# deal.II Implementation of a Two-Field Finite Element Solver for Poroelasticity

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## Abstract

This paper presents a finite element solver for poroelasticity in the 2-field approach and its implementation on the deal.II platform. Numerical experiments on benchmarks are presented to demonstrate the accuracy and efficiency of this new solver.

## Keywords

Darcy flow deal.II Finite element methods Hexahedral meshes Poroelasticity Quadrilateral meshes Weak Galerkin## 1 Introduction

*p*through the following partial differential equations (PDEs)

*s*a known fluid source, \( \alpha \) (usually close to 1) the Biot-Williams constant, and \( c_0 \ge 0 \) the constrained storage capacity. Appropriate boundary and initial conditions are posed to close the system.

*2-field*: Solid displacement and fluid pressure are to be solved;*3-field*: Solid displacement, fluid pressure and velocity are to be solved;*4-field*: Solid stress & displacement, fluid pressure & velocity are to be solved.

A major issue in numerical solvers for poroelasticity is the poroelasticity locking, which usually appears as nonphysical pressure oscillations. This happens when the porous media are low-permeable or low-compressible [12, 28, 36].

Early on, the continuous Galerkin (CG) FEMs were applied respectively to solve for displacement and pressure. But it was soon recognized that such solvers were subject to poroelasticity locking and the 2-field approach was nearly abandoned. The mixed finite element methods can be used to solve for pressure and velocity simultaneously and meanwhile coupled with a FEM for linear elasticity that is free of Poisson-locking. Therefore, the 3-field approach has been the main stream [5, 25, 26, 27, 33, 34]. The 4-field approach is certainly worth of investigation, but it just involves too many unknowns (degrees of freedom) [35].

The weak Galerkin (WG) finite element methods [31] have emerged as a new class of numerical methods with nice features that can be applied to a wide variety of problems including Darcy flow and linear elasticity [14, 18, 24, 30]. Certainly, WG solvers can be developed for linear poroelasticity [17], they are free of poroelasticity locking but may involve a lot of degrees of freedom.

Recently, our efforts have been devoted to reviving the 2-field approach for development of efficient and robust finite element solvers for poroelasticity [13]. This may involve incorporation of WG FEMs with WG FEMs or classical FEMs. In this paper, we continue such efforts to develop a poroelasticity solver that couples the WG finite elements for Darcy flow and the classical Lagrangian elements with reduced integration for linear elasticity. Moreover, we provide an accessible efficient implementation of this new solver on deal.II, a popular finite element package [3].

## 2 Discretization of Linear Elasticity by Lagrangian Elements with Reduced Integration

*E*is the elasticity modulus and \( \nu \in (0,\frac{1}{2}) \) is Poisson’s ratio.

One major issue in finite element solvers for linear elasticity is that as the elastic material becomes nearly incompressible or \( \nu \rightarrow \frac{1}{2} \), mathematically as \( \lambda \rightarrow \infty \), a FE solver may fail to produce correct results. This often appears as loss of convergence rates in displacement errors or spurious behaviors in numerical stress and dilation (divergence of displacement). This is the so-called “Poisson locking” [6]. It is well known that the classical linear (bilinear, trilinear) Lagrangian finite elements are subject to Poisson locking.

Many remedies for Poisson locking have been developed. Reduced integration is probably the easiest technique aiming at a quick fix for the classical Lagrangian elements, although the theory was less elegant [7, 9, 22].

In this paper, we adopt the remedy in [9] and extend it to 3-dim. In other words, we consider vector-valued Lagrangian bilinear and trilinear finite elements with reduced integration CG.\(Q_1^d\) (R.I.) (here \( d=2,3 \)) for solving linear elasticity and provide deal.II implementation of these solvers. Specifically, the 1-point Gaussian quadrature is employed for handling the dilation term.

