An Extended Finite Element Method for Two-Phase Fluids

Abstract

An extended finite element method with arbitrary interior discontinuous gradients is applied to two-phase immiscible flow problems. The discontinuity in the derivative of the velocity field is introduced by an enrichment with an extended basis whose gradient is discontinuous across the interface. Therefore, the finite element approximation can capture the discontinuities at the interface without requiring the mesh to conform to the interface, eliminating the need for remeshing. The equations for incompressible flow are solved by a fractional step method where the advection terms are stabilized by a characteristic Galerkin method. The phase interfaces are tracked by level set functions which are discretized by the same finite element mesh and are updated via a stabilized conservation law. The method is demonstrated in several examples.

Derivative recovery and a posteriori error estimate for extended finite elements

“…For problems with discontinuities in the primal dependent variable (cracks [29,28,13], fluid-structure interaction [26]), discontinuous functions are added. To handle discontinuities in the dual unknown (multi-materials [41], solidification [16], biofilms [35], multi-phase flow [15], Stokes flow [47]), functions with discontinuous derivatives enhance the standard basis. These additional functions allow to model the discontinuous features without recourse to a conforming mesh, while retaining an exact representation of the discontinuities, to which the mesh need not conform.…”

A new fast finite element method for dislocations based on interior discontinuities

“…The method was extended to three dimensions by Sukumar et al [11] and to nonlinear fracture mechanics by Moës and Belytschko [12]. XFEM has been applied to shear bands by Song et al [13], inclusions and holes by Sukumar et al [14], and two-phase flow by Chessa and Belytschko [15]. Some other recent applications are [16][17][18][19][20][21][22]; of particular similarity are applications to shear bands [23,24].…”

A locally extended finite element method for the simulation of multi-fluid flows using the Particle Level Set method

“…Similar to [15], these enrichments are condensed before assembly. Beside local enrichment, global one, and in particular the XFEM have been also widely used to model multi-fluid flow [17][18][19][20]. Except for the intrinsic XFEM [21], all versions of the XFEM add enrichments to the global system and therefore the graph of the system needs to be updated as the interface moves.…”