# Versatile stabilized finite element formulations for nearly and fully incompressible solid mechanics

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## Abstract

Computational formulations for large strain, polyconvex, nearly incompressible elasticity have been extensively studied, but research on enhancing solution schemes that offer better tradeoffs between accuracy, robustness, and computational efficiency remains to be highly relevant. In this paper, we present two methods to overcome locking phenomena, one based on a displacement-pressure formulation using a stable finite element pairing with bubble functions, and another one using a simple pressure-projection stabilized \(\mathbb {P}_1 - \mathbb {P}_1\) finite element pair. A key advantage is the versatility of the proposed methods: with minor adjustments they are applicable to all kinds of finite elements and generalize easily to transient dynamics. The proposed methods are compared to and verified with standard benchmarks previously reported in the literature. Benchmark results demonstrate that both approaches provide a robust and computationally efficient way of simulating nearly and fully incompressible materials.

## Keywords

Incompressible elasticity Large strain elasticity Mixed finite elements Piecewise linear interpolation Transient dynamics## 1 Introduction

Locking phenomena, caused by ill-conditioned global stiffness matrices in finite element analyses, are an often observed and extensively studied issue when modeling nearly incompressible, hyperelastic materials [10, 18, 46, 84, 87]. Typically, methods based on Lagrange multipliers are applied to enforce incompressibility. A common approach is the split of the deformation gradient into a volumetric and an isochoric part [38]. Here, locking commonly arises when unstable standard displacement formulations are used that rely on linear shape functions to approximate the displacement field \({\MakeLowercase {\mathbf {u}}}\) and piecewise-constant finite elements combined with static condensation of the hydrostatic pressure \(p\), e.g., \(\mathbb {P}_1 - \mathbb {P}_0\) elements. It is well known that in such cases solution algorithms may exhibit very low convergence rates and that variables of interest such as stresses can be inaccurate [41].

From mathematical theory it is well known that approximation spaces for the primal variable \({\MakeLowercase {\mathbf {u}}}\) and \(p\) have to be well chosen to fulfill the Ladyzhenskaya–Babuŝka–Brezzi (LBB) or *inf–sup* condition [9, 19, 26] to guarantee stability. A classical stable approximation pair is the Taylor–Hood element [78], however, this requires quadratic ansatz functions for the displacement part. For certain types of problems higher order interpolations can improve efficiency as higher accuracy is already reached with coarser discretizations [25, 57]. In many applications though, where geometries are fitted to, e.g., capture fine structural features, this is not beneficial due to a possible increase in degrees of freedom and consequently a higher computational burden. Also for coupled problems such as electromechanical or fluid–structure–interaction models high-resolution grids for mechanical problems are sometimes required when interpolations between grids are not desired [5, 51]. As a remedy for these kind of applications quasi Taylor–Hood elements with an order of \(\tfrac{3}{2}\) have been considered, see [62], as well as equal order linear pairs of ansatz functions which has been a field of intensive research in the last decades, see [6, 48] and references therein. Unfortunately, equal order pairings do not fulfill the LBB conditions and hence a stabilization of the element is of crucial importance. There is a significant body of literature devoted to stabilized finite elements for the Stokes and Navier–Stokes equations. Many of those methods were extended to incompressible elasticity, amongst other approaches by Hughes, Franca, Balestra, and collaborators [39, 47]. Masud and co-authors followed an idea by means of variational multiscale (VMS) methods [58, 59, 60, 85], a technique that was recently extended to dynamic problems (D-VMS) [66, 71]. Further stabilizations of equal order finite elements include orthogonal sub-scale methods [24, 27, 30, 54] and methods based on pressure projections [33, 86]. Different classes of methods to avoid locking for nearly incompressible elasticity were conceived by introducing nonconforming finite elements such as the Crouzeix–Raviart element [32, 37] and Discontinuous Galerkin methods [49, 80]. Enhanced strain formulations [64, 79] have been considered as well as formulations based on multi-field variational principles [17, 68, 69].

In this study we introduce a novel variant of the MINI element for accurately solving nearly and fully incompressible elasticity problems. The MINI element was originally established for computational fluid dynamics problems [3] and pure tetrahedral meshes and previously used in the large strain regime, e.g. in [25, 56]. We extend the MINI element definition for hexahedral meshes by introducing two bubble functions in the element and provide a novel proof of stability and well-posedness in the case of linear elasticity. The support of the bubble functions is restricted to the element and can thus be eliminated from the system using static condensation. This also allows for a straightforward inclusion in combination with existing finite element codes since all required implementations are purely on the element level. Additionally, we introduce a pressure-projection stabilization method originally published for the Stokes equations [14, 33] and previously used for large strain nearly incompressible elasticity in the field of particle finite element methods and plasticity [22, 65]. Due to its *simplicity*, this type of stabilization is especially attractive from an implementation point of view.

