# A stabilized finite element method for finite-strain three-field poroelasticity

## Abstract

We construct a stabilized finite-element method to compute flow and finite-strain deformations in an incompressible poroelastic medium. We employ a three-field mixed formulation to calculate displacement, fluid flux and pressure directly and introduce a Lagrange multiplier to enforce flux boundary conditions. We use a low order approximation, namely, continuous piecewise-linear approximation for the displacements and fluid flux, and piecewise-constant approximation for the pressure. This results in a simple matrix structure with low bandwidth. The method is stable in both the limiting cases of small and large permeability. Moreover, the discontinuous pressure space enables efficient approximation of steep gradients such as those occurring due to rapidly changing material coefficients or boundary conditions, both of which are commonly seen in physical and biological applications.

## 1 Introduction

Poroelasticity theory assumes a superposition of solid and fluid components to capture complex interactions between a deformable porous medium and the fluid flow within it, and was originally developed to study geophysical applications such as reservoir geomechanics [26, 28, 41]. Fully saturated, incompressible poroelastic models have since been used to model a variety of biological tissues and processes. Biological examples include the coupling of flow in coronary vessels with the mechanical deformation of myocardial tissue to create a poroelastic model of coronary perfusion [13, 15]. Other examples include modelling tissue deformation and the ventilation in the lungs [6], protein-based hydrogels embedded within cells [22], brain oedema and hydrocephalus [39, 56], microcirculation of blood and interstitial fluid in the liver lobule [36], and interstitial fluid and tissue in articular cartilage and intervertebral discs [21, 24, 42].

When using the finite element method to solve the poroelastic equations the main challenge is to ensure convergence of the method and prevent numerical instabilities that often manifest themselves in the form of spurious oscillations in the pressure field. It has been suggested that this problem is caused by the saddle point structure in the coupled equations resulting in a violation of the famous Ladyzhenskaya–Babuska–Brezzi (LBB) condition, thus highlighting the need for a stable combination of mixed finite elements [23].

In addition, there is a need for methods that do not give rise to localised pressure oscillations when seeking to approximate steep pressure gradients in the solution. For example, when modelling the diseased lung, abrupt changes in tissue properties and heterogeneous airway narrowing are possible. This can result in a patchy ventilation and pressure distribution [51]. In this situation methods that solve the poroelastic equations using a continuous pressure approximation struggle to capture the steep gradients in pressure and produce localised oscillations in the pressure [46]. Steep pressure gradients can also result from imposed Dirichlet pressure boundary conditions such as those in Terzaghi’s problem [43, 54]. The method presented here is able to overcome these types of pressure instability.

### 1.1 Two variable versus three variable formulations

- 1.
It allows for greater accuracy in the fluid velocity field. This can be of particular interest when a poroelastic model is coupled with an advection diffusion equation, e.g., to account for gas exchange, thermal effects, contaminant transport or the transport of nutrients or drugs within a porous tissue [30].

- 2.
Physically meaningful boundary conditions can be applied at the interface when modelling the interaction between a fluid and a poroelastic structure [5].

- 3.
It allows for an easy extension of the fluid model from a Darcy to a Brinkman flow model, for which there are numerous applications in modelling biological tissues [30].

- 4.
It avoids the calculation of the fluid flux in post-processing.

### 1.2 Previous results: infinitesimal strain

Error estimates for finite element solutions of the linear three-field problem, using continuous piecewise linear approximations for displacements and mixed low-order Raviart–Thomas elements for the fluid flux and pressure variables, are presented in [44, 45]. However this method was found to be susceptible to spurious pressure oscillations [47]. In an effort to overcome these pressure oscillations, a discontinuous linear three-field method was analysed in [38] with moderate success, and a linear non-conforming three-field method was analysed in [58]. However no implementation of these methods in 3D has yet been presented.

Due to the size of the discrete systems resulting from a three field approach, there has been considerable work on operating splitting (iterative) approaches in which the poroelastic equations are separated into a fluid problem and deformation problem [19, 28, 32, 53]. These methods are often able to take advantage of existing finite element software for elasticity and fluid flow. Matrix assembly for discontinuous and non-conforming finite elements in 3D can be complicated and calculating stresses using these methods can be particularly challenging. Methods that use standard and simple to implement elements are very appealing [55]. In [28], a linear three-field mixed finite element method using lowest order Raviart–Thomas elements was shown to overcome Dirichlet boundary type pressure instabilities.

