NCSA Training Events

A multiscale finite element framework that is applicable to both fluids and solids is presented. This framework allows to embed fine scales of the problem directly into the mathematical formulation. Modeling of the fine scales leads to multiscale-stabilized formulations. The strength of multiscale formulations, especially for multi-physics problems, is highlighted. The limitations of conventional finite element approaches applied to multi-physics problems are outlined.

In fluid dynamics, multiscale method is applied to the incompressible Navier-Stokes equations. The resulting formulation shows superior mathematical and numerical properties that are illustrated via a set of test problems on fluid-structure interactions. In solid mechanics and multiscale material modeling, a two-level scale separation is presented that leads to a mathematically consistent method for bridging molecular mechanics and quasi-continuum mechanics. Contrary to the commonly practiced multigrid/computational nesting of information from smaller scales into the larger ones, a mathematical nesting of scales is proposed via evolution equations for the bridging-scales. Inter-atomic interactions are incorporated through nanoscale based material moduli with internal variables that are functions of the local state of deformation. Point defects are modeled by multi-body inter-atomic potentials. Representative numerical examples are shown to validate the models and demonstrate their range of applicability.