A reduced integration solid-shell finite element based on the EAS and the ANS concept - geometrically linear problems. (English) Zbl 1183.74315
Summary: In this paper a new reduced integration eight-node solid-shell finite element is presented. The enhanced assumed strain (EAS) concept based on the Hu-Washizu variational principle requires only one EAS degree-of-freedom to cure volumetric and Poisson thickness locking. One key point of the derivation is the Taylor expansion of the inverse Jacobian with respect to the element center, which closely approximates the element shape and allows us to implement the assumed natural strain (ANS) concept to eliminate the curvature thickness and the transverse shear locking. The second crucial point is a combined Taylor expansion of the compatible strain with respect to the center of the element and the normal through the element center leading to an efficient and locking-free hourglass stabilization without rank deficiency. Hence, the element requires only a single integration point in the shell plane and at least two integration points in thickness direction. The formulation fulfills both the membrane and the bending patch test exactly, which has, to the authors’ knowledge, not yet been achieved for reduced integration eight-node solid-shell elements in the literature. Owing to the three-dimensional modeling of the structure, fully three-dimensional material models can be implemented without additional assumptions.
Keywords:
solid-shell element; enhanced assumed strain; assumed natural strain; hourglass stabilization; Taylor expansionSoftware:
FEAPReferences:
| [1] | Bathe, A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation, International Journal for Numerical Methods in Engineering 21 pp 367– (1985) · Zbl 0551.73072 · doi:10.1002/nme.1620210213 |
| [2] | Simo, On a stress resultant geometrically exact shell model. Part V. Nonlinear plasticity: formulation and integration algorithms, Computer Methods in Applied Mechanics and Engineering 96 pp 133– (1992) · Zbl 0754.73042 · doi:10.1016/0045-7825(92)90129-8 |
| [3] | César de Sá, Development of shear locking-free shell elements using an enhanced assumed strain formulation, International Journal for Numerical Methods in Engineering 53 pp 1721– (2002) · Zbl 1114.74484 · doi:10.1002/nme.360 |
| [4] | Belytschko, Explicit algorithms for the nonlinear dynamics of shells, Computer Methods in Applied Mechanics and Engineering 42 pp 225– (1984) · Zbl 0512.73073 · doi:10.1016/0045-7825(84)90026-4 |
| [5] | LS-Dyna, User’s Manual (2007) |
| [6] | Betsch, A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains, Computer Methods in Applied Mechanics and Engineering 130 pp 57– (1996) · Zbl 0861.73068 · doi:10.1016/0045-7825(95)00920-5 |
| [7] | Bischoff, Shear deformable shell elements for large strains and rotations, International Journal for Numerical Methods in Engineering 40 pp 4427– (1997) · Zbl 0892.73054 · doi:10.1002/(SICI)1097-0207(19971215)40:23<4427::AID-NME268>3.0.CO;2-9 |
| [8] | Brank, Nonlinear shell problem formulation accounting for through-the-thickness stretching and its finite element implementation, Computers and Structures 80 pp 699– (2002) · doi:10.1016/S0045-7949(02)00042-1 |
| [9] | Cardoso, One point quadrature shell element with through-thickness stretch, Computer Methods in Applied Mechanics and Engineering 194 pp 1161– (2005) · Zbl 1106.74056 · doi:10.1016/j.cma.2004.06.017 |
| [10] | Lu, Development of a new quadratic shell element considering the normal stress in the thickness direction for simulating sheet methal forming, Journal of Materials Processing Technology 171 pp 341– (2006) · doi:10.1016/j.jmatprotec.2005.06.081 |
| [11] | Parisch, A continuum-based shell theory for non-linear applications, International Journal for Numerical Methods in Engineering 38 pp 1855– (1995) · Zbl 0826.73041 · doi:10.1002/nme.1620381105 |
| [12] | Miehe, A theoretical and computational model for isotropic elastoplastic stress analysis in shells at large strains, Computer Methods in Applied Mechanics and Engineering 155 pp 193– (1998) · Zbl 0970.74043 · doi:10.1016/S0045-7825(97)00149-7 |
| [13] | Hauptmann, A systematic development of ’solid-shell’ element formulations for linear and non-linear analyses employing only displacement degrees of freedom, International Journal for Numerical Methods in Engineering 42 pp 49– (1998) · Zbl 0917.73067 · doi:10.1002/(SICI)1097-0207(19980515)42:1<49::AID-NME349>3.0.CO;2-2 |