The finite volume method has been used by [31, 32] to discretize the flow. This results in a discontinuous approximation of the fluid pressure which is able to overcome localised pressure oscillations due to steep pressure gradients in the solution.

Introducing a displacement stress field, e.g. see [49], reduces the regularity requirements on the displacement field, thereby allowing for the implementation of a four-field conforming Raviart–Thomas element, but consequently greatly increasing the overall size and complexity of the problem.

### 1.3 Previous results: finite strain

Monolithic approaches for solving the quasi-static two-field incompressible finite-strain deformation equations are outlined in [1]. Two different approaches are advocated, a mixed-penalty formulation, in which the continuity condition is imposed using a penalty approach, and a mixed solid velocity–pressure formulation, where the linear momentum for the fluid is used to eliminate the fluid velocity in the remaining equations. The solid velocity–pressure formulation is similar to the commonly used reduced \((\varvec{u}/p)\) formulation in [4]. Two-field formulations require a stable mixed element pair such as the popular Taylor-Hood element to satisfy the LBB inf-sup stability requirement. However, using a continuous pressure element means that jumps in material coefficients may introduce large solution gradients across the interface, requiring severe mesh refinement or failing to reliably capture jumps in the pressure solution [54]. An operator splitting (iterative) approach for a near incompressible model is described by [13].

A three-field (displacement, fluid flux, pressure) formulation has been outlined in [37], however this method uses a low-order mixed finite element approximation without any stabilisation and therefore is not inf-sup stable. A three-field finite element using a continuous pressure approximation has been implemented in [52].

For both two-field and three-fields formulations and for any choices of finite element, implementation, construction and linearization of the nonlinear equations and convergence of the nonlinear mechanics problem using Newton’s method or other iterative procedure is also nontrivial [50].

### 1.4 Contributions of the current work

In [7], we developed a stabilized, low-order, three-field mixed finite element method for the fully saturated, incompressible, small deformation case for which a linearly elastic model is sufficient. Low-order finite element methods are relatively easy to implement and allow for efficient preconditioning [20, 27, 55]. Rigorous theoretical results for the stability and optimal convergence rate for linear poroelasticity were presented in [7]. The stabilization term requires only a small amount of additional computational work and can be assembled locally on each element using standard finite element information, leading to a symmetric addition to the original system matrix and preserving any existing symmetry. The effect of the stabilization on the conservation of mass is minimal in 3D, and decreases as the mesh is refined, see [7].

- 1.
A method to solve finite-strain fully incompressible poroelasticity using a stabilized discontinuous pressure approximation. [Note that for the linear (infinitesimal strain) equations other methods that use a discontinuous pressure approximation have been previously presented [7, 28, 38, 58].]

- 2.
A method for finite-strain fully incompressible poroelasticity that is both inf-sup stable and is able to overcome localized pressure oscillations.

- 3.
A finite element method that is robust within all modelling regimes. Large differences in permeability within the computational domain can result in regions in which Darcy flow dominates over elastic effects and regions in which elastic effects are dominant. This low-order element is reliable in both scenarios, providing an effective numerical approach for problems in which heterogeneity presents computational challenges.

- 4.
A finite element method that results in a discrete system with blocks arising from simple linear finite elements allowing fast solver approaches and preconditioning techniques to be easily implemented.

- 5.
A finite element method for a finite-strain poroelastic model that resolves steep pressure gradients without localized oscillations.

## 2 Poroelasticity theory

Two complementary approaches have been developed for modelling a deformable porous medium. Mixture theory, also known as the Theory of Porous Media (TPM) [8, 10, 11], has its roots in the classical theories of gas mixtures and makes use of a volume fraction concept in which the porous medium is represented by spatially superposed interacting media. An alternative, purely macroscopic approach is mainly associated with the work of Biot. A comprehensive development of the macroscopic theory appears in [16]. Relationships between the two theories are explored in [14, 17]. As is most common in biological applications, we use the mixture theory for poroelasticity as outlined in [8] and recently summarized in [52].

### 2.1 Kinematics

### 2.2 Volume fractions

### 2.3 The model

*g*is a general source or sink term and \(\varvec{\sigma }_{e}\) is the stress tensor given by

It is important to recognize that \(\nabla ( \cdot )= \partial /\partial \varvec{x} (\cdot ) \) denotes the partial derivative with respect to the *deformed* configuration. We will use \(\nabla \) to denote the spatial gradient in \(\varOmega (t)\) rather than the more explicit \(\nabla _{\varvec{x}=\varvec{\chi }(\varvec{X},t)}\). The latter more clearly indicates the dependency of the gradient operator on the deformation \(\varvec{\chi }(\varvec{X},t)\) and highlights the inherent nonlinearity that arises due to the fact that the deformation \(\varvec{\chi }(\varvec{X},t)\) is one of the unknowns. Similarly the deformed domain \(\varOmega (t)\) in which equations (8) pertain, is a function of the deformation map \(\varvec{\chi }\), and therefore incorporates another important nonlinearity.

