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 displacementpressure formulation using a stable finite element pairing with bubble functions, and another one using a simple pressureprojection 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 dynamics1 Introduction
Locking phenomena, caused by illconditioned 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 piecewiseconstant 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 highresolution 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 coauthors followed an idea by means of variational multiscale (VMS) methods [58, 59, 60, 85], a technique that was recently extended to dynamic problems (DVMS) [66, 71]. Further stabilizations of equal order finite elements include orthogonal subscale 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 multifield 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 wellposedness 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 pressureprojection 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 pressureprojection 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 highlyresolved 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 saddlepoint 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
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)
3 Finite element approximation and stabilization
3.1 Nearly incompressible linear elasticity
3.2 The pressureprojection 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 MINIelements
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 socalled macroelement technique proposed by Stenberg [76].
Theorem 3
 (M1)
for each \(M_i \in \mathcal {E}_j\), \(j=1,\ldots ,q\), the space \(N_\mathrm {M}\) is onedimensional 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 macroelements of \(\mathcal {M}_h\).
 (M4)
each \(E \in \varGamma _h\) is contained in the interior of at least one and not more than L macroelements of \(\mathcal {M}_h\).
Remark 1
Contrary to the twodimensional 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
Concerning wellposedness 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 wellposed in general. However, it remains wellposed for strictly divergencefree 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 indepth discussion we refer the interested reader to [7, 8]. A detailed discussion on Herrmanntype pressure in the nonlinear case is presented in [72, 73].

\(\mu ^*=\mu \) for neoHookean 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
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  – 
For a further analysis regarding computational costs of the MINI element and the pressureprojection 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 Cooktype 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.
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 projectionbased 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 quasiincompressible formulation in [69].
4.4 Twisting column test
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 saddlepoint 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 projectionbased stabilization; \(283\,\hbox {min}\) for tetrahedral meshes and MINI elements; \(449\,\hbox {min}\) for hexahedral meshes and projectionbased 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 lowerright 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 wellsuited for simple singlecore 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 H2020MSCAIF2016 InsiliCardio, GA No. 750835 to CMA. Additionally, the research was supported by the Grants F3210N18 and I2760B30 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 BioTechMedGraz (Grant No. Flagship Project: ILearnHeart).
References
 1.Aguirre M, Gil AJ, Bonet J, Arranz Carreño A (2014) A vertex centred finite volume Jameson–Schmidt–Turkel (JST) algorithm for a mixed conservation formulation in solid dynamics. J Comput Phys 259:672–699MathSciNetzbMATHGoogle Scholar
