The figures show level-set functions whose zero-levels are interpreted as discontinuities in a two-dimensional domain. Fig. 1 shows a closed (circular) discontinity that can be defined by one level-set function. In contrast, open discontinuities end in the domain (e.g. cracks), and an additional level-set function is required for their definition, see Fig. 2.
Cracks are a typical application field of the XFEM. Figs. 3 and 4 show a crack mode I and mode II solution. By means of the XFEM, optimal convergence rates are obtained for crack simulations on regular meshes. The crack surfaces do not need to align with the element edges and no mesh refinement is needed near the crack tips.
Tank sloshing of two incompressible fluids. The density and viscosity changes discontinuously across the interface between the two fluids. This leads to velocity and pressure fields involving strong and weak discontinuities. The interface is moving throughout the whole simulation, however, the mesh stays fixed and considers the inner-element discontinuities by the enrichment.
Figs. 6 and 7 display a rising bubble in an incompressible fluid. The density and viscosity ratio is different in the two pictures. Surface tension is considered, as a result, the pressure field has a jump at the interface. In Fig. 7, the topology of the interface changes and makes the application of an interface tracking algorithm in the context of the classical FEM impossible. In contrast, an interface capturing scheme and the use of the XFEM does not cause problems.
Application of the XFEM in the context of fluid-structure interaction. A wave hits an elastic structure. The fluid domain is seperated into two areas (water and air) which differ in density and viscosity. The discontinuos change of field quantities across the interface is considered implicetly by the XFEM. The interface between the fluids and the structure is explicitely meshed and considered in the standard way.
Another example of fluid-structure interaction where the membrane is modelled as an infinitely thin line where the velocity differs on each side. The membrane is described implicitely by the level-set concept.
Thomas-Peter Fries (email@example.com)