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

Let \(\varOmega _{s0}\in {\mathbb {R}}^2\) be the original bounded domains of the solid field. The original domain is transformed to the current (gel) domain \(\varOmega \) by:where \({\mathbf {x}}\in \varOmega \), \({\mathbf {X}}\in \varOmega _{s0}\), \({\chi }_s \) is an invertible and continuously differentiable mapping from \(\varOmega _{s0}\) to \(\varOmega \). As we are interested in the deformation of the solid field, without explicit indication, the original domain is referred to as \(\varOmega _0=\varOmega _{s0}\). The velocity of the solid phase is given by:Time derivative \(\frac{D}{Dt}\) is a material time derivative with respect to the original solid field and is defined as:The deformation gradient \({\mathbf {F}}\), right Cauchy-Green strain tensor \({\mathbf {C}}\) and volume ratio J, are defined as:We define the apparent density of \(a-\)constituent \(\rho _a\) as the mass of constituent 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:The assumption that the gel is fully saturated yields a constraint:for \({\mathbf {x}}\in \varOmega , t\ge 0\). Adding up the mass conservation equation for individual constituents Eq. (7) and making use of the relation Eq. (8), the mass conservation for the mixture at the current configuration is derived:This relationship is the basis of the derivation of mass conservation in Sect. 3 (Field equations).

Next, we postulate that Helmholtz free energy per initial mixture volume W consists of two independent parts: ionic part and elastic part. Namely, we have:The form of the elastic part \(W_{elastic}\) is taken from Wilson et al. [31]. Explicitly, we have:where 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:Substituting Eq. (12) into (11), the final form of \(W_{elastic}\) is derived:The ionic part \(W_{ionic}\) represents the energy related to the affinity between the ions and the water. The form the ionic part is given by:where 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:

In the biphasic swelling model [6], it is hypothesized that Donnan osmosis equilibrium is achieved instantly. Under the condition that the activity coefficient of the mobile ions is the same in the gel and in the equilibrium solution, the Donnan equilibrium ion concentrations inside the gel are derived [8]:where \(c^{fc}\) denotes the (deformation-dependent) fixed charged density and \(c^{ex}\) the (constant) ion concentration in the external solution. The osmotic pressure \(\pi \) inside of the gel follows the van ’t Hoff law:where R is universal gas constant, \(\varGamma \) is the osmotic coefficient and T is absolute temperature.

By incorporating general balance principles (mass conservation and linear momentum conservation) and incompressibility constraints into the entropy inequality, we derive the following set of equations Eqs. (19)–(21) that a mixture continuum needs to obey. They are “pulled back” to the reference (initial) configuration [32], since we deal with finite deformation in the current work.The divergence operator with “0” subscript indicates the divergence is taken with respect to the initial configuration.

Equation (19) represents linear momentum balance in the reference configuration. \({\mathbf {T}}\) denotes the first Piola-Kirchhoff stress of the mixture and is calculated by:where 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): Recall that G is shear modulus and \(\phi _{s,0}\) is the initial solid volume fraction.

Equation (20) represents Darcy’s law in an extended sense. \({\mathbf {Q}}\) denotes the flux in the initial configuration and is related to the variables in current configuration by:\({\mathbf {K}}\) is the hydraulic permeability tensor written in the reference configuration and is related to the hydraulic permeability k in the current configuration by:Chemical potential \(\mu \) inside of the gel consists of two parts: hydraulic pressure p and osmotic pressure \(\pi \). Specifically we have [33]:where \(\pi \) is given in Eq. (18). Fixed charge is considered to be part of solid matrix. Its density \(c^{fc}\) is therefore decreases as the solvent molecules enter the gel. In other words, the fixed charge density is deformation dependent:where \(c^{fc}_0\) denotes the initial fixed charge density and \(\phi _{f,0}\) denotes the initial fluid volume fraction.

At last, Eq. (21) represents the fluid content balance. It is basically Eq. (9) written in the initial configuration. Based on the molecular incompressibility assumption [34], the volume of the solid stays unchanged during swelling. We have thus:which leads toConsequently, we have:

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.

Firstly, we need to determine proper search/test function spaces for the approximation of the prime variables. For the position field, the continuity requirement demands \({\mathbf {x}}\) at least continuously differentiable. We have thus \({\mathbf {x}}({\mathbf {X}},t)\in (H^1_D(\varOmega ))^2\), wherewhere \(\varGamma ^u_D\) denotes the Dirichlet boundary for \({\mathbf {x}}\). For \(\mu \), we set \(\mu \in L^2(\varOmega )\). At last, we let \({\mathbf {Q}}\in H_N(div,\varOmega )\), wherewhere \(\varGamma ^\mu _N\) denotes the Neumann boundary for \(\mu \).

