Abstract
This paper investigates the capacity of solid finite elements with independent interpolations for displacements and strains to address shear, membrane and volumetric locking in the analysis of beam, plate and shell structures. The performance of the proposed strain/displacement formulation is compared to the standard one through a set of eleven benchmark problems. In addition to the relative performance of both finite element formulations, the paper studies the effect of discretization and material characteristics. The first refers to different solid element typologies (hexahedra, prisms) and shapes (regular, skewed, warped configurations). The second refers to isotropic, orthotropic and layered materials, and nearly incompressible states. For the analysis of nearly incompressible cases, the B-bar method is employed in both standard and strain/displacement formulations. Numerical results show the enhanced accuracy of the proposed strain/displacement formulation in predicting stresses and displacements, as well as producing locking-free discrete solutions, which converge asymptotically to the corresponding continuous problems.
Similar content being viewed by others
References
Prathap G (1985) The poor bending response of the four-node plane stress quadrilateral. Int J Numer Methods Eng 21(5):825–835. https://doi.org/10.1002/nme.1620210505
Bathe KJ (1996) Finite element procedures. Englewood Cliffs, New Jersey
Crisfield MA, Tassoulas JL (1993) Non-Linear finite element analysis of solids and structures, Volume 1. J Eng Mech 119(7):1504–1505. https://doi.org/10.1061/(asce)0733-9399(1993)119:7(1504)
Heyman J (1966) The stone skeleton. Int J Solids Struct 2(2):249–279. https://doi.org/10.1016/0020-7683(66)90018-7
Tralli A, Alessandri C, Milani G (2014) Computational methods for masonry vaults: a review of recent results. Open Civ Eng J 8:272–287
Feizolahbeigi A, Lourenço PB, Golabchi M, Ortega J, Rezazadeh M (2021) Discussion of the role of geometry, proportion and construction techniques in the seismic behavior of 16th to 18th century bulbous discontinuous double shell domes in central Iran. J Build Eng 33:101575. https://doi.org/10.1016/j.jobe.2020.101575
Zienkiewicz OC, Taylor RL, Too JM (1971) Reduced integration technique in general analysis of plates and shells. Int J Numer Methods Eng 3:275–290. https://doi.org/10.1002/nme.1620030211
Stolarski H, Belytschko T (1983) Shear and membrane locking in curved C0 elements. Comput Methods Appl Mech Eng 41(3):279–296. https://doi.org/10.1016/0045-7825(83)90010-5
Belytschko T, Stolarski H, Liu WK, Carpenter N, Ong JS (1985) Stress projection for membrane and shear locking in shell finite elements. Comput Methods Appl Mech Eng 51(1–3):221–258. https://doi.org/10.1016/0045-7825(85)90035-0
Pitkäranta J (1992) The problem of membrane locking in finite element analysis of cylindrical shells. Numerische Mathematik 61(1):523–542. https://doi.org/10.1007/BF01385524
Wriggers P, Eberlein R, Reese S (1996) A comparison of three-dimensional continuum and shell elements for finite plasticity. Int J Solids Struct 33:3309–3326. https://doi.org/10.1016/0020-7683(95)00262-6
Korelc J, Wriggers P (1996) An efficient 3D enhanced strain element with Taylor expansion of the shape functions. Comput Mech 19(2):30–40. https://doi.org/10.1007/bf02757781
Wriggers P, Korelc J (1996) On enhanced strain methods for small and finite deformations of solids. Comput Mech 18(6):413–428. https://doi.org/10.1007/BF00350250
Hauptmann R, Schweizerhof K (1998) A systematic development of ‘solid-shell’ element formulations for linear and non-linear analyses employing only displacement degrees of freedom. Int J Numer Methods Eng 42(1):49–69. https://doi.org/10.1002/(SICI)1097-0207(19980515)42:1<49::AID-NME349>3.0.CO;2-2
