PhaseVelocity Code

PhaseVelocity is a FreeFEM++ Code to compute the phase velocity of an ultrasonic pulse $g(t)$, while it travels along a transversally annular isotropic plate, see Figure 1

Figure 1: Annular isotropic plate.

The pulse (depending on the frequency $f_0$) is applied on the black boundary of the plate and it is given by (see Figure 2):

$g(t)=\phi\ \sin \left( 2\ \pi\ {\it f_0}\ t \right){{e}^{-0.9134221857\ {{\it f_0}}^{2} \left( t-\ \frac{9}{4 {\it f_0}} \right) ^{2}}},\ \ for\ \ t\geq 0$

Figure 2: Pulse $g(t)$ for $\it f_0 = 0.5\ MHz$ and $\phi = 1.0\times 10^{-3}m$.

PhaseVelocity code implements a combination of the velocity method (PVM) ¹ with the finite element method (FEM). The FEM-PVM ² is based on the numerical solution of the wave propagation equations, which is computed using a second order difference scheme for the temporal variable and FEM triangular quadratic Lagrange discretization for spatial variables. The phase velocity for a given frequency is obtained from the computed displacements at few points on the top of the plate, in a similar way as the case of thin plane plates ³. Running the code for several frequency values, an array of frequency vs phase velocity points is obtained. The phase velocity dispersion curve can be computed fitting these points.

Requirements

PhaseVelocity Code has been tested with FreeFem++ 3.56 and above and should run on most personal laptops and desktop computers.

Input

• $f_0$ in the interval $[1,15] \times 10^5 Hz\ (100kHz - 1.5MHz)$

• Curvature κ in the interval $[0.01,90]\ m^{-1} $

Functionality

• Computation of the displacements of points of the plate.
• Plot of the wave propagation.
• Computation of the phase velocity.

Graphical Output

Figure 3: Vertical displacements of 6 selected points on the top of the plate

Figure 4: Deformation of the curved steel plate for a fixed time after emitting a pulse. Colors correspond to the intensity of the norm of the displacement field.

Note

To execute the code the funcions.edp file and PhaseVelocity.edp file must be in the same folder. Run the main file PhaseVelocity.edp

Bibliography

Footnotes

1. [E. Moreno, N. Galarza, B. Rubio and J. A. Otero, Phase velocity method for guided wave measurements in composite plates, in UIA Symposium 2014, Physics Procedia, Vol. 63, 54--60, 2015.] ↩

2. M. Cruz, V. Hernández, J. Estrada, E. Moreno, A. Mansur, Phase velocity method for computing dispersion curves in curved plate waveguides, to appear in journal REVISTA INVESTIGACIÓN OPERACIONAL, 2023. ↩

3. M. Cruz, A. Mansur, E. Moreno, V. Hernández and J. Estrada, Wave Propagation Code with FreeFEM, https://github.com/MathICIMAF/Wave-Propagation-Code-with-FreeFEM. ↩