A Galerkin least-square stabilisation technique for hyperelastic biphasic soft tissue

Abstract

An hyperelastic biphasic model is presented. For slow-draining problems (permeability less than 1 × 10⁻² mm⁴ N⁻¹ s⁻¹), numerical instabilities in the form of non-physical oscillations in the pressure field are observed in 3D problems using tetrahedral Taylor–Hood finite elements. As an alternative to considerable mesh refinement, a Galerkin least-square stabilisation framework is proposed. This technique drastically reduces the pressure discrepancies and prevents these oscillations from propagating towards the centre of the medium. The performance and robustness of this technique are demonstrated on a 3D numerical example.

Introduction

The theory of porous media (TPM) is a powerful and yet simple tool to model multi-phase media, primarily comprising a solid and a fluid (which are referred to as “constituents”). Due to its modularity, this theory also represents a practical framework to include other effects such as mass exchange, chemical reactions and electrochemical phenomena. There is no restriction as to which material models can be used, e.g. anisotropy or viscosity and is therefore well suited for the analysis of hydrogels, polymeric foams and hydrated biological soft tissues (e.g. cartilage and intervertebral disc).

The concepts behind the TPM find their roots in diffusion and soil mechanics problems formulated in the nineteenth century (for historical developments, see [3], [9]). The TPM is a homogenised macroscopic representation of the porous media. It is a continuum-based model which fully couples the fluid and the solid (see Fig. 1). It overcomes the difficulty of obtaining an accurate geometrical description of the microstructure by using the concept of volume fractions to “smear” the constituent properties over a control space to obtain properties of the overall mixture. This is described in the following section.

The authors have employed TPM in the development of an hyperelastic biphasic swelling model for modelling the intervertebral disc. This application illustrates slow-draining problems, where a porous medium with low permeability (in the range 1 × 10⁻² − 1 × 10⁻³ mm⁴ N⁻¹ s⁻¹) is rapidly loaded. Such low permeabilities are typical for soft tissues (e.g. cartilage, brain tissue), mortar or homogeneous clays [2], [13], [17].

This model has been implemented in a finite element framework, employing Taylor–Hood (quadratic shape functions for the solid displacement, linear shape functions for the pressure) tetrahedral elements, aiming to fulfil the inf–sup condition (see [6], [5]). However, numerical instabilities manifest in the form of non-physical oscillations in the pressure field, which is a shortcoming already observed in the past (see for example [16] for a recent review of biological applications). Vermeer and Verruijt [21] explain that these instabilities occur because loads applied to free-flow boundary conditions may lead to singularities in the derivatives of the pressure field. They also derive a lower bound critical time-step for one-dimensional problems, suggesting the requirement for large time-steps to overcome this issue, often incompatible with fast loading rates.

Several stabilisation techniques have been proposed in the context of Biot’s consolidation problems for small deformations (e.g. [12] using least-squares mixed finite element methods, and [1] by perturbation of the flow equation). The current work proposes to stabilise the pressure oscillations in the context of TPM for finite deformation problems, using a Galerkin Least-Square (GLS) formulation based on [19]. In order to focus only on the stabilisation aspect, only a biphasic mixture is considered, that is a porous medium composed of two constituents α: an isotropic, non viscous hyperelastic solid (α = S) and an ideal fluid (α = F).