A CBS-type stabilizing algorithm for the consolidation of saturated porous media

Abstract

The presented method stems from the works by Zienkiewicz and co-workers for coupled fluid/thermal problems starting from the early 1990s. They propose algorithms to overcome the difficulties connected to the application of the FEM to the area of fluid mechanics, which include the problems of singular behaviour in incompressibility and the problems connected to convective terms. The major step forward was to introduce the concept of characteristic lines (the particle paths in a simple convection situation): for a class of problems with a single scalar variable, the equations in the characteristic co-ordinates regain self-adjointness. The procedure is called characteristic based split algorithm (CBS). We use here a CBS-type procedure for a saturated deformable elastic porous medium, in which the fluid velocity is governed by Darcy's equation (which comes directly from Navier-Stokes ones). The physical-mathematical model is a fully coupled one and is here used to study an incompressible flow inside a continuum with incompressible solid grains. The power of the adopted algorithm is to treat the basic equations in their strong form and to transform a usual 'u-p' problem into a 4u-v-p one, where it generally indicates the displacement of the solid matrix and p and v the pressure and velocity of the fluid, respectively. Attention is focused on the expression of Darcy's velocity which is considered as the starting point of the algorithm. The accuracy of the scheme is checked by comparing the present predictions in a typical consolidation test with available analytical and numerical u-p Solutions. A good fitting arnong different results has been obtained. It is further shown that the procedure eliminates the oscillations at the onset of consolidation, typical for many schemes. The FEM code Ed-Multifield has been used for implementing and testing the procedure

Stabilized‐coupled modelling of creep phenomena for saturated porous media

“…A CBS-type procedure has been applied [6] to the basic equations governing flow problems in saturated porous media, where the fluid velocity is no more governed by the NS equations but by the generalized seepage (Darcy) one [26]: this aspect does not invalidate the adoption of the CBS method and its rearrangement but, on the contrary, it supports our choice because it is known [27,28] that the Darcy's equation can be obtained from the NS ones.…”

“…A smoothing matrix was proposed in Reference [29] to eliminate these spatial oscillations. However, it is to be noted that our problem is a pressure stabilization one and it is not influenced by the characteristic convection stabilization terms [6].…”

“…The algorithm is briefly recalled here in its general lines; the reader is referred to Reference [6] for full details.…”

“…code for the analysis of consolidation of porous media through a u-v-p (displacement, fluid velocity, pressure) formulation. The basic numerical algorithm comes from an innovative application of the characteristic-basedsplit method [6] to the generalized Darcy's equation (defining filtration phenomena) in order to produce a new pressure-stabilized algorithm. Hence, the addition of suitable creep models inside such a code, already effective for reproducing standard consolidation problems without spurious numerical oscillations, have increased the domain of its prediction capabilities.…”

A two‐step Taylor‐characteristic‐based Galerkin method for incompressible flows and its application to flow over triangular cylinder with different incidence angles

“…Simultaneously, the CBS scheme and its extension have been applied widely for the solution of fluid and solid dynamic problems encountered in engineering, including general compressible and incompressible flows [22,[26][27][28][29][30], turbulent flows [18,31], shallow-water flows [14,32], thermal flows [33,34], porous medium flows [35][36][37][38], viscoelastic flows [39], solid dynamics [40] and bulk metal forming [41]. However, the application of the CBS algorithm to the flow over triangular cylinder with different incidences, which has fundamental fluid mechanics interest with respect to many technical applications, is not yet found 1183 in the existing literature, and that motivates us to numerically simulate this problem by using the characteristic-based method.…”