Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fully-Mixed Virtual Element Method for the Biot Problem

Michele Botti, Daniele Prada, Anna Scotti, Michele Visinoni

TL;DR

Biot poroelasticity couples deformation and fluid flow, posing numerical challenges for accurate Darcy velocity and stress approximation. The authors develop a fully mixed four-field formulation discretized with the lowest-order Virtual Element Method, embedding stress symmetry directly in the discrete spaces to avoid extra Lagrange multipliers. They provide a complete a priori analysis proving stability and convergence that are robust with respect to material parameters, including incompressible limits, and validate the approach with 3D numerical tests on polytopal meshes and a footing benchmark. The result is a geometry-flexible, mass-conserving framework for 3D poroelasticity with potential extensions to fractures and hybrid-dimensional models.

Abstract

Poroelasticity describes the interaction of deformation and fluid flow in saturated porous media. A fully-mixed formulation of Biot's poroelasticity problem has the advantage of producing a better approximation of the Darcy velocity and stress field, as well as satisfying local mass and momentum conservation. In this work, we focus on a novel four-fields Virtual Element discretization of Biot's equations. The stress symmetry is strongly imposed in the definition of the discrete space, thus avoiding the use of an additional Lagrange multiplier. A complete a priori analysis is performed, showing the robustness of the proposed numerical method with respect to limiting material properties. The first order convergence of the lowest-order fully-discrete numerical method, which is obtained by coupling the spatial approximation with the backward Euler time-advancing scheme, is confirmed by a complete 3D numerical validation. A well known poroelasticity benchmark is also considered to assess the robustness properties and computational performance.

Fully-Mixed Virtual Element Method for the Biot Problem

TL;DR

Biot poroelasticity couples deformation and fluid flow, posing numerical challenges for accurate Darcy velocity and stress approximation. The authors develop a fully mixed four-field formulation discretized with the lowest-order Virtual Element Method, embedding stress symmetry directly in the discrete spaces to avoid extra Lagrange multipliers. They provide a complete a priori analysis proving stability and convergence that are robust with respect to material parameters, including incompressible limits, and validate the approach with 3D numerical tests on polytopal meshes and a footing benchmark. The result is a geometry-flexible, mass-conserving framework for 3D poroelasticity with potential extensions to fractures and hybrid-dimensional models.

Abstract

Poroelasticity describes the interaction of deformation and fluid flow in saturated porous media. A fully-mixed formulation of Biot's poroelasticity problem has the advantage of producing a better approximation of the Darcy velocity and stress field, as well as satisfying local mass and momentum conservation. In this work, we focus on a novel four-fields Virtual Element discretization of Biot's equations. The stress symmetry is strongly imposed in the definition of the discrete space, thus avoiding the use of an additional Lagrange multiplier. A complete a priori analysis is performed, showing the robustness of the proposed numerical method with respect to limiting material properties. The first order convergence of the lowest-order fully-discrete numerical method, which is obtained by coupling the spatial approximation with the backward Euler time-advancing scheme, is confirmed by a complete 3D numerical validation. A well known poroelasticity benchmark is also considered to assess the robustness properties and computational performance.

Paper Structure

This paper contains 24 sections, 4 theorems, 93 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model Problem
  3. Weak formulation
  4. Virtual Element Discretization
  5. Mesh assumptions
  6. Discrete spaces
  7. Space $\mathbf{RM}(E)$
  8. Space $\mathbf{T}_h(f)$
  9. Stress space
  10. Displacement space
  11. Velocity space
  12. Pressure space
  13. The local forms
  14. Global Semi--Discrete Formulation
  15. Step 1: energy estimate
  16. ...and 9 more sections

Key Result

Theorem 3.1

Let $(\bfsigma_h(t),\mathbf{u}_h(t),\mathbf{w}_h(t),p_h(t))$ solve problem:semidiscrete_formulation for each $t\in(t_0,t_f]$ and assume the problem data to be regular enough to define Then, there exists a constant $C>0$ independent of $s_0$, $\lambda$, the length of the time interval $(t_f-t_0)$, and the mesh size $h$, such that The estimate remains valid in the incompressible limits $s_0\righta

Figures (6)

  • Figure 1: From left to right, the degrees of freedom are denoted as: blue sphere for displacement, green square for stress, red sphere for pressure, and orange square for velocity.
  • Figure 2: Overview of adopted meshes for convergence assessment numerical tests.
  • Figure 3: Test 1: a compressible material.
  • Figure 4: Test 2: a nearly-incompressible material with null storage coefficient.
  • Figure 5: Scheme for poroelasticity footing: $\Gamma_1$ is the upper boundary face and $\Gamma_2$ (the green area) is the area where we apply the uniform load force (green arrows).
  • ...and 1 more figures

Theorems & Definitions (9)

  • Remark 1: Solutions at $t=t_0$
  • Remark 2
  • Theorem 3.1: Stability
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4: Error estimate
  • proof