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Simulating soft robots in cluttered environments remains an open problem due to the challenge of capturing complex dynamics and interactions with the environment. Furthermore, fast simulation is desired for quickly exploring robot behaviors in the context of motion planning. In this paper, we examine a particular class of inflated-beam soft growing robots called "vine robots", and present a dynamics simulator that captures general behaviors, handles robot-object interactions, and runs faster than real time. The simulator framework uses a simplified multi-link, rigid-body model with contact constraints. To narrow the sim-to-real gap, we develop methods for fitting model parameters based on video data of a robot in motion and in contact with an environment. We provide examples of simulations, including several with fit parameters, to show the qualitative and quantitative agreement between simulated and real behaviors. Our work demonstrates the capabilities of this high-speed dynamics simulator and its potential for use in the control of soft robots.