Hauptmann R, Doll S, Harnau M, Schweizerhof K (2001) ‘Solid-shell’ elements with linear and quadratic shape functions at large deformations with nearly incompressible materials. Comput Struct 79(18):1671–1685. https://doi.org/10.1016/S0045-7949(01)00103-1
Sze KY, Yao LQ, Yi S (2000) A hybrid stress ANS solid-shell element and its generalization for smart structure modelling. Part II - Smart structure modelling, International Journal for Numerical Methods in Engineering 48(4):565–582. https://doi.org/10.1002/(SICI)1097-0207(20000610)48:4$<$565::AID-NME890$>$3.0.CO;2-U
Simo JC, Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29(8):1595–1638. https://doi.org/10.1002/nme.1620290802
Simo JC, Armero F (1992) Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int J Numer Methods Eng 33(7):1413–1449. https://doi.org/10.1002/nme.1620330705
Simo JC, Armero F, Taylor RL (1993) Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems. Comput Methods Appl Mech Eng 110(3–4):359–386. https://doi.org/10.1016/0045-7825(93)90215-J
Kasper EP, Taylor RL (2000) Mixed-enhanced strain method. Part I: geometrically linear problems. Comput Struct 75(3):237–250. https://doi.org/10.1016/S0045-7949(99)00134-0
Kim KD, Liu GZ, Han SC (2005) A resultant 8-node solid-shell element for geometrically nonlinear analysis. Comput Mech 35(5):315–331. https://doi.org/10.1007/s00466-004-0606-9
Schwarze M Reese (2009) A reduced integration solid-shell finite element based on the EAS and the ANS concept-Geometrically linear problems. Int J Numer Methods Eng 8:1322–1355. https://doi.org/10.1002/nme
Huang J, Cen S, Li Z, Li CF (2018) An unsymmetric 8-node hexahedral solid-shell element with high distortion tolerance: Linear formulations. Int J Numer Methods Eng 116(12–13):759–783. https://doi.org/10.1002/nme.5945
Reese S, Wriggers P, Reddy BD (2000) A new locking-free brick element technique for large deformation problems in elasticity. Comput Struct 75(3):291–304. https://doi.org/10.1016/S0045-7949(99)00137-6
Areias PM, de Sé JM, António CA (2003) Analysis of 3D problems using a new enhanced strain hexahedral element. Int J Numer Methods Eng 58(11):1637–1682. https://doi.org/10.1002/nme.835
Wriggers P, Eberlein R, Reese S (1996) Continuum Shell Elements Finite Plasticity 33(20):3309–3326. https://doi.org/10.1016/0020-7683(95)00262-6
Vlachakis G, Cervera M, Barbat GB, Saloustros S (2019) Out-of-plane seismic response and failure mechanism of masonry structures using finite elements with enhanced strain accuracy. Eng Failure Anal 97:534–555. https://doi.org/10.1016/J.ENGFAILANAL.2019.01.017
Malkus DS, Hughes TJ (1978) Mixed finite element methods—reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng 15(1):63–81. https://doi.org/10.1016/0045-7825(78)90005-1
Babuška I, Melenk JM (1997) The partition of unity method. International Journal for Numerical Methods in Engineering 40(4):727–758. https://doi.org/10.1002/(SICI)1097-0207(19970228)40:4$<$727::AID-NME86$>$3.0.CO;2-N
Boffi D, Brezzi F, Fortin M (2013) Mixed finite element methods and applications. In: Series in Computational Mathematics (vol 44). Springer, Heidelberg
Lafontaine NM, Rossi R, Cervera M, Chiumenti M (2015) Explicit mixed strain-displacement finite element for dynamic geometrically non-linear solid mechanics. Comput Mech 55(3):543–559. https://doi.org/10.1007/s00466-015-1121-x
Nagtegaal JC, Parks DM, Rice JR (1974) On numerically accurate finite element solutions in the fully plastic range. Comput Methods Appl Mech Eng 4(2):153–177. https://doi.org/10.1016/0045-7825(74)90032-2
