Investigation of conforming mixed finite elements with quadratic pressure approximations
Date
2023
Authors
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Publisher
University of Delaware
Abstract
To simulate the response of saturated geomaterials requires a numerical approach such as the finite element method that employs a generalized Biot formulation. Under isothermal conditions, the primary dependent variables are displacements (u) and pore pressure (ϑ), which are suitably coupled. The resulting finite elements are mixed and are historically referred to as “u-p” elements. Traditionally, such elements employ quadratic and linear approximation for u and ϑ, respectively. Since, according to Darcys law, the velocity of flow of the pore fluid is proportional to the gradient of the pore pressure, the approximate velocity is thus constant over an element. In slow “consolidation-type” problems the accurate computation of such velocities is not crucial, as they are generally very small. The need to accurately simulate the effects associated with rising sea levels, rapidly rising flood waters, etc., however, now requires that accurate approximate velocities be computed. This is particularly true in higher permeability soils. Consequently, robust, and computationally efficient higher-order u-p elements need to be used in simulating such problems. ☐ The aim of the present study was to investigate a mixed finite element method with quadratic pressure approximation. The elements of interests are the T15P6c triangle and the Q17P8c quadrilateral. These elements are collectively known as higher-order elements (HO). For reference and performance assessment, the so-called Taylor hood (TH) elements, bubble function (BF) elements and equal order interpolation (EOI) elements are also used in the analysis. ☐ The HO element performance related to the specific temporal integration algorithm used to integrate the finite element equations in time is first assessed. When considering a soil with a relatively high permeability, for a temporal integration parameter of 0.500, all TH, BF and EOI investigated produced oscillations in the excess pore pressure at nodes near the permeable boundary. By contrast, the T15P6c elements exhibited only very minor oscillations and no oscillation were noticed for the Q17P8c elements. Oscillations disappear when the temporal integration parameter is changed to 1.00. For simulations involving relatively low permeabilities, oscillations appear only in the first few time steps. ☐ Secondly the stability of HO elements during initial response is assessed and compared to TH, EOI and BF elements. TH, BF and EOI elements exhibit varying degrees of pore pressure oscillations very shortly after the application of an instantaneously applied pressure that is maintained constant. Oscillations associated with the Q4P4c (EOI element) were significantly greater than those associated with the TH and BF elements. The initial response of the HO elements was superior to that of the TH, BF and EOI elements investigated. After the first-time step, the response of all the TH, BF, EOI and HO elements was oscillation-free. ☐ Finally, the HO elements were assessed using one-dimensional singly drained consolidation problem with a single consolidating layer and then with two consolidating layers. For the former all the TH, BF, EOI and HO elements considered in this study accurately simulate the problem of one-dimensional consolidation of a single soil layer with single drainage. Relatively coarse meshes of T15P6c and Q17P8c elements produced results that were very similar to those obtained using finer meshes of TH, BF and EOI elements with linear pressure approximation. The Q8P4c, Q9P4c, T6P3c, and T7P3c exhibited some slight oscillations in the normalized excess pore pressure response for the dimensionless time factor Tv = 0.001. The HO elements did not overestimate the analytical solution. For the two-layer soil problem T15P6C and Q17P8C did not overshoot the analytical solution for a value of Tv = 0.001 and both exhibited no oscillations or other inconsistencies at the interface between the two soil layers.
Description
Keywords
Biot formulation, Conforming finite elements, Mixed finite elements, Quadratic pore pressure approximation