Robustness and performance of both the MINI element and the pressure-projection approach are verified and compared to standard benchmarks reported previously in literature. A key advantage of the proposed methods is their *high versatility*: first, they are readily applicable to nearly and fully incompressible solid mechanics; second, with little adjustments the stabilization techniques can be applied to all kinds of finite elements, in this study we investigate the performance for hexahedral and tetrahedral meshes; and third, the methods generalize easily to transient dynamics.

Real world applications often require highly-resolved meshes and thus efficient and massively parallel solution algorithms for the linearized system of equations become an important factor to deal with the resulting computational load. We solve the arising saddle-point systems by using a GMRES method with a block preconditioner based on an algebraic multigrid (AMG) approach. Extending our previous implementations [5] we performed the numerical simulations with the software *Cardiac Arrhythmia Research Package* (CARP) [82] which relies on the MPI based library *PETSc* [12] and the incorporated solver suite *hypre/BoomerAMG* [43]. The combination of these advanced solving algorithms with the proposed stable elements which only rely on linear shape functions proves to be *very efficient* and renders feasible simulations on grids with high structural detail.

The paper is outlined as follows: Sect. 2 summarizes in brief the background on the methods. In Sect. 3, we introduce the finite element discretization and discuss stability. Subsequently, Sect. 4 documents benchmark problems where our proposed elements are applied and compared to results published in the literature. Finally, Sect. 5 concludes the paper with a discussion of the results and a brief summary.

## 2 Continuum mechanics

### 2.1 Nearly incompressible nonlinear elasticity

*deformation gradient*\(\varvec{F}\) and the Jacobian

*J*as

*left Cauchy–Green tensor*as \(\varvec{C} := \varvec{F}^\top \varvec{F}\). Here, \({\text {Grad}}\,(\bullet )\) denotes the gradient with respect to the reference coordinates \(\underline{X} \in \varOmega _0\). The displacement field \({\MakeLowercase {\mathbf {u}}}\) is sought as infimizer of the functional

*bulk modulus*. The function \(U(J)\) acts as a penalization of incompressibility and we require that it is strictly convex and twice continuously differentiable. Additionally, a constitutive model for \(U(J)\) should fulfill that (i) it vanishes in the reference configuration and that (ii) an infinite amount of energy is required to shrink the body to a point or to expand it indefinitely, i.e.,

### 2.2 Consistent linearization

### 2.3 Review on solvability of the linearized problem

- (i)The
*inf–sup condition*: there exists \(c_1 > 0\) such that$$\begin{aligned} \underset{q \in Q}{\inf } \ \underset{{\MakeLowercase {\mathbf {v}}} \in V_{{\MakeLowercase {\mathbf {0}}}}}{\sup } \frac{b_k(q,{\MakeLowercase {\mathbf {v}}})}{{\left||{{\MakeLowercase {\mathbf {v}}}}\right||}_{V_{{\MakeLowercase {\mathbf {0}}}}} {\left||{q}\right||}_{Q}} \ge c_1. \end{aligned}$$(24) - (ii)The
*coercivity on the kernel condition*: there exists \(c_2 > 0\) such thatwhere$$\begin{aligned} a_k({\MakeLowercase {\mathbf {v}}}, {\MakeLowercase {\mathbf {v}}}) \ge c_2 {\left||{{\MakeLowercase {\mathbf {v}}}}\right||}_{V_{{\MakeLowercase {\mathbf {0}}}}}^2&\text {for all } {\MakeLowercase {\mathbf {v}}} \in \ker B, \end{aligned}$$(25)$$\begin{aligned} \ker B := \left\{ {\MakeLowercase {\mathbf {v}}} \in V_{{\MakeLowercase {\mathbf {0}}}} : b_k(q, {\MakeLowercase {\mathbf {v}}}) = 0 ~\text {for all } q \in Q \right\} . \end{aligned}$$ - (iii)
*Positivity of*\(c\): it holds$$\begin{aligned} c(q,q) \ge 0 \ \text {for all }q\in Q. \end{aligned}$$(26)

*inf–sup*condition from standard arguments, see [83, Section 5.2]. The positivity of the bilinear form

*c*is always fulfilled. However, it is not possible to show the coercivity condition (25) for a general hyperelastic material or load configuration. Nevertheless, for some special cases it is possible to establish a result. We refer to [7, 8, 83] for a more detailed discussion. Henceforth, we will assume that our given input data is such that we stay in the range of stability of the problem. Examples for cases in which bilinear form \(a_k\) lacks coercivity can be found in [83, Chapter 9] and [7, Section 4].