Sloan SW, Randolph MF (1982) Numerical prediction of collapse loads using finite element methods. Intl J Numer Anal Methods Geomech 6(1):47–76. https://doi.org/10.1002/nag.1610060105
Hughes TJ (1980) Generalization of selective integration procedures to anisotropic and nonlinear media. Int J Numer Methods Eng 15(9):1413–1418. https://doi.org/10.1002/nme.1620150914
Brezzi F (1974) On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev Fr Autom Inf Rech Oper 8:129–151. https://doi.org/10.1051/m2an/197408R201291
Arnold DN, Winther R (2002) Mixed finite elements for elasticity. Numerische Mathematik 92(3):401–419. https://doi.org/10.1007/s002110100348
Mijuca D (2004) On hexahedral finite element HC8/27 in elasticity. Comput Mech 33(6):466–480. https://doi.org/10.1007/s00466-003-0546-9
Arnold DN, Awanou G, Winther R (2008) Finite elements for symmetric tensors in three dimensions. Math Comput. https://doi.org/10.1090/s0025-5718-08-02071-1
Cervera M, Lafontaine N, Rossi R, Chiumenti M (2016) Explicit mixed strain-displacement finite elements for compressible and quasi-incompressible elasticity and plasticity. Comput Mech 58(3):511–532. https://doi.org/10.1007/s00466-016-1305-z
Cervera M, Barbat GB, Chiumenti M (2017) Finite element modeling of quasi-brittle cracks in 2D and 3D with enhanced strain accuracy. Comput Mech 60(5):767–796. https://doi.org/10.1007/s00466-017-1438-8
Macneal RH, Harder RL (1985) A proposed standard set of problems to test finite element accuracy. Finite Elements Anal Design 1(1):3–20. https://doi.org/10.1016/0168-874X(85)90003-4
Lo SH, Ling C (2000) Improvement on the 10-node tetrahedral element for three-dimensional problems. Comput Methods Appl Mech Eng 189(3):961–974. https://doi.org/10.1016/S0045-7825(99)00410-7
White DW, Abel JF (1989) Testing of shell finite element accuracy and robustness. Finite Elements Anal Design 6(2):129–151. https://doi.org/10.1016/0168-874X(89)90040-1
Dvorkin EN, Bathe KJ (1984) A continuum mechanics based four-node shell element for general nonlinear analysis. Eng Comput 1(1):77–88. https://doi.org/10.1108/eb023562
Büchter N, Ramm E, Roehl D (1994) Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept. Int J Numer Methods Eng 37(15):2551–2568. https://doi.org/10.1002/nme.1620371504
Nguyen P, Doškár M, Pakravan A, Krysl P (2018) Modification of the quadratic 10-node tetrahedron for thin structures and stiff materials under large-strain hyperelastic deformation. Int J Numer Methods Eng 114(6):619–636. https://doi.org/10.1002/nme.5757
Scordelis A, Lo K Computer Analysis of Cylindrical Shells. ACI J Proc. https://doi.org/10.14359/7796
Heyman J (1966) The stone skeleton. Int J Solids Struct 2(2):249–279. https://doi.org/10.1016/0020-7683(66)90018-7
Izzuddin BA, Liang Y (2020) A hierarchic optimisation approach towards locking-free shell finite elements. Comput Struct 232:105839. https://doi.org/10.1016/j.compstruc.2017.08.010
Klinkel S, Gruttmann F, Wagner W (2006) A robust non-linear solid shell element based on a mixed variational formulation. Comput Methods Appl Mech Eng 195(1–3):179–201. https://doi.org/10.1016/j.cma.2005.01.013
Reese S (2007) A large deformation solid-shell concept based on reduced integration with hourglass stabilization. Int J Numer Methods Eng 69:1671–1716. https://doi.org/10.1002/nme.1827
Lindberg G, Olson M, Copwer G (1969) New developments in the finite element analysis of shells. Quart Bull Div Mech Eng Natl Aeronaut Establish 4:1–38
Flügge W (1973) Stresses in shells. Springer, Berlin. https://doi.org/10.1007/978-3-662-01028-0
Reese S (2012) A large deformation solid-shell concept based on reduced integration with hourglass stabilization. Int J Nu 69:1971–1716. https://doi.org/10.1002/nme.3279/full
