1. Linear Elastodynamics:A rectangular plate with a crack is stressed on two opposite sides. The resulting shock wave travels through the plate and reflects off the crack tip. (Only the upper right quadrant of plate is simulated, with symmetric boundary conditions.) The spacetime mesh computed by our algorithm, shown here with time as the vertical axis, accurately captures the passage of shear and pressure waves through the plate. Research objective: A wide variety of wavelike physical phenomena are described by systems of hyperbolic partial differential equations. Spacetime discontinuous Galerkin finite element methods developed at UIUC can be used to efficiently compute accurate solutions for many hyperbolic systems, given an appropriate unstructured mesh of the simulation domain in spacetime. We are developing efficient finite-element meshing algorithms to meet the unique requirements of these new numerical methods. Approach: Extending our earlier work on spacetime meshing, we develop a new adaptive advancing-front mesh generation algorithm. Our algorithm adds tetrahedral elements to an evolving unstructured mesh in small patches; the solution within each patch is computed as soon as the patch is created. By responding to numerical error estimates, our algorithm adapts the size and shape of spacetime elements to local geometric and physical features of the solution. Significant results: The meshes generated by our algorithm effectively resolve shocks and other interesting features of the solution, using smaller elements near these features and larger elements everywhere else. This adaptivity allows us to compute numerically accurate solutions several orders of magnitude faster than using a fine mesh everywhere. Broader impact: Unlike standard mesh generation in space, meshing directly in spacetime presents unique theoretical challenges that are solved and validated by experiments for linear and nonlinear systems. In concert with new numerical methods, our meshing algorithms promise much more efficient and accurate simulations for a wide variety of physical phenomena of interest to materials scientists and manufacturers. Appeared at 20th Annual ACM Symposium on Computational Geometry (SoCG), June 2004.