# A mixed hybrid finite element framework for the simulation of swelling ionized hydrogels

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## Abstract

Ionized hydrogels, as osmoelastic media, swell enormously (1000 times its original volume in unionized water) due to the osmotic pressure difference caused by the presence of the negatively charged ion groups attached to the solid matrix (polymer chains). The coupling between the extremely large deformations (induced by swelling) and fluid permeation is a field of application that regular poroelasticity formulations cannot handle. In this work, we present a mixed hybrid finite element (MHFE) computational framework featuring a three-field (deformation-chemical potential-flux) formulation. This formulation guarantees that mass conservation is preserved both locally and globally. The impact of such a property on the swelling simulations is demonstrated by four numerical examples in 2D. This paper focuses on the implementation aspects of the MHFE model and shows that it stays robust and accurate for a volume increase of more than 3000%.

## Keywords

Finite deformation Swelling model Mixed formulation Numerical methods Hydrogel## 1 Introduction

Hydrogels are water-swollen and cross-linked polymeric networks, produced by the simple reaction of one or more monomers. Hydrogels exhibit ability to swell and retain a significant fraction of water within its structure without dissolving. Various environmental stimuli can induce the swelling of a hydrogel. The common stimuli are pH, ion concentration in the solution, electric potential and temperature [1]. Hydrogels have received considerable attention in the past 50 years due to their outstanding application in wide range of fields including biomedicine [2].

There are several mechanisms leading to the swelling of SAPs. First of all, the backbone of the polymer chains (carboxylate acid group) is hydrophilic. As a result, when the polymer is in contact with water, water molecules are attracted to the polymer. Hydration happens and hydrogen bonds are formed as these activities increase the entropy of the system. This process is known as mixing. Secondly, besides mixing, for a partially neutralized gel the positive sodium ions are able to move relatively freely inside the gel as soon as the water molecules weaken the bonding force between them and negatively charged carboxylate groups. Consequently, the gel acts like a semi-permeable membrane and the difference in osmotic pressure arises in and outside of the gel. It has been proven that compared to the mixing part, the difference in osmotic pressure is the main drive contributing to the swelling of a SAPs gel [5].

To study the mechanical behavior of a hydrogel during swelling, a reliable swelling model is needed. Under the framework of mixture theory, Lanir [6] developed an osmoelastic model arguing that the ionic concentrations are always in equilibrium with the external solution. In essence, this is a biphasic (solid and fluid phase) swelling model. Later, Lai et al. [7] developed a triphasic (solid, fluid and ion phases) theory to model the swelling and deformation behavior in articular cartilage. Later, Huyghe and Janssen [8] proposed a finite deformation quadriphasic model where solid, fluid, cation and ion phases are all independently considered for incompressible porous media.

Considering a SAPs gel particle facing a gush of urine which can be approximated by the physiological salt solution, i.e. a solution of Na+ and Cl− of concentration 0.154 mol/L, chemical potential responds much faster to the local ions concentration change than solvent permeating into the gel. This claim is justified by the following calculation. We have, on one hand, the diffusion coefficient \(d_{ion}\) of ions in free water of the magnitude \(10^{-9}\,\mathrm{m}^2/\mathrm{s}\) [9]. On the other hand, the hydraulic pressure diffusion coefficient \(d_p\) is estimated by the multiplication of hydraulic permeability and Young’s modulus. In a SAPs gel particle, its hydraulic permeability is of order \(10^{-3}\, \mathrm{mm}^4/(\mathrm{Ns})\) and Young’s modulus is of the order \(10^{-2} \, \mathrm{N}/\mathrm{mm}^2\). Therefore, after some simple calculation, one finds that \(d_{ion}/d_p=10^2\). For this reason, we assume ions respond infinitely fast to the presence of outer solution compared to the fluid. In other words, Lanir’s osmoelastic model was found sufficient to characterize the swelling mechanism.