| [14] | Hauptmann, Extension of the ’solid-shell’ concept for application to large elastic and large elastoplastic deformations, International Journal for Numerical Methods in Engineering 49 pp 1121– (2000) · Zbl 1048.74041 · doi:10.1002/1097-0207(20001130)49:9<1121::AID-NME130>3.0.CO;2-F |
| [15] | Sze, A hybrid stress ANS solid-shell element and its generalization for smart structure modelling. Part I-solid-shell element formulation, International Journal for Numerical Methods in Engineering 48 pp 545– (2000) · Zbl 0990.74073 · doi:10.1002/(SICI)1097-0207(20000610)48:4<545::AID-NME889>3.0.CO;2-6 |
| [16] | Simo, A class of mixed assumed strain methods and the method of incompatible modes, International Journal for Numerical Methods in Engineering 29 pp 1595– (1990) · Zbl 0724.73222 · doi:10.1002/nme.1620290802 |
| [17] | Simo, Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes, International Journal for Numerical Methods in Engineering 33 pp 1413– (1992) · Zbl 0768.73082 · doi:10.1002/nme.1620330705 |
| [18] | Simo, Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems, Computer Methods in Applied Mechanics and Engineering 110 pp 359– (1993) · Zbl 0846.73068 · doi:10.1016/0045-7825(93)90215-J |
| [19] | Andelfinger, EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements, International Journal for Numerical Methods in Engineering 36 pp 1311– (1993) · Zbl 0772.73071 · doi:10.1002/nme.1620360805 |
| [20] | Korelc, An efficient 3D enhanced strain element with Taylor expansion of the shape functions, Computational Mechanics 19 pp 30– (1996) · Zbl 0888.73062 · doi:10.1007/BF02757781 |
| [21] | Alves de Sousa, A new volumetric and shear locking-free 3D enhanced strain element, Engineering Computations 20 pp 896– (2003) · Zbl 1063.74537 · doi:10.1108/02644400310502036 |
| [22] | Fontes Valente, An enhanced strain 3D element for large deformation elastoplastic thin-shell applications, Computational Mechanics 34 pp 38– (2004) · Zbl 1141.74367 |
| [23] | de Souza Neto, Remarks on the stability of enhanced strain elements in finite elasticity and elastoplasticity, Communications in Numerical Methods in Engineering 11 pp 951– (1995) · Zbl 0836.73066 · doi:10.1002/cnm.1640111109 |
| [24] | Wriggers, A note on enhanced strain methods for large deformations, Computer Methods in Applied Mechanics and Engineering 135 pp 201– (1996) · Zbl 0893.73072 · doi:10.1016/0045-7825(96)01037-7 |
| [25] | Doll, On volumetric locking of low-order solid and solid-shell elements for finite elastoviscoplastic deformations and selective reduced integration, Engineering Computations 17 pp 874– (2000) · Zbl 1012.74069 · doi:10.1108/02644400010355871 |
| [26] | Harnau, About linear and quadratic ’Solid-Shell’ elements at large deformations, Computers and Structures 80 pp 805– (2002) · doi:10.1016/S0045-7949(02)00048-2 |
| [27] | Hughes, Generalization of selective integration procedures to anisotropic and nonlinear media, International Journal for Numerical Methods in Engineering 15 pp 1413– (1980) · Zbl 0437.73053 · doi:10.1002/nme.1620150914 |
| [28] | Liu, A multiple-quadrature eight-node hexahedral finite element for large deformation elastoplastic analysis, Computer Methods in Applied Mechanics and Engineering 154 pp 69– (1998) · Zbl 0935.74068 · doi:10.1016/S0045-7825(97)00106-0 |
| [29] | Duarte Filho, Geometrically nonlinear static and dynamic analysis of shells and plates using the eight-node hexahedral element with one-point quadrature, Finite Elements in Analysis and Design 40 pp 1297– (2004) · doi:10.1016/j.finel.2003.08.012 |
| [30] | Masud, A stabilized 3-D co-rotational formulation for geometrically nonlinear analysis of multi-layered composite shells, Computational Mechanics 26 pp 1– (2000) · Zbl 0980.74066 · doi:10.1007/s004660000144 |
| [31] | Alves de Sousa, A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness-Part II: nonlinear applications, International Journal for Numerical Methods in Engineering 67 pp 160– (2006) · Zbl 1110.74840 · doi:10.1002/nme.1609 |
| [32] | Cardoso, Enhanced assumed strain (EAS) and assumed natural strain (ANS) methods for one-point quadrature solid-shell elements, International Journal for Numerical Methods in Engineering 75 pp 156– (2008) · Zbl 1195.74165 · doi:10.1002/nme.2250 |