 2.Alnæs MS, Blechta J, Hake J, Johansson A, Kehlet B, Logg A, Richardson C, Ring J, Rognes ME, Wells G N (2015) The FEniCS Project Version 1.5, Archive of Numerical Software 3.100Google Scholar
 3.Arnold DN, Brezzi F, Fortin M (1984) A stable finite element for the Stokes equations. Calcolo 21:337–344MathSciNetzbMATHGoogle Scholar
 4.Atluri SN, Reissner E (1989) On the formulation of variational theorems involving volume constraints. Comput Mech 5(5):337–344zbMATHGoogle Scholar
 5.Augustin CM, Neic A, Liebmann M, Prassl AJ, Niederer SA, Haase G, Plank G (2016) Anatomically accurate high resolution modeling of cardiac electromechanics: a strongly scalable algebraic multigrid solver method for nonlinear deformation. J Comput Phys 305:622–646MathSciNetzbMATHGoogle Scholar
 6.Auricchio F, Veiga LB, Brezzi F, Lovadina C (2017) Mixed finite element methods. In: Stein E, Borst R, Hughes TJ (eds) Encyclopedia of computational mechanics, 2nd edn. https://doi.org/10.1002/9781119176817.ecm2004
 7.Auricchio F, Da Veiga LB, Lovadina C, Reali A (2010) The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity: mixed FEMs versus NURBSbased approximations. Comput Methods Appl Mech Eng 199(5):314–323MathSciNetzbMATHGoogle Scholar
 8.Auricchio F, da Veiga LB, Lovadina C, Reali A (2005) A stability study of some mixed finite elements for large deformation elasticity problems. Comput Methods Appl Mech Eng 194(9):1075–1092MathSciNetzbMATHGoogle Scholar
 9.Babuška I (1973) The finite element method with Lagrangian multipliers. Numer Math 20:179–192MathSciNetzbMATHGoogle Scholar
 10.Babuška I, Suri M (1992) Locking effects in the finite element approximation of elasticity problems. Numer Math 62(1):439–463MathSciNetzbMATHGoogle Scholar
 11.Bai W (1997) The quadrilateral ‘Mini’ finite element for the Stokes problem. Comput Methods Appl Mech Eng 143(1):41–47MathSciNetzbMATHGoogle Scholar
 12.Balay S, Abhyankar S, Adams M, Brown J, Brune P, Buschelman K, Dalcin L, Dener A, Eijkhout V, Gropp W, Kaushik D, Knepley M, Dave AM, McInnes LC, Mills RT, Munson T, Rupp K, Sanan P, Smith B, Zampini S, Zhang H, Zhang H (2018) PETSc Users Manual, Technical report. ANL95/11  Revision 3.10, Argonne National LaboratoryGoogle Scholar
 13.Ball JM (1976) Convexity conditions and existence theorems in nonlinear elasticity. Arch Ration Mech Anal 63(4):337–403MathSciNetzbMATHGoogle Scholar
 14.Bochev P, Dohrmann C, Gunzburger M (2006) Stabilization of loworder mixed finite elements for the Stokes equations. SIAM J Numer Anal 44(1):82–101MathSciNetzbMATHGoogle Scholar
 15.Boffi D, Brezzi F, Fortin M (2013) Mixed finite element methods and applications. Springer, BerlinzbMATHGoogle Scholar
 16.Boffi D, Stenberg R (2017) A remark on finite element schemes for nearly incompressible elasticity. Comput Math Appl 74(9):2047–2055MathSciNetzbMATHGoogle Scholar
 17.Bonet J, Gil AJ, Ortigosa R (2015) A computational framework for polyconvex large strain elasticity. Comput Methods Appl Mech Eng 283:1061–1094MathSciNetzbMATHGoogle Scholar
 18.Braess D (2007) Finite elements. Cambridge University Press, CambridgezbMATHGoogle Scholar
 19.Brezzi F (1974) On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrangian multipliers. Rech Opér Anal Numér 8:129–151MathSciNetzbMATHGoogle Scholar
 20.Brezzi F, Bristeau MO, Franca LP, Mallet M, Rogé G (1992) A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput Methods Appl Mech Eng 96(1):117–129MathSciNetzbMATHGoogle Scholar
 21.Brink U, Stein E (1996) On some mixed finite element methods for incompressible and nearly incompressible finite elasticity. Comput Mech 19(1):105–119zbMATHGoogle Scholar
 22.Cante J, Dávalos C, Hernández JA, Oliver J, Jonsén P, Gustafsson G, Häggblad HÅ (2014) PFEMbased modeling of industrial granular flows. Comput Part Mech 1(1):47–70Google Scholar
 23.Caylak I, Mahnken R (2012) Stabilization of mixed tetrahedral elements at large deformations. Int J Numer Methods Eng 90(2):218–242MathSciNetzbMATHGoogle Scholar