In order to simulate such a fluid permeation and solid deformation coupled problem, proper numerical schemes need to be implemented. Due to the essential focus on solid deformation, FEM (Finite Element Method) has been a natural choice for such swelling simulations. Lots of effort in numerical implementation are made by various groups over the years. Frijns et al. [10] implemented the quadriphasic model in one dimension to simulate the swelling and shrinking of biological tissues. The extension to the 3D implementation of quadriphasic model in finite deformation is done by Van Loon et al. [11]. Limiting to gel swelling simulations, Hong et al. [12] and Kuang and Huang [13] presented a finite element swelling model for inhomogeneous swelling at equilibrium state. A number of works ([14, 15, 16]) take a different approach to model a swelling hydrogel. Namely the swelling front is treated as a sharp transition surface between the solution phase and the gel phase (characterized by different chemical potential). The literature listed above only dealt with spherical gel specimen. The generalization to an arbitrary gel shape is not a trivial job. The application of eXtended Finite Element method in such simulations have received wide attentions [17, 18, 19]. Extended Finite element is applied to simulate the crack propagation in hydrogel induced by swelling [20].

However, to achieve satisfactory simulation results, it is essential to deploy a discretization method that takes the physics of the problem into account. In swelling simulations, (local) solid deformation is directly related to (local) net in/out flow of the fluid. As a result, it is sensible to value the accurate calculation of the flux field in swelling simulations. In FEM, the flux field is calculated by numerical differentiation of the chemical potential field, which leads to a serious loss of precision [21]. On the other hand, MHFEM (Mixed Hybrid Finite Element Method), which approximates flux as an independent variable using Raviart–Thomas element and resolves the resulting indefinite coefficient matrix by means of hybridization procedure, possesses local mass conservation property and has proven to be effective in solving Darcy type equations [22, 23]. Moreover, similar discretization method is applied to solve Biot consolidation problem in geomechanics. Specifically, Jha and Juanes [24] approximated displacement, pressure and flux using linear (\(P_1\)), lowest order Raviart–Thoams (\(RT_0\)) and element-wise constant (\(P_0\)) elements respectively and successfully performed reservoir simulations. Ferronato et al. [25] adopted the same discretization method in solving three dimensional Biot consolidation and showed that the mixed formulation alleviates the pressure oscillations at the interface between different permeability.

Extension from linear poroelasticity to finite strain is also investigated. Berger et al. [26] presented a three-field finite strain poroelasticity framework where the pressure and flux are approximated using \(P_1/P_0\) elements combined with a stabilization technique to avoid spurious pressure oscillation. In the field of biomechanics, where large deformations of biological tissues are often expected, Wall et al. [27] and Levenston et al. [28] presented their own mixed finite element formulations under the theoretical framework of mixture theory.

Motived by the success of MHFEM in solving Darcy’s type equations and Biot consolidation problems, we apply MHFEM in swelling simulations in order to achieve more reliable and satisfactory results. Malakpoor et al. [29] applied MHFEM in the simulation of the swelling of cartilaginous tissues. However, his simulations were limited to small deformations. To our knowledge, there is so far no research available that apply MHFEM to the finite swelling of gels. In this study, a nonlinear system of partial differential equations modeling the swelling of SAPs particles is derived and solved using MHFEM. The model is first discretized in time using the first order implicit Euler finite difference scheme and then linearized using the Newton–Raphson strategy. Space discretization is achieved using the lowest order Raviart–Thomas space followed by a hybridization procedure. The numerical model is validated in one dimension by comparing to a semi-analytical solution obtained using MATLAB. By means of numerical examples in two dimensions, we demonstrate that MHFEM is a robust and accurate method for swelling simulations involving large deformations.

## 2 Relevant theory

In the framework of mixture theory [30], SAPs are treated as the superposition of two constituents that occupy the same physical domain: fluid phase and solid phase (\(a\in \{s,f\}\)). We assume that the fluid phase and the solid phase are nonreactive and incompressible. Body force and inertia are ignored.

### 2.1 Preliminaries

*J*, are defined as:

*a*per unit volume of the mixture, in contrast to the true density of \(a-\)constituent \(\gamma _a\), which is defined as the mass per unit volume of the constituent. An important dimensionless quantity in mixture theory of immiscible constituents is volume fraction \(\phi _a(x,t):=\rho _a/\gamma _a\). The physical meaning of \(\phi _a(x,t)\) is the volume of the \(a-\)constituent per unit volume of the mixture. In the case of incompressible solid and fluid (\(\gamma _a\) is constant), mass conservation of constituent

*a*in the current configuration is derived as:

### 2.2 Helmholtz free energy function

*W*consists of two independent parts: ionic part and elastic part. Namely, we have:

*K*is the bulk modulus and

*G*is the shear modulus. One can recognize that it represents the compressible Neo-Hookean law. Further, we include the following relationship between Poisson’s ratio \(\nu \) and solid volume fraction \(\phi _f\): \(\nu =0.5\phi _{s}\). This relation is justified by the assumption that at the dry state (\(\phi _s=1\)), the polymer network becomes incompressible (\(\nu =0.5\)). Making use of the relationship between the current solid volume fraction and the initial volume fraction \(\phi _s=\frac{\phi _{s,0}}{J}\), the bulk modulus

*K*can be rewritten in terms of

*G*and \(\phi _{s,0}\) as:

*R*is the universal gas constant,

*T*is the absolute temperature, \(\varGamma \) is the osmotic coefficient and

*c*is the molar concentration of the fluid phase and \(\varPhi \) is the fluid volume fraction in initial configuration (\(\varPhi =J\phi _f\)). It is related to the Donnan osmotic pressure \(\pi \) via the relation:

### 2.3 Biphasic swelling model

*R*is universal gas constant, \(\varGamma \) is the osmotic coefficient and

*T*is absolute temperature.

## 3 Field equations

In Sect. 2, we have presented some preliminaries of mixture theory and the derivation of the osmotic pressure. In this section, we formulate our model problem mathematically using the related theory presented in Sect. 2. The isothermal condition, hyperelasticity and isotropy of the material are assumed. The governing equations contain linear momentum balance, fluid content balance and extended Darcy’s law.

### 3.1 Governing equations and constitutive relations

*p*is hydraulic pressure and \(\mathbf \sigma _{eff}\) is the effective stress. Assuming proper constitutive relations for the gel, the effective stress is derived from the elastic part of the Helmholtz energy function \(W_{elastic}\) (Eq. 13):

*G*is shear modulus and \(\phi _{s,0}\) is the initial solid volume fraction.

*k*in the current configuration by:

*p*and osmotic pressure \(\pi \). Specifically we have [33]:

### 3.2 Three-field weak formulation

We adopt mixed formulation to describe fluid permeation: position \({\mathbf {x}}\), chemical potential \(\mu \) and flux \({\mathbf {Q}}\) are chosen to be prime variables in the weak formulation. In this section, the corresponding weak form of the governing equations is presented.

Note that \({\mathbf {f}}_{ext}\) denotes external surface tension applied to the Neumann part boundary \(\varGamma ^u_N\). \(\mu _{ext}\) denotes the chemical potential in the external solution at the Dirichlet boundary part \(\varGamma ^\mu _D\). We have \(\mu _{ext}=-2RTc^{ex}\).

## 4 Mixed hybrid finite element method

Given the weak form (35)–(37), we present in this section how MHFEM was applied to solve the system. The model is first discretized in time using the first order implicit Euler finite difference scheme and then linearized using the Newton–Raphson strategy. Space discretization is achieved using the lowest order Raviart–Thomas space followed by a hybridization procedure.

### 4.1 Time discretization and linearization

*n*.

### 4.2 Spatial discretized form

*X*,

*Y*) follows Piola transformation:

### Remark

In poroelasticity, locking (which often manifests itself as spurious pressure oscillation) receives a considerable amount of attention over the years. Locking has been proved to be related to the violation of inf-sup condition for the coupling discrete element spaces [36]. Numerous work has been done to unveil the cause of locking and its numerical remedy [37, 38, 39, 40]. As Haga et al. [40] argued that for (more than two fields) mixed formulation, the coupling between stable element spaces should satisfy individual problems. In our three-field formulation, there are two pairs of coupling in consideration: chemical potential-flux and displacement-chemical potential. For the chemical potential-flux pair, we have chosen the well-known stable pair \(P_0/RT_0\) for Darcy flow problem, which conveniently also possesses local mass conservation property. As to the displacement-chemical potential pair, as heuristically explained by Philips and Wheeler [41], care needs to be taken at the beginning of the deformation of a porous medium when a small time step and low permeability are considered in poroelasticity. Basically, the divergence of the displacement is close to zero due to the incompressibility constraint on the solid matrix. Extension from infinitesimal strain to finite strain is straightforward.

