Mixed Finite Element Formulation for 2D Problems Analysis Based on Analytical Solutions of Deferential Equation

Document Type : Research Article

Authors

1 Assistant Professor, Department of Civil Engineering, Faculty of Engineering, Islamic Azad University, Larestan Branch, Iran.

2 Assistant Professor, Civil Engineering and Architectural Department, Faculty of Engineering, University of Torbat Heydarieh, Torbat Heydarieh, Iran.

Abstract

In this paper, a high-order eight-node element based on the analytical response of the governing differential equation is proposed for the analysis of plane structures. The formulation of the proposed element is based on the Hellinger-Reisner mixed functional and the analytical response of the compatibility equation governing plane problems. It is worth noting that in order to formulate finite elements with the Hellinger-Reisner functional, two independent stress and displacement fields are required. For this purpose, Airy stress functions are first made available by the analytical solution of the compatibility equation. By utilizing these stress functions, the stress field within the element is obtained. Also, the quadratic displacement field of the isoparametric element is used for intra-element displacement. By applying the Hellinger-Reisner mixed functional and stationary of this functional relative to the independent stress and displacement fields, the stiffness matrix, and the element node force vector are made available. Finally, with various numerical tests, the accuracy and efficiency of the proposed element are evaluated. These tests prove the high accuracy of the proposed element in the analysis of plane structures.

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Main Subjects

References

[1] N.S. Lee, K.J. Bathe, Effects of element distortions on the performance of isoparametric elements, International Journal for numerical Methods in engineering, 36(20) (1993) 3553-3576.
[2] T.H. Pian, State-of-the-art development of hybrid/mixed finite element method, Finite elements in analysis and design, 21(1-2) (1995) 5-20.
[3] C. Felippa, Advanced finite element methods, Institute of Theoretical Physics, Faculty of Mathematics and Physics …, 2000.
[4] T.H. Pian, Derivation of element stiffness matrices by assumed stress distributions, AIAA journal, 2(7) (1964) 1333-1336.
[5] R.L. Spilker, S. Maskeri, E. Kania, Plane isoparametric hybrid‐stress elements: invariance and optimal sampling, International Journal for Numerical Methods in Engineering, 17(10) (1981) 1469-1496.
[6] T.H. Pian, K. Sumihara, Rational approach for assumed stress finite elements, International Journal for Numerical Methods in Engineering, 20(9) (1984) 1685-1695.
[7] X.R. Fu, S. Cen, C.F. Li, X.M. Chen, Analytical trial function method for development of new 8‐node plane element based on the variational principle containing Airy stress function, Engineering Computations, 27(4) (2010) 442-463.
[8] S. Cen, M.-J. Zhou, X.-R. Fu, 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) (2011) 517-528.
[9] S. Cen, X.-R. Fu, M.-J. Zhou, 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) (2011) 2321-2336.
[10] C. Wang, Y. Wang, C. Yang, X. Zhang, P. Hu, 8-node and 12-node plane elements based on assumed stress quasi-conforming method immune to distorted mesh, Engineering Computations, 34(8) (2017) 2731-2751.
[11] S. Cen, G.H. Zhou, X.R. Fu, A shape‐free 8‐node plane element unsymmetric analytical trial function method, International Journal for Numerical Methods in Engineering, 91(2) (2012) 158-185.
[12] S. Cen, X. Fu, G. Zhou, M. Zhou, C. Li, Shape-free finite element method: the plane hybrid stress-function (HS-F) element method for anisotropic materials, Science China Physics, Mechanics and Astronomy, 54(4) (2011) 653-665.
[13] Y.-T. Zhao, M.-Z. Wang, Y. Chen, Y. Su, Polynomial stress functions of anisotropic plane problems and their applications in hybrid finite elements, Acta Mechanica, 223(3) (2012) 493-503.
[14] I. Herrera, Boundary methods: an algebraic theory, Pitman Advanced Publishing Program, 1984.
[15] Q.-H. Qin, Trefftz Finite Element Method and Its Applications, Applied Mechanics Reviews, 58(5) (2005) 316-337.
[16] M. Rezaiee-Pajand, M. Karkon, An effective membrane element based on analytical solution, European Journal of Mechanics-A/Solids, 39 (2013) 268-279.
[17] S. Ai-Kah, L. Yuqiu, C. Song, Development of eight-node quadrilateral membrane elements using the area coordinates method, Computational Mechanics, 25(4) (2000) 376-384.
[18] S. Cen, X.-M. Chen, X.-R. Fu, Quadrilateral membrane element family formulated by the quadrilateral area coordinate method, Computer Methods in Applied Mechanics and Engineering, 196(41-44) (2007) 4337-4353.
[19] G. Zhang, J. Xiang, Eight‐node conforming straight‐side quadrilateral element with high‐order completeness (QH8‐C1), International Journal for Numerical Methods in Engineering, 121(15) (2020) 3339-3361.
[20] S. Rajendran, K. Liew, 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) (2003) 1713-1748.
[21] G. Zhang, M. Wang, Development of eight-node curved-side quadrilateral membrane element using chain direct integration scheme (SCDI) in area coordinates (MHCQ8-DI), Arabian Journal for Science and Engineering, 44(5) (2019) 4703-4724.
[22] L. Yuqiu, X. Yin, Generalized conforming triangular membrane element with vertex rigid rotational freedoms, Finite elements in analysis and design, 17(4) (1994) 259-271.
[23] S.P. Timoshenko, J.N. Goodier, Theory of elasticity,  (1951).
[24] R. Piltner, R. Taylor, A quadrilateral mixed finite element with two enhanced strain modes, International Journal for Numerical Methods in Engineering, 38(11) (1995) 1783-1808.
Amirkabir Journal of Mechanical Engineering
Volume 54, Issue 6
September 2022
Pages 1443-1458

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