  We present a new method for two-material Lagrangian hydrodynamics, which
combines the Shifted Interface Method (SIM) with a high-order Finite Element
Method. Our approach relies on an exact (or sharp) material interface
representation, that is, it uses the precise location of the material
interface. The interface is represented by the zero level-set of a continuous
high-order finite element function that moves with the material velocity. This
strategy allows to evolve curved material interfaces inside curved elements. By
reformulating the original interface problem over a surrogate (approximate)
interface, located in proximity of the true interface, the SIM avoids cut cells
and the associated problematic issues regarding implementation, numerical
stability, and matrix conditioning. Accuracy is maintained by modifying the
original interface conditions using Taylor expansions. We demonstrate the
performance of the proposed algorithms on established numerical benchmarks in
one, two and three dimensions.