Hughes TJ, Tezduyar TE (1981) Finite elements based upon mindlin plate theory with particular reference to the four-node bilinear isoparametric element. J Appl Mech Trans ASME 48(3):587–596. https://doi.org/10.1115/1.3157679
Piltner R, Joseph DS (2001) An accurate low order plate bending element with thickness change and enhanced strains. Comput Mech 27(5):353–359. https://doi.org/10.1007/s004660100247
Bathe KJ, Iosilevich A, Chapelle D (2000) Inf-sup test for shell finite elements. Comput Struct 75(5):439–456. https://doi.org/10.1016/S0045-7949(99)00213-8
Chama A, Reddy BD (2013) New stable mixed finite element approximations for problems in linear elasticity. Comput Methods Appl Mech Eng 256:211–223. https://doi.org/10.1016/j.cma.2012.12.006
Simo JC, Oliver J, Armero F (1993) An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput Mech 12(5):277–296. https://doi.org/10.1007/BF00372173
Timoshenko SP, Goodier JN, Abramson HN, Theory of Elasticity (3rd ed.), Journal of Applied Mechanics. https://doi.org/10.1115/1.3408648
Codina R (2000) Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods. Comput Methods Appl Mech Eng 190(13–14):1579–1599. https://doi.org/10.1016/S0045-7825(00)00254-1
Hughes TJ, 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 equal-order interpolations. Comput Methods Appl Mech Eng 59(1):85–99. https://doi.org/10.1016/0045-7825(86)90025-3
Hughes TJ, Feijóo GR, Mazzei L, Quincy JB (1998) The variational multiscale method—a paradigm for computational mechanics. Comput Methods Appl Mech Eng 166(1–2):3–24. https://doi.org/10.1016/S0045-7825(98)00079-6
Badia S, Codina R (2009) Unified stabilized finite element formulations for the stokes and the darcy problems. SIAM J Numer Anal 47(3):1971–2000. https://doi.org/10.1137/08072632x
Cervera M, Chiumenti M, Codina R (2010) Mixed stabilized finite element methods in nonlinear solid mechanics. Part I: formulation. Comput Methods Appl Mech Eng 199(37–40):2559–2570. https://doi.org/10.1016/j.cma.2010.04.006
de Sousa R J Alves, Cardoso RP, Valente R A Fontes, Yoon JW, Grácio JJ, Jorge R M Natal (2005) A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness: Part I - Geometrically linear applications. Int J Numer Methods Eng 62(7):952–977. https://doi.org/10.1002/nme.1226
Acknowledgements
The authors gratefully acknowledge the financial support from the Ministry of Science, Innovation and Universities (MCIU) via: the ADaMANT project (Computational Framework for Additive Manufacturing of Titanium Alloy, Proyectos de I+D -Excelencia-, ref. num. DPI2017-85998-P); the SEVERUS project (Multilevel evaluation of seismic vulnerability and risk mitigation of masonry buildings in resilient historical urban centres, ref. num. RTI2018-099589-B-I00); and the Severo Ochoa Programme for Centres of Excellence in R&D (CEX2018-000797-S). Sungchul Kim gratefully acknowledges the support received from the Agència de Gestió d’Ajuts Universitaris i de Recerca (AGAUR) and the European Social Fund (ESF) through the predoctoral FI grants (ref. num. 2019FI_B00727).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Vectors and Matrices
Displacements \( \varvec{u} \), strains \(\varvec{\varepsilon } \), stresses \(\varvec{\sigma }\) and forces \(\varvec{f}\) are represented following Voigt’s notation as vectors
The differential symmetric gradient operator relating the displacements with the strains has the following form
The projection matrix, introduced in Eq. (7), is
where \( \varvec{n}=\left( n_x,n_y, n_z\right) ^T\) is the outward normal vector at the boundary of the analysed domain \( \varGamma _t \).
