We develop an optimal transportation meshfree (OTM) method for simulating general
solid and fluid flows, including fluid-structure interaction. The method combines
concepts from optimal transportation theory with material-point sampling and max-ent
meshfree interpolation. The proposed OTM method generalizes the Benamou-Brenier
differential formulation of optimal mass transportation problems to problems
including arbitrary geometries and constitutive behavior. The OTM method enforces
mass transport and essential boundary conditions exactly and is free from tension
instabilities. The OTM method exactly conserves linear and angular momentum and its
convergence characteristics are verified in standard benchmark problems. We
illustrate the range and scope of the method by means of two examples of
application: the bouncing of a gas-filled balloon off a rigid wall; and the
classical Taylor-Anvil benchmark test extended to the hypervelocity range.