Consistent pressure Laplacian stabilization for incompressible continua via higher‐order finite calculus

Published in Int. Journal for Numerical Methods in Engineering Vol. 87 (1-5), pp. 171-195, 2011
doi: 10.1002/nme.3021

Abstract

We present a stabilized numerical formulation for incompressible continua based on a higher‐order Finite Calculus (FIC) approach and the finite element method. The focus of the paper is on the derivation of a stabilized form for the mass balance (incompressibility) equation. The simpler form of the momentum equations neglecting the non‐linear convective terms, which is typical for incompressible solids, Stokes flows and Lagrangian flows is used for the sake of clarity. The discretized stabilized mass balance equation adds to the standard divergence of velocity term a pressure Laplacian and an additional boundary term. The boundary term is relevant for the accuracy of the numerical solution, especially for free surface flow problems. The Laplacian and boundary stabilization terms are multiplied by non‐linear parameters that have an extremely simple expression in terms of element sizes, the pressure and the discrete residuals of the incompressibility equation and the momentum equations, thus ensuring the consistency of the method. The stabilized formulation allows solving the incompressible problem iteratively using an equal‐order interpolation for the velocities (or displacements) and the pressure, which are the only unknowns. The use of additional pressure gradient projection variables, typical of many stabilized methods, is unnecessary.

The formulation is particularly useful for heterogeneous incompressible materials with discontinuous material properties, as it allows computing all the stabilization matrices at the element level. Details of the finite element formulation are given. The good behaviour of the new pressure Laplacian stabilization (PLS) technique is shown in simple but demonstrative examples of application. A very accurate solution was obtained in all cases in 2–3 iterations.