This project investigates computational techniques for modeling dissipative physical systems experiencing contact, impact, friction and plasticity. The attendant optimization problems are strongly nonlinear, nonsmooth, and nonconvex in nature, necessitating the development of novel numerical methods. The research grounds the development of these methods by building on discrete geometric mechanics and variational integrators, which restate the fundamental principles of physics in a discrete, hence immediately computable form. The research examines five core behaviors arising from the Principle (momentum/energy/symmetry preservation, breaking contact, and non-interpenetration) and thereby derives computer algorithms whose outputs capture these physical behaviors. The project success is measured by (a) the production of novel computer algorithms that are able to simulate dissipative physical phenomena more accurately and efficiently than ever before, (b) new theoretical results on what can be expected of computer algorithms that simulate these phenomena, and (c) the successful adoption of the novel techniques by industry partners. The project results are publicly disseminated via articles in journals, release of source code on the world wide web, and transfer of technological expertise and data to industrial partners. Improving computational techniques for the simulation of contact, impact, and dissipative phenomena helps make engineering safety analyses, biomechanical models, computer visualizations, surgical training tools, and industrial manufacturing simulations that better predict reality. Algorithms that accurately capture the physics help to gain important new insights into many open questions that influence our understanding of large-scale geophysics of earthquakes and calving of icebergs, small-scale dissipation in high-frequency micro- and nano-electronic mechanical devices, prosaic domestic phenomena such as the chattering of chalk on a board and even the excitation of violin strings.