Abstract: We describe an Eulerian-on-Lagrangian solid simulator that reduces or eliminates many of the problems experienced by fully Eulerian methods but retains their advantages. Our method does not require the construction of an explicit object discretization and the fixed nature of the simulation mesh avoids tangling during large deformations. By introducing Lagrangian modes to the simulation we enable unbounded simulation domains and reduce the time-step restrictions which can plague Eulerian simulations. Our method features a new solver that can resolve contact between multiple objects while simultaneously distributing motion between the Lagrangian and Eulerian modes in a least-squares fashion. Our method successfully bridges the gap between Lagrangian and Eulerian simulation methodologies without having to abandon either one.