Abstract
A high-performance shape-free polygonal hybrid displacement-function finite-element method is proposed for analyses of Mindlin–Reissner plates. The analytical solutions of displacement functions are employed to construct element resultant fields, and the three-node Timoshenko’s beam formulae are adopted to simulate the boundary displacements. Then, the element stiffness matrix is obtained by the modified principle of minimum complementary energy. With a simple division, the integration of all the necessary matrices can be performed within polygonal element region. Five new polygonal plate elements containing a mid-side node on each element edge are developed, in which element HDF-PE is for general case, while the other four, HDF-PE-SS1, HDF-PE-Free, IHDF-PE-SS1, and IHDF-PE-Free, are for the edge effects at different boundary types. Furthermore, the shapes of these new elements are quite free, i.e., there is almost no limitation on the element shape and the number of element sides. Numerical examples show that the new elements are insensitive to mesh distortions, possess excellent and much better performance and flexibility in dealing with challenging problems with edge effects, complicated loading, and material distributions.
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References
Cen S, Shang Y (2015) Developments of Mindlin–Reissner Plate elements. Math Probl Eng 2015:12. https://doi.org/10.1155/2015/456740
Nguyen-Xuan H (2017) A polygonal finite element method for plate analysis. Comput Struct 188:45–62. https://doi.org/10.1016/j.compstruc.2017.04.002
Perumal L (2018) A brief review on polygonal/polyhedral finite element methods. Math Probl Eng 2018:22. https://doi.org/10.1155/2018/5792372
Wachspress EL (1971) A rational basis for function approximation. IMA J Appl Math 8(1):223–252
Ghosh S, Mukhopadhyay SN (1993) A material based finite-element analysis of heterogeneous media involving Dirichlet Tessellations. Comput Methods Appl Mech Eng 104(2):211–247. https://doi.org/10.1016/0045-7825(93)90198-7
Zhang J, Katsube N (1997) A polygonal element approach to random heterogeneous media with rigid ellipses or elliptical voids. Comput Methods Appl Mech Eng 148(3–4):225–234. https://doi.org/10.1016/s0045-7825(97)00062-5
Meyer M, Barr A, Lee H, Desbrun M (2002) Generalized barycentric coordinates on irregular polygons. J Graph Tools 7(1):13–22. https://doi.org/10.1080/10867651.2002.10487551
Sukumar N, Tabarraei A (2004) Conforming polygonal finite elements. Int J Numer Methods Eng 61(12):2045–2066. https://doi.org/10.1002/nme.1141
Dai KY, Liu GR, Nguyen TT (2007) An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics. Finite Elem Anal Des 43(11–12):847–860. https://doi.org/10.1016/j.finel.2007.05.009
Song C, Wolf JP (1997) The scaled boundary finite-element method—alias consistent infinitesimal finite-element cell method—for elastodynamics. Comput Methods Appl Mech Eng 147(3–4):329–355. https://doi.org/10.1016/s0045-7825(97)00021-2
Zhou PL, Cen S (2015) A novel shape-free plane quadratic polygonal hybrid stress-function element. Math Probl Eng 2015:1–13. https://doi.org/10.1155/2015/491325
Zhou MJ, Cen S, Bao Y, Li CF (2014) A quasi-static crack propagation simulation based on shape-free hybrid stress-function finite elements with simple remeshing. Comput Methods Appl Mech Eng 275:159–188. https://doi.org/10.1016/j.cma.2014.03.006
Cen S, Bao Y, Li CF (2016) Quasi-static crack propagation modeling using shape-free hybrid stress-function elements with drilling degrees of freedom. Int J Comput Methods 13(03):1650014. https://doi.org/10.1142/s0219876216500146
Peng Y, Zhang L, Pu J, Guo Q (2014) A two-dimensional base force element method using concave polygonal mesh. Eng Anal Bound Elem 42:45–50. https://doi.org/10.1016/j.enganabound.2013.09.002
Videla J, Natarajan S, Bordas SPA (2019) A new locking-free polygonal plate element for thin and thick plates based on Reissner-Mindlin plate theory and assumed shear strain fields. Comput Struct 220:32–42. https://doi.org/10.1016/j.compstruc.2019.04.009