*k*), the fluid mass conservation equation is reduced to a constraint on the position field \({\varvec{x}}\) at the beginning stage of the swelling simulation:

### 4.3 Hybridization procedure

So far, the search function space we applied in the discretization for the flux field is \(RT_{-1}^0\) (bigger) instead of \(RT_0^0\) (smaller). For a function that is in \(RT_{-1}^0\) to be also in \(RT_0^0\), a necessary and sufficient condition is that the normal flow across the edge between the neighboring elements is continuous [21]. There are several ways to implement such a constraint [42]. Here we adopt the Lagrange multiplier method. By introducing a new variable \(\lambda \) with the physical meaning chemical potential on edges as Lagrange multiplier, the constraint mentioned above is enforced. One of the advantages of such an implementation is that by means of static condensation the total number of unknowns is eventually reduced from (\({\mathbf {x}}, {\mathbf {Q}}, \mu ,\lambda \)) to (\({\mathbf {x}}, \lambda \)), which leads to improvement in computational efficiency. Besides, for such an implementation no extra information like edge orientations is needed and therefore is more desirable when mesh structure is complex. In what follows, we show how the system becomes “hybrid” by introducing a Lagrange multiplier \(\lambda \).

*e*of the decomposition \({\mathcal {Q}}_h\). Next, to incorporate boundary conditions, we define

## 5 Solution verification

So far, we have presented various aspects of the numerical simulation engine. In this section, we focus on the verification of such a simulation engine. Namely, given the partial differential equation system we would like to solve, we calculate the solution in a different way and then compare results with the proposed simulation results.

Since there is no analytical solution available for the transient swelling or consolidation simulation with finite deformation even in one dimension. Simulation results are compared with semi-analytical results. These semi-analytical results are solutions calculated by a MATLAB internal partial differential equation solver (“pdepe”) in one dimension. On the other hand, the equilibrium state of a circular swollen gel can be calculated analytically (homogeneous swelling) we also compared our simulation results with that.

*y*-direction). Basically, deformation tensor \({\mathbf {F}}\) is reduced to volume ratio

*J*, and the simplified equations in terms of

*J*and \(\mu \) are:

*J*, given by:

*J*is derived:

Model parameters

Parameter | Value | Unit |
---|---|---|

Shear modulus | 0.015 | \(\mathrm{N}/\mathrm{mm}^2\) |

Hydraulic permeability | \(10^{-3}\) | \(\mathrm{mm}^4/(\mathrm{Ns})\) |

Initial porosity \(\phi _{f,0}\) | 0.83 | |

Osmotic coefficient \(\varGamma \) | 0.99999 | |

Initial fixed charge density \(c^{fc}_0\) | \(3.32\times 10^{-4}\) | mol/ml |

Parameters given in Table 1 are chosen carefully so that they are within the industrially relevant regime. These parameters are used throughout all simulations presented in this paper unless otherwise indicated. In this section we set \(c^{ex}\) to be physiological salt concentration, thus equals \(1.54\times 10^{-4} \;\mathrm{mol}/\mathrm{ml}\). The applied (downward) pressure \(f_{ext}\) in the consolidation simulation is 0.02 MPa. Mesh size and time steps are chosen to be the same for MHFE and MATLAB solutions.

The evolution of swelling ratio and dimensionless chemical potential on the top surface are plotted for both consolidation and swelling simulations (Figs. 2, 3). The characteristic time scale \(\tau \) in this problem equals \(l^2m_p/RTk\), where *l* is the characteristic length (dry size), \(m_p\) is the molar volume of the polymer (taken to be 105 \(\mathrm{cm}^3/\mathrm{mol}\)). The dimensionless time is found to be \(t_d=t/\tau \). Similarly, the dimensionless chemical potential is derived as \(\mu _d=\mu /(\frac{RT}{m_p})\).

Figure 2a shows that the chemical potential at the top surface monotonically decreases over time with some delay at the beginning of the consolidation simulation. This delay can be explained by the fact that it takes time for the fluid inside the gel to drain through the bottom (\(y=0\)). We conclude from the figures that MHFEM solutions match MATLAB solutions very well for both variables of interest in both simulations.