The discrete strain-displacement matrix (or discrete symmetric gradient operator) is expressed as
for \( 1 \le i \le n_n \), with \( n_n \) being the number of nodes in the element. The submatrix \( \varvec{B}_{u_{i}} \) and its volumetric part \( \varvec{B}_{u_{i}}^{vol} \) are expressed in Voigt’s notation as
where \( N_i \) is the shape function of node i and \( \partial N_{i,j} \) is its derivative with respect to the jth Cartesian coordinate (\( j=[1:3] \)). The deviatoric part is obtained by
Principle of virtual Work
This Appendix presents the derivation of equation (7) from equation (5) in two steps. First, Eq. (5) is premultiplied by an arbitrary virtual displacement \(\delta \varvec{u}\) and integrated over the spatial domain \(\varOmega \)
Then, the Divergence Theorem is applied on the first term of the above equation yielding
In the previous derivation, Eq. (3) is used on the integral over \( \varGamma \) and adopted the assumption that the prescribed displacements vanish on the boundary \( \varGamma _u \). Finally, substituting Eq. (46) into Eq. (45) the final version of the Principle of Virtual Work for the mixed \( \varepsilon /u \) formulation is obtained
presented in equation (7).
Variational multiscale stabilization method
This section presents the stabilization procedure leading to the final system of Eq. (24) of the \( \varepsilon /u \) indepedent interpolation formulation. The stabilisation procedure adopted herein consists in the modification of the discrete variational form using the Orthogonal Subscales Method, introduced in [61] within the framework of the Variational Multiscale Stabilization methods [62, 63].
The stabilization of the problem is achieved by substituting the approximated strains in Eq. (9) with the following form
where \( \tau _\varepsilon =\left[ 0,1\right] \) is a stabilization parameter. Observe that for \( \tau _\varepsilon = 0\) the stabilization effect is lost, while for \( \tau _\varepsilon = 1\) the strain interpolation of the standard irreducible formulation is recovered
The use of equation (48) in equations (6)-(7) gives the final stabilized set of equations for the mixed \( \varepsilon /u \) FE formulation
Residual-based stabilisation procedures, like the one in (48) used herein, do not introduce any additional approximation nor any consistency error. For this, the stabilisation technique is variationally consistent, meaning that converging values of the unknowns \( \varvec{\varepsilon }\) ad \( \varvec{u}\) satisfying the Galerkin system (16)–(17) also satisfy the stabilized form (50)–(51). In particular, considering a converged solution, when the size of the element h tends to zero, \( h \rightarrow 0 \), \( \varvec{\varepsilon }\rightarrow \varvec{N}_\varepsilon \varvec{E} = \varvec{B}_u \varvec{U}\) and the stabilization term vanishes. Considering a non-converged situation, the added terms \( \tau _\varepsilon (\varvec{B}_u\varvec{U}-\varvec{N}_\varepsilon \varvec{E}) \) are small, as they depend on the difference between two approximations of different order to the same quantity. This means that for a given FE mesh, using different values of the stabilization procedure yields slightly different results (see Appendix D). Nevertheless, the consistency of the residual-based stabilization guarantees that the discrete problem converges to the unique solution. The use of different stabilization parameters on the same mesh is analogous to the use of different FE interpolations of the same order of convergence with the same nodal arrangement.
As shown in [64, 65], the optimal convergence rate in linear problems is obtained reducing the stabilization on mesh refinement, such that
where \( c_\varepsilon \) stands for a positive number of the order \( c_\varepsilon =O(1)\), h for the finite element size and \( L_0 \) is the characteristic size of the problem.
Following the above, the stabilized system of equations becomes
with
Influence of parameter \(\tau _\varepsilon \)
This Appendix investigates the influence of the stabilization parameter \(\tau _\varepsilon \) in the numerical results obtained with the \(\varepsilon /u\) FE formulation. The parameter \(\tau _\varepsilon \) is defined in all the studied cases through the equation (52), in which intervenes the parameter c aside with the parameters h and \(L_0\), associated with the finite element size and the characteristic size of the problem, respectively. Here, we investigate the influence of parameter c, with regard to the case of the clamped square plate.
