Shock hydrodynamics algorithms for fluid dynamics applications have traditionally performed poorly on triangular or tetrahedral meshes. However, simplex-type meshes are very important in industrial computations, in which automatic mesh generation can provide a considerable reduction in overall analysis and design turn-around time. In addition, simplicial meshes are usually preferred in problems involving radiation effects. A newly developed method in the context of SUPG stabilization for nodal-based finite element discretizations has been successfully applied in Lagrangian computations [1,2]. Numerical results show very good agreement with state-of-the-art methods on quadrilateral or hexahedral meshes, as well as triangular meshes. Details on recent advances on the next page.