×

zbMATH — the first resource for mathematics

An unsymmetric 4-node, 8-DOF plane membrane element perfectly breaking through MacNeal’s theorem. (English) Zbl 1352.74158
Summary: Among numerous finite element techniques, few models can perfectly (without any numerical problems) break through MacNeal’s theorem: any 4-node, 8-DOF membrane element will either lock in in-plane bending or fail to pass a \(C_{0}\) patch test when the element’s shape is an isosceles trapezoid. In this paper, a 4-node plane quadrilateral membrane element is developed following the unsymmetric formulation concept, which means two different sets of interpolation functions for displacement fields are simultaneously used. The first set employs the shape functions of the traditional 4-node bilinear isoparametric element, while the second set adopts a novel composite coordinate interpolation scheme with analytical trail function method, in which the Cartesian coordinates (\(x\),\(y\)) and the second form of quadrilateral area coordinates (QACM-II) (\(S\),\(T\)) are applied together. The resulting element US-ATFQ4 exhibits amazing performance in rigorous numerical tests. It is insensitive to various serious mesh distortions, free of trapezoidal locking, and can satisfy both the classical first-order patch test and the second-order patch test for pure bending. Furthermore, because of usage of the second form of quadrilateral area coordinates (QACM-II), the new element provides the invariance for the coordinate rotation. It seems that the behaviors of the present model are beyond the well-known contradiction defined by MacNeal’s theorem.

MSC:
74K15 Membranes
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] MacNeal, A theorem regarding the locking of tapered four-noded membrane elements, International Journal for Numerical Methods in Engineering 24 (9) pp 1793– (1987) · Zbl 0625.73083
[2] MacNeal, On the limits of element perfectability, International Journal for Numerical Methods in Engineering 35 (8) pp 1589– (1992) · Zbl 0768.73076
[3] MacNeal, A proposed standard set of problems to test finite element accuracy, Finite Elements in Analysis and Design 1 (1) pp 3– (1985)
[4] Wilson, Numerical and Computer Methods in Structural Mechanics pp 43– (1973)
[5] Taylor, A non-conforming element for stress analysis, International Journal for Numerical Methods in Engineering 10 (6) pp 1211– (1976) · Zbl 0338.73041
[6] Abaqus 6.9 HTML Documentation (2009)
[7] Simo, A class of mixed assumed strain methods and the method of incompatible modes, International Journal for Numerical Methods in Engineering 29 (8) pp 1595– (1990) · Zbl 0724.73222
[8] Pian, Rational approach for assumed stress finite elements, International Journal for Numerical Methods in Engineering 20 (9) pp 1685– (1984) · Zbl 0544.73095
[9] Pian, Hybrid and Incompatible Finite Element Methods (2006) · Zbl 1110.65003
[10] Wu, Consistency condition and convergence criteria of incompatible elements: general formulation of incompatible functions and its application, Computers & Structures 27 (5) pp 639– (1987) · Zbl 0623.73077
[11] Yeo, New stress assumption for hybrid stress elements and refined four-node plane and eight-node brick elements, International Journal for Numerical Methods in Engineering 40 (16) pp 2933– (1997) · Zbl 0905.73073
[12] Sze, On immunizing five-beta hybrid stress element models from ’trapezoidal locking’ in practical analyses, International Journal for Numerical Methods in Engineering 47 (4) pp 907– (2000) · Zbl 0956.74065
[13] Cen, A hybrid-stress element based on Hamilton principle, Acta Mechanica Sinica 26 (4) pp 625– (2010) · Zbl 1269.74200
[14] Long, Advanced Finite Element Method in Structural Engineering (2009) · Zbl 1168.74005
[15] Chen, Isoparametric quasi-conforming element, Journal of Dalian University of Technology 20 (1) pp 63– (1981)
[16] Piltner, A quadrilateral mixed finite element with two enhanced strain modes, International Journal for Numerical Methods in Engineering 38 (11) pp 1783– (1995) · Zbl 0824.73073
[17] Korelc, Improved enhanced strain four-node element with Taylor expansion of the shape functions, International Journal for Numerical Methods in Engineering 40 (3) pp 407– (1997)
[18] Lautersztajn, Further discussion on four-node isoparametric quadrilateral elements in plane bending, International Journal for Numerical Methods in Engineering 47 (1-3) pp 129– (2000) · Zbl 0994.74072
[19] Piltner, A systematic constructions of B-bar functions for linear and nonlinear mixed-enhanced finite elements for plane elasticity problems, International Journal for Numerical Methods in Engineering 44 (5) pp 615– (1999) · Zbl 0947.74067
