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This work aims to propose an implicit algorithm for solving the magnetoelastic wave propagation in a magnetostrictive actuator. This algorithm is built by combining meshless methods with the Newmark implicit time scheme. The mathematical formulation of vibrations of the actuator is performed by modeling the experimental actuator device by a monodimensional magnetostrictive cylindrical rod attached to an elastic spring, subjected to an axial magnetic field. The formulation of the problem is carried out using the rod model. The study is carried out taking into account mechanical and magnetic effects described by a linear model of strain-stress magnetoelastic relations, whose coefficients depend on the square of the magnetic field, elastic, magnetostrictive, and magnetoelastic characteristics of the elastomagnetic material of which the rod is made. The dynamic equation which describes the magnetoelastic wave propagation in the actuator is obtained by applying the Lagrangian formalism based on the least action principle. The wave propagation is governed by a spatio-temporal partial differential equation whose coefficients are a function of on elastic, magnetostrictive, and magnetic characteristics and implicitly depends on time. The resolution of the dynamic equation is made by adopting two numerical approaches developed by combining the mesh-free collocation methods, the Moving Least Squares (MLS), and the Radial Point Interpolation Method (RPIM) with the Newmark implicit scheme temporal integration. By performing the spatial discretization of the displacement, we obtain a system of linear second-order ordinary differential equations (ODEs), then transform it into a system of first-order ordinary differential equations after integrating Newmark’s implicit time scheme. A comparison of results computed by the proposed numerical approaches RPIM and MLS with experimental data available in the literature is presented. The stability and the solutions qualities of two meshless methods, RPIM and MLS, are also studied. The influence of magnetical and mechanical parameters as well as the coupling coefficients on the performance of the magnetostrictive actuator is also made.

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