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A novel iterative time integration scheme for linear poroelasticity

R. Altmann, M. Deiml

TL;DR

The paper tackles efficient time integration for linear poroelasticity (Biot consolidation) by introducing a damped iterative scheme that decouples flow and mechanics while enforcing a fixed number of inner iterations. An explicit bound on the required inner iterations, $K$, is derived as a function of the coupling parameter $\omega$, ensuring first-order convergence. Numerical experiments, including brain-tissue simulations and a sharpness study, validate the theory and show competitive performance relative to fixed-stress and semi-explicit schemes for moderate coupling. The method offers a tunable, robust alternative to fully implicit time stepping without the need for problem-specific stabilization.

Abstract

Within this paper, we introduce and analyze a novel time stepping scheme for linear poroelasticity. In each time frame, we iteratively solve the flow and mechanics equations with an additional damping step for the pressure variable. Depending on the coupling strength of the two equations, we explicitly quantify the needed number of inner iteration steps to guarantee first-order convergence. Within a number of numerical experiments, we confirm the theoretical results and study the dependence of inner iteration steps in terms of the coupling strength. Moreover, we compare our method to the well-known fixed-stress scheme.

A novel iterative time integration scheme for linear poroelasticity

TL;DR

The paper tackles efficient time integration for linear poroelasticity (Biot consolidation) by introducing a damped iterative scheme that decouples flow and mechanics while enforcing a fixed number of inner iterations. An explicit bound on the required inner iterations, , is derived as a function of the coupling parameter , ensuring first-order convergence. Numerical experiments, including brain-tissue simulations and a sharpness study, validate the theory and show competitive performance relative to fixed-stress and semi-explicit schemes for moderate coupling. The method offers a tunable, robust alternative to fully implicit time stepping without the need for problem-specific stabilization.

Abstract

Within this paper, we introduce and analyze a novel time stepping scheme for linear poroelasticity. In each time frame, we iteratively solve the flow and mechanics equations with an additional damping step for the pressure variable. Depending on the coupling strength of the two equations, we explicitly quantify the needed number of inner iteration steps to guarantee first-order convergence. Within a number of numerical experiments, we confirm the theoretical results and study the dependence of inner iteration steps in terms of the coupling strength. Moreover, we compare our method to the well-known fixed-stress scheme.
Paper Structure (16 sections, 2 theorems, 74 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 16 sections, 2 theorems, 74 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear poroelasticity and its weak formulation
  4. Coupling parameter
  5. Spatial discretization
  6. Time Stepping Schemes
  7. Implicit Euler scheme
  8. Iterative schemes
  9. Semi-explicit Euler scheme
  10. A novel iterative scheme with damping
  11. Convergence Analysis
  12. Numerical Examples
  13. Sharpness of the iteration bound
  14. Application to brain tissue
  15. Implementation details and runtime comparison
  16. ...and 1 more sections

Key Result

Proposition 4.3

Let the right-hand sides of the semi-discrete problem eq:semiDiscretePoro satisfy the smoothness conditions $f \in C^2([0,T], \mathbb{R}^{n_u})$ and $g \in C^1([0,T], \mathbb{R}^{n_p})$. Further assume consistent initial data, Assumption ass:initialStaticPressure, and Then the iterates of the scheme eq:scheme are stable in the sense that where $C>0$ is a constant that depends only on the $C^2([0

Figures (5)

  • Figure 5.1: Relative error (measured in the Euclidean norm) of the proposed scheme applied to the model problem of Section \ref{['sec:numerics:sharpness']} in comparison to the implicit Euler solution at final time $T = 1$.
  • Figure 5.2: Comparison of the proven iteration bound \ref{['eq:iterationBound']} and experimental iteration bounds for the model problem of Section \ref{['sec:numerics:sharpness']}.
  • Figure 5.3: Numerical results for the brain model of Section \ref{['sec:numerics:brain']}.
  • Figure 5.4: Comparison of runtimes of the fixed-stress, the implicit Euler and the novel iterative scheme \ref{['eq:scheme']} for different numbers of inner iterations and $\omega = 2.8$.
  • Figure 5.5: Comparison of runtimes of the three schemes for different numbers of inner iterations and $\omega=10$ (left) and $\omega=100$ (right).

Theorems & Definitions (12)

  • Remark 2.1: Differential--algebraic structure
  • Remark 2.2: Coupling strength
  • Remark 3.1
  • Remark 3.2: No convergence with final relaxation
  • Remark 3.3: Interpretation as matrix splitting
  • Remark 4.2
  • Proposition 4.3
  • proof
  • Theorem 4.4: First-order convergence
  • proof
  • ...and 2 more