# Iterative solvers for Biot model under small and large deformation

## TL;DR

This paper develops and analyzes iterative solvers, including Newton and L-scheme methods, for the Biot model under both small and large deformations, incorporating nonlinear fluid compressibility and Saint Venant-Kirchoff law.

## Contribution

It introduces a unified framework for linearization schemes applicable to any spatial discretization of the Biot model with nonlinear fluid and large deformation considerations.

## Key findings

- Convergence of schemes proven analytically for small deformation cases.
- Numerical examples demonstrate effectiveness for large deformation scenarios.
- Schemes are flexible for different discretization methods.

## Abstract

We consider L-scheme and Newton based solvers for Biot model under small or large deformation. The mechanical deformation follows the Saint Venant-Kirchoff constitutive law. Further, the fluid compressibility is assumed to be nonlinear. A Lagrangian frame of reference is used to keep track of the deformation. We perform an implicit discretization in time (backward Euler) and propose two linearization schemes for solving the nonlinear problems appearing within each time step: Newton's method and L-scheme. The linearizations are used monolithically or in combination with a splitting algorithm. The resulting schemes can be applied for any spatial discretization. The convergences of all schemes are shown analytically for cases under small deformation. Illustrative numerical examples are presented to confirm the applicability of the schemes, in particular, for large deformation.