zbMATH — the first resource for mathematics

Smoothed finite element method implemented in a resultant eight-node solid-shell element for geometrical linear analysis. (English) Zbl 1311.74119
Summary: A smoothed finite element method formulation for the resultant eight-node solid-shell element is presented in this paper for geometrical linear analysis. The smoothing process is successfully performed on the element mid-surface to deal with the membrane and bending effects of the stiffness matrix. The strain smoothing process allows replacing the Cartesian derivatives of shape functions by the product of shape functions with normal vectors to the element mid-surface boundaries. The present formulation remains competitive when compared to the classical finite element formulations since no inverse of the Jacobian matrix is calculated. The three dimensional resultant shell theory allows the element kinematics to be defined only with the displacement degrees of freedom. The assumed natural strain method is used not only to eliminate the transverse shear locking problem encountered in thin-walled structures, but also to reduce trapezoidal effects. The efficiency of the present element is presented and compared with that of standard solid-shell elements through various benchmark problems including some with highly distorted meshes.

MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74A05 Kinematics of deformation 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 74K25 Shells
ABAQUS; XFEM
Full Text:
References:
 [1] Betsch, P; Gruttmann, F; Stein, E, A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains, Comput Methods Appl Mech Eng, 130, 57-79, (1996) · Zbl 0861.73068 [2] Bischoff, M; Ramm, E, Shear deformable shell elements for large strains and rotations, Int J Numer Methods Eng, 40, 4427-4449, (1997) · Zbl 0892.73054 [3] Cardoso, RPR; Yoon, JW, One point quadrature shell element with through-thickness stretch, Comput Methods Appl Mech Eng, 194, 1161-1199, (2005) · Zbl 1106.74056 [4] Ahmad, S; Irons, BM; Zienkiewicz, O, Analysis of thick and thin shell structures by curved finite elements, Int J Numer Methods Eng, 2, 419-451, (1970) [5] Hughes, TJR; Liu, WK, Nonlinear finite element analysis of shells: part I. three-dimensional shells, Comput Methods Appl Mech Eng, 26, 331-362, (1981) · Zbl 0461.73061 [6] Hughes, TJR; Liu, WK, Nonlinear finite element analysis of shells: part II. two-dimensional shells, Comput Methods Appl Mech Eng, 27, 167-181, (1981) · Zbl 0474.73093 [7] Parisch, H, Geometrical nonlinear analysis of shells, Comput Methods Appl Mech Eng, 14, 159-178, (1978) · Zbl 0379.73083 [8] Dvorkin, EN; Bathe, K-J, A continuum mechanics based four-node shell element for general non-linear analysis, Eng Comput, 1, 77-88, (1984) [9] Belytschko, T; Lin, JI; Chen-Shyh, T, Explicit algorithms for the nonlinear dynamics of shells, Comput Methods Appl Mech Eng, 42, 225-251, (1984) · Zbl 0512.73073 [10] Bathe, K-J; Dvorkin, EN, A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation, Int J Numer Methods Eng, 21, 367-383, (1985) · Zbl 0551.73072 [11] Simo, JC; Fox, DD, On a stress resultant geometrically exact shell model. part I: formulation and optimal parametrization, Comput Methods Appl Mech Eng, 72, 267-304, (1989) · Zbl 0692.73062 [12] Simo, JC; Rifai, MS, A class of mixed assumed strain methods and the method of incompatible modes, Int J Numer, 29, 1595-1638, (1990) · Zbl 0724.73222 [13] Belytschko T, Liu WK, Moran B (2000) Nonlinear finite elements for continua and structures, 1st edn. Wiley, New York · Zbl 0959.74001 [14] César de Sá, JMA; Natal Jorge, RM; Fontes Valente, RA; Almeida Areias, PM, Development of shear locking-free shell elements using an enhanced assumed strain formulation, Int J Numer Methods Eng, 53, 1721-1750, (2002) · Zbl 1114.74484 [15] Parisch, H, A continuum-based shell theory for non-linear applications, Int J Numer Methods Eng, 38, 1855-1883, (1995) · Zbl 0826.73041 [16] Hauptmann, R; Schweizerhof, K, 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, 49-69, (1998) · Zbl 0917.73067 [17] Hauptmann, R; Schweizerhof, K; Doll, S, Extension of the “solid shell” concept for application to large elastic and large elastoplastic deformations, Int J Numer Methods Eng, 49, 1121-1141, (2000) · Zbl 1048.74041 [18] Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells, 2nd edn. Mcgraw-Hill College, New York · Zbl 0114.40801 [19] Klinkel, S; Gruttmann, F; Wagner, W, A robust non-linear solid shell element based on a mixed variational formulation, Comput Methods Appl Mech Eng, 195, 179-201, (2006) · Zbl 1106.74058 [20] Kim, KD; Liu, GZ; Han, SC, A resultant 8-node solid-shell element for geometrically nonlinear analysis, Comput Mech, 35, 315-331, (2005) · Zbl 1109.74360 [21] Hannachi M (2007) Formulation d’éléments finis volumiques adaptés à l’analyse, linéaire et non linéaire, et à l’optimisation de coques isotropes et composites. Université de technologie de Compiègne, UTC · Zbl 1183.74261 [22] Cardoso, RPR; Yoon, J-W; Mahardika, M; Choudhry, S, Enhanced assumed strain (EAS) and assumed natural strain (ANS) methods for one point quadrature solid shell elements, Int J Numer Methods Eng, 75, 156-187, (2008) · Zbl 1195.74165 [23] Schwarze, M; Reese, S, A reduced integration solid-shell finite element based on the EAS and the ANS concept-geometrically linear problems, Int J Numer Methods Eng, 80, 1322-1355, (2009) · Zbl 1183.74315 [24] Nguyen NH (2009) Development of solid-shell elements for large deformation simulation and springback prediction. Université de Liège, Liège [25] Andelfinger, U; Ramm, E, EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements, Int J Numer Methods Eng, 36, 1311-1337, (1993) · Zbl 0772.73071 [26] Alves de Sousa, RJ; Natal Jorge, RM; Fontes Valente, RA; César De Sá, JMA, A new volumetric and shear locking-free 3D enhanced strain element, Eng Comput, 20, 896-925, (2003) · Zbl 1063.74537 [27] Wriggers, P; Eberlein, R; Reese, S, A comparison of three-dimensional continuum and shell elements for finite plasticity, Int J Solids Struct, 33, 3309-3326, (1996) · Zbl 0913.73071 [28] Alves de Sousa, RJ; Cardoso, RPR; Yoon, J-W; Grácio, JJ, 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, 952-977, (2005) · Zbl 1161.74487 [29] Li, LM; Peng, YH; Li, DY, A stabilized underintegrated enhanced assumed strain solid-shell element for geometrically nonlinear plate/shell analysis, Finite Elem Anal Des, 47, 511-518, (2011) [30] Fontes Valente, RA; Alves de Sousa, RJ; Natal Jorge, RM, An enhanced strain 3D element for large deformation elastoplastic thin-shell applications, Comput Mech, 34, 38-52, (2004) · Zbl 1141.74367 [31] Büchter, N; Ramm, E; Roehl, D, Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept, Int J Numer Methods Eng, 37, 2551-2568, (1994) · Zbl 0808.73046 [32] Miehe, C, A theoretical and computational model for isotropic elastoplastic stress analysis in shells at large strains, Comput Methods Appl Mech Eng, 155, 193-233, (1998) · Zbl 0970.74043 [33] Lyly, M; Stenberg, R; Vihinen, T, A stable bilinear element for the Reissner-Mindlin plate model, Comput Methods Appl Mech Eng, 110, 343-357, (1993) · Zbl 0846.73065 [34] Kouhia, R, On stabilized finite element methods for the Reissner-Mindlin plate model, Int J Numer Methods Eng, 74, 1314-1328, (2008) · Zbl 1159.74429 [35] Liu, GR; Dai, KY; Nguyen, TT, A smoothed finite element method for mechanics problems, Comput Mech, 39, 859-877, (2007) · Zbl 1169.74047 [36] Liu, GR; Nguyen, TT; Dai, KY; Lam, KY, Theoretical aspects of the smoothed finite element method (SFEM), Int J Numer Methods Eng, 71, 902-930, (2007) · Zbl 1194.74432 [37] Chen, J-S; Wu, C-T; Yoon, S; You, Y, A stabilized conforming nodal integration for Galerkin mesh-free methods, Int J Numer Methods Eng, 50, 435-466, (2001) · Zbl 1011.74081 [38] Simo, JC; Hughes, TJR, On the variational foundations of assumed strain methods, J Appl Mech, 53, 51-54, (1986) · Zbl 0592.73019 [39] Nguyen-Xuan, H; Bordas, S; Nguyen-Dang, H, Smooth finite element methods: convergence, accuracy and properties, Int J Numer Methods Eng, 74, 175-208, (2008) · Zbl 1159.74435 [40] Bordas, SPA; Natarajan, S, On the approximation in the smoothed finite element method (SFEM), Int J Numer Methods Eng, 81, 660-670, (2009) · Zbl 1183.74261 [41] Dai KY, Liu GR (2007) Smoothed finite element method, CE006. http://hdl.handle.net/1721.1/35825 · Zbl 0913.73071 [42] Dai, KY; Liu, GR, Free and forced vibration analysis using the smoothed finite element method (SFEM), J Sound Vib, 301, 803-820, (2007) [43] Dai, KY; Liu, GR; Nguyen, TT, An n-sided polygonal smoothed finite element method (nsfem) for solid mechanics, Finite Elem Anal Des, 43, 847-860, (2007) [44] Liu, GR; Nguyen-Thoi, T; Nguyen-Xuan, H; Lam, KY, A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems, Comput Struct, 87, 