*E*be a convex quadrilateral with vertices \( P_i(x_i,y_i) (i=1,2,3,4) \) that are oriented counterclockwise. A bilinear mapping

*F*from \( (\hat{x},\hat{y}) \) in the reference element \( \hat{E}=[0,1]^2 \) to \( (x,y) \in E \) is established. Its Jacobian determinant is denoted as \( J(\hat{x}, \hat{y}) \). On \( \hat{E} \), we have 4 scalar-valued bilinear basis functions

*E*as rational functions of

*x*,

*y*:

*E*, we have 8 node-based vector-valued local basis functions:

*E*| is the volume of

*E*.

## 3 WG Finite Element Discretization for Darcy Flow

This section briefly discusses the weak Galerkin finite element discretization for Darcy flow that is needed for our new 2-field solver for linear poroelasticity.

*k*separately defined in element interiors and on edges/faces. Their discrete weak gradients are reconstructed in the unmapped Raviart-Thomas spaces \( RT_{[k]} \) and used to approximate the classical gradient in the variational form. The WG Darcy solvers based on these novel notions

- (i)
are locally mass-conservative;

- (ii)
provide continuous normal fluxes;

- (iii)
result in SPD linear systems that are easy to be solved.

In [32], we discussed deal.II implementation of such WG Darcy solvers for \( 0 \le k \le 5 \). The numerical tests on SPE10 Model 2 have demonstrated the aforementioned nice features and practical usefulness of the novel WG methodology.

*p*the primal unknown pressure, \( \mathbf {u} \) the Darcy velocity, \( \mathbf {K} \) conductivity tensor (medium permeability divided fluid dynamic viscosity) that is uniformly SPD over the domain,

*s*a known source, \( p_D \) a Dirichlet boundary condition, \( u_N \) a Neumann boundary condition, \( \mathbf {n} \) the outward unit normal vector on \( \partial \varOmega \), which has a nonoverlapping decomposition \( \varGamma ^D \cup \varGamma ^N \).

*E*, we consider 5 discrete weak functions \( \phi _i (0 \le i \le 4) \) as follows:

\( \phi _0 \) for element interior: It takes value 1 in the interior \( E^\circ \) but 0 on the boundary \( E^\partial \);

\( \phi _i (1 \le i \le 4) \) for the four sides respectively: Each takes value 1 on the

*i*-th edge but 0 on all other three edges and in the interior.

*E*becomes a rectangle \( [x_1,x_2] \times [y_1,y_2] \) with \( \varDelta x = x_2 - x_1 \), \( \varDelta y = y_2 - y_1 \), one can obtain these discrete weak gradients explicitly:

## 4 Coupling WG\((Q_0,Q_0;RT_{[0]})\) and CG.\(Q_1^2\) (R.I.) for Poroelasticity

In this section, the continuous Galerkin \( Q_1^d \) (\( d=2,3 \)) elements with reduced integration and the weak Galerkin WG\((Q_0,Q_0;RT_{[0]})\) elements are combined with the implicit Euler temporal discretization to solve linear poroelasticity problems.

*T*], let

Let \( \mathbf {V}_h \) and \( \mathbf {V}_h^0 \) be the spaces of vector-valued shape functions based on the first-order CG elements. Let \( \mathbf {u}_h^{(n)}, \mathbf {u}_h^{(n-1)} \in \mathbf {V}_h \) be the approximations to solid displacement at time moments \( t^{(n)} \) and \( t^{(n-1)} \), respectively.

