In this work, the propagation of an ultrasonic pulse in a thin plate is computed solving the differential equations modeling this problem. To solve these equations finite differences are used to discretize the temporal variable, while spacial variables are discretized using Finite Element method. The variational formulation of the problem corresponding to a fixed value of time is obtained and the existence an uniqueness of the solution is proved. The proposed approach leads to a sequence of linear systems with the same sparse, symmetric and positive defined matrix. The free software FreeFem++ is used to compute the approximated solution using polynomial triangular elements. Numerical experiments show that velocities computed using the approximated displacements for different frequency values are in good correspondence with analytical dispersion curves for the phase velocity.
In this work, we use the phase velocity method in combination with finite element method to compute the dispersion curve for phase velocity of an ultrasonic pulse traveling in a thin isotropic plate. This method is based on the numerical solution of the wave propagation equations for several selected frequencies. To solve these equations, a second order difference scheme is used to discretize the temporal variable, while spatial variables are discretized using the finite element method. The variational formulation of the problem corresponding to a fixed value of time is obtained and the existence and uniqueness of the solution is proved. A priori error estimates in the energy norm and in the [Formula: see text] norm are also obtained. The open software FreeFem++ is used with quadratic triangular elements to compute the displacements. Numerical experiments show that the velocities computed from the approximated displacements for different frequency values are in good agreement with analytical dispersion curve. This confirms that the proposed symbiosis between finite element and phase velocity method is suitable for computing dispersion curves in more general wave propagation problems, where the geometry is complex and the material is anisotropic.
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