A moving boundary finite element method-based numerical approach for the solution of one-dimensional problems in shape memory alloys

A moving boundary finite element method (MBFEM)-based numerical approach is proposed to solve one-dimensional (1-D) thermomechanical problems in shape memory alloys (SMAs) with a moving phase boundary. The Newton-Raphson method and recursive iterations, respectively, are used to address the non-linearity and coupling in the system of equations. At variance with other MBFEM approaches, the 1-D temperature field is modeled here by introducing a grid node (at the phase boundary and moving with it) in a 1-D mesh, which otherwise remains unchanged from one time step to another. The approach is demonstrated to be accurate, unconditionally stable and robust when compared to an analytical solution for purely thermal transformations, and also compares favorably with the predictions of a finite difference method (S.J. Kim, R. Abeyaratne, Continuum Mechanics and Thermodynamics 7 (1995) 311-332) for stress-induced transformations.