| [33] | Areias, Analysis of 3D problems using a new enhanced strain hexahedral element, International Journal for Numerical Methods in Engineering 58 pp 1637– (2003) · Zbl 1032.74661 · doi:10.1002/nme.835 |
| [34] | Hughes, Finite elements based upon Mindlin plate theory with particular references to the four node bilinear isoparametric element, Journal of Applied Mechanics 48 pp 587– (1981) · Zbl 0459.73069 · doi:10.1115/1.3157679 |
| [35] | Wempner, A simple and efficient approximation of shells via finite quadrilateral elements, Journal of Applied Mechanics 49 pp 115– (1982) · doi:10.1115/1.3161951 |
| [36] | Dvorkin, A continuum mechanics based four-node shell element for general non-linear analysis, Engineering Computations 1 pp 77– (1984) · doi:10.1108/eb023562 |
| [37] | Simo, On the variational foundations of assumed strain methods, Journal of Applied Mechanics 53 pp 51– (1986) · Zbl 0592.73019 · doi:10.1115/1.3171737 |
| [38] | Belytschko, Physical stabilization of the 4-node shell element with one point quadrature, Computer Methods in Applied Mechanics and Engineering 113 pp 321– (1994) · Zbl 0846.73058 · doi:10.1016/0045-7825(94)90052-3 |
| [39] | Vu-Quoc, Optimal solid shells for non-linear analyses of multilayer composites. I. Statics, Computer Methods in Applied Mechanics and Engineering 192 pp 975– (2003) · Zbl 1091.74524 · doi:10.1016/S0045-7825(02)00435-8 |
| [40] | Kim, A resultant 8-node solid-shell element for geometrically nonlinear analysis, Computational Mechanics 35 pp 315– (2005) · Zbl 1109.74360 · doi:10.1007/s00466-004-0606-9 |
| [41] | Klinkel, A robust non-linear solid shell element based on a mixed variational formulation, Computer Methods in Applied Mechanics and Engineering 195 pp 179– (2006) · Zbl 1106.74058 · doi:10.1016/j.cma.2005.01.013 |
| [42] | Betsch, An assumed strain approach avoiding artificial thickness straining for a non-linear 4-node shell element, Communications in Numerical Methods in Engineering 11 pp 899– (1995) · Zbl 0833.73051 · doi:10.1002/cnm.1640111104 |
| [43] | Belytschko, Advances in one-point quadrature shell elements, Computer Methods in Applied Mechanics and Engineering 96 pp 93– (1992) · Zbl 0760.73061 · doi:10.1016/0045-7825(92)90100-X |
| [44] | Zeng, A new one-point quadrature, general non-linear quadrilateral shell element with physical stabilization, International Journal for Numerical Methods in Engineering 42 pp 1307– (1998) · Zbl 0904.73070 · doi:10.1002/(SICI)1097-0207(19980815)42:7<1307::AID-NME444>3.0.CO;2-# |
| [45] | Geyer, On reduced integration and locking of flat shell finite elements with drilling rotations, Communications in Numerical Methods in Engineering 19 pp 85– (2003) · Zbl 1024.74040 · doi:10.1002/cnm.567 |
| [46] | Gruttmann, A linear quadrilateral shell element with fast stiffness computation, Computer Methods in Applied Mechanics and Engineering 194 pp 4279– (2005) · Zbl 1151.74418 · doi:10.1016/j.cma.2004.11.005 |
| [47] | Cardoso, Enhanced one-point quadrature shell element for nonlinear applications, International Journal for Numerical Methods in Engineering 69 pp 627– (2007) · Zbl 1194.74372 · doi:10.1002/nme.1784 |
| [48] | Abed-Meraim, SHB8PS-a new adaptive, assumed-strain continuum mechanics shell element for impact analysis, Computers and Structures 80 pp 791– (2002) · doi:10.1016/S0045-7949(02)00047-0 |
| [49] | Legay, Elastoplastic stability analysis of shells using the physically stabilized finite element SHB8PS, International Journal for Numerical Methods in Engineering 57 pp 1299– (2003) · Zbl 1062.74619 · doi:10.1002/nme.728 |
| [50] | Flanagan, A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, International Journal for Numerical Methods in Engineering 17 pp 679– (1981) · Zbl 0478.73049 · doi:10.1002/nme.1620170504 |
| [51] | Belytschko, Assumed strain stabilization of the eight node hexahedral element, Computer Methods in Applied Mechanics and Engineering 105 pp 225– (1993) · Zbl 0781.73061 · doi:10.1016/0045-7825(93)90124-G |
| [52] | Alves de Sousa, A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness: Part I-geometrically linear applications, International Journal for Numerical Methods in Engineering 62 pp 952– (2005) · Zbl 1161.74487 · doi:10.1002/nme.1226 |