To achieve the symmetry of the system, we use the second Piola-Kirchhoff stress \({\mathbf {S}}\) and Green strain tensor \({\mathbf {E}}\) pair to calculate the virtual work. We make use of the following relation:where the bar “-” denotes that the variable is a virtual variable and we defineThe three-field variational form of the system is given as follows. Find \(({\mathbf {x}},{\mathbf {Q}},\mu )\in ((H^1_D(\varOmega ))^2,H_N(div,\varOmega ),L^2(\varOmega ))\), such that the following set of equations are satisfiedfor any \((\bar{{\mathbf {x}}},\bar{{\mathbf {Q}}},{\bar{\mu }})\in ((H^1_D(\varOmega ))^2,H_N(div,\varOmega ),L^2(\varOmega ))\).

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 \(\mu _{ext}=-2RTc^{ex}\).

First of all, equations are discretized in time. To achieve unconditional stability, first order implicit Euler time discretization scheme is implemented. The time-discretized system is derived as follows:where subscript \(n/n-1\) denotes variables at time step \(n/n-1\); \(\varDelta t\) denotes the time step size. From now on, we will skip subscript n.

The non-linearity stems from both the gel constitutive law (geometric and material non-linearity) and the nonlinear dependency of osmotic pressure on deformation (Eq. 27). Newton–Raphson strategy is applied to solve such a nonlinear system. “\(\delta \)” in front of a variable indicates the increment of the variable and the tilde above the variables indicates the current (known) value of the variable so far. Linearization is performed as follows: find \((\delta {\mathbf {x}},\delta {\mathbf {Q}},\delta \mu )\in ((H^1_D(\varOmega ))^2,H_N(div,\varOmega ),L^2(\varOmega ))\), such thatwherefor any \((\overline{\delta {\mathbf {x}}},\overline{\delta {\mathbf {Q}}},\overline{\delta \mu })\in ((H^1_D(\varOmega ))^2,H_N(div,\varOmega ), L^2(\varOmega ))\), where \({\mathbb {C}}\) is the fourth order elasticity tensor.

Given the linearized system of equations (Eqs. 41–43), we further discretize the spatial domain \(\varOmega _0\). In this work, we assume two dimensional domain and we use four-node linear quadrilateral elements to discretize the domain. \({\mathcal {Q}}_h\) denotes a quadrilateral domain discretization. Finite dimensional approximations of the search function spaces \((H^1_D(\varOmega ))^2\), \(H_N(div,\varOmega )\), and \(L^2(\varOmega )\) are introduced. Specifically, we have:where The finite dimensional function spaces are defined as:Note that \(P^1(\varOmega _e)\) denotes the space of polynomials of degree less or equal than 1 on one single element domain \(\varOmega _e\). We have:\(RT^0(\varOmega _e)\) stands for the lowest order Raviart–Thomas element [35] and is defined on one single element \(\varOmega _e\) as\(M^0(\varOmega _e)\) denotes the function space that is constant on element \(\varOmega _e\). Therefore, the spatial discretizations on one single element \(\varOmega _e\) are derived:where \(\varphi _i\) and \({\upsilon }_i\) are basis functions of the function spaces \(P_0^1({\mathcal {Q}}_h)\) and \(RT_{-1}^0({\mathcal {Q}}_h)\) and \(\varPsi \) is an arbitrary constant.

The isoparametric concept has been invoked for the basis functions \(\varphi _i\) and \({\upsilon }_i\). The basis functions on the reference domain \(({\hat{x}},{\hat{y}})\) are given as:For Raviart–Thomas element basis function \({\upsilon }_i\), the transformation between the reference domain \(({\hat{x}},{\hat{y}})\) and the real domain (X, Y) follows Piola transformation:where \({\mathbf {M}}\) is the affine mapping matrix from the reference domain to the real domain. Such a transformation guarantees that the integration of flux on each edges stays the same as in the reference domain and in the real domain [21].

Finally, based on the discretization details we presented above, the matrix form of the system on the element level is derived as:where for \(i,j=1,\ldots ,4\)

In our formulation, the incompressibility constraint (Eqs. 28) has been implicitly included in the fluid mass conservation equation. As a matter of fact, the infinitesimal volume change of gel \(\delta J\) is approximated by \(\nabla \cdot {\varvec{x}}\). Combining with a small time step (\(\delta t\)) and low permeability (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:It is well-known that for such a constraint, the current discrete element space combination \(P_1/P_0\) is unstable [21, 36]. Hence, to avoid locking, different element spaces combinations (for example, \(P_2/P_0\) or \(P_2/P_1\)) or stabilization techniques need to be applied. However, since finite swelling simulations stay quite far away from incompressibility constraint except for the very beginning stage, we assess the impact of such an unstable pair on the simulation by means of numerical example (Sect. 6.1).

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 \(\lambda \).