## 3 The stabilized finite element method

We extend the method of [7] from the linear, small deformation poroelastic case to finite-strain poroelasticity. For ease of presentation, we will assume all Dirichlet boundary conditions are homogeneous, ie., \(\varvec{u}_{D} = {\varvec{0}}, {q_{D}} = 0, p_{D}=0\).

### 3.1 Weak formulation

### 3.2 The fully discrete model

*K*, where

*h*denotes the size of the largest element in \(\mathcal {T}^{h}\). We then define the following finite element spaces,

*K*respectively. We define the combined solution space \(\mathcal {U}_h(t) = \mathbf {W}_h^{E}(\varOmega (0))\times \mathbf {W}_h^{D}(\varOmega (t))\times Q_h(\varOmega (t))\).

The discretization in time is given by partitioning [0, *T*] into *N* evenly spaced non-overlapping regions \((t_{n-1}, t_n]\), \(n=1,2,\dots , N\) , where \(t_n-t_{n-1} = \varDelta t\). For any sufficiently smooth function *v*(*t*, *x*) we define \(v^n(x) = v(t_n,x)\) and the discrete time derivative by \(v_{\varDelta t}^{n} := \frac{v^{n}-v^{n-1}}{\varDelta t}\).

*h*and \(\varDelta t \). Here \(h_{\partial K}\) denotes the size (diameter) of an element edge in 2D or face in 3D, and \(\llbracket \cdot \rrbracket \) is the jump across an edge or face (taken on the interior edges only). The stabilization term has been introduced here to add stability and ensure a well-posed fully-discrete model. It has been shown that the convergence is insensitive to \(\varUpsilon \), e.g. see in [7, 12, 27] .

### 3.3 Solution via quasi-Newton iteration at \(t_n, n=1, \dots , N\).

*DG*is the directional derivative of

*G*, at \(\overline{\mathfrak {u}}_{h}^n\), in the direction \({\delta \mathfrak {u}_{h}}\).

#### 3.3.1 Approximation of \(DG^n\).

*G*by assuming the nonlinear elasticity term is the dominant nonlinearity and ignoring the other nonlinearities [50, 54]. Let

## 4 Implementation details

### 4.1 Matrix assembly for the Newton iteration

*i*th step within the Newton method at time \(t_{n}\). The Newton algorithm at a particular time step

*n*, is given in Algorithm 1.

*M*. Details of the matrices \(\varvec{D}\) and \(\varvec{E}\) appear in “Appendix 2”. Note that the matrix equations are integrated in the deformed configuration obtained from the previous Newton step. This update Lagrangian approach overcomes complex linearisation otherwise needed when using a total Lagrangian approach [50]. The Newton iteration was found to be robust with respect to the stabilization parameter. More precisely, in all calculations fewer than four Newton iterations were required independently of the size of the stabilization parameter. Tables 2 and 3 in Sect. 5.1 show the Newton convergence for two choices of the stabilization parameter two orders of magnitude apart for a 3D stress relaxation text problem. In practice, the stabilization parameter was chosen so as to be as small as possible without producing oscillations, unless otherwise stated.

### 4.2 Stabilization matrix assembly

*K*. We define \(\mathcal {A}(K)\) to be the set of elements \(L \in \mathcal {T}_{h}\) neighboring

*K*.

### 4.3 Fluid-flux boundary condition

When solving the equations for Darcy flow using the Raviart–Thomas element (RT-P0), the fluid-flux boundary condition is enforced naturally by this divergence free element. Unfortunately this is not possible using our proposed P1-P1-P0-stabilized element. However, solving the poroelastic equations (8) using a piecewise linear approximation for the deformation and Raviart–Thomas element for the fluid (P1-RT-P0) does not satisfy the discrete inf-sup condition and can yield spurious pressure oscillations, see [46, 47] for details.