## 3 Finite element approximation and stabilization

### 3.1 Nearly incompressible linear elasticity

*discrete inf–sup condition*, reading

*inf–sup*condition puts constraints on the choice of the spaces \(V_{h,0}\) and \(Q_h\). A finite element pairing fulfilling (35) is called a

*stable pair*. A classic example for tetrahedral meshes would be the Taylor–Hood element. In this paper, we will focus on two different finite element pairings, the MINI element and a stabilized equal order element. The stabilized equal order pairing has been used in this context for pure tetrahedral meshes, see [22, 65]. To the best of the authors knowledge those elements have not been used in the present context for general tesselations.

### 3.2 The pressure-projection stabilized equal order pair

In the following, we present a stabilized lowest equal order finite element pairing, adapted to nonlinear elasticity from the pairing originally introduced by Dohrmann and Bochev [14, 33] for the Stokes equations.

### Theorem 1

There exists a unique bounded solution to the discrete problem (36).

### Theorem 2

### Proof

Due to the similarity of the linear elasticity and the Stokes problem the proof follows from [14, Theorem 4.1, Theorem 5.1 and Corollary 5.2]. \(\square \)

### 3.3 Discretization with MINI-elements

#### 3.3.1 Tetrahedral elements

#### 3.3.2 Hexahedral meshes

In the literature mostly two dimensional quadrilateral tessellations, see for example [11, 15, 55], were considered for MINI element discretizations. In this case, the proof of stability relies on the so-called *macro-element technique* proposed by Stenberg [76].

*M*is a connected set of elements in \(\mathcal {T}_h\). Moreover, two macro-elements \(M_1\) and \(M_2\) are said to be equivalent if and only if they can be mapped continuously onto each other. Additionally, for a macro element

*M*we define the spaces

### Theorem 3

*L*, and a macro-element partition \(\mathcal {M}_h\) such that

- (M1)
for each \(M_i \in \mathcal {E}_j\), \(j=1,\ldots ,q\), the space \(N_\mathrm {M}\) is one-dimensional consisting of functions that are constant on

*M*; - (M2)
each \(M \in \mathcal {M}_h\) belongs to one of the classes \(\mathcal {E}_i\), \(i=1,2,\ldots ,q\);

- (M3)
each \(K \in \mathcal {T}_h\) is contained in at least one and not more than

*L*macro-elements of \(\mathcal {M}_h\). - (M4)
each \(E \in \varGamma _h\) is contained in the interior of at least one and not more than

*L*macro-elements of \(\mathcal {M}_h\).

*Mathematica*\(^{\mathrm {TM}}\) and further analyzed. We can conclude that the rank of \(\varvec{D}\) is 26 and thus (M1) holds and we can apply Theorem 3. A

*Mathematica*\(^{\mathrm {TM}}\) notebook containing computations discussed in this section is available upon request.

### Remark 1

Contrary to the two-dimensional case studied in [11, 55] it is not sufficient to enrich the standard isoparametric finite element space for hexahedrons with only one bubble function. In this case both the spaces \(\varvec{V}_{0,\mathrm {M}_i}\) and \(P_{\mathrm {M}_i}\) have a dimension of 27, however, matrix \(\varvec{D}\) has only rank 24.

### Remark 2

Although not mentioned explicity, the stability of the MINI element holds also for mixed discretizations.

### 3.4 Changes and limitations in the nonlinear case

*p*as remarked in [16]. Consider, as an example, the strain energy function for a nearly incompressible neo-Hookean material where

*Herrmann pressure*[44], above formulation (44) and (45) uses the so-called

*hydrostatic pressure*.