 24.Cervera M, Chiumenti M, Valverde Q, Agelet de Saracibar C (2003) Mixed linear/linear simplicial elements for incompressible elasticity and plasticity. Comput Methods Appl Mech Eng 192(49–50):5249–5263zbMATHGoogle Scholar
 25.Chamberland É, Fortin A, Fortin M (2010) Comparison of the performance of some finite element discretizations for large deformation elasticity problems. Comput Struct 88(11–12):664–673Google Scholar
 26.Chapelle D, Bathe KJ (1993) The inf–sup test. Comput Struct 47:537–545MathSciNetzbMATHGoogle Scholar
 27.Chiumenti M, Cervera M, Codina R (2015) A mixed threefield FE formulation for stress accurate analysis including the incompressible limit. Comput Methods Appl Mech Eng 283:1095–1116MathSciNetzbMATHGoogle Scholar
 28.Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized\(\alpha \) method. J Appl Mech 60:371MathSciNetzbMATHGoogle Scholar
 29.Ciarlet PG (2002) The finite element method for elliptic problems, vol 40. SIAM, PhiladelphiazbMATHGoogle Scholar
 30.Codina R (2000) Stabilization of incompressibility and convection through orthogonal subscales in finite element methods. Comput Methods Appl Mech Eng 190(13–14):1579–1599MathSciNetzbMATHGoogle Scholar
 31.Deuflhard P (2011) Newton methods for nonlinear problems: affine invariance and adaptive algorithms, vol 35. Springer, BerlinzbMATHGoogle Scholar
 32.Di Pietro DA, Lemaire S (2014) An extension of the Crouzeix–Raviart space to general meshes with application to quasiincompressible linear elasticity and Stokes flow. Math Comput 84(291):1–31MathSciNetzbMATHGoogle Scholar
 33.Dohrmann CR, Bochev PB (2004) A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int J Numer Methods Fluids 46(2):183–201MathSciNetzbMATHGoogle Scholar
 34.Doll S, Schweizerhof K (2000) On the development of volumetric strain energy functions. J Appl Mech 67(1):17zbMATHGoogle Scholar
 35.Elguedj T, Bazilevs Y, Calo V, Hughes T (2008) B and F projection methods for nearly incompressible linear and nonlinear elasticity and plasticity using higherorder NURBS elements. Comput Methods Appl Mech Eng 197(33):2732–2762zbMATHGoogle Scholar
 36.Ern A, Guermond JL (2013) Theory and practice of finite elements, vol 159. Springer, BerlinzbMATHGoogle Scholar
 37.Falk RS (1991) Nonconforming finite element methods for the equations of linear elasticity. Math Comput 57(196):529MathSciNetzbMATHGoogle Scholar
 38.Flory P (1961) Thermodynamic relations for high elastic materials. Trans Faraday Soc 57:829–838MathSciNetGoogle Scholar
 39.Franca LP, Hughes TJR, Loula AFD, Miranda I (1988) A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov–Galerkin finite element formulation. Numer Math 53(1):123–141MathSciNetzbMATHGoogle Scholar
 40.Gil AJ, Lee CH, Bonet J, Aguirre M (2014) A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics. Comput Methods Appl Mech Eng 276:659–690MathSciNetzbMATHGoogle Scholar
 41.Gültekin O, Dal H, Holzapfel GA (2018) On the quasiincompressible finite element analysis of anisotropic hyperelastic materials. Comput Mech 63:443–453MathSciNetzbMATHGoogle Scholar
 42.Hartmann S, Neff P (2003) Polyconvexity of generalized polynomialtype hyperelastic strain energy functions for nearincompressibility. In J Solids Struct 40(11):2767–2791MathSciNetzbMATHGoogle Scholar
 43.Henson VE, Yang UM (2002) BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl Numer Math 41:155–177MathSciNetzbMATHGoogle Scholar
 44.Herrmann LR (1965) Elasticity equations for incompressible and nearly incompressible materials by a variational theorem. AIAA J 3(10):1896–1900MathSciNetGoogle Scholar
 45.Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley, ChichesterzbMATHGoogle Scholar
 46.Hughes TJR (1987) The finite element method, linear static and dynamic finite element analysis. PrenticeHall, Englewood CliffszbMATHGoogle Scholar
 47.Hughes TJR, Franca LP, Balestra M (1986) A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška–Brezzi condition: a stable Petrov–Galerkin formulation of the Stokes problem accommodating equalorder interpolations. Comput Methods Appl Mech Eng 59:85–99zbMATHGoogle Scholar
