Comparing mixed hybrid finite element method with standard FEM in swelling simulations involving extremely large deformations

Hydrogels swell by the permeation of solvent (water) molecules into polymer networks. Due to the existence of cross-links in the polymer networks, water permeation leads to the expansion and rearrangement of polymer chains as a whole. There are various external stimuli that induce the swelling of a hydrogel: temperature, pH value and ionic strength in the outer solution [39]. The responsiveness and bio-compatibility of hydrogels make them popular materials for diverse applications including carriers for drug delivery, tissue engineering matrices [21], actuator and sensors [2].

Over the years, many works are dedicated to the development of theoretical frameworks that enable the coupling between fluid permeation and solid deformation. Following the linear and finite poroelasticity approach established by Biot [4, 5], Hong et al. [23] proposed a theoretical framework that enables finite deformations with combination of Flory theory [19] to model the swelling of a polymeric gel. Many more works have been done to extend this framework to include features like non-Gaussian constitutive relations for solid [12], viscoelasticity [11, 43] and temperature-sensitivity [13, 16]. Besides the thermodynamic approach, which treats the gel as one single phase, mixture theory [8], which allows the study of multiple phases (fluid, solid, ions and other substances) at the same time, gains popularity in the biomechanical studies. A biphasic swelling model is proposed by Lanir [31] assuming the existence of fixed charge groups and the instantaneous equilibrium of the ion phase. Later, Lai et al. [29] developed a triphasic theory in which fluid, solid and ion phases were considered separately. A further separation of cation and anion phases leads to the quadriphasic theory [25].

Development in the theoretical frameworks fostered many numerical investigations into the swelling process. Hong et al. [22] and Kuang and Huang [27] simulated the inhomogeneous swelling of a gel at equilibrium state. Later on, transient swelling simulations are the topic of many studies [7, 14, 32, 34, 40, 46]. We note that all the studies listed above adopted standard FEM in which deformation and chemical potential were chosen to be prime variables implemented in commercial softwares like ABAQUS or COMSOL. Böger et al. [6] proposed an alternative variational formulation which treated deformation and flux as prime variables. On the other hand, studies [3, 23] on the solvent molecules migration kinetics suggest that the mobility tensor (the coefficient tensor in the Darcy’s equation) is not homogeneous during transient swelling. Basically, the part of a gel that gets in touch with the external solvent earlier increases dramatically its hydraulic permeability. This inhomogeneity in the mobility tensor may require special attention in terms of numerical treatment. As pointed out by Kaasschieter and Huijben [26] and Mosé et al. [36], in a Darcy flow problem with heterogeneous domain, the standard FEM suffers poor accuracy in computing flux field due to the violation of local mass conservation. In contrast, a mixed formulation that treats flux and pressure as separate independent variables greatly improves the accuracy of the flux field as mass conservation is preserved locally and globally. As a matter of fact, using proper numerical methods to solve Darcy flow problem is a well-established topic in porous media studies [15, 33]. Inspired by the success of mixed formulations in Darcy flow problems, a mixed hybrid finite element method (MHFEM) was implemented to simulate the transient swelling of a hydrogel in one of our previous works [45].

In this work, we aim to evaluate the effectiveness of MHFEM comparing with standard FEM in terms of robustness, accuracy and efficiency the solution method when applied to simulate the finite transient swelling of hydrogels. As the theoretical basis of the MHFEM is elaborated in [45], in this work, the focus lies on comparing and furthermore demonstrating the advantages of MHFEM over standard FEM in swelling simulations involving large deformations. Results and discussions of numerical examples form the key part of this work. Depending on the hydraulic permeability laws we apply, our numerical examples fall into three categories: homogeneous permeability (the mobility tensor is constant), heterogeneous permeability (the mobility tensor is a strain-dependent, thus non-uniform but change continuous during transient swelling) and heterogeneous with interface of discontinuous permeability (mobility tensor is thus discontinuous). The effectiveness of MHFEM is evaluated for all three categories. The paper is organized as follows. In Sect. 2, we recap the swelling mechanism and the resulting governing equations. In Sect. 3 and 4, the weak form and discretization details of standard FEM and MHFEM are described respectively. Section 5 presents the comparison results and discussions of four selected numerical examples. At last, conclusions are drawn in Sect. 6.

