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In this paper we describe a numerical algorithm for the study of shear-band formation and growth in a two-dimensional antiplane shear of granular materials. The algorithm combines front-tracking techniques and adaptive mesh refinement. Tracking provides a more careful evolution of the band when coupled with special techniques to advance the ends of the shear band in the presence of a loss of hyperbolicity. The adaptive mesh refinement allows the computational effort to be concentrated in important areas of the deformation, such as the shear band and the elastic relief wave. The main challenges are the problems related to shear bands that extend across several grid patches and the effects that a nonhyperbolic growth rate of the shear bands has in the refinement process. We give examples of the success of the algorithm for various levels of refinement.