Parameter Robust Isogeometric Methods for a Four-Field Formulation of Biot's Consolidation Model

TL;DR

This work targets Biot's consolidation model using a novel four-field formulation with unknowns $\mathbf{u}$, $\psi$, $\mathbf{w}$, and $p$, solved via mixed isogeometric spaces built from B-splines. The authors develop a backward-Euler fully discrete scheme that is provably stable and yields parameter-robust, high-order error estimates, remaining effective as the storage coefficient $c_0$ vanishes or the Lamé parameter $\lambda$ grows large. They prove inf-sup stability for the isogeometric Taylor–Hood pairing, establish existence and uniqueness of the discrete solutions, and derive error bounds $O(h^{\gamma})$ with $\gamma=\min\{p_{\mathbf v},p_p+1\}$, uniform in $\lambda$ and $c_0$ under relevant regimes. Numerical experiments in 2D and 3D confirm the theoretical rates, demonstrate robustness to material parameters, suppress pressure oscillations, and show superior accuracy of isogeometric discretizations (including $h$-, $p$-, and $k$-refinement) over conventional FEM for Biot-type poroelastic problems.

Abstract

In this paper, a novel isogeometric method for Biot's consolidation model is constructed and analyzed, using a four-field formulation where the unknown variables are the solid displacement, solid pressure, fluid flux, and fluid pressure. Mixed isogeometric spaces based on B-splines basis functions are employed in the space discretization, allowing a smooth representation of the problem geometry and solution fields. The main result of the paper is the proof of optimal error estimates that are robust with respect to material parameters for all solution fields, particularly in the case of nearly incompressible materials. The analysis does not require a uniformly positive storage coefficient. The results of numerical experiments in two and three dimensions confirm the theoretical error estimates and high-order convergence rates attained by the proposed isogeometric Biot discretization and assess its performance with respect to the mesh size, spline polynomial degree, spline regularity, and material parameters.

Figures (7)

• Figure 1: Displacement errors $\mathbf{E}_{\mathbf u}$ from Table 1 as a function of: (left) the mesh size $h$ for fixed $p_{\mathbf v}$; (right) the polynomial degree $p_{\mathbf v}$ for fixed $h$
• Figure 2: Errors for Test 2 on an L-shaped domain with different spline polynomial degree and regularity. a) $p_{\mathbf v}=2$; b-c) $p_{\mathbf v}=3$; d-e-f) $p_{\mathbf v}=4$; g-h-i-j) $p_{\mathbf v}=5$
• Figure 3: 2D Pressure field for Test 4 at time $T=0.001$ for the cantilever bracket problem; surface plot(left), top view (right)
• Figure 4: Four solution fields for the cantilever bracket problem of Test 4 in a 3D horseshoe domain at time $T=0.001$. Solid displacement and pressure (top panels), fluid flux and pressure (bottom panels)
• Figure 5: Time discretization errors for the backward Euler method.

Theorems & Definitions (11)

• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Theorem 3.1
• Theorem 3.2
• Lemma 4.1
• Theorem 4.1