Through a mathematical and computational model of the physical behavior of shape memory alloy wires, this study shows that localized heating and cooling of such materials provides an effective means of damping vibrational energy. The thermally induced pseudo-elastic behavior of a shape memory wire is modeled using a continuum thermodynamic description based on an improved Landau-Devonshire potential. Our construction of the potential function allows the model to account for particular alloys as well as the general solid-state phase transformation, improving over traditional potentials that idealize many of the material properties or focus only on individual processes. The material's thermodynamic response is modeled using a nonlinear conservation of momentum and a nonlinear heat equation. The heat equation introduces an inhomogeneous version of the Fourier heat flux, thereby describing the discontinuous heat flow associated with shape memory materials more thoroughly than standard, continuous heat dissipation mechanisms do. This continuum thermodynamic model is then solved computationally to determine the resulting state of the wire in time. Continuous time Galerkin methods and affine finite elements treat the temporal and spatial discretizations of the model, respectively. Traditional methods for solution of the resulting finite-dimensional, nonlinear, nonconvex system of equations must introduce a significant artificial dissipation to achieve existence of solutions. The solution of the discrete system here uses a novel combination of the damped Newton method and a homotopy method for minimizing the artificial dissipation. This combination, inspired by the well-known Method of Vanishing Viscosity for the solution of scalar hyperbolic conservation laws, reduces the artificial dissipation errors introduced by traditional approaches for such nonlinear, nonconvex thermomechanical models. Computational tests show that the proposed model successfully describes the relevant physical processes inherent in shape memory alloy behavior. Further computational experiments then confirm that up to 80% of an initial shock of vibrational energy can be eliminated at the onset of a thermally-induced phase transformation.