## Three-field mixed finite element methods for nonlinear elasticity.(English)Zbl 07415023

Summary: In this paper, we extend the tangential-displacement normal-normal-stress continuous (TDNNS) method from [A. Pechstein and J. Schöberl, Math. Models Methods Appl. Sci. 21, No. 8, 1761–1782 (2011; Zbl 1237.74187)] to nonlinear elasticity. By means of the Hu-Washizu principle, the distributional derivatives of the displacement vector are lifted to a regular strain tensor. We introduce three different methods, where either the deformation gradient, the Cauchy-Green strain tensor, or both of them are used as independent variables. Within the linear sub-problems, all stress and strain variables can be locally eliminated leading to an equation system in displacement variables, only. The good performance and accuracy of the presented methods are demonstrated by means of several numerical examples.

### MSC:

 74B20 Nonlinear elasticity 74S05 Finite element methods applied to problems in solid mechanics 35Q74 PDEs in connection with mechanics of deformable solids 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Zbl 1237.74187

Netgen; NGSolve
Full Text:

### References:

 [1] Boffi, D.; Brezzi, F.; Fortin, M., Mixed Finite Element Methods and Applications, Vol. 44 (2013), Springer: Springer Berlin, Heidelberg [2] Washizu, K., Variational Methods in Elasticity and Plasticity, Vol. 3 (1975), Pergamon Press: Pergamon Press Oxford · Zbl 0164.26001 [3] Arnold, D. N.; Winther, R., Mixed finite elements for elasticity, Numer. Math., 92, 3, 401-419 (2002) · Zbl 1090.74051 [4] Arnold, D.; Awanou, G.; Winther, R., Finite elements for symmetric tensors in three dimensions, Math. Comp., 77, 263, 1229-1251 (2008) · Zbl 1285.74013 [5] Arnold, D. N.; Brezzi, F.; Douglas, J., PEERS: A new mixed finite element for plane elasticity, Japan J. Appl. Math., 1, 2, 347 (1984) · Zbl 0633.73074 [6] Stenberg, R., A family of mixed finite elements for the elasticity problem, Numer. Math., 53, 5, 513-538 (1988) · Zbl 0632.73063 [7] Arnold, D.; Falk, R.; Winther, R., Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp., 76, 260, 1699-1723 (2007) · Zbl 1118.74046 [8] Pechstein, A.; Schöberl, J., Tangential-displacement and normal-normal-stress continuous mixed finite elements for elasticity, Math. Models Methods Appl. Sci., 21, 8, 1761-1782 (2011) · Zbl 1237.74187 [9] Pechstein, A.; Schöberl, J., Anisotropic mixed finite elements for elasticity, Internat. J. Numer. Methods Engrg., 90, 2, 196-217 (2012) · Zbl 1242.74148 [10] Pechstein, A.; Schöberl, J., An analysis of the TDNNS method using natural norms, Numer. Math., 139, 1, 93-120 (2018) · Zbl 1412.65224 [11] Braess, D., Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory (2007), Cambridge University Press · Zbl 1180.65146 [12] Simo, J. C.; Rifai, M. S., A class of mixed assumed strain methods and the method of incompatible modes, Internat. J. Numer. Methods Engrg., 29, 8, 1595-1638 (1990) · Zbl 0724.73222 [13] Kasper, E. P.; Taylor, R. L., A mixed-enhanced strain method: Part I: Geometrically linear problems, Comput. Struct., 75, 3, 237-250 (2000) [14] Reddy, B. D.; Simo, J. C., Stability and convergence of a class of enhanced strain methods, SIAM J. Numer. Anal., 32, 6, 1705-1728 (1995) · Zbl 0855.73073 [15] Simo, J. C.; Armero, F., Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes, Internat. J. Numer. Methods Engrg., 33, 7, 1413-1449 (1992) · Zbl 0768.73082 [16] Simo, J. C.; Armero, F.; Taylor, R. L., Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems, Comput. Methods