*inf–suf*condition for this linear problem remains the same as for (31) and (32). For the extension of the

*inf–suf*condition to the nonlinear case we already stated earlier in Eq. (27) that

Concerning well-posedness of (44)–(45), it was noted in [16], that the coercivity on the kernel condition (25) does not hold in general, which makes the formulation with hydrostatic pressure not well-posed in general. However, it remains well-posed for strictly divergence-free finite elements or pure Dirichlet boundary conditions. This has also been observed by other authors, see [52, 81]. Even if the coercivity on the kernel condition can be shown for the hydrostatic, nearly incompressible linear elastic case this result may not transfer to the nonlinear case. Here, this condition is highly dependent on the chosen nonlinear material law and for the presented benchmark examples (Sect. 4) we did not observe any numerical instabilities.

For an in-depth discussion we refer the interested reader to [7, 8]. A detailed discussion on Herrmann-type pressure in the nonlinear case is presented in [72, 73].

*Mathematica*\(^{\mathrm {TM}}\) notebook containing the computations discussed is available upon request.

\(\mu ^*=\mu \) for neo-Hookean materials and

\(\mu ^*=c_1\) for Mooney–Rivlin materials

The considerable advantage of the MINI element is that there are no modifications needed and that no additional stabilization parameters are introduced into the system.

### 3.5 Changes and limitations in the transient case

## 4 Numerical examples

^{1}package from the library

*PETSc*[12] and the incorporated solver suite

*hypre/BoomerAMG*[43]. By extending our previous work [5] we implemented the methods in the finite element code

*Cardiac Arrhythmia Research Package*(CARP) [82].

### 4.1 Analytic solution

Properties of *cube meshes* used in Sect. 4.2

Hexahedral meshes | Tetrahedral meshes | ||||
---|---|---|---|---|---|

\(\ell \) | Elements | Nodes | \(\ell \) | Elements | Nodes |

1 | 512 | 729 | 1 | 3072 | 729 |

2 | 4096 | 4913 | 2 | 24,576 | 4913 |

3 | 32,768 | 35,937 | 3 | 196,608 | 35,937 |

4 | 262,144 | 274,625 | 4 | 1,572,864 | 274,625 |

5 | 2,097,152 | 2,146,689 | 5 | 12,582,912 | 2,146,689 |

### 4.2 Block under compression

*Block under compression:* comparison of computational times for different discretizations. Timings were obtained using (a) 48 cores and (b) 192 cores on ARCHER, UK. Coarser grids, see Table 1, are used for Taylor–Hood elements \(\mathbb {P}_2- \mathbb {P}_1\) to compare computational times for a similar number of degrees of freedom (DOF)

Discretization | Grid | DOF (Mio.) | Tet. (s) | Hex. (s) |
---|---|---|---|---|

(a) | ||||

Projection | \(\ell =4\) | 1.098 | 330 | 438 |

MINI | \(\ell =4\) | 1.098 | 873 | 655 |

\(\mathbb {P}_2- \mathbb {P}_1\) | \(\ell =3\) | 0.860 | 1202 | – |

(b) | ||||

Projection | \(\ell =5\) | 8.587 | 2488 | 2192 |

MINI | \(\ell =5\) | 8.587 | 3505 | 4640 |

\(\mathbb {P}_2 - \mathbb {P}_1\) | \(\ell =4\) | 6.715 | 27,154 | – |

**A**of the quarter of the cube, is plotted in Fig. 7. Small discrepancies can be attributed to differences in the meshes for tetrahedral and hexahedral grids, however, the different stabilization techniques yield almost the same results for finer grids. Note, that the displacements at the edge point

**A**obtained using the simple \(\mathbb {Q}_1 - \mathbb {P}_0\) hexahedral and \(\mathbb {P}_1 - \mathbb {P}_0\) tetrahedral elements seem to be in a similar range compared to the other approaches. The overall displacement field, however, was totally inaccurate rendering \(\mathbb {Q}_1 - \mathbb {P}_0\) and \(\mathbb {P}_1 - \mathbb {P}_0\) elements an inadequate choice for this benchmark problem. The solution for Taylor–Hood (\(\mathbb {P}_2 - \mathbb {P}_1\)) tetrahedral elements was obtained using the FEniCS project [2]. Here, as a linear solver, we used a GMRES solver with preconditioning similar to the MINI and projection-based approach, see first paragraph of Sect. 4. The PCFIELDSPLIT and

