain and the Perfectly Matched Layers surround the computational domain suppressing the reflections from the outer boundary. An acoustic pulse is generated by an initial Gaussian distribution at the center of the computational domain. An analytical solution to the problem exists and is used to validate the solution. The model shows how to set up and use the PMLs.

For more information about PMLs in acoustics, see the section Modeling with the Pressure Acoustics Branch (FEM-Based Interfaces) in the Acoustics Module User’s Guide.

Model Definition

The computational domain with perfectly matched layers is depicted in Figure 1. The model is set up in a dimensionless system where the speed of sound c0 = 1 and the density ρ0 = 1. The Gaussian pulse is emitted at the origin x = 0 with initial values

(1)

where α = ln(2)/9 and β = 0.04. Since the velocity u = (u, v) does not enter the wave equation explicitly, the two last initial values are reformulated in terms of the first time derivative of the pressure. The use of the mass conservation law yields

The Gaussian pulse parameters and the expressions for the initial values are included to the model as parameters and variables.

The analytical solution to Equation 1 is given by (see Ref. 1)

In the model, the integrate() operator is used to express the analytical solution. The integration is performed on a finite interval.

Figure 1: Geometry: physical domain and perfectly matched layers.

Results and Discussion

The propagation of the acoustic pulse is depicted in Figure 2. It shows the distribution of the acoustic pressure across the computational domain at four time steps. By the time t = 30, the pulse has entered the PML domain without any reflections from the interface between the physical domain and the PMLs. The pressure at the final simulated time t = 120 is depicted in Figure 3. By this time, the pulse has left the computational domain. There are no visible signals propagating back to the physical domain, which asserts the long-time stability of the perfectly matched layers.

In the next two figures, the simulated results are compared with the analytical solution. In Figure 4, the pressure at point (20, 10) is depicted as function of time. In Figure 5, the pressure is depicted along the x-axis at t = 40. Both show very good agreement with the analytical solution.

Figure 6 shows the frequency spectrum of the signal at point (20, 10).