## 6 Numerical examples

Edge EF error with \(G=0.15\) MPa

Mesh sizes (mm) | 0.0833 | 0.0417 | 0.0167 | 0.0083 |
---|---|---|---|---|

\(e_{max}^{FEM}\) | 0.0137 | 0.0042 | 9.78e−4 | 2.85e−4 |

\(e_{max}^{MHFEM}\) | 0.0045 | 0.0035 | 0.0012 | 2.85e−4 |

\(e_{avg}^{FEM}\) | 0.0044 | 0.0011 | 1.81e−4 | 5.04e−5 |

\(e_{avg}^{MHFEM}\) | 3.49e−4 | 1.39e−4 | 1.90e−5 | 2.36e−6 |

Edge EF error with \(G=0.025\) MPa

Mesh sizes (mm) | 0.0833 | 0.0417 | 0.0167 | 0.0083 |
---|---|---|---|---|

\(e_{max}^{FEM}\) | 0.0291 | 0.0139 | 0.0033 | 6.59e−4 |

\(e_{max}^{MHFEM}\) | 0.0155 | 0.0076 | 0.0013 | 6.59e−4 |

\(e_{avg}^{FEM}\) | 0.0085 | 0.0042 | 7.25e−4 | 9.85e−5 |

\(e_{avg}^{MHFEM}\) | 0.0012 | 3.05e−4 | 2.15e−5 | 5.44e−6 |

### 6.1 Swelling with a low-permeable stripe

Inspired by a numerical example presented in [22], a heterogeneously permeable domain is introduced (Fig. 4). Simulations are carried out using both MHFEM and standard FEM with different mesh sizes (\(0.0833\,\mathrm{mm}\), \(0.0417\,\mathrm{mm}\), \(0.0167\,\mathrm{mm}\), \(0.0083\,\mathrm{mm}\)). Note that regardless of mesh refinement, the area and location of the low permeable stripe is kept unchanged. Due to the existence of the low permeability stripe, the top edge EF is expected to form a curve as the swelling starts and the curve is used to characterize the deformation of the gel (Figs. 5, 6).

Both Tables 2 and 3 show that MHFEM produces more accurate results than FEM given the same mesh sizes. As a matter of fact, the average error indicates that MHFEM produced more accurate results than FEM with the mesh 2.5 times coarser when shear modulus \(G=0.15\) MPa. The accuracy advantages of MHFEM over FEM is more pronounced when the shear modulus is reduced. Table 3 shows that at mesh sizes \(0.0833\,\mathrm{mm}\) and \(0.0417\,\mathrm{mm}\) the maximum error of FEM is almost twice as much as the one of MHFEM.

Convergence of position and flux field

Mesh sizes (mm) | 0.1000 | 0.0500 | 0.0333 | 0.0025 | 0.020 | 0.0167 |
---|---|---|---|---|---|---|

\(e_{position}\) (mm) | 0.0318 | 0.0177 | 0.0011 | 0.0063 | 0.0035 | 0.0015 |

\(e_{flux}\) | 1.343e−3 | 1.277e−3 | 9.753e−4 | 6.852e−4 | 4.270e−4 | 2.002e−4 |

To investigate the impact of an unstable pair (\(P_1/P_0\)) on the swelling simulation, we slightly adapt boundary conditions in Fig. 4. Instead of edge HG in touch with the outer solution, we let EF directly in touch and the rest of edges are with no flow conditions. Also the domain size is changed to \(1\,\mathrm{mm}\times 1\,\mathrm{mm}\). This way we generate a sharp gradient in chemical potential especially for the low permeability area. Fig. 8 shows the chemical potential contour of both an unstable pair (\(P_1/P_0\)) and a stable pair (\(P_2/P_0\)) for displacement-chemical potential coupling. We observe that indeed the unstable pair led to a checkerboard distribution of pressure while the stable pair not. Also due to the higher degree of approximation for displacement, the deformations generated by the two pairs are slightly different. However, we notice that the oscillation in the pressure field dissipates as the swelling goes on and eventually disappears. Local element refinement also helps with the alleviation of checkerboard distribution. Moreover, the use of higher order element also causes higher computational cost. Hence, unless great importance is attached to the calculation of chemical potential at the initial stage, we keep using \(P_1/RT_0/P_0\) pair in our swelling simulations potentially combined with local mesh refinement scheme.

### 6.2 Free swelling of a quarter of a square

The chemical potential contour plots over time are given in Fig. 10. The color indicates the magnitude of the chemical potential. It shows that fluid permeation starts from the two edges (AB and CB) gradually reaching the core. The square shape of OABC is temporarily lost due to the faster swelling area near node B comparing to node A and C. At the equilibrium state (\(t_d\rightarrow \infty \)) the square shape is recovered (Fig. 10).

