A Lagrange-multiplier-based XFEM to solve pressure Poisson equations in problems with quasi-static interfaces

The XFEM (extended finite element method) has a lot of advantages over other numerical methods to resolve discontinuities across quasi-static interfaces due to the jump in fluidic parameters or surface tension. However, singularities corresponding to enriched degrees of freedom (DOFs) embedded in XFEM arise in the discrete pressure Poisson equations. In this paper, constraints on these DOFs are derived from the interfacial equilibrium condition and introduced in terms of stabilized Lagrange multipliers designed for non-boundary-fitted meshes to address this issue. Numerical results show that the weak and strong discontinuities in pressure with straight and circular interfaces are accurately reproduced by the constraints. Comparisons with the SUPG/PSPG (streamline upwind/pressure stabilizing Petrov-Galerkin) method without Lagrange multipliers validate the applicability and flexibility of the proposed constrained algorithm to model problems with quasi-static interfaces.

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