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

TL;DR

This paper reformulates the dynamic Biot-Allard poro-elasticity system without convolution integrals using frequency domain series, proving its well-posedness and enabling efficient numerical methods.

Contribution

It introduces an equivalent form of the dynamic Biot-Allard model that simplifies analysis and computation, and establishes its well-posedness using abstract evolutionary problem theory.

Findings

Reformulation eliminates convolution integrals for computational efficiency

Proves well-posedness of the reformulated system

Provides a foundation for numerical approximation schemes

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.