- \(L_2([0,T];L_2(\varOmega ))\)
**-norm for interior pressure errors**$$\begin{aligned} \Vert p-p_h^\circ \Vert _{L_2(L_2)}^2 = {\sum _{n=1}^{N} \varDelta t_n \Vert p^{(n)} -p_h^{((n),\circ )} \Vert _{L_2(\varOmega )}^{2}}, \end{aligned}$$(22) - \(L_2([0,T];L_2(\varOmega ))\)
**-norm for displacement errors**$$\begin{aligned} \Vert \mathbf {u}-\mathbf {u}_h\Vert _{L_2(L_2)}^2 = {\sum _{n=1}^{N} \varDelta t_n \Vert \mathbf {u}^{(n)} - \mathbf {u}_h^{(n)} \Vert _{L_2(\varOmega )}^{2}}, \end{aligned}$$(23) - \(L_2([0,T];H^1(\varOmega ))\)
**-norm for displacement errors**$$\begin{aligned} \Vert \mathbf {u}-\mathbf {u}_h\Vert _{L_2(H^1)}^2 = {\sum _{n=1}^{N} \varDelta t_n \Vert \nabla \mathbf {u}^{(n)} - \nabla \mathbf {u}_h^{(n)} \Vert _{L_2(\varOmega )}^{2}}, \end{aligned}$$(24) - \(L_2([0,T];L_2(\varOmega ))\)
**-norm for stress errors**$$\begin{aligned} \Vert \sigma -\sigma _h\Vert _{L_2(L_2)}^2 = {\sum _{n=1}^{N} \varDelta t_n \Vert \sigma ^{(n)} - \sigma _h^{(n)}\Vert _{L_2(\varOmega )}^{2}}. \end{aligned}$$(25)

## 5 Code Excerpts with Comments

This section provides some code excerpts with comments. More details can be found in our code modules for deal.II (subject to minor changes). We want to point that the elasticity discretization can also be replaced by the so-called \( EQ_1 \) or \( BR_1 \) elements [3, 16], which are now available in deal.II Version 9.1.

### 5.1 Code Excerpts for WG\((Q_0,Q_0;RT_{[0]})\)

### 5.2 Code Excerpts for CG.\( Q_1^2 \) with Reduced Integration

### 5.3 Code Excerpts for Coupled Discretizations for Poroelasticity

## 6 Numerical Experiments

This section presents numerical examples to demonstrate the accuracy and robustness of this new finite element solver for poroelasticity.

### Example 1

**(A 2-dim smooth example for convergence rates).**Here our domain is \(\varOmega = (0,1)^2 \). Analytical solutions for solid displacement and fluid pressure are given as

Ex.1 with \( \lambda =1 \): Numerical results of CG.\(Q_1^2\)(R.I.) + WG\((Q_0,Q_0;RT_{[0]})\) solver on rectangular meshes

1/ | \( 1/\varDelta t \) | \( \Vert p-p_h^\circ \Vert _{L_2(L_2)} \) | \( \Vert \mathbf {u}-\mathbf {u}_h\Vert _{L_2(L_2)} \) | \( |\mathbf {u}-\mathbf {u}_h|_{L_2(H^1)} \) | \( \Vert \sigma - \sigma _h \Vert _{L_2(L_2)} \) |
---|---|---|---|---|---|

4 | 16 | 5.07478E−1 | 1.78798E−1 | 2.35598E−0 | 4.44080E−0 |

8 | 64 | 2.52365E−1 | 4.54880E−2 | 1.15497E−0 | 2.29855E−0 |

16 | 256 | 1.25983E−1 | 1.14071E−2 | 5.74435E−1 | 1.15784E−0 |

32 | 1024 | 6.29657E−2 | 2.85375E−3 | 2.86836E−1 | 5.79949E−1 |

Conv. rate | 1.00 | 1.98 | 1.01 | 0.97 |

Ex.1 with \( \lambda =10^6 \): Numerical results of CG.\(Q_1^2\)(R.I.) + WG\((Q_0,Q_0;RT_{[0]})\) solver on rectangular meshes

1/ | \( 1/\varDelta t \) | \( \Vert p-p_h^\circ \Vert _{L_2(L_2)} \) | \( \Vert \mathbf {u}-\mathbf {u}_h\Vert _{L_2(L_2)} \) | \( |\mathbf {u}-\mathbf {u}_h|_{L_2(H^1)} \) | \( \Vert \sigma - \sigma _h \Vert _{L_2(L_2)} \) |
---|---|---|---|---|---|