Figure 47 presents the results obtained using three different values of \(c = 5; 1; 1/5\) in equation (52). A value of \(c=1\) corresponds to the reference value used for this case \(\tau _{\varepsilon ,ref}=h/L_0\). The results show that the convergence rate is very similar for all the selected values of \(\tau _\varepsilon \), as analytically predicted [64, 65]. The fact that using different values for c (i.e. different \(\tau _\varepsilon \)) produces different approximate solutions can be seen as similar to getting different approximate solutions by using meshes with different layouts, as already mentioned in Appendix C. Nevertheless, convergence to the solution, and optimal rate of convergence, are independent from the choice of parameter c.
As can be observed, for the same mesh, the use of a higher value of \(\tau _\varepsilon \) results in an increase of the estimated error. This is to be expected, as for the limit value of \( \tau _\varepsilon = 1\) the standard irreducible formulation is recovered. On the other end, very small values of \(\tau _\varepsilon \) fail to effectively stabilize the \(\varepsilon /u\) formulation.
Comparison with solid-shell and EAS FEs
This Appendix presents a comparison between the numerical results of the standard displacement based linear hexahedron (referred in the tables as Q1), the proposed \(\varepsilon /u\) FEs (referred in the tables as Q1Q1) and the reported results of several successful solid-shell and EAS elements for three benchmark shell problems: the Scordelis-Lo roof (Table 5), the hemispherical shell (Table 6) and the pinched cylinder (Table 7).
The following solid-shell and EAS elements are considered:
-
Wriggers and Koralc QS/E9 [13]: 3D solid-shell enhanced strain element with 9 enhanced modes based on Taylor expansion with exact symbolic integration.
-
Wriggers and Koralc QS/E12 [13]: 3D solid-shell enhanced strain element with 12 enhanced modes based on Taylor expansion with exact symbolic integration.
-
Reese [51]: EAS solid-shell based on reduced integration with hourglass stabilization (QISPs).
-
Kim et al. [21]: ANS solid-shell with plane stress assumption (XSolid85).
-
Alves de Sousa et al. [66] : EAS solid-shell with reduced (in-plane) integration (RESS).
-
Areias et al. [25]: EAS solid element with penalty stabilization.
-
Kasper and Taylor [20]: Mixed-enhanced strain element with nine enhanced modes (H1/ME9).
-
Schwarze and Reese [22]: Reduced integration solid-shell based on the EAS and the ANS concepts.
-
Huang et al. [23]: unsymmetric 8-node hexahedral solid-shell (US-ATFHS8).
-
Sze et al. [16]: hybrid stress ANS solid-shell.
It is observed that:
-
1
The standard general purpose FEs lock in the tested curved thick shell situations, while the proposed \(\varepsilon /u\) FEs and the solid-shell elements do not.
-
2
The special purpose solid-shell elements, enhanced with higher order bending modes, are more accurate than the general purpose \( \varepsilon /u \) finite elements. However, their corresponding displacement convergence rate is the same.
-
3
Even if the stable solid-shell elements are notoriously more accurate than the corresponding underlying linear element, the asymptotic rate of convergence of displacements is the same as they do not interpolate with the full second order polynomial needed to achieve higher order convergence.
-
4
Only displacement results are reported in the literature for the solid-shell elements. The mixed \(\varepsilon /u\) FEs are devised to yield enhanced strain and stress order of convergence.
-
5
All the reported tests are performed on hexahedral elements, as this is the shape of the solid-shell elements. Mixed \(\varepsilon /u\) FEs can be equally shaped as prisms or tetras, without loss of convergence rate.
-
6
All the reported tests are performed in regular meshes. EAS elements often underperform in distored meshes. Huang et al. [23] solve this quaint at the expense of using an unsymmetrical element.
Rights and permissions
About this article
Cite this article
Saloustros, S., Cervera, M., Kim, S. et al. Accurate and locking-free analysis of beams, plates and shells using solid elements. Comput Mech 67, 883–914 (2021). https://doi.org/10.1007/s00466-020-01969-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-020-01969-0