[20] Rajendran, A ”FE-meshfree” QUAD4 element based on partition of unity, Computer Methods in Applied Mechanics and Engineering 197 (1-4) pp 128– (2007) · Zbl 1169.74628
[21] Xu, A partition-of-unity based ’FE-Meshfree’ QUAD4 element with radial-polynomial basis functions for static analyses, Computer Methods in Applied Mechanics and Engineering 200 (47-48) pp 3309– (2011) · Zbl 1230.74203
[22] Lee, Effects of element distortion on the performance of isoparametric elements, International Journal for Numerical Methods in Engineering 36 (20) pp 3553– (1993) · Zbl 0800.73465
[23] Felippa, Supernatural QUAD4: a template formulation, Computer Methods in Applied Mechanics and Engineering 195 (41-43) pp 5316– (2006) · Zbl 1120.74049
[24] Fotiu, On shape sensitivity and patch test requirements of incompatible quadrilateral elements in physical coordinates, Acta Mechanica 226 (1) pp 55– (2015) · Zbl 1326.74119
[25] Dasgupta, Incompressible and locking free finite elements from Rayleigh mode vectors: quadratic polynomial displacement fields, Acta Mechanica 223 (8) pp 1645– (2012) · Zbl 1401.74270
[26] Dasgupta, Locking free compressible quadrilateral finite elements: Poisson’s ratio-dependent vector interpolants, Acta Mechanica 225 (1) pp 309– (2014) · Zbl 1401.74271
[27] Long, Area co-ordinates used in quadrilateral elements, Communications in Numerical Methods in Engineering 15 (8) pp 533– (1999) · Zbl 0959.74068
[28] Long, Some basic formulae for area co-ordinates used in quadrilateral elements, Communications in Numerical Methods in Engineering 15 (10) pp 841– (1999) · Zbl 0965.74070
[29] Chen, A new quadrilateral area coordinate method (QACM-II) for developing quadrilateral finite element models, International Journal for Numerical Methods in Engineering 73 (11) pp 1911– (2008) · Zbl 1195.74168
[30] Long, The third form of the quadrilateral area coordinate method (QACM-III): theory, application, and scheme of composite coordinate interpolation, Finite Elements in Analysis and Design 46 (10) pp 805– (2010)
[31] Chen, Membrane elements insensitive to distortion using the quadrilateral area coordinate method, Computers & Structures 82 (1) pp 35– (2004)
[32] Cen, Quadrilateral membrane element family formulated by the quadrilateral area coordinate method, Computer Methods in Applied Mechanics and Engineering 196 (41-44) pp 4337– (2007) · Zbl 1173.74403
[33] Cen, The analytical element stiffness matrix of a recent 4-node membrane element formulated by the quadrilateral area coordinate method, Communications in Numerical Methods in Engineering 23 (10) pp 1095– (2007) · Zbl 1388.74094
[34] Du, Geometrically nonlinear analysis with a 4-node membrane element formulated by the quadrilateral area coordinate method, Finite Elements in Analysis and Design 44 (8) pp 427– (2008)
[35] Cen, Quadrilateral membrane elements with analytical element stiffness matrices formulated by the new quadrilateral area coordinate method (QACM-II), International Journal for Numerical Methods in Engineering 77 (8) pp 1172– (2009) · Zbl 1156.74376
[36] Li, A four-node plane parametric element based on quadrilateral area coordinate and its application to coupled solid-deformation/fluid-flow simulation for porous geomaterials, International Journal for Numerical and Analytical Methods in Geomechanics 39 (3) pp 251– (2015)
[37] Cardoso, A new approach to reduce membrane and transverse shear locking for one-point quadrature shell elements: linear formulation, International Journal for Numerical Methods in Engineering 66 (2) pp 1207– (2006) · Zbl 1110.74844
[38] Cardoso, Enhanced one-point quadrature shell element for nonlinear applications, International Journal for Numerical Methods in Engineering 69 (3) pp 627– (2007) · Zbl 1194.74372
[39] Yoon, Puncture fracture in an aluminum beverage can, International Journal of Impact Engineering 37 (2) pp 150– (2010)
[40] Wang, Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness, Acta Mechanica Sinica 30 (3) pp 418– (2014) · Zbl 1346.74177
[41] Flajs, On convergence of nonconforming convex quadrilateral finite elements AGQ6, Computer Methods in Applied Mechanics and Engineering 199 (25-28) pp 1816– (2010) · Zbl 1231.74408
[42] Cardoso, One point quadrature shell elements: a study on convergence and patch tests, Computational Mechanics 40 (5) pp 871– (2007) · Zbl 1178.74162
[43] Prathap, Making sense of the quadrilateral area coordinate membrane elements, Computer Methods in Applied Mechanics and Engineering 197 (49-50) pp 4379– (2008) · Zbl 1194.74466
[44] Chen, Several treatments on non-conforming element failed in the strict patch test, Mathematical Problems in Engineering 2013 (90) pp 1495– (2013)