4 | 16 | 5.07481E−7 | 1.76096E−1 | 2.30126E−0 | 1.36770E+6 |

8 | 64 | 2.52367E−7 | 4.48677E−2 | 1.12759E−0 | 7.66388E+5 |

16 | 256 | 1.25984E−7 | 1.12553E−2 | 5.60529E−1 | 3.92554E+5 |

32 | 1024 | 6.29658E−8 | 2.81600E−3 | 2.79849E−1 | 1.97411E+5 |

Conv. rate | 1.00 | 1.98 | 1.01 | 0.93 |

### Example 2

**(A 3-dim example with a sandwiched low permeability layer)**. The domain is the unit cube \( \varOmega = (0,1)^3 \). The permeability is \( \mathbf {K} = \kappa \mathbf {I} \). Specifically, the middle region \( 0.25 \le z \le 0.75 \) has a low permeability \( \kappa = 10^{-8} \), whereas \( \kappa = 1 \) in other parts, see Fig. 1(a). There is no body force for solid or source for fluid. Other parameters are \( \lambda =1 \), \( \mu =1 \), \( \alpha =1 \), \( c_0=0 \).

The boundary conditions are as follows.

- (i)
For the solid, a downward traction (Neumann) condition \( \mathbf {t}_N = (0,0,-1)^T \) is posed on the top face, whereas all five other faces are clamped, i.e., \( \mathbf {u} = \mathbf {0} \);

- (ii)
For the fluid, the top face (\( z=1 \)) has a Dirichlet condition \( p=0 \); whereas all five other faces have a no-flow condition, in other words, zero Neumann boundary condition.

A similar 2-dim problem has been tested in [12, 13, 17]. But we shall observe richer features in this 3-dim problem.

For numerical simulations, we use uniform rectangular meshes and a uniform temporal partition. Specifically, \( h=\frac{1}{32} \) and \( \varDelta t = 10^{-3} \) so that \( \varDelta t \approx h^2 \). The final time is \( T=0.01 \), which means 10 time steps for simulation. Shown in Fig. 1 are the profiles of numerical pressure and velocity along with solid displacement magnitude. There is no pressure oscillation, even though there is a layer with a very low permeability. A pressure steep front is observed near \( z=0.75 \). The low permeability layer provides some kind of insulation. There is basically no solid deformation or fluid pressure change below this layer.

## 7 Concluding Remarks

A new finite element solver for poroelasticity is presented and proven numerically to be locking-free. This new solver is in the 2-field approach, i.e., only solid displacement and fluid pressure are treated as unknowns. Specifically, the new solver discretizes displacement using the classical Lagrangian *Q*-type elements with reduced integration, whereas the pressure is approximated by piecewise constants respectively defined inside elements and on inter-element boundaries. Discrete weak gradients of such piecewise constant shape functions are established in the unmapped lowest-order Raviart-Thomas spaces on quadrilaterals and hexahedra, which are required to be asymptotically parallelogram or parallelopiped. This new solver has been implemented in the dimension-independent paradigm on the deal.II platform. Our code modules are openly accessible.

The new solver in this paper is different than the one presented in [13]. Now the elasticity part is discretized using the classical Lagrangian *Q*-type elements with reduced integration. This results in even less degrees of freedom.

There are several directions one can go from here.

- (i)
Code optimization, especially, preconditioning and parallelization, shall make this new solver more efficient;

- (ii)
A rigorous analysis on this new solver is to be established for the locking-free property and convergence rates;

- (iii)
A similar solver can be developed for simplicial (triangular and tetrahedral) meshes; Implementation on FEniCS or FreeFEM++ platforms are surely attractive for scientific computing tasks;

- (iv)
To remove the restriction

*asymptotically parallelogram or parallelopiped*, we could utilize the newly developed Arbogast-Correa and Abogast-Tao elements [1, 2] for more general convex quadrilaterals and cuboidal hexahedra. Again deal.II implementation will be attractive.

These are under our investigation and will be reported in our future work.

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