Appl. Mech. Engrg., 110, 3, 359-386 (1993) · Zbl 0846.73068 [17] Simo, J. C.; Taylor, R. L.; Pister, K. S., Variational and projection methods for the volume constraint in finite deformation elasto-plasticity, Comput. Methods Appl. Mech. Engrg., 51, 1, 177-208 (1985) · Zbl 0554.73036 [18] Kasper, E. P.; Taylor, R. L., A mixed-enhanced strain method: Part II: Geometrically nonlinear problems, Comput. Struct., 75, 3, 251-260 (2000) [19] Viebahn, N.; Schröder, J.; Wriggers, P., An extension of assumed stress finite elements to a general hyperelastic framework, Adv. Model. Simul. Eng. Sci., 6, 1, 1-22 (2019) [20] Pfefferkorn, R.; Betsch, P., Extension of the enhanced assumed strain method based on the structure of polyconvex strain-energy functions, Internat. J. Numer. Methods Engrg., 121, 8, 1695-1737 (2020) [21] Schröder, J.; Wriggers, P.; Balzani, D., A new mixed finite element based on different approximations of the minors of deformation tensors, Comput. Methods Appl. Mech. Engrg., 200, 49, 3583-3600 (2011) · Zbl 1239.74012 [22] Bonet, J.; Gil, A. J.; Ortigosa, R., A computational framework for polyconvex large strain elasticity, Comput. Methods Appl. Mech. Engrg., 283, 1061-1094 (2015) · Zbl 1423.74122 [23] Bonet, J.; Gil, A. J.; Ortigosa, R., On a tensor cross product based formulation of large strain solid mechanics, Int. J. Solids Struct., 84, 49-63 (2016) [24] Reese, S.; Wriggers, P.; Reddy, B. D., A new locking-free brick element technique for large deformation problems in elasticity, Comput. Struct., 75, 3, 291-304 (2000) [25] Reese, S., A large deformation solid-shell concept based on reduced integration with hourglass stabilization, Internat. J. Numer. Methods Engrg., 69, 8, 1671-1716 (2007) · Zbl 1194.74469 [26] Wulfinghoff, S.; Bayat, H. R.; Alipour, A.; Reese, S., A low-order locking-free hybrid discontinuous Galerkin element formulation for large deformations, Comput. Methods Appl. Mech. Engrg., 323, 353-372 (2017) · Zbl 1439.74476 [27] Bayat, H. R.; Krämer, J.; Wunderlich, L.; Wulfinghoff, S.; Reese, S.; Wohlmuth, B.; Wieners, C., Numerical evaluation of discontinuous and nonconforming finite element methods in nonlinear solid mechanics, Comput. Mech., 62, 6, 1413-1427 (2018) · Zbl 1462.74147 [28] Reese, S.; Bayat, H. R.; Wulfinghoff, S., On an equivalence between a discontinuous Galerkin method and reduced integration with hourglass stabilization for finite elasticity, Comput. Methods Appl. Mech. Engrg., 325, 175-197 (2017) · Zbl 1439.74052 [29] Angoshtari, A.; Shojaei, M. F.; Yavari, A., Compatible-strain mixed finite element methods for 2D compressible nonlinear elasticity, Comput. Methods Appl. Mech. Engrg., 313, 596-631 (2017) · Zbl 1439.74383 [30] Shojaei, M. F.; Yavari, A., Compatible-strain mixed finite element methods for incompressible nonlinear elasticity, J. Comput. Phys., 361, 247-279 (2018) · Zbl 1390.74177 [31] Shojaei, M. F.; Yavari, A., Compatible-strain mixed finite element methods for 3D compressible and incompressible nonlinear elasticity, Comput. Methods Appl. Mech. Engrg., 357, Article 112610 pp. (2019) · Zbl 1442.74034 [32] Angoshtari, A.; Yavari, A., Hilbert complexes of nonlinear elasticity, Z. Angew. Math. Phys., 67, 6, 143 (2016) · Zbl 1358.35185 [33] Beirão da Veiga, L.; Lovadina, C.; Mora, D., A virtual element method for elastic and inelastic problems on polytope meshes, Comput. Methods Appl. Mech. Engrg., 295, 327-346 (2015) · Zbl 1423.74120 [34] Artioli, E.; Da Veiga, L. B.; Lovadina, C.; Sacco, E., Arbitrary order 2D virtual elements for polygonal meshes: Part