Firstly, we define the discretized approximation space for \(\lambda \):where \({\mathfrak {L}}_0^0({\mathcal {Q}}_h)\) denotes the set of functions that are constant on each edge e of the decomposition \({\mathcal {Q}}_h\). Next, to incorporate boundary conditions, we define which indicates that \(\lambda _h\) has to satisfy the Dirichlet boundary condition of \(\mu \). The rationale of hybridization can be summarized in the following result: let \(({\mathbf {Q}}_h,\mu _h)\in (RT_0^0({\mathcal {Q}}_h),M_{-1}^0({\mathcal {Q}}_h))\) be the solution of the system:for any given function \(J({\mathbf {X}},t)\) that satisfies \(\frac{DJ}{Dt}({\mathbf {X}},t)\in L^2(\varOmega )\), \(\forall t\). Then, according to [21] Theorem 1.1 p.180, there exists a unique \(\lambda _h\in \varLambda _{\mu _{ext},D}\) satisfying:for any \(\bar{{\mathbf {Q}}}_h\in RT_{-1}^0({\mathcal {Q}}_h)\). Moreover, \(({\mathbf {Q}}_h, \mu _h,\lambda _h) \in (RT_{-1}^0({\mathcal {Q}}_h), M_{-1}^0({\mathcal {Q}}_h),\varLambda _{\mu _{ext},D})\) is the solution of the following “hybrid” system:for any (\(\bar{{\mathbf {Q}}}_h, {\bar{\mu }}_h, {\bar{\lambda }}_h)\in (RT_{-1}^0({\mathcal {T}}_h), M_{-1}^0({\mathcal {T}}_h),\varLambda _{0,D}\)) is also the solution of (82)–(83). Therefore, by introducing the Lagrange multiplier \(\lambda \), the continuity of the normal flow is guaranteed and we are assured that the solution is found in the correct function space.

In terms of implementation, given the approximation of \(\delta \lambda _h\) on one single element \(\varOmega _e\):where \(\eta _i\)’s are constant on \(e_i\in \partial \varOmega _e\), the system of equations (69) is extended towhereAfter applying static condensation, we end end up solving the following (much smaller) system:whereNote that \({\mathfrak {A}}_1\) and \({\mathfrak {A}}_3\) are proven to be symmetric positive definite matrices [29].

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).

We conclude that both methods show convergence (to the same curve) as the mesh size decreases. Moreover, with a smaller shear modulus (Fig. 6), where the larger degree of swelling appears, differences in edge EF geometry evolution with the decreasing mesh sizes between the two methods becomes obvious. The quantitative results of Figs. 5 and 6 are summarized in Tables 2 and 3. Using the mean of standard FEM solution and MHFEM solution with the finest mesh (\(0.0083\,mm\)) as the canonical solution, two types of errors are presented. \(e_{max}^{FEM/MHFEM}\) denotes the maximum of the absolute value of the difference between the given solution and the canonical solution; and \(e_{avg}^{FEM/MHFEM}\) denotes the average (over the nodes) of the absolute value of the difference between the given solution and the canonical solution.

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.

As it is argued that MHFEM possesses local mass conservation property, which should lead to more accurate calculation of flux, we plot the flux vectors and corresponding streamlines originating from the bottom edge for both methods at a given moment Fig. 7) in order to trace back the deformation differences presented above. Note that in standard FEM chemical potential is approximated by linear node-based interpolation functions, as a result of which the flux vector is element-wise constant. As to MHFEM, fluxes are derived edge-wise. Adding up flux on each edge, the net flux of a given element is derived for MHFEM solution and thus also element-wise constant.

A close check on the streamlines near the low permeability area reveals that the streamlines are not completely symmetric in the FEM solution although geometry, permeability distribution and boundary conditions are strictly symmetric (Fig. 7a). On the other hand, the streamlines in MHFEM solution stay strictly symmetric (Fig. 7b). The difference in streamline distribution leads to visible deformation difference. We notice that elements at the corner of the low permeability area in Fig. 7a exhibit non-physical distortions; while the MHFEM solution (Fig. 7b) presents a much more physical deformation. Such non-physical distortions not only affect the accuracy of deformation calculation but also cause converging issues. In fact, FEM simulation failed to converge to the next step in this simulation while MHFEM succeeded. In our simulation practice, it is observed that FEM is more demanding than MHFEM in terms of choosing modeling and discretization parameters to reach convergence.

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.

In this section, we apply MHFEM to simulate the free swelling of a quarter of a square OABC (Fig. 9). The boundary conditions prescribed for free swelling are given as follows:where \(\mu _{ext}\) denotes the chemical potential in the external solution and equals \(-2RTc^{ex}\); \(\varGamma ^u_D\) represents OA in y-direction and OC in x-direction; \(\varGamma ^u_N\) contains AB and BC; \(\varGamma ^\mu _D\) contains AB and BC; \(\varGamma ^\mu _N\) contains OA and OC. The initial conditions are:where \(\pi _0=-RT\varGamma \sqrt{(c_0^{fc})^2+4(c^{ex})^2}\). The outer solution concentration \(c^{ex}\) is set to be \(4.25\times 10^{-5}\) mol/ml. The domain is discretized by \(20\times 20\) elements and we use an adaptive time step size.

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.

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.

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.