On Mixed Isogeometric Analysis of Poroelasticity

TL;DR

This work introduces a mixed isogeometric formulation for Biot poroelasticity to address pressure oscillations at small time steps. By leveraging the high-continuity basis of isogeometric analysis and a mixed discretization, it demonstrates improved stability and accuracy over equal-order formulations in Terzaghi’s consolidation and a layered medium with a low-permeability layer. Numerical results show that increasing polynomial degree and adopting graded meshes can substantially reduce oscillations, though some residual effects persist, highlighting the potential of adaptive IGA refinement for poroelastic problems. The findings underscore the practical value of mixed IGA in accurately simulating coupled poroelastic processes in geomechanics and related fields.

Abstract

Pressure oscillations at small time steps have been known to be an issue in poroelasticity simulations. A review of proposed approaches to overcome this problem is presented. Critical time steps are specified to alleviate this in finite element analyses. We present a mixed isogeometric formulation here with a view to assessing the results at very small time steps. Numerical studies are performed on Terzaghi's problem and consolidation of a layered porous medium with a very low permeability layer for varying polynomial degrees, continuities across knot spans and spatial discretizations. Comparisons are made with equal order simulations.

Figures (9)

• Figure 1: Different continuities across knotspans, after cottrell2009iga.
• Figure 2: Number of control points for a given element on a simple B-Spline surface with different polynomial degrees. The element is highlighted and the blue squares represent control points.
• Figure 3: Terzaghi's problem: Domain and boundary conditions.
• Figure 4: Numerical solution to the Terzhagi problem with $p_{\mathrm{p}}=1$, $p_{\mathrm{u}}=2$ and and $N_e=72$ using critical time step.
• Figure 5: Numerical solution to the Terzaghi problem with $N_{\mathrm{e}}=72$ using a sub-critical time step of $~\Delta t = 0.1\Delta t_{\mathrm{c}}$ for different polynomial degrees. All plots are shown for the first time step.