Tracking Evolving Discontinuities in Spacetime Solutions
Shocktube problem: Spacetime mesh refinement captures shock moving rapidly to right and expansion wave opening left. But coarsened mesh tracks a contact discontinuity moving slowly to right (arrow). 
Objective
Evolving surfaces of discontinuity arise in continuum physics as shocks, moving phase boundaries, cracks and dislocations; their discrete representation is one of the most challenging problems in computational science. Conventional methods use expensive grid refinement and error-inducing stabilization to capture these features. We seek more effective finite elemen methods that explicitly track the trajectories of singular surfaces.
Approach
Adaptive meshing aligns spacetime element faces with solution-dependent trajectories of singular surfaces; our discontinuous Galerkin model admits solution discontinuities and enforces necessary jump conditions. Spacetime smoothing and Delaunay edge flips ensure mesh quality.
Impact
Our adaptive spacetime discontinuous Galerkin approach provides more robust tracking capabilities and more accurate solutions with substantial cost reductions. These techniques can be used to model evolving microstructures, to predict the paths of dynamic fractures and to model vascular and cellular mechanics in biomedical applications.
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