 48.Hughes TJ, Scovazzi G, Franca LP (2017) Multiscale and stabilized methods. In: Stein E, Borst R, Hughes TJ (eds) Encyclopedia of computational mechanics, 2nd edn. https://doi.org/10.1002/9781119176817.ecm2051
 49.Kabaria H, Lew A, Cockburn B (2015) A hybridizable discontinuous Galerkin formulation for nonlinear elasticity. Comput Methods Appl Mech Eng 283:303–329zbMATHGoogle Scholar
 50.Kadapa C, Dettmer W, Peric D (2017) On the advantages of using the firstorder generalisedalpha scheme for structural dynamic problems. Comput Struct 193:226–238Google Scholar
 51.Karabelas E, Gsell MAF, Augustin CM, Marx L, Neic A, Prassl AJ, Goubergrits L, Kuehne T, Plank G (2018) Towards a computational framework for modeling the impact of aortic coarctations upon left ventricular load. Front Physiol 9(May):1–20Google Scholar
 52.Khan A, Powell CE, Silvester DJ (2019) Robust a posteriori error estimators for mixed approximation of nearly incompressible elasticity. Int J Numer Methods Eng 119(1):18–37MathSciNetGoogle Scholar
 53.Knabner P, Korotov S, Summ G (2003) Conditions for the invertibility of the isoparametric mapping for hexahedral finite elements. Finite Elements Anal Des 40(2):159–172MathSciNetGoogle Scholar
 54.Lafontaine N, Rossi R, Cervera M, Chiumenti M (2015) Explicit mixed strain–displacement finite element for dynamic geometrically nonlinear solid mechanics. Comput Mech 55(3):543–559MathSciNetzbMATHGoogle Scholar
 55.Lamichhane BP (2017) A quadrilateral ‘MINI’ finite element for the Stokes problem using a single bubble function. Int J Numer Anal Model 14(6):869–878MathSciNetzbMATHGoogle Scholar
 56.Lamichhane BP (2009) A mixed finite element method for nonlinear and nearly incompressible elasticity based on biorthogonal systems. Int J Numer Methods Eng 79(7):870–886MathSciNetzbMATHGoogle Scholar
 57.Land S, Gurev V, Arens S, Augustin CM, Baron L, Blake R, Bradley C, Castro S, Crozier A, Favino M, Fastl TE, Fritz T, Gao H, Gizzi A, Griffith BE, Hurtado DE, Krause R, Luo X, Nash MP, Pezzuto S, Plank G, Rossi S, Ruprecht D, Seemann G, Smith NP, Sundnes J, Rice JJ, Trayanova N, Wang D, JennyWang Z, Niederer SA (2015) Verification of cardiac mechanics software: benchmark problems and solutions for testing active and passive material behaviour. Proc R Soc A Math Phys Eng Sci 471(2184):20150641Google Scholar
 58.Masud A, Truster TJ (2013) A framework for residualbased stabilization of incompressible finite elasticity: stabilized formulations and F methods for linear triangles and tetrahedra. Comput Methods Appl Mech Eng 267:359–399MathSciNetzbMATHGoogle Scholar
 59.Masud A, Xia K (2005) A stabilized mixed finite element method for nearly incompressible elasticity. J Appl Mech 72(5):711MathSciNetzbMATHGoogle Scholar
 60.Nakshatrala KB, Masud A, Hjelmstad KD (2007) On finite element formulations for nearly incompressible linear elasticity. Comput Mech 41(4):547–561MathSciNetzbMATHGoogle Scholar
 61.Newmark NM (1959) A method of computation for structural dynamics. J Eng Mech Div 85(3):67–94Google Scholar
 62.Quaglino A, Favino M, Krause R (2017) Quasiquadratic elements for nonlinear compressible and incompressible elasticity. Comput Mech 62:213–231MathSciNetzbMATHGoogle Scholar
 63.Reese S, Wriggers P, Reddy B (2000) A new lockingfree brick element technique for large deformation problems in elasticity. Comput Struct 75(3):291–304Google Scholar
 64.Reese S, Wriggers P, Reddy BD (1998) A new lockingfree brick element formulation for continuous large deformation problems. In: Computational mechanics, new trends and applications, proceedings of the fourth world congress on computational mechanics WCCM IV Buenos Aires, CIMNE (Centro Internacional de Métodes Numéricos in Enginería), BarcelonaGoogle Scholar
 65.Rodriguez JM, Carbonell JM, Cante JC, Oliver J (2016) The particle finite element method (PFEM) in thermomechanical problems. Int J Numer Methods Eng 107(9):733–785MathSciNetzbMATHGoogle Scholar