Katili I, Maknun IJ, Katili AM, Bordas SPA, Natarajan S (2019) A unified polygonal locking-free thin/thick smoothed plate element. Compos Struct 219:147–157. https://doi.org/10.1016/j.compstruct.2019.03.020
Cen S, Wu CJ, Li Z, Shang Y, Li CF (2019) Some advances in high-performance finite element methods. Eng Comput 36(8):2811–2834. https://doi.org/10.1108/ec-10-2018-0479
Arnold DN, Falk RS (1989) Edge effects in the Reissner–Mindlin plate theory. Analytical and Computational Models for Shells, pp 71–90
Cen S, Shang Y, Li CF, Li HG (2014) Hybrid displacement function element method: a simple hybrid-Trefftz stress element method for analysis of Mindlin–Reissner plate. Int J Numer Methods Eng 98(3):203–234. https://doi.org/10.1002/nme.4632
Shang Y, Cen S, Li CF, Huang JB (2015) An effective hybrid displacement function element method for solving the edge effect of Mindlin–Reissner plate. Int J Numer Methods Eng 102(8):1449–1487. https://doi.org/10.1002/nme.4843
Shang Y, Cen S, Li Z, Li CF (2017) Improved hybrid displacement function (IHDF) element scheme for analysis of Mindlin–Reissner plate with edge effect. Int J Numer Methods Eng 111(12):1120–1169. https://doi.org/10.1002/nme.5496
Bao Y, Cen S, Li CF (2017) Distortion-resistant and locking-free eight-node elements effectively capturing the edge effects of Mindlin–Reissner plates. Eng Comput 34(2):548–586. https://doi.org/10.1108/ec-04-2016-0143
Huang J-B, Cen S, Shang Y, Li C-F (2017) A new triangular hybrid displacement function element for static and free vibration analyses of Mindlin–Reissner plate. Lat Am J Solids Strut 14(5):765–804
Hu H (1984) Variational principles of theory of elasticity with applications. CRC Press, Boca Raton
Shang Y, Li CF, Zhou MJ (2019) A novel displacement-based Trefftz plate element with high distortion tolerance for orthotropic thick plates. Eng Anal Bound Elem 106:452–461. https://doi.org/10.1016/j.enganabound.2019.06.002
Shang Y, Cen S, Ouyan WG (2018) New hybrid-Trefftz Mindlin–Reissner plate elements for efficiently modeling the edge zones near free/SS1 edges. Eng Comput 35(1):136–156. https://doi.org/10.1108/ec-04-2017-0123
Jelenic G, Papa E (2011) Exact solution of 3D Timoshenko beam problem using linked interpolation of arbitrary order. Arch Appl Mech 81(2):171–183. https://doi.org/10.1007/s00419-009-0403-1
Ayad R, Dhatt G, Batoz JL (1998) A new hybrid-mixed variational approach for Reissner-Mindlin plates. The MiSP model. Int J Numer Methods Eng 42(7):1149–1179. https://doi.org/10.1002/(sici)1097-0207(19980815)42:7%3c1149:aid-nme391%3e3.0.co;2-2
Ayad R, Rigolot A (2002) An improved four-node hybrid-mixed element based upon Mindlin’s plate theory. Int J Numer Methods Eng 55(6):705–731. https://doi.org/10.1002/nme.528
Morley LSD (1963) Skew plates and structures. Pergamon Press, Oxford (distributed in the Western Hemisphere by Macmillan, New York)
Babuska I, Scapolla T (1989) Benchmark computation and performance evaluation for a rhombic plate bending problem. Int J Numer Methods Eng 28(1):155–179. https://doi.org/10.1002/nme.1620280112
Abaqus 6.9 (2009) HTML Documentation. Dassault Systèmes Simulia Corp., Providence
Kant T, Hinton E (1983) Mindlin plate analysis by segmentation method. J Eng Mech 109(2):537–556. https://doi.org/10.1061/(asce)0733-9399(1983)109:2(537)
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The authors would like to thank for the financial supports from the National Natural Science Foundation of China (11872229, 11702133) and the Natural Science Foundation of Jiangsu Province (BK20170772).
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Wu, Cj., Cen, S. & Shang, Y. Shape-free polygonal hybrid displacement-function element method for analyses of Mindlin–Reissner plates. Engineering with Computers 37, 1975–1998 (2021). https://doi.org/10.1007/s00366-019-00922-x
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DOI: https://doi.org/10.1007/s00366-019-00922-x