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The simplest node-based equilibrium finite element for 2D Poisson equation with exploration of the equivalence among hybrid, flux-based and stress-function-based equilibrium elements

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Abstract

Getting a pair of compatible and equilibrated solutions is a prerequisite for dual analysis. Generally, compatible solution is obtained by the conventional displacement-based finite element method (FEM), while equilibrated solution can be achieved via the equilibrium finite element method (EFEM). However, the existing EFEM involves more complex construction of the equilibrated field or more degrees of freedom (DOFs), which to a great extent hinders the widespread popularity of the EFEM. Aiming to overcome this limitation, a simplest node-based equilibrium element for general 2D Poisson equation with simply- or multiply-connected domain is proposed in this paper. This element is directly derived from simple nodal interpolation of Prandtl stress function for a simply-connected domain, while specialized hole-dependent discontinuous stress functions are additionally introduced to enable representation of non-zero resultant hole fluxes in a multiply-connected domain. As a result, the proposed equilibrium formulation has almost the same amount of DOFs and the same exponential ratio of the condition number (of system matrix) to the mesh size when compared to the conventional FEM. Further analysis shows that hybrid, flux-based and stress-function-based (or the proposed node-based) equilibrium elements are equivalent to each other, but the node-based element is the most efficient. Numerical examples are studied, and the results indeed verify the accuracy, the convergence and the ability to strictly bounding the error of the proposed node-based EFEM.

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All data, models, and code generated or used during the study appear in the article.

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Acknowledgements

This work is supported by the National Key Research and Development Program of China (No. 2020YFC2201101), the National Natural Science Foundation of China (No. 11702336 and No. 11972380), and the Natural Science Foundation of Guangdong Province (No. 2018B030311001).

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  1. School of Aeronautics and Astronautics, Sun Yat-sen University, Shenzhen Campus, Shenzhen, People’s Republic of China

    Guanfu Lin, Zhong-Rong Lu, Jike Liu & Li Wang

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  1. Guanfu Lin
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Correspondence to Li Wang.

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Lin, G., Lu, ZR., Liu, J. et al. The simplest node-based equilibrium finite element for 2D Poisson equation with exploration of the equivalence among hybrid, flux-based and stress-function-based equilibrium elements. Engineering with Computers 40, 637–660 (2024). https://doi.org/10.1007/s00366-023-01810-1

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