We limit ourselves to one specific type of hydrogels: superabsorbent polymers (SAP) or in other words, partially neutralized sodium polyacrylate hydrogels. A schematic illustration of its structure is given in Fig. 1. Once the particle gets in touch with the outer solution, the water interacts (1) with the hydrophillic polychains and (2) with the solium ions electrostatically linked to those polychains. The latter interaction is represented by the ionic part of the free energy while the former is represented by the mixing part of the free energy. Moreover, it has been suggested that in the case of monovalent solvent, the contribution from the mixing energy can be ignored [24].

We make the following assumptions in the swelling model. The gel is placed in isothermal condition and responsive only to the ion concentration change in the outer solution due to Donnan osmosis. Both the external and the solid skeleton are assumed to be incompressible and the gel as a whole is assumed to be hyperelastic. The ion equilibrium in and outside of the gel is established instantly, as a consequent of which the biphasic swelling model [31] applies.

The swelling model consists of two governing equations (linear momentum balance and (fluid) mass conservation) plus constitutive relation (a Darcy type equation which relates flux and chemical potential gradient). Since deformation of the solid field is of our interest and is in the finite deformation regime, the equations are given in Lagrangian form. First of all, the linear momentum balance in Lagrangian form is given as:where \({{\varvec{T}}}\) is the first Piola-Kirchoff stress tensor. It is related to Cauchy stress tensor \({\varvec{\sigma }}\) by the relation: \({{\varvec{T}}}=J{\varvec{\sigma }}{\varvec{F}}^{-T}\), where J is the volume ratio \(J=\det ({\varvec{F}})\) and \({\varvec{F}}\) is the deformation tensor. Cauchy stress in a hydrogel following the convention in poroelasticity is given by:where \({\varvec{\sigma }}_{eff}\) denotes the effective stress and p stands for the hydraulic pressure in the gel. We further specify the concrete form of \({\varvec{\sigma }}_{eff}\) following the work of Wilson et al. [44]. Although the solid skeleton and fluid phase are both assumed to be incompressible, the gel experiences large volume change by imbibing external fluid. Therefore, the material law of the gel is assumed to be compressible Neo-Hookean. Besides, we make the assumption that Poisson ratio \(\nu \) is proportional to the solid volume fraction \(\phi _s\), \(\nu =0.5\phi _{s}\). This assumption ensures that when there is no fluid (\(\phi _s=1\)), the incompressibility of the solid skeleton is recovered. Following the derivation presented in [44], the final form of \({\varvec{\sigma }}_{eff}\) is given by:where G is shear modulus, which indicates the softness of the gel and \(\phi _{s,0}\) denotes the initial solid volume fraction and related to \(\phi _s\) by \(\phi _s=\phi _{s,0}/J\). The other part of Cauchy stress, p is derived by the relation [3]:where \(\mu \) denotes the chemical potential and \(\varPi \) is the osmotic pressure due to Donnan osmosis. Assuming biphasic swelling model [31], the osmotic pressure is related to fixed charge density \(c^{fc}\) as:where R and T are universal gas constant and absolute temperature respectively; \({\bar{c}}\) is the external ion concentration. The current fixed charge density \(c^{fc}\) is related to the initial fixed charge density \(c^{fc}_0\) by the relation: \(c^{fc}=c^{fc}_0\phi _{f,0}/(J-\phi _{s,0})\) [44].

The second equation represents mass balance of the fluid phase:where D/Dt denotes the material time derivative of a given material field variable. \(\varPhi _f\) denotes the nominal porosity and related to the porosity at the current configuration \(\phi _f\) by \(\varPhi _f=J\phi _f\). \({\varvec{Q}}\) denotes the flux vector in Lagrangian form. Due to the incompressibility conditions, the conservation of the solid phase can be written as:Therefore, Eq. (6) can be rewritten as:At last, we relate the flux Q with the chemical potential \(\mu \) with a Darcy’s type equation:where \({\varvec{K}}\) is the Lagrangian form of the intrinsic permeability k. They are related to each other by \(\varvec{K}=J{\varvec{F}}^{-1}{\varvec{F}}^{-T}k/\eta \) [35], where \(\eta \) is the viscosity of the fluid. The intrinsic permeability k is assumed to be constant, strain-dependent or discontinuous in this work. The specific values of k and strain-dependent forms of k are discussed in Sect. 5. Unlike the original Darcy’s law, Eq. (9) features the chemical potential instead of pressure. It has two contributions: hydraulic pressure p and osmotic pressure \(\varPi \), as given in Eq. (4).

