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Finite element simulations are used to calculate the rate of whisker growth due to intermetallic formation in a Sn film with columnar grain structure on a Cu substrate. The simulations account for plastic flow by dislocation motion within the grains, as well as diffusion along grain boundaries. Grains with a slightly lower yield stress than their neighbors are shown to act as sinks for material and are progressively extruded from the film. A simple analytical model is developed to estimate the resulting whisker growth rate, and is shown to be in good agreement with experimental measurements.