*hypre/BoomerAMG*settings were slightly adapted to optimize computational performance for quadratic ansatz functions. We comparing simulations with about the same number of degrees of freedom, not accuracy as, e.g., in [25]. For coarser grids computational times were in the same time range for all approaches; see, e.g., the cases with approximately \(10^6\) degrees of freedom and target pressure of \(p_0=320\,\hbox {mmHg}\) in Table 2(a). For the simulations with the finest grids with approximately \(10^7\) degrees of freedom, however, we could not find a setting for the Taylor–Hood elements that was competitive to MINI and pressure-projection stabilizations. The computational times to arrive at the target pressure of \(p_0=320\,\hbox {mmHg}\) using 192 cores on ARCHER, UK were about 10 times higher for Taylor–Hood elements using FEniCS, see Table 2(b). We attribute that to a higher communication load and higher memory requirements of the Taylor–Hood elements: memory to store the block stiffness matrices was approximately 2.5 times higher for Taylor–Hood elements compared to MINI and projection-stabilization approaches (measured using the MatGetInfo

^{2}function provided by PETSc). Note, that although we used the same linear solvers, the time comparisons are not totally just as results were obtained using two different finite element solvers, CARP and FEniCS. Note also, that timings are usually very problem dependent and for this block under compression benchmark high accuracy was already achieved with coarse grids for hexahedral and Taylor–Hood discretizations.

For a further analysis regarding computational costs of the MINI element and the pressure-projection stabilization, see Sect. 4.4.

Properties of *cantilever meshes* used in Sect. 4.3

Hexahedral meshes | Tetrahedral meshes | ||||
---|---|---|---|---|---|

\(\ell \) | Elements | Nodes | \(\ell \) | Elements | Nodes |

1 | 324 | 500 | 1 | 1944 | 500 |

2 | 2592 | 3249 | 2 | 15,552 | 3249 |

3 | 20,736 | 23,273 | 3 | 124,416 | 23,273 |

4 | 165,888 | 175,857 | 4 | 995,328 | 175,857 |

### 4.3 Cook-type cantilever problem

Displacements \(u_x\), \(u_y\), and \(u_z\) at point \(\mathbf {C}\) are shown in Fig. 10. The proposed stabilization techniques give comparable displacements in all three directions and also match results published in [17, 69]. Mesh convergence can also be observed for the stresses \(\sigma _{xx}\) at point \(\mathbf {A}\) and \(\mathbf {B}\) and \(\sigma _{yy}\) at point \(\mathbf {B}\), see Fig. 11. Again, results match well those presented in [17, 69]. Small discrepancies can be attributed to the fully incompressible formulation used in our work and differences in grid construction.

In Figs. 12 and 13a the distribution of \(J=\det (\varvec{F})\) is shown to provide an estimate of how accurately the incompressibility constraint is fulfilled by the proposed stabilization techniques. For most parts of the computational domains the values of *J* are close to 1, however, hexahedral meshes and here in particular the MINI element maintain the condition of \(J \approx 1\) more accurately on the element level. Note, that for all discretizations the overall volume of the cantilever remained unchanged at \(14{,}400\,\hbox {mm}^3\), rendering the material fully incompressible on the domain level.

*J*in Fig. 13a. A similar checkerboard pattern is present for the projection based stabilization.

In the third row of Fig. 13 we compare the stresses \(\sigma _{xx}\) for the different stabilization techniques. We can observe slight oscillations for the the projection-based approach, whereas the MINI element gives a smoother solution. Compared to results in [69, Figure 10] the \(\sigma _{xx}\) stresses have a similar contour but are slightly higher. As before, we attribute that to the fully incompressible formulation in our paper compared to the quasi-incompressible formulation in [69].

### 4.4 Twisting column test

*z*-axes by an angle of \(\theta =5.2{^{\circ }}\).

**D**is analyzed. While differences at lower levels of refinement \(\ell =1,2\) are severe, the displacements converge for higher levels of refinement \(\ell =3,4,5\). For finer grids the curves for tetrahedral and hexahedral elements are almost indistinguishable, see also Fig. 16, and the results are in good agreement with those presented in [71]. While this figure was produced using MINI elements we also observed a similar behavior of mesh convergence for the projection-based stabilization. In fact, for the finest grid, all the proposed stabilization techniques and elements gave virtually identical results, see Fig. 16. Further, as already observed by Scovazzi et al. [71], the fully and nearly incompressible formulations gave almost identical deformations, see Fig. 17.