II, inelastic problem, Comput. Mech., 60, 4, 643-657 (2017) · Zbl 1386.74133 [35] Chi, H.; da Veiga, L. B.; Paulino, G., Some basic formulations of the virtual element method (VEM) for finite deformations, Comput. Methods Appl. Mech. Engrg., 318, 148-192 (2017) · Zbl 1439.74397 [36] Wriggers, P.; Reddy, B. D.; Rust, W.; Hudobivnik, B., Efficient virtual element formulations for compressible and incompressible finite deformations, Comput. Mech., 60, 2, 253-268 (2017) · Zbl 1386.74146 [37] Sinwel, A., A New Family of Mixed Finite Elements for Elasticity (2009), Johannes Kepler University Linz, https://www.numa.uni-linz.ac.at/Teaching/PhD/Finished/sinwel-diss.pdf [38] Pechstein, A. S., Large deformation mixed finite elements for smart structures, Mech. Adv. Mater. Struct., 27, 23, 1983-1993 (2020) [39] Regge, T., General relativity without coordinates, Il Nuovo Cimento (1955-1965), 19, 3, 558-571 (1961) [40] Cheeger, J.; Müller, W.; Schrader, R., Kinematic and tube formulas for piecewise linear spaces, Indiana Univ. Math. J., 35, 4, 737-754 (1986) · Zbl 0615.53058 [41] Christiansen, S. H., On the linearization of Regge calculus, Numer. Math., 119, 4, 613-640 (2011) · Zbl 1269.83022 [42] Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2000), Dover Publications Inc.: Dover Publications Inc. Mineola, New York · Zbl 1191.74002 [43] Neunteufel, M., Mixed Finite Element Methods for Nonlinear Continuum Mechanics and Shells (2021), TU Wien, (Ph.D. thesis) [44] Monk, P., (Finite Element Methods for Maxwell’s Equations. Finite Element Methods for Maxwell’s Equations, Numerical Mathematics and Scientific Computation (2003), Oxford University Press: Oxford University Press New York), xiv+450, 2059447 · Zbl 1024.78009 [45] Cockburn, B.; Gopalakrishnan, J.; Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47, 2, 1319-1365 (2009) · Zbl 1205.65312 [46] Zienkiewicz, O.; Taylor, R., The Finite Element Method. Vol. 1: The Basis (2000), Butterworth-Heinemann · Zbl 0991.74002 [47] Nédélec, J. C., A new family of mixed finite elements in $$\mathbb{R}^3$$, Numer. Math., 50, 1, 57-81 (1986) · Zbl 0625.65107 [48] Raviart, P.-A.; Thomas, J.-M., A mixed finite element method for 2-nd order elliptic problems, (Mathematical Aspects of Finite Element Methods, Vol. 66 (1977), Springer), 292-315 [49] Brezzi, F.; Douglas, J.; Marini, L. D., Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47, 2, 217-235 (1985) · Zbl 0599.65072 [50] Meindlhumer, M.; Pechstein, A., 3D mixed finite elements for curved, flat piezoelectric structures, Int. J. Smart Nano Mater., 10, 4, 249-267 (2019) [51] Li, L., Regge Finite Elements with Applications in Solid Mechanics and Relativity (2018), University of Minnesota, http://hdl.handle.net/11299/199048 [52] Zaglmayr, S., High Order Finite Element Methods for Electromagnetic Field Computation (2006), Johannes Kepler Universität Linz, https://www.numerik.math.tugraz.at/ zaglmayr/pub/szthesis.pdf [53] Schöberl, J., NETGEN an advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Sci., 1, 1, 41-52 (1997) · Zbl 0883.68130 [54] Schöberl, J., C++ 11 Implementation of Finite Elements in NGSolve (2014), Institute for Analysis and Scientific Computing, Vienna University of Technology, https://www.asc.tuwien.ac.at/ schoeberl/wiki/publications/ngs-cpp11.pdf [55] Reese, S., On the equivalent of mixed element formulations and the concept of reduced integration in large deformation problems, Int. J. Nonlinear Sci. Numer. Simul., 3, 1, 1-34 (2002) · Zbl 1079.74057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.