14-26, (2009) [45] Nguyen-Thoi, T; Liu, GR; Nguyen-Xuan, H, An n-sided polygonal edge-based smoothed finite element method (nes-FEM) for solid mechanics, Int J Numer Methods Biomed Eng, 27, 1446-1472, (2011) · Zbl 1248.74043 [46] Nguyen-Thanh, T; Liu, GR; Dai, KY; Lam, KY, Selective smoothed finite element method, Tsinghua Sci Technol, 12, 497-508, (2007) [47] Hughes, TJR, Generalization of selective integration procedures to anisotropic and nonlinear media, Int J Numer Methods Eng, 15, 1413-1418, (1980) · Zbl 0437.73053 [48] Nguyen-Xuan, H; Rabczuk, T; Bordas, S; Debongnie, J, A smoothed finite element method for plate analysis, Comput Methods Appl Mech Eng, 197, 1184-1203, (2008) · Zbl 1159.74434 [49] Nguyen-Thanh, N; Rabczuk, T; Nguyenxuan, H; Bordas, S, A smoothed finite element method for shell analysis, Comput Methods Appl Mech Eng, 198, 165-177, (2008) · Zbl 1194.74453 [50] Wu, CT; Wang, HP, An enhanced cell-based smoothed finite element method for the analysis of Reissner-Mindlin plate bending problems involving distorted mesh, Int J Numer Methods Eng, 95, 288-312, (2013) · Zbl 1352.74452 [51] Liu, GR; Nguyen-Thoi, T; Lam, KY, An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids, J Sound Vib, 320, 1100-1130, (2009) [52] Nguyen-Xuan H (2008) A strain smoothing method in finite elements for structural analysis. Université de Liège, Liège [53] Bordas, SPA; Rabczuk, T; Hung, N-X; Nguyen, VP; Natarajan, S; Bog, T; Quan, DM; Hiep, NV, Strain smoothing in FEM and XFEM, Comput Struct, 88, 1419-1443, (2010) [54] Puso, MA; Solberg, J, A stabilized nodally integrated tetrahedral, Int J Numer Methods Eng, 67, 841-867, (2006) · Zbl 1113.74075 [55] Cui, X; Liu, G; Li, GY; Zhao, X; Nguyen, TT; Sun, GY, A smoothed finite element method (SFEM) for linear and geometrically nonlinear analysis of plates and shells, Comput Model Eng Sci, 28, 109-125, (2008) · Zbl 1232.74099 [56] Liu, SJ; Wang, H; Zhang, H, Smoothed finite elements large deformation analysis, Int J Comput Methods, 7, 513-524, (2010) · Zbl 1267.74113 [57] Élie-Dit-Cosaque X, Gakwaya A, Lévesque J, Guillot M (2011) Smoothed finite element method for the resultant eight-node solid shell element analysis. In: Simulia customer conference, Barcelona · Zbl 1311.74119 [58] Rashid, MM; Sadri, A, The partitioned element method in computational solid mechanics, Comput Methods Appl Mech Eng, 237-240, 152-165, (2012) · Zbl 1253.74113 [59] Simo, JC; Armero, F; Taylor, RL, Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems, Comput Methods Appl Mech Eng, 110, 359-386, (1993) · Zbl 0846.73068 [60] Vu-Quoc, L; Tan, XG, Optimal solid shells for non-linear analyses of multilayer composites. I. statics, Comput Methods Appl Mech Eng, 192, 975-1016, (2003) · Zbl 1091.74524 [61] Domissy E (1997) Formulation et évaluation d’éléments finis volumiques modifiés pour l’analyse linéaire et non linéaire des coques. Université de technologie de Compiègne, UTC · Zbl 0861.73068 [62] Sze, KY; Yao, L, A hybrid-stress ANS solid-shell element and its generalization for smart structure modelling—part I: solid-shell element formulation, Int J Numer Methods Eng, 48, 545-564, (2000) · Zbl 0990.74073 [63] Hughes, TJR; Tezduyar, TE, Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element, J Appl Mech, 48, 587-596, (1981) · Zbl 0459.73069 [64] Batoz J-L, Dhatt G (1992) Modélisation des structures par éléments finis: coques, Les Presses de l’Université Laval · Zbl 0273.49066 [65] Oden, JT; Reddy, JN, On dual-complementary variational principles in mathematical physics, Int J Eng Sci, 12, 1-29, (1974) · Zbl 0273.49066 [66] Hodge PG (1970) Continuum mechanics: an introductory text for engineers. McGraw-Hill, New York [67] Hao S, Liu WK, Belytschko T (2004) Moving particle finite element method with global smoothness. Int J Numer Methods Eng 59(7):1007-1020. doi:10.1002/nme.999 · Zbl 1065.74608 [68] Zienkiewicz, OC; Taylor, RL; Too, JM, Reduced integration technique in general analysis of plates and shells, Int J Numer Methods Eng, 3, 275-290, (1971) · Zbl 0253.73048 [69] Abaqus user’s manual, ver 6.8,(2007) Abaqus user’s manual. Dassault Systèmes, Simulia · Zbl 0724.73222 [70] Scordelis, AC; Lo, KS, Computer analysis of cylindrical shells, J Am Concr Inst, 69, 539-559, (1964) [71] Macneal, RH; Harder, RL, A proposed standard set of problems to test finite element accuracy, Finite Elem Anal Des, 1, 3-20, (1985)
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.