New Two 20-Node High-Order Finite Elements Based on the SFR Concept for Analyzing 3D Elasticity Problems

Document Type : Research Paper

Authors

1 Laboratoire de G´enie Energ´etique et Mat´eriaux (LGEM), University of Biskra, Biskra, Algeria

2 Laboratoire de G´enie Energ´etique et Mat´eriaux (LGEM), University of Biskra, Biskra, Algeria---- University of Batna 2, Faculty of Technology, Department of Mechanics, Batna, Algeria

3 Laboratoire de G´enie M´ecanique (LGM), University of Biskra, Biskra, Algeria

10.22034/jsm.2021.1942740.1718

Abstract

This paper proposes conforming and nonconforming 20-node hexahedral finite elements. The elements’ formulation stems from the so-called Space Fiber Rotation (SFR) concept, allowing a spatial rotation of three-dimensional virtual fiber within the elements. Adding rotational degrees of freedom results in six degrees of freedom per node (three rotations and three translations) which enhances the approximation of the classical displacement field. The incompatible modes approach has been adopted in the nonconforming element formulation in order to avoid numerical deficiencies associated with the Poisson’s ratio locking phenomenon. The accuracy of the proposed elements is examined through a series of three-dimensions linear elastic benchmarks including beam, plates, and shell structures. The proposed elements were shown to give better results than the standard 20-node hexahedron especially when mesh distortion is applied.  This confirms that the two proposed elements are less sensitive to mesh distortion. The elements also show good performance when compared with analytical and numerical solutions from the literature.