[45] Prathap, Stay Cartesian, or go natural? A comment on the article ”Supernatural QUAD4: a template formulation” by C. A. Felippa [Comput. Methods Appl. Mech. Engrg., 195 (2006) 5316-5342], Computer Methods in Applied Mechanics and Engineering 196 (9-12) pp 1847– (2007) · Zbl 1173.74438
[46] Rajendran, A novel unsymmetric 8-node plane element immune to mesh distortion under a quadratic displacement field, International Journal for Numerical Methods in Engineering 58 (11) pp 1713– (2003) · Zbl 1032.74680
[47] Rajendran, A technique to develop mesh-distortion immune finite elements, Computer Methods in Applied Mechanics and Engineering 199 (17-20) pp 1044– (2010) · Zbl 1227.74090
[48] Liew, A quadratic plane triangular element immune to quadratic mesh distortions under quadratic displacement fields, Computer Methods in Applied Mechanics and Engineering 195 (9-12) pp 1207– (2006) · Zbl 1115.74050
[49] Ooi, A 20-node hexahedral element with enhanced distortion tolerance, International Journal for Numerical Methods in Engineering 60 (14) pp 2501– (2004) · Zbl 1075.74665
[50] Ooi, Extension of unsymmetric finite elements US-QUAD8 and US-HEXA20 for geometric nonlinear analyses, Engineering Computations 24 (4) pp 407– (2007) · Zbl 1198.74101
[51] Rajendran, Mesh-distortion immunity assessment of QUAD8 elements by strong-form patch tests, Communications in Numerical Methods in Engineering 23 (2) pp 157– (2007) · Zbl 1107.74046
[52] Ooi, Remedies to rotational frame dependence and interpolation failure of US-QUAD8 element, Communications in Numerical Methods in Engineering 24 (11) pp 1203– (2008) · Zbl 1153.74048
[53] Cen, A shape-free 8-node plane element unsymmetric analytical trial function method, International Journal for Numerical Methods in Engineering 91 (2) pp 158– (2012) · Zbl 1246.74057
[54] Qin, Trefftz finite element method and its applications, Applied Mechanics Reviews 58 (5) pp 316– (2005)
[55] Cen, 8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shapes, Computer Methods in Applied Mechanics and Engineering 200 (29-32) pp 2321– (2011) · Zbl 1230.74173
[56] Cen, Shape-free finite element method: the plane hybrid stress-function (HS-F) element method for anisotropic materials, Science China Physics, Mechanics & Astronomy 54 (4) pp 653– (2011)
[57] Cen, A 4-node hybrid stress-function (HS-F) plane element with drilling degrees of freedom less sensitive to severe mesh distortions, Computers & Structures 89 (5-6) pp 517– (2011)
[58] Cen, Shape-free finite element method: another way between mesh and mesh-free methods, Mathematical Problems in Engineering 2013 (49) pp 1626– (2013)
[59] Zhou, A quasi-static crack propagation simulation based on shape-free hybrid stress-function finite elements with simple remeshing, Computer Methods in Applied Mechanics and Engineering 275 pp 159– (2014) · Zbl 1296.74134
[60] Zhou, A novel shape-free plane quadratic polygonal hybrid stress-function element, Mathematical Problems in Engineering 2015 (49) pp 1325– (2015)
[61] Cook, Improved two-dimension finite element, Journal of the Structural Division (ASCE) 100ST9 pp 1851– (1974)
[62] 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 (8) pp 1311– (1993) · Zbl 0772.73071
[63] MacNeal, A refined four-noded membrane element with rotation degrees of freedom, Computers & Structures 28 (1) pp 75– (1988) · Zbl 0631.73064
[64] Ibrahimgovic, A robust quadrilateral membrane element with rotational degrees of freedom, International Journal for Numerical Methods in Engineering 30 (3) pp 445– (1990) · Zbl 0729.73207
[65] Timoshenko, Theory of Elasticity, 3. ed. (1934) · JFM 60.1350.01
[66] Spilker, Plane isoparametric hybrid-stress elements: invariance and optimal sampling, International Journal for Numerical Methods in Engineering 17 (10) pp 1469– (1981) · Zbl 0462.73050
[67] Cook, Concepts and Applications of Finite Element Analysis, 3. ed. (1989)
[68] INTESIM 2014 Theory Manual (2014)
[69] Cowan, Rotationally invariant distortion resistant finite-elements, Computer Methods in Applied Mechanics and Engineering 275 pp 189– (2014) · Zbl 1296.74107
[70] Qin, Nonlinear analysis of thick plates by HT FE approach, Computers & Structures 61 (2) pp 271– (1996) · Zbl 0900.73822
[71] Qin, Formulation of hybrid Trefftz finite element method for elastoplasticity, Applied Mathematical Modelling 29 (3) pp 235– (2005) · Zbl 1081.74043
[72] Li, Hexahedral volume coordinate method (HVCM) and improvements on 3D Wilson hexahedral element, Computer Methods in Applied Mechanics and Engineering 197 (51-52) pp 4531– (2008) · Zbl 1194.74427
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.