## 5 Numerical results

*E*and the Poisson ratio \(\nu \) by \(\mu =E/(2(1+\nu ))\) and \(\lambda =(E\nu )/((1+\nu )(1-2\nu ))\). Details of the effective stress tensor and fourth-order spatial tangent modulus for this particular law can be found in “Appendix 2”. For the permeability law we chose

### 5.1 3D unconfined compression problem

Parameters used for the 3D unconfined compression test problem

Parameter | Description | Value |
---|---|---|

\(\phi _{0}\) | Initial fluid volume fraction | 0.9 |

\(k_{0}\) | Dynamic permeability | \(10^{-3} \; \text{ m }^{3}\,\text{ s }\,\text{ kg }^{-1}\) |

\(\nu \) | Poisson’s ratio | 0.15 |

| Young’s modulus | 1000 \(\text{ kg }\,\text{ m }^{-1}\,\text{ s }^{-2}\) |

\(\varDelta t\) | Time step used in the simulation | \(4\,\text{ s }\) |

| Final time of the simulation | \(1000\,\text{ s }\) |

\(\varUpsilon \) | Stabilization parameter | \( 10^{-3}\) |

*u*is the radial displacement, \(\epsilon _{0}\) is the amplitude of the applied axial strain and

*a*is the radius of the cylinder. Here \(\alpha _n\) are the solutions to the characteristic equation, given by \(J_{1}(x)-(1-\nu )xJ_{0}(x)/(1-2\nu )=0,\) where \(J_{0}\) and \(J_{1}\) are Bessel functions. The characteristic time of diffusion \(t_{g}\) is given by \(t_{g}= a^{2}/M k\), where \(M=\lambda + 2\mu \) is the P-wave modulus of the elastic solid skeleton,

*k*is the permeability.

For small axial compression the computed radial displacement shown in Fig. 3 is in good agreement with the analytical solution, indicating that the nonlinear poroelastic model is accurate in the small strain limit. As the axial compression becomes large, the numerical finite strain solution departs from the analytical linear small deformation solution as expected.

Convergence of the Newton iteration for the 3D unconfined compression problem with \(\varUpsilon =10^{-1}\) at \(t=4\,\text{ s }\)

Newton iteration | \(||\mathfrak {u}_{i}^{n}-\mathfrak {u}^{n}_{i-1}||\) | \(||\varvec{R}(\mathfrak {u}_{i}^{n},\mathfrak {u}^{n-1})||\) |
---|---|---|

1 | 0.81 | 0.023202 |

2 | 2.81699e−04 | 0.011276 |

3 | 6.93986e−08 | 1.34048e−06 |

4 | 4.10726e−10 | 7.64882e−09 |

Convergence of the Newton iteration for the 3D unconfined compression problem with \(\varUpsilon =10^{-3}\) at \(t=4\,\text{ s }\)

Newton iteration | \(||\mathfrak {u}_{i}^{n}-\mathfrak {u}^{n}_{i-1}||\) | \(||\varvec{R}(\mathfrak {u}_{i}^{n},\mathfrak {u}^{n-1})||\) |
---|---|---|

1 | 0.81 | 0.0235609 |

2 | 1.28528e−05 | 1.99541e−04 |

3 | 7.71658e−08 | 1.49304e−06 |

4 | 4.6844e−10 | 8.17681e−09 |

### 5.2 Terzaghi’s problem

Parameters used for Terzaghi’s problem

Parameter | Description | Value |
---|---|---|

\(\phi _{0}\) | Initial fluid volume fraction | 0.9 |

\(k_{0}\) | Dynamic permeability | \(10^{-5} \; \text{ m }^{3}\,\text{ s }\,\text{ kg }^{-1}\) |

\(\nu \) | Poisson ratio | 0.25 |

| Young’s modulus | 100 \(\text{ kg }\,\text{ m }^{-1}\,\text{ s }^{-2}\) |

\(\varDelta t\) | Time step used in the simulation | \(0.01\,\text{ s }\) |

| Final time of the simulation | \(1\,\text{ s }\) |

\(\varUpsilon \) | Stabilization parameter | \(2\times 10^{-5}\) |

### 5.3 Swelling test

Given a unit cube of material, a fluid pressure gradient is imposed between the two opposite faces at \(X=0\) and \(X=1\). The pressure \(p_{D}\) on the inlet face \(X = 0\) is increased very rapidly from zero to a limiting value of \(10 \text{ kPa }\), i.e., \(p_{D} = 10^{4} (1-\text{ exp }(-t^{2}/0.25)) \;\text{ Pa })\). On the outlet face \(X = 1\), the pressure is fixed to be zero, \(p_{D} =0\). There are no sources of sinks of fluid. A zero flux condition is applied for the fluid velocity on the four other faces (\(Y=0,1, \;Z=0,1\)). Normal displacements are required to be zero on the planes \(X = 0, \ Y = 0\) and \(Z = 0\). The permeability of the cube \(0< X< 0.5, \ 0.5< Y< 1, \ 0< Z <0.5\), i.e., 1/8 of the volume, is smaller than in the rest of the unit cube by a factor of 500. The computational domain is shown in Fig. 6a, highlighting the region of reduced permeability. The parameters chosen for this test problem are given in Table 5. This problem is similar to the one in [13] and highlights the method’s ability to reliably capture steep gradients in the pressure solution due to rapid changes in material parameters.

