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An aspect ratio dependent lumped mass formulation for serendipity finite elements with severe side-length discrepancy

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Abstract

The frequency solutions of finite elements may significantly deteriorate as the mesh aspect ratios become large, which implies a severe element side-length discrepancy. In this work, an aspect ratio dependent lumped mass (ARLM) formulation is proposed for serendipity elements, i.e., the two-dimensional eight-node and three dimensional twenty-node quadratic elements for linear problems. In particular, a generalized parametric lumped mass matrix template taking into account the mesh aspect ratios is introduced to examine the frequency accuracy of serendipity elements. This generalized lumped mass matrix template completely meets the mass conservation and non-negativity requirements. Subsequently, analytical frequency error estimates are developed for serendipity elements, which clearly illustrate the relationship between the frequency accuracy and element aspect ratios. Accordingly, optimal mass parameters are obtained as the functions of element aspect ratios through solving a constrained optimization problem for frequency accuracy. It turns out that the resulting aspect ratio dependent lumped mass matrices yield much more accurate frequency solutions, in comparison to the diagonal scaling lumped mass (HRZ) matrices and the mid-node lumped mass (MNLM) matrices without consideration of the element aspect ratios, especially for finite element discretizations with severe element side-length discrepancy. The superior accuracy and robustness of the proposed ARLM over HRZ and MNLM are consistently demonstrated by numerical examples.

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Acknowledgements

The support of this work by the National Natural Science Foundation of China (12372201, 12072302) and the Natural Science Foundation of Fujian Province of China (2021J02003) is gratefully acknowledged.

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Authors and Affiliations

  1. Fujian Key Laboratory of Digital Simulations for Coastal Civil Engineering, Department of Civil Engineering, Xiamen University, Xiamen, 361005, Fujian, China

    Songyang Hou, Zhiwei Lin & Dongdong Wang

  2. College of Safety Engineering and Emergency Management, Shijiazhuang Tiedao University, Shijiazhuang, 050043, Hebei, China

    Xiwei Li

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  1. Songyang Hou
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Correspondence to Dongdong Wang.

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Appendix

Appendix

In this Appendix, the critical times steps for all numerical examples with different meshes are listed in Tables 2, 3, 4 and 5, which coherently indicate that the critical times steps corresponding to the proposed method of ARLM are larger than those of HRZ for both 2D and 3D cases.

Table 2 Comparison of critical time steps for the 2D plane strain elasticity problem
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Table 3 Comparison of critical time steps for the 2D elastic bar problem
Full size table
Table 4 Comparison of critical time steps for the 2D one-third annular membrane problem
Full size table
Table 5 Comparison of critical time steps for the 3D cylindrical cavity problem
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Hou, S., Li, X., Lin, Z. et al. An aspect ratio dependent lumped mass formulation for serendipity finite elements with severe side-length discrepancy. Comput Mech (2024). https://doi.org/10.1007/s00466-024-02457-5

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