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The Biot-Allard poro-elasticity system: equivalent forms and well-posedness

Jakob S. Stokke, Markus Bause, Nils Margenberg, Florin A. Radu

TL;DR

The paper addresses poroelasticity with memory (the Biot--Allard model) by converting the convolution-based memory term into an auxiliary differential equation (ADE) framework through a frequency-domain series approximation of the dynamic permeability. It proves well-posedness within the abstract Picard framework for evolutionary problems, first for a single ADE and then generalizing to an $N$-pole expansion, establishing per-term positivity conditions that ensure stability. This convolution-free reformulation yields a coupled first-order evolutionary system that preserves memory effects while enabling efficient time-domain numerical schemes. The approach promises reduced memory requirements and faster simulations for dynamic poroelastic problems across geophysical and bio-mechanical contexts.

Abstract

We consider the fully dynamic Biot-Allard model, which includes memory effects. Convolution integrals in time model the history of the porous medium. We use a series representation of the dynamic permeability in the frequency domain to rewrite the equations in a coupled system without convolution integrals, suitable for the design of efficient numerical approximation schemes. The main result is the well-posedness of the system, proved by the abstract theory of R. Picard for evolutionary problems.

The Biot-Allard poro-elasticity system: equivalent forms and well-posedness

TL;DR

The paper addresses poroelasticity with memory (the Biot--Allard model) by converting the convolution-based memory term into an auxiliary differential equation (ADE) framework through a frequency-domain series approximation of the dynamic permeability. It proves well-posedness within the abstract Picard framework for evolutionary problems, first for a single ADE and then generalizing to an -pole expansion, establishing per-term positivity conditions that ensure stability. This convolution-free reformulation yields a coupled first-order evolutionary system that preserves memory effects while enabling efficient time-domain numerical schemes. The approach promises reduced memory requirements and faster simulations for dynamic poroelastic problems across geophysical and bio-mechanical contexts.

Abstract

We consider the fully dynamic Biot-Allard model, which includes memory effects. Convolution integrals in time model the history of the porous medium. We use a series representation of the dynamic permeability in the frequency domain to rewrite the equations in a coupled system without convolution integrals, suitable for the design of efficient numerical approximation schemes. The main result is the well-posedness of the system, proved by the abstract theory of R. Picard for evolutionary problems.
Paper Structure (5 sections, 3 theorems, 34 equations)

This paper contains 5 sections, 3 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Biot--Allard system with an auxiliary differential equation
  3. Solution theory for evolutionary problems
  4. General approximation of the dynamic permeability with $N>1$ in Eq. \ref{['relation:dynamicpermeability']}
  5. Conclusions

Key Result

Lemma 3.2

For $\nu\in\mathbb{R}_{>0}$ there holds that $\partial_t=\mathcal{L}_{\nu}^{*}(\nu+\normalfont{\color{red}\hbox{i}} t)\mathcal{L}_{\nu}$.

Theorems & Definitions (11)

  • Remark 1
  • Definition 3.1
  • Lemma 3.2: trostorff2015
  • Definition 3.3
  • Theorem 3.4: seifert2022
  • Definition 3.5
  • Definition 3.6
  • Theorem 3.7
  • proof
  • Remark 2
  • ...and 1 more