S-PAL: A stabilized finite element formulation for computing viscoplastic flows

Abstract

• Stabilized finite element method for viscoplastic flows using equal order polynomials. • Computational cost is considerably reduced. • Increased numerical stability in critical flow conditions, i.e., bubble entrapment • Numerical results in quantitative agreement with other popular methods Ideal viscoplastic models present an inherent discontinuity at the yielded/unyielded interface, which poses great numerical difficulties. In this work, the recently proposed method for the solution of viscoplastic flows, the Penalized Augmented Lagrangian (PAL) method (Dimakopoulos et al. (2018)) is coupled with a new projection-based finite element formulation that allows for equal order interpolants in all variables. The accuracy of the new finite element formulation is assessed by comparing its numerical results to those of the literature in three problems: (a) the buoyancy-driven bubble rise through a viscoplastic medium, (b) the steady-state lid-driven cavity and its transient counterpart, and (c) the transient gas-assisted displacement of a viscoplastic material in straight and corrugated tubes. The method presents increased numerical stability and reduced computational cost without hindering the accuracy of the solution. By changing the solution procedure, we are able to obtain solutions that are as accurate as those obtained with the Augmented Lagrangian (AL) method. In addition, we show that the optimal value of the penalty parameter is correlated with the mesh and the critical dimensionless numbers that correspond in each problem.