To predict material failure (for example, the initiation of fractures), magnitude, directions and locations of the maximum and minimum stresses must be identified. Between the two parts of stress (effective stress and hydraulic pressure), only the effective stress contributes to the initiation of fractures. Using the proposed numerical model, the magnitude of the maximum and minimum normal stresses and maximum shear stress distribution over the sample can be calculated (Fig. 11). It is observed that the maximum normal stress appears near node A and C along the edges that are in touch with the outer solution. Similar observation can be made for minimum normal stress. The maximum normal stresses are tensile stress (positive values) and the minimum stresses are compressive stresses (negative values). As to the maximum shear stresses, near nodes C and A the sample sustains the highest shear stresses and near origin O and node B the lowest.

The convergence of the simulation as the mesh sizes decreases is demonstrated in Table 4. The y-coordinate of the outer node B and the influx on edge AB at a transient moment (2.48 s) are recorded. Using the calculated values at high mesh density (\(70\times 70\)) as canonical solution, the error yielded by different meshes are given in the table (\(e_{position}\) and \(e_{flux}\)). As the mesh size decreases, the errors converges towards zero.

### 6.3 Free swelling of a quarter of a circle

Next, we investigate gels with a circular shape. Only a quarter of the circle (OAB) is simulated due to symmetry reasons (Fig. 12). The boundary conditions are the same as for the square except that \(\varGamma ^u_D\) represents OA in x-direction and OB in y-direction; \(\varGamma ^u_N\) and \(\varGamma ^\mu _D\) represent curve AB; \(\varGamma ^\mu _N\) contains OA and OB. The domain is discretized by 163 quadrilateral elements with time step taken to be same as for the square.

Unlike in the square simulation, the shape of OAB stays unchanged during swelling (Fig. 13). The extremal effective stresses plot (Fig. 14) shows that at the transient state the maximum normal and shear stresses are largest along the outer boundary and gradually decrease towards the core. The opposite holds true for minimum normal stress, where the largest minimum normal stress is at the core and decreases as the radius increases.

### 6.4 Swelling-induced bifurcation

A swelling bifurcation experiment is reported by Zhang et al. [43]. By exposing a hydrogel membrane with periodic circular hole array to a solvent, the holes deform into ellipses and the interaction between them yield a “diamond plate” pattern. In this subsection, we apply the proposed numerical model to simulate the swelling of such a membrane aiming to replicate the deformation of holes qualitatively as reported in the experimental work by Zhang et al. [43].

The domain (Fig. 15) is discretized by 2926 linear quadrilateral elements and the time step is 0.025 s. The outer solution concentration \(c^{ex}\) set to be \(5.54\times 10^{-4}\) mol/ml. The chemical potential contour plots (Fig. 16) show that the deformation of the originally circular holes. Firstly, the circular holes collapsed into ellipses of alternating directions (Fig. 16a). Next, as the swelling went on, the elliptic holes further elongated themselves in their major axis direction (Fig. 16b). At last, a narrowing “waist” of the elongated slits appeared (Fig. 16c). The simulation using the developed numerical model goes on until contact between the elements happens, since contact was not defined in the numerical model. However, the calculated deformations of the holes by MHFEM model are in good agreement with the experimental results.

## 7 Conclusions

In this study, we have developed a MHFE model under the theoretical framework of mixture theory for the simulation of swelling ionized hydrogels in two dimensions. Newton–Raphson strategy was applied to handle the non-linearity arising from hyperelasticity and nonlinear osmotic pressure term. We deployed the lowest order Raviart–Thomas element to approximate flux as an independent variable. Then hybridization procedure was introduced to guarantee the continuity of normal flux across neighboring elements so that the lowest order Raviart–Thomas elements were properly implemented. The calculated solution using proposed model was verified by comparing with semi-analytical solutions calculated by MATLAB in one dimension.

Local mass conservation property of the proposed model guarantees more accurate calculation of flux and deformation, which is crucial for the simulation of extremely large deformation induced by swelling as shown in the low-permeable stripe simulations. Next, we continue to simulate the free swelling of a square and a circular-shaped gel using MHFEM. In both simulations, the swelling ratio is more than 30. Chemical potential contours and extremal stresses distributions are presented. Chemical potential contours give us a good idea about the fluid permeation progress during swelling and extremal stress distribution plots are useful for failure mechanics. At last, we carried out a swelling-bifurcation simulation. The simulation results are verified against experimental results and showed good agreement qualitatively.

## Notes

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