Numerical solution of the wave propagation problem in a plate

TL;DR

This paper presents a numerical method combining finite differences and finite element techniques to simulate ultrasonic wave propagation in thin plates, validated by numerical experiments matching analytical dispersion curves.

Contribution

It introduces a variational formulation and a computational approach using FreeFem++ for solving wave equations in plates, with proofs of existence and uniqueness.

Findings

Velocities match analytical dispersion curves

Method produces sparse, symmetric, positive definite matrices

Numerical results validate the approach

Abstract

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.