SUPG-stabilized shock hydrodynamics (more, downloads)
Typical finite element algorithms for Lagrangian shock hydrodynamics are effective only on quadrilaterals in two dimensions and hexahedral in three dimensions. A newly developed approach, based on SUPG stabilization, shows results comparable with, and in some cases superior to state-of-the-art computations, on quadrilateral/hexahedral meshes, as well as triangular/tetrahedral meshes. An Arbitrary Lagrangian Eulerian method is currently under development.

Invariance analysis of SUPG-stabilized methods (more, downloads)
Galilean invariance is one of the key requirements of many physical models adopted in theoretical and computational mechanics. Recent research in shock hydrodynamics computations revealed the need for a detailed invariance analysis for SUPG operators, due to catastrophic instabilities arising in Lagrangian computations. A new perspective is proposed on stabilization, by means of an arbitrary Lagrangian-Eulerian (ALE) framework. Stabilization operators for Lagrangian and Eulerian mesh computations are obtained as limits cases of the underlying ALE formulation.

Multi-scale hourglass stabilization (more, downloads)
A new multi-scale, stabilized method for Q1/P0 finite element computations of Lagrangian shock hydrodynamics is under investigation. Instabilities (of hourglass type) are controlled by a stabilizing operator derived using the variational multi-scale analysis paradigm. The resulting stabilizing term takes the form of a residual-based pressure correction. The stabilizing residual embeds a discrete form of the Clausius-Duhem inequality. Effectively, the proposed stabilization samples and acts to counter the production of entropy due to numerical instabilities. The proposed technique is applicable to materials with no shear strength, for which there exists a caloric equation of state.

Multiscale discontinuous Galerkin methods (more, downloads)
A new class of Discontinuous Galerkin (DG) methods using variational multiscale ideas. This novel approach hinges on an additive decomposition of the discontinuous finite element space into continuous (coarse) and discontinuous (fine) components. Variational multiscale analysis is used to define an interscale transfer operator that associates coarse and fine scale functions. Composition of this operator with a donor DG method yields a new formulation that combines the advantages of DG methods with the attractive and more efficient computational structure of a continuous Galerkin method.

Variariational multiscale modeling of turbulent flows (more,downloads)
The paradigm of variational multiscale analysis has been successfully applied to large eddy simulation (LES) of turbulent flows in recent years. In a newly proposed approach, the SUPG stabilization operator is used as a turbulence model, and applied in combination with a version of the dynamic Smagorinsky model acting on the highest wavenumbers of the energy spectrum.