| [53] | Reese, A large deformation solid-shell concept based on reduced integration with hourglass stabilization, International Journal for Numerical Methods in Engineering 69 pp 1671– (2007) · Zbl 1194.74469 · doi:10.1002/nme.1827 |
| [54] | MacNeal, A proposed standard set of problems to test finite element accuracy, Finite Elements in Analysis and Design 1 pp 3– (1985) · doi:10.1016/0168-874X(85)90003-4 |
| [55] | Belytschko, Hourglass control in linear and nonlinear problems, Computer Methods in Applied Mechanics and Engineering 43 pp 251– (1984) · Zbl 0522.73063 · doi:10.1016/0045-7825(84)90067-7 |
| [56] | Hauptmann, ’Solid shell’ elements with linear and quadratic shape functions at large deformations with nearly incompressible materials, Computers and Structures 79 pp 1671– (2001) · doi:10.1016/S0045-7949(01)00103-1 |
| [57] | Puso, A highly efficient enhanced assumed strain physically stabilized hexahedral element, International Journal for Numerical Methods in Engineering 49 pp 1029– (2000) · Zbl 0994.74075 · doi:10.1002/1097-0207(20001120)49:8<1029::AID-NME990>3.0.CO;2-3 |
| [58] | Zhang, Numerical integration with Taylor truncations for the quadrilateral and hexahedral finite elements, Journal of Computational and Applied Mathematics 205 pp 325– (2007) · Zbl 1128.65100 · doi:10.1016/j.cam.2006.05.007 |
| [59] | Küssner, The equivalent parallelogram and parallelepiped, and their application to stabilized finite elements in two and three dimensions, Computer Methods in Applied Mechanics and Engineering 190 pp 1967– (2001) · Zbl 1049.74047 · doi:10.1016/S0045-7825(00)00217-6 |
| [60] | Arunakirinathar, A stable affine-approximate finite element method, SIAM Journal on Numerical Analysis 40 pp 180– (2002) · Zbl 1215.74077 · doi:10.1137/S0036142900382442 |
| [61] | Hu, A one-point quadrature eight-node brick element with hourglass control, Computers and Structures 65 pp 893– (1997) · Zbl 0933.74538 · doi:10.1016/S0045-7949(96)00088-0 |
| [62] | Taylor, Version 8.2, User Manual (2008) |
| [63] | Kasper, A mixed-enhanced strain method part I: geometrically linear problems, Computers and Structures 75 pp 237– (2000) · doi:10.1016/S0045-7949(99)00134-0 |
| [64] | Gruttmann, A stabilized one-point integrated quadrilateral Reissner-Mindlin plate element, International Journal for Numerical Methods in Engineering 61 pp 2273– (2004) · Zbl 1075.74646 · doi:10.1002/nme.1148 |
| [65] | Cho, Development of geometrically exact new shell elements based on general curvilinear co-ordinates, International Journal for Numerical Methods in Engineering 56 pp 81– (2003) · Zbl 1026.74070 · doi:10.1002/nme.546 |
| [66] | Cardoso, Development of a one point quadrature shell element for nonlinear applications with contact and anisotropy, Computer Methods in Applied Mechanics and Engineering 191 pp 5177– (2002) · Zbl 1083.74583 · doi:10.1016/S0045-7825(02)00455-3 |
| [67] | Simo, On a stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization, Computer Methods in Applied Mechanics and Engineering 72 pp 267– (1989) · Zbl 0692.73062 · doi:10.1016/0045-7825(89)90002-9 |
| [68] | Lowan, Table of the zeros of the Legendre polynomials of order 1-16 and the weight coefficients for Gauss’ mechanical quadrature formula, Bulletin of the American Mathematical Society 48 pp 739– (1942) · Zbl 0061.30406 · doi:10.1090/S0002-9904-1942-07771-8 |
| [69] | Dhatt, The Finite Element Method Displayed (1984) · Zbl 0553.65070 |
| [70] | Taylor, The patch test-a condition for assessing FEM convergence, International Journal for Numerical Methods in Engineering 22 pp 39– (1986) · Zbl 0593.73072 · doi:10.1002/nme.1620220105 |
| [71] | Zienkiewicz, The finite element patch test revisited a computer test for convergence, validation and error estimates, Computer Methods in Applied Mechanics and Engineering 149 pp 223– (1997) · Zbl 0918.73134 · doi:10.1016/S0045-7825(97)00085-6 |
| [72] | Timoshenko, Theory of Plates and Shells (1959) |
| [73] | Scordelis, Computer analysis of cylindrical shells, Journal of American Concrete Institute 61 pp 539– (1969) |
| [74] | Simo, On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects, Computer Methods in Applied Mechanics and Engineering 73 pp 53– (1989) · Zbl 0724.73138 · doi:10.1016/0045-7825(89)90098-4 |
| [75] | Ramm, Consistent linearization in elasto-plastic shell analysis, Engineering Computations 5 pp 289– (1988) · doi:10.1108/eb023748 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
