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Generalized mixed finite element method for 3D elasticity problems

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Abstract

Without applying any stable element techniques in the mixed methods, two simple generalized mixed element (GME) formulations were derived by combining the minimum potential energy principle and Hellinger–Reissner (H–R) variational principle. The main features of the GME formulations are that the common \(C_{0}\)-continuous polynomial shape functions for displacement methods are used to express both displacement and stress variables, and the coefficient matrix of these formulations is not only automatically symmetric but also invertible. Hence, the numerical results of the generalized mixed methods based on the GME formulations are stable. Displacement as well as stress results can be obtained directly from the algebraic system for finite element analysis after introducing stress and displacement boundary conditions simultaneously. Numerical examples show that displacement and stress results retain the same accuracy. The results of the noncompatible generalized mixed method proposed herein are more accurate than those of the standard noncompatible displacement method. The noncompatible generalized mixed element is less sensitive to element geometric distortions.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant 11502286).

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Authors and Affiliations

  1. College of Aeronautical Engineering, Civil Aviation University of China, Tianjin, 300300, China

    Guanghui Qing, Junhui Mao & Yanhong Liu

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  1. Guanghui Qing
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  2. Junhui Mao
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  3. Yanhong Liu
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Correspondence to Guanghui Qing.

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Qing, G., Mao, J. & Liu, Y. Generalized mixed finite element method for 3D elasticity problems. Acta Mech. Sin. 34, 371–380 (2018). https://doi.org/10.1007/s10409-017-0690-7

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