Parameters used for the swelling test problem

Parameter | Description | Value |
---|---|---|

\(\phi _{0}\) | Initial fluid volume fraction | 0.9 |

\(k_{0}\) | Dynamic permeability | \(10^{-5} \; \text{ m }^{3}\,\text{ s }\,\text{ kg }^{-1}\) |

\(\nu \) | Poisson ratio | 0.3 |

| Young’s modulus | 8000 \(\text{ kg }\,\text{ m }^{-1}\,\text{ s }^{-2}\) |

\(\varDelta t\) | Time step used in the simulation | \(0.02\,\text{ s }\) |

| Final time of the simulation | \(20\,\text{ s }\) |

\(\varUpsilon \) | Stabilization parameter | \(10^{-4}\) |

Figure 8 shows the pressure solution for this test problem with (\(\varUpsilon =10^{-4}\) in Fig. 8a) and without stabilization (\(\varUpsilon =0\) in Fig. 8b) at \(t=0.02\,\text{ s }\). The computation was performed using 512 hexahedral elements. Note that without any stabilization pressure an instability in the pressure is observed.

To further investigate and demonstrate that a lack of stabilization will result in a loss of inf-sup stability and thus result in a spurious chequer board pressure solution, we run the same swelling test problem, but with a homogeneous permeability permeability set at \(k_{0}=10^{-5}\). Furthermore we solve this problem on a 12,288 element tetrahedral mesh, which has a smaller ratio of fluid and displacement nodes to pressure nodes, compared to the previously used hexahedral mesh, thus worsening the inf-sup instability properties [18]. Figure 9 shows the pressure solution with (\(\varUpsilon =10^{-4}\) in Fig. 9a) and (\(\varUpsilon =\varUpsilon =10^{-12}\) in Fig. 9b) at \(t=0.02\,\text{ s }\). Note that without sufficient stabilization the lack of inf-sup stability can clearly be observed in the form of a spurious pressure checkerboard solution. These numerical examples demonstrate that the stabilization scheme is very robust to ensuring inf-sup stability by allowing a large range of stabilization parameters to be used, before spurious pressure oscillations caused by a loss of inf-sup stability are observed. However when wishing to capture pressure boundary layer type solutions, care does need to be taken to ensure that the pressure solution is not overly smoothed. As a practical guide we recommend first choosing a large stabilization parameter and then repeatedly lowering the stabilization parameter by an order of magnitude until the lowest parameter is found that leads to a pressure solution without any oscillations.

## 6 Conclusions

Stabilized low-order methods can offer significant computational advantages over higher order approaches. In particular, one can employ meshes with fewer degrees of freedom, fewer Gauss points, and simpler data structures. The additional stabilization terms can also improve the convergence properties of iterative solvers.

The main contribution of this paper is to extend the local pressure jump stabilization method [12] already applied to three-field linear poroelasticity in [7], to the finite strain case. Thus, the proposed scheme is built on an existing scheme for which rigorous theoretical results addressing the stability and optimal convergence have been proven, and for which numerical experiments have demonstrated its ability to overcome spurious pressure oscillations. Owing to the discontinuous pressure approximation, sharp pressure gradients due to changes in material coefficients or boundary layers can be captured reliably, circumventing the need for severe mesh refinement. The addition of the stabilization term introduces minimal additional computational work, can be assembled locally on each element using standard element information, and leads to a symmetric addition to the original system matrix, thus preserving any existing symmetry. As the numerical examples have demonstrated, the stabilization scheme is robust and leads to high-quality solutions.

## Notes

### Acknowledgements

Funding was provided by Engineering and Physical Sciences Research Council. Lorenz Berger was funded by the EPSRC via the Life Sciences Interface Doctoral Training Centre, Oxford University and Rafel Bordas was funded via the EU FP7 AirPROM project (grant agreement no. 270194).

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