Keywords

  1.  Babuska I., Aziz K., 1972, Survey lectures on the mathematical foundations of the finite element method,The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations 3: ‏
  2.  Musa S.M., 2012, Computational Finite Element Methods in Nanotechnology, CRC Press.
  3.  Macneal R.H., Harder R.L., 1985, A proposed standard set of problems to test finite element accuracy,Finite Elements in Analysis and Design 1(1): 3-20.‏
  4.  Hughes T.J., Cohen M., Haroun M., 1978, Reduced and selective integration techniques in the finite element analysis of plates,Nuclear Engineering and Design 46(1): 203-222.‏
  5.  Zienkiewicz O.C., Taylor R.L., Too J., 1971, Reduced integration technique in general analysis of plates and shells,International Journal for Numerical Methods in Engineering 3(2): 275-290.‏
  6.  Wilson E.L., Taylor R.L., Doherty W.P., Ghaboussi J., 1973, Incompatible displacement models,Numerical and Computer Methods in Structural Mechanics, Academic Press.‏
  7.  Taylor R.L., Beresford P.J., Wilson E.L., 1976, A non‐conforming element for stress analysis,International Journal for Numerical Methods in Engineering 10(6): 1211-1219.‏
  8.  Simo J.C., Rifai M., 1990, A class of mixed assumed strain methods and the method of incompatible modes,International Journal for Numerical Methods in Engineering 29(8): 1595-1638.‏
  9.  Andelfinger U., Ramm E., 1993, 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): 1311-1337.‏
  10.  Simo J.C., Armero F., 1992, Geometrically non‐linear enhanced strain mixed methods and the method of incompatible modes,International Journal for Numerical Methods in Engineering 33(7): 1413-1449.‏
  11.  Simo J.C., Armero F., Taylor R., 1993, Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems,Computer Methods in Applied Mechanics and Engineering 110(3-4): 359-386.‏
  12.  Fredriksson M., Ottosen N.S., 2007, Accurate eight‐node hexahedral element, International Journal for Numerical Methods in Engineering 72(6): 631-657.‏
  13.  De Sousa R.A., Jorge R.N., Valente R.F., De Sá J.C., 2003, A new volumetric and shear locking‐free 3D enhanced strain element,Engineering Computations 20: 896-925.‏
  14.  Pian T.H., 1995, State-of-the-art development of hybrid/mixed finite element method,Finite Elements in Analysis and Design 21(1-2): 5-20.‏
  15.  Pian T.H.H., 1978, A historical note about ‘hybrid elements’,International Journal for Numerical Methods in Engineering 12(5): 891-892.‏
  16.  Pian T.H., Sumihara K., 1984, Rational approach for assumed stress finite elements,International Journal for Numerical Methods in Engineering 20(9): 1685-1695.‏
  17. Pian T.H., Tong P., 1986, Relations between incompatible displacement model and hybrid stress model,International Journal for Numerical Methods in Engineering 22(1): 173-181.
  18.  Sze K.Y., Ghali A., 1993, Hybrid hexahedral element for solids, plates, shells and beams by selective scaling,International Journal for Numerical Methods in Engineering 36(9): 1519-1540.‏
  19.  Bussamra F.L., Pimenta P.D.M., Freitas J.A.T.D., 2001, Hybrid-Trefftz stress elements for three-dimensional elastoplasticity,Computer Assisted Mechanics and Engineering Sciences 8(2-3): 235-246.‏
  20.  Bussamra F.L.S., Neto E.L., Raimundo Jr D.S., 2012, Hybrid quasi-Trefftz 3-D finite elements for laminated composite plates,Computers & Structures 92: 185-192.‏
  21.  Allman D., 1984, A compatible triangular element including vertex rotations for plane elasticity analysis,Computers & Structures 19(1-2): 1-8.‏
  22.  Yunus S.M., Saigal S., Cook R.D., 1989, On improved hybrid finite elements with rotational degrees of freedom, International Journal for Numerical Methods in Engineering28(4): 785-800.‏
  23.  Yunus S.M., Pawlak T.P., Cook R.D., 1991, Solid elements with rotational degrees of freedom: Part 1-hexahedron elements,International Journal for Numerical Methods in Engineering 31(3): 573-592.‏
  24.  Pawlak T.P., Yunus S.M., Cook R.D., 1991, Solid elements with rotational degrees of freedom: Part II-tetrahedron elements,International Journal for Numerical Methods in Engineering 31(3): 593-610.‏
  25.  Ayad R., 2002, Contribution à la Modélisation Numérique Pour l’Analyse des Solides et des Structures, et Pour la Mise en Forme des Fluides Non-Newtoniens, Application a des Matériaux d’emballage, Habilitation to conduct researches, University of Reims, Reims, France.
  26.  Shang Y., Li C.F., Jia K.Y., 2020, 8‐node hexahedral unsymmetric element with rotation degrees of freedom for modified couple stress elasticity,International Journal for Numerical Methods in Engineering 121(12): 2683-2700.‏
  27.  Shang Y., Cen S., Zhou M.J., 2018, 8-node unsymmetric distortion-immune element based on Airy stress solutions for plane orthotropic problems,Acta Mechanica 229(12): 5031-5049.‏
  28.  Meftah K., Ayad R., Hecini M., 2013, A new 3D 6-node solid finite element based upon the “Space Fibre Rotation” concept,European Journal of Computational Mechanics/Revue Européenne de Mécanique Numérique 22(1): 1-29.‏
  29. Ayad R., Zouari W., Meftah K., Zineb T. B., Benjeddou A., 2013, Enrichment of linear hexahedral finite elements using rotations of a virtual space fiber,International Journal for Numerical Methods in Engineering 95(1): 46-70.‏
  30.  Meftah K., Zouari W., Sedira L., Ayad R., 2016, Geometric non-linear hexahedral elements with rotational DOFs,Computational Mechanics 57(1): 37-53.‏
  31.  Meftah K., Sedira L., Zouari W., Ayad R., Hecini M., 2015, A multilayered 3D hexahedral finite element with rotational DOFs,European Journal of Computational Mechanics 24(3): 107-128.‏
  32.  Meftah K., Sedira L., 2019, A four-node tetrahedral finite element based on space fiber rotation concept,Acta Universitatis Sapientiae, Electrical and Mechanical Engineering 11(1): 67-78.‏
  33. Ayadi A., Meftah K., Sedira L., Djahara H., 2019, An eight-node hexahedral finite element with rotational DOFs for elastoplastic applications,Acta Universitatis Sapientiae, Electrical and Mechanical Engineering 11(1): 54-66.‏
  34. Ayadi A., Meftah K., Sedira L., 2020, Elastoplastic analysis of plane structures using improved membrane finite element with rotational DOFs: Elastoplastic analysis of plane structures,Frattura ed Integrità Strutturale 14(52): 148-162.‏
  35.  Zhang H., Kuang J.S., 2008, Eight‐node membrane element with drilling degrees of freedom for analysis of in‐plane stiffness of thick floor plates,International Journal for Numerical Methods in Engineering 76(13): 2117-2136.‏
  36.  Madeo A., Casciaro R., Zagari G., Zinno R., Zucco G., 2014, A mixed isostatic 16 dof quadrilateral membrane element with drilling rotations, based on Airy stresses,Finite Elements in Analysis and Design 89: 52-66.‏
  37.  Nodargi N. A., Bisegna, P., 2017, A novel high-performance mixed membrane finite element for the analysis of inelastic structures,Computers & Structures 182: 337-353.‏
  38.  Ooi E.T., Rajendran S., Yeo J.H., 2004, A 20‐node hexahedron element with enhanced distortion tolerance,International Journal for Numerical Methods in Engineering 60(15): 2501-2530.‏
  39.  Ooi E.T., Rajendran S., Yeo, J.H., 2007, Extension of unsymmetric finite elements US‐QUAD8 and US‐HEXA20 for geometric nonlinear analyses,Engineering Computations 24: 407-431.‏
  40.  Li C.J., Chen J., Chen W.J., 2011, A 3D hexahedral spline element,Computers & Structures 89(23-24): 2303-2308.‏
  41.  Batoz J.L., Dhatt G., 1990, Modélisation des Structures par Eléments Finis : Solides Elastiques, Presses Université Laval.‏
  42.  Abed-Meraim F., Trinh V.D., Combescure A., 2013, New quadratic solid–shell elements and their evaluation on linear benchmark problems,Computing 95(5): 373-394.‏
  43. ABAQUS, 2010, Analysis User’s Manual, V.6.11.
  44.  Kohnke P., 1997, ANSYS: Theory Reference Release 5.4. ANSYS, Inc: Canonsburg, PA.
  45.  Bathe K.J., 2006, Finite Element Procedures, Klaus-Jurgen Bathe.‏
  46.  Zienkiewicz O.C., Taylor R.L., Nithiarasu P., Zhu J.Z., 1977, The Finite Element Method, London, McGraw-H‏
  47.  Legay A., Combescure A., 2003, Elastoplastic stability analysis of shells using the physically stabilized finite element SHB8PS,International Journal for Numerical Methods in Engineering 57(9): 1299-1322.‏
  48.  Areias P.M.A., César de Sá J.M.A., António C.C., Fernandes A.A., 2003, Analysis of 3D problems using a new enhanced strain hexahedral element,International Journal for Numerical Methods in Engineering 58(11): 1637-1682.‏
  49.  Timoshenko S., Woinowsky-Krieger S., 1959, Theory of Plates and Shells, McGraw-H
  50.  Rabczuk T., Areias P.M.A., Belytschko T., 2007, A meshfree thin shell method for non‐linear dynamic fracture,International Journal for Numerical Methods in Engineering 72(5): 524-548.‏
  51.  Flügge W., 2013, Stresses in Shells, Springer Science & Business Media.