We describe a GPU-based algorithmic framework for optimizing the shape of elements in a simplicial volume. Optimization is done on a per-vertex basis using only local neighborhood information in order to exploit the massive fine-grained parallelism on modern GPU hardware. We propose and apply three optimization methods which have potential to be suited for local optimization of element shape and present a framework which may be generalized to other methods. Experiments which compare our method to state-of-the-art algorithms show a more than ten-fold performance increase for a similar final quality in both test and practical real-world meshes.