 66.Rossi S, Abboud N, Scovazzi G (2016) Implicit finite incompressible elastodynamics with linear finite elements: a stabilized method in rate form. Comput Methods Appl Mech Eng 311:208–249MathSciNetGoogle Scholar
 67.Rüter M, Stein E (2000) Analysis, finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity. Comput Methods Appl Mech Eng 190(5–7):519–541MathSciNetzbMATHGoogle Scholar
 68.Schröder J, Viebahn N, Balzani D, Wriggers P (2016) A novel mixed finite element for finite anisotropic elasticity; the SKAelement Simplified Kinematics for Anisotropy. Comput Methods Appl Mech Eng 310:475–494MathSciNetGoogle Scholar
 69.Schröder J, Wriggers P, Balzani D (2011) A new mixed finite element based on different approximations of the minors of deformation tensors. Comput Methods Appl Mech Eng 200(49–52):3583–3600MathSciNetzbMATHGoogle Scholar
 70.Schwab C (1998) p and hpfinite element methods, theory and applications in solid and fluid mechanics, vol 6. Clarendon Press, OxfordzbMATHGoogle Scholar
 71.Scovazzi G, Carnes B, Zeng X, Rossi S (2016) A simple, stable, and accurate linear tetrahedral finite element for transient, nearly, and fully incompressible solid dynamics: a dynamic variational multiscale approach. Int J Numer Methods Eng 106(10):799–839MathSciNetzbMATHGoogle Scholar
 72.Shariff MHBM (1997) An extension of Herrmann’s principle to nonlinear elasticity. Appl Math Model 21(2):97–107zbMATHGoogle Scholar
 73.Shariff MHBM, Parker DF (2000) An extension of Key’s principle to nonlinear elasticity. J Eng Math 37(1):171–190MathSciNetzbMATHGoogle Scholar
 74.Soulaimani A, Fortin M, Ouellet Y, Dhatt G, Bertrand F (1987) Simple continuous pressure elements for two and threedimensional incompressible flows. Comput Methods Appl Mech Eng 62(1):47–69zbMATHGoogle Scholar
 75.Steinbach O (2008) Numerical approximation methods for elliptic boundary value problems. Springer, New YorkzbMATHGoogle Scholar
 76.Stenberg R (1990) Error analysis of some finite element methods for the Stokes problem. Math Comput 54(190):495–508MathSciNetzbMATHGoogle Scholar
 77.Sussman T, Bathe KJ (1987) A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Comput Struct 26(1–2):357–409zbMATHGoogle Scholar
 78.Taylor C, Hood P (1973) A numerical solution of the Navier–Stokes equations using the finite element technique. Comput Fluids 1(1):73–100MathSciNetzbMATHGoogle Scholar
 79.Taylor RL (2000) A mixedenhanced formulation for tetrahedral finite elements. Int J Numer Methods Eng 47(1–3):205–227MathSciNetzbMATHGoogle Scholar
 80.Ten Eyck A, Lew A (2006) Discontinuous Galerkin methods for nonlinear elasticity. Int J Numer Methods Eng 67(9):1204–1243MathSciNetzbMATHGoogle Scholar
 81.Viebahn N, Steeger K, Schröder J (2018) A simple and efficient Hellinger–Reissner type mixed finite element for nearly incompressible elasticity. Comput Methods Appl Mech Eng 340:278–295MathSciNetGoogle Scholar
 82.Vigmond E, Weber dos Santos R, Prassl A, Deo M, Plank G (2008) Solvers for the cardiac bidomain equations. Prog Biophys Mol Biol 96(1):3–18Google Scholar
 83.Weise M (2014) Elastic incompressibility and large deformations: numerical simulation with adaptive mixed FEM. PhD Thesis, Department of Mathematics, Technische Universität ChemnitzGoogle Scholar
 84.Wriggers P (2008) Nonlinear finite element methods. Springer, Berlin, pp 1–559zbMATHGoogle Scholar
 85.Xia K, Masud A (2009) A stabilized finite element formulation for finite deformation elastoplasticity in geomechanics. Comput Geotech 36(3):396–405Google Scholar
 86.Zienkiewicz OC, Rojek J, Taylor RL, Pastor M (1998) Triangles and tetrahedra in explicit dynamic codes for solids. Int J Numer Methods Eng 43(3):565–583MathSciNetzbMATHGoogle Scholar
 87.Zienkiewicz OC, Taylor RL, Taylor RL (2000) The finite element method: solid mechanics, vol 2. ButterworthHeinemann, OxfordzbMATHGoogle Scholar
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