Properties of *column meshes* used in Sect. 4.4

Hexahedral meshes | Tetrahedral meshes | ||||
---|---|---|---|---|---|

\(\ell \) | Elements | Nodes | \(\ell \) | Elements | Nodes |

1 | 48 | 117 | 1 | 240 | 117 |

2 | 384 | 625 | 2 | 1920 | 625 |

3 | 3072 | 3969 | 3 | 15,360 | 3969 |

4 | 24,576 | 28,033 | 4 | 122,880 | 28,033 |

5 | 196,608 | 210,177 | 5 | 983,040 | 210,177 |

In Fig. 18 stress \(\sigma _{yy}\) and pressure *p* contours are plotted on the deformed configuration for the incompressible case at time instant \(t=0.3\,\hbox {s}\). Minor pressure oscillations can be observed for tetrahedral elements. Again, results match well those presented in [71, Figure 22].

Finally, in Fig. 19, we compare the magnitude of velocity and acceleration at time instant \(t=0.3\,\hbox {s}\). Results for these variables are very smooth and hardly distinguishable for all the different approaches.

The *computational costs* for this nonlinear elasticity problem were significant due to the required solution of a saddle-point problem in each Newton step and a large number of time steps. However, this challenge can be addressed by using a massively parallel iterative solving method and exploiting potential of modern HPC hardware. The most expensive simulations were the fully incompressible cases for the finest grids with a total of 840,708 degrees of freedom and 400 time steps. These computations were executed at the national HPC computing facility ARCHER in the United Kingdom using 96 cores. Computational times were as follows: \(239\,\hbox {min}\) for tetrahedral meshes and projection-based stabilization; \(283\,\hbox {min}\) for tetrahedral meshes and MINI elements; \(449\,\hbox {min}\) for hexahedral meshes and projection-based stabilization; and \(752.5\,\hbox {min}\) for hexahedral meshes and MINI elements. Simulation times for nearly incompressible problems were lower, ranging from 177 to \(492\,\hbox {min}\). This is due to the additional matrix on the lower-right side of the block stiffness matrix which led to a smaller number of linear iterations. Simulations with hexahedral meshes were, in general, computationally more expensive compared to simulations with tetrahedral grids; the reason being mainly a higher number of linear iterations. Computational burden for MINI elements was larger due to higher matrix assembly times. However, this assembly time is highly scalable as there is almost no communication cost involved in this process.

## 5 Conclusion

In this study we described methodology for modeling nearly and fully incompressible solid mechanics for a large variety of different scenarios. A stable MINI element was presented which can serve as an excellent choice for applied problems where the use of higher order element types is not desired, e.g., due to fitting accuracy of the problem domain. We also proposed an easily implementable and computationally cheap technique based on a local pressure projection. Both approaches can be applied to stationary as well as transient problems without modifications and perform excellent with both hexahedral and tetrahedral grids. Both approaches allow a straightforward inclusion in combination with existing finite element codes since all required implementations are purely on the element level and are well-suited for simple single-core simulations as well as HPC computing. Numerical results demonstrate the robustness of the formulations, exhibiting a great accuracy for selected benchmark problems from the literature.

While the proposed projection method works well for relatively stiff materials as considered in this paper, the setting of the parameter \(\mu ^*\) has to be adjusted for soft materials such as biological tissues. A further limitation is that both formulations render the need of solving a block system, which is computationally more demanding and suitable preconditioning is not trivial. However, the MINI element approach can be used without further tweaking of artificial stabilization coefficients and preliminary results suggested robustness, even for very soft materials. Consistent linearization as presented ensures that quadratic convergence of the Newton–Raphson algorithm was achieved for all the problems considered. Note that all computations for forming the tangent matrices and also the right hand side residual vectors are kept local to each element. This benefits scaling properties of parallel codes and also enables seamless implementation in standard finite element software.

The excellent performance of the methods along with their high versatility ensure that this framework serves as a solid platform for simulating nearly and fully incompressible phenomena in stationary and transient solid mechanics. In future studies, we plan to extend the formulation to anisotropic materials with stiff fibers as they appear for example in the simulation of cardiac tissue and arterial walls.

## Footnotes

## Notes

### Acknowledgements

Open access funding provided by Medical University of Graz. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska–Curie Action H2020-MSCA-IF-2016 InsiliCardio, GA No. 750835 to CMA. Additionally, the research was supported by the Grants F3210-N18 and I2760-B30 from the Austrian Science Fund (FWF), and a BioTechMed award to GP. We acknowledge PRACE for awarding us access to resource ARCHER based in the UK at EPCC. This study was supported by BioTechMed-Graz (Grant No. Flagship Project: ILearnHeart).

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