We discuss multiple strategies to perform general computations on unstructured grids using a GPU, with specific application to the assembly of systems of equations in finite element methods (FEMs). For each method, we discuss the GPU hardware's limiting resources, optimizations, key data structures, and dependence of the performance with respect to problem size, element size, and GPU hardware generation. These methods are applied to a nonlinear hyperelastic material model to develop a large-scale real-time interactive elastodynamic visualization. By performing the assembly, solution, update, and visualization stages solely on the GPU, the similuation benefits from speed-ups in each stage and avoids costly GPU-CPU transfers of data.