PSI - Issue 80

Luke Wyatt et al. / Procedia Structural Integrity 80 (2026) 31–42

41

L. Wyatt et. al. / Structural Integrity Procedia 00 (2023) 000–000

11

4

VEMQuad VEMVor Reference

VEMQuad VEMVor Reference

2 . 5

3

K 1 B / ( M 0 √ π a )

K 1 B / ( M 0 √ π a )

2

2

1 . 5

1

0

0 . 2 0 . 4 0 . 6 0 . 8

1

0 . 2 0 . 4 0 . 6 0 . 8

1

a / c

a / c

(a) t / c = 0 . 1

(b) t / c = 0 . 5

Fig. 9. Converged VEM results of K 1 B for crack problem 2 with varying crack size

investigation into the impact of the scaling of the stability part of the VEM sti ff ness matrix on the results of higher modes of vibration and buckling could improve the method further. The stability part of other VEM formulations has previously been calibrated to improve performance in certain situations, so a similar procedure could be used to improve the element response in vibration and buckling [19]. More of the advantages that VEM provides such a hanging nodes could be further explored to improve the modelling of the crack tip without significantly increasing the size of the analysis, as has been explored for other singularities [4].

References

[1] T. J. Hughes, M. Cohen, M. Haroun, Reduced and selective integration techniques in the finite element analysis of plates, Nuclear Engineering and Design 46 (1978) 203–222. doi: 10.1016/0029-5493(78)90184-X . [2] L. Beira˜o Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, A. Russo, Basic principles of virtual element methods, Mathematical Models and Methods in Applied Sciences 23 (2013) 199–214. URL: /doi/pdf/10.1142/S0218202512500492?download=true . doi: 10. 1142/S0218202512500492 . [3] C. Chinosi, Virtual Elements for the Reissner-Mindlin plate problem, Numerical Methods for Partial Di ff erential Equations 34 (2018) 1117–1144. doi: 10.1002/NUM.22248 . [4] L. Beira˜o da Veiga, D. Mora, G. Rivera, Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates, Mathematics of Computa tion 88 (2019) 149–178. URL: https://www.ams.org/mcom/2019-88-315/S0025-5718-2018-03331-2/ . doi: 10.1090/MCOM/3331 . [5] X. Du, G. Zhao, W. Wang, J. Yang, M. Guo, R. Zhang, A Numerical Study of Static Bending of Reissner-Mindlin plate Using a Virtual Element Method, Computer-Aided Design and Applications (2022). URL: https://doi.org/10.14733/cadaps.2022.510-521 . doi: 10.14733/ cadaps.2022.510-521 . [6] A. M. D’Altri, L. Patruno, S. de Miranda, E. Sacco, First-order VEM for Reissner–Mindlin plates, Computational Mechanics 69 (2022) 315–333. URL: https://link.springer.com/article/10.1007/s00466-021-02095-1 . doi: 10.1007/S00466-021-02095-1 . [7] L. Wyatt, M. Aliabadi, Z. S. Khodaei, Reduced Order Shear Projection Virtual Element Method for Mindlin Plates [Preprint] (2025). URL: https://papers.ssrn.com/abstract=5362898 . doi: https://dx.doi.org/10.2139/ssrn.5362898 . [8] J. N. Reddy, Theory