A stabilized formulation for the solution of the incompressible unsteady Stokes equations in the frequency domain

Abstract

A stabilized finite element method is introduced for the simulation of time-periodic creeping flows, such as those found in the cardiorespiratory systems. The new technique, which is formulated in the frequency rather than time domain, strictly uses real arithmetics and permits the use of similar shape functions for pressure and velocity for ease of implementation. It involves the addition of the Laplacian of pressure to the continuity equation with a complex-valued stabilization parameter that is derived systematically from the momentum equation. The numerical experiments show the excellent accuracy and robustness of the proposed method in simulating flows in complex and canonical geometries for a wide range of conditions. The present method significantly outperforms a traditional solver in terms of both computational cost and scalability, which lowers the overall solution turnover time by several orders of magnitude.