Standard finite element formulation is by far the most widely used formulation in swelling simulations. Commercial software such as ABAQUS or COMSOL is used in combination with user-defined subroutines. Regardless of the swelling mechanisms and constitutive relations of the gel, flux field \({\varvec{Q}}\) in the mass conservation equation (8) is substituted using Darcy type equation (9) in this formulation, yielding a system of equations of \(({\varvec{x}},\mu )\):Due to geometric and material nonlinearity in the material law for the gel, Newton–Raphson procedure is carried out. Therefore, instead of solving \(({\varvec{x}}, \mu )\) directly, its linear increment \((\delta {\varvec{x}},\delta \mu )\) is solved from the linearized version of Eqs. (10) and (11). Let \(\varGamma _D^x\) and \(\varGamma _D^\mu \) denote the Dirichlet boundary for \({\varvec{x}}\) and \(\mu \). We define the function space \(H_D^1(\varOmega )\):The linearized weak form of Eqs. (10) and (11) is thus derived: find \((\delta {\varvec{x}},\delta \mu )\in ((H_D^1(\varOmega ))^2,H_D^1(\varOmega ))\), such that for any \((\bar{{\varvec{x}}}.{\bar{\mu }})\in ((H_D^1(\varOmega ))^2,H_D^1(\varOmega ))\),withwhere \({\varvec{S}}\) is the second Piola-Kirchoff stress tensor; \({\mathbb {C}}\) is the fourth-order elasticity tensor; \(\bar{{\varvec{E}}}\) and \(\delta {\varvec{E}}\) are the Green strain tensor based on \(\bar{{\varvec{x}}}\) and \(\delta {\varvec{x}}\). Specifically, we have:The finite element method is introduced by a partition \({\mathcal {T}}_h=\{\varOmega ^e\}\) of the initial domain \(\varOmega \). A natural choice for the discretized sub-spaces for \(H_0^1(\varOmega )\) iswhere \(P^1(\varOmega ^e)\) denotes constant or linear polynomial on \(\varOmega ^e\). Let \(\varphi _i\) be the basis interpolation functions in \(P_0^1({\mathcal {T}}_h)\), we have:where i runs over all the nodes introduced by the spatial discretization and \({\varvec{x}}_i\) and \(\mu _i\) denotes the value of the position and chemical potential at node i.

To avoid any instability due to time discretization, we choose backward Euler scheme for time discretization. Let subscript \(n/n-1\) denote the variable at time step \(n/n-1\) and \(\varDelta t\) the time step size, the fully discretized version of Eqs. (13) and (14) is:For each time step, flux field \({\varvec{Q}}\) is computed by numerical differentiation of the chemical potential field. As a result, the fluid mass balance is only satisfied in a weak sense. At the element level, this leads to the violation of mass conservation.

To circumvent the violation of local mass conservation, a mixed formulation is proposed. In the mixed formulation, flux field and the chemical potential are solved simultaneously. Raviart–Thomas elements [37] guarantees the continuity of flux in the normal direction between neighboring elements; therefore, mass conservation is preserved locally and globally. Further, to implement Raviart–Thomas elements, a Lagrange multiplier \(\lambda \) is introduced [1]. Finally, a static condensation technique is applied which reduces the number of degree of freedom from four to two. To begin with, the system of equations of \((\varvec{x},\mu ,{\varvec{Q}})\) is given by:We introduce the function space \(H_0(\text {div},\varOmega )\):where \(\varGamma _N^\mu \) denotes the Neumann boundary for the chemical potential. Like in the standard FEM, the linearized weak form of Eqs. (27), (28) and (29) is derived: find \((\delta {\varvec{x}},\delta \mu , \delta \varvec{Q})\in ((H_D^1(\varOmega ))^2,L^2(\varOmega ),H_0(\text {div},\varOmega ))\),for any \((\bar{{\varvec{x}}},{\bar{\mu }},\bar{{\varvec{Q}}})\in ((H_D^1(\varOmega ))^2,L^2(\varOmega ),H_0(\text {div},\varOmega ))\). The form of a, b and \(r^1\) has been given in Eqs. (15), (16) and (18). Here we give the specific form of \(d,e,r^3\) and \(r^4\):where \({\tilde{\mu }}\) is the prescribed value for chemical potential at the Dirichlet boundary condition, \({\varvec{n}}\) is the outward normal vector at \(\varGamma _N^\mu \). The spatial discretization spaces are chosen to be \((\delta {\varvec{x}}_h,\delta \mu _h,\delta {\varvec{Q}}_h)\in (P_0^1({\mathcal {T}}_h), M_{-1}^0({\mathcal {T}}_h), RT_0^0({\mathcal {T}}_h))\), whereNote that the combination of approximation spaces \(M_{-1}^0({\mathcal {T}}_h)\) and \(RT_0^0({\mathcal {T}}_h)\) are proven to satisfy the LBB condition and are thus a stable combination [9]. However, to facilitate the implementation, we first consider a bigger function space \(RT_{-1}^0({\mathcal {T}}_h)=\{{\varvec{v}}\in L^2(\varOmega ):\varvec{v}\vert _{\varOmega ^e}\in RT^0({\mathcal {T}}_h)\}\) and later limit it to the desired space \(RT_{0}^0({\mathcal {T}}_h)\) by introducing a Lagrange multiplier. This way, we do not need to design certain procedure to make sure the orientation of the edges are compatible. The interpolation of the independent variables is given by:where i, k and j loop over each node, element and edge respectively. Again, we use backward Euler as time discretization scheme for stability consideration. The procedure of introducing the Lagrange multiplier and static condensation is known as hybridization technique. Its relevant details can be found in [45]. The result of hybridization is the simple and efficient implementation of lowest-order Raviart–Thomas elements with a reduction of degrees of freedom. In the end, the system of linear equations we solve is:The chemical potential and flux field can be reconstructed from \({\varvec{x}}\) and \(\lambda \) without loss of accuracy.

In this section, four numerical examples solved by standard FEM as well as MHFEM are presented. Depending on different permeability distributions over the domain, the four examples can be categorized into three cases. In the first example, we consider constant permeability k over the whole domain. The second example treats permeability k to be strain-dependent and continuously varying its value over the domain as soon as the swelling starts. Example 3 and 4 deal with permeability with discontinuous jumps. Specifically, the swelling is confined to one direction in the third example and a low permeability stripe is added to the geometry. Example 4 allows swelling in two dimensions and represents a soft high-permeable core surrounded partly by a stiff low-permeable shell structure.

For all four examples, comprehensive comparisons are carried out between the two numerical methods in terms of and robustness, accuracy and computational cost. First, we look into the solution convergence behavior of both methods. In other words, the robustness of both methods for a given range of material and simulation parameters is investigated. Next, we examine the accuracy of the deformation field by means of comparing mesh convergence rates (provided that solution convergence is achieved). In addition, we present the contour plots of the chemical potential with streamlines on top to gain good insights on the accuracy of the chemical potential and flux field calculation at transient states. At last, computational cost is summarized in tables with focus on the computing time and the number of degree of freedoms. All examples are two dimensional with the standard set of parameters given in Table 1 (note the very low shear modulus of 15 kPa [17]). The simulation time span covers the whole swelling process (0 s to \(10^{6}\) s) and is divided into 48 adaptive steps with smaller time step size at the initial stage and much larger step size near equilibrium. All the simulations are done using MATLAB 2016b on a windows machine with Intel Core 17-6700 CPU and 64GB RAM.

In this work, we first reviewed the Donnan osmosis based swelling mechanism. Then two computation formulations was introduced in detail. The standard FE formulation uses deformation \({\varvec{x}}\) and chemical potential \(\mu \) as prime variables. The flux \({\varvec{Q}}\) is calculated by numerical differentiation. On the other hand, MHFEM adopts mixed formulation in which \({\varvec{Q}}\) and \(\mu \) are considered as independent variables. Lowest order Raviart–Thoams element was used in the mixed formulation, which guarantees the conservation of mass locally. Such a property has proven to be important in the calculation of flux field solving Darcy’s type equations with heterogeneous permeability [26, 36]. By examining four numerical examples, we evaluate the importance of using MHFEM in swelling simulations involving extremely large deformations.

The four numerical examples falls into three categories (depending on the permeability distributions): constant permeability, strain-dependent permeability and permeability with discontinuous jumps. For each example, aspects such as robustness in solution convergence, accuracy (mesh convergence rate) and computational cost are investigated.

To summarize, the local mass conservation property of MHFEM has been proven to improve the accuracy of the solution in swelling simulations when there is discontinuities in permeability present. Moreover, MHFEM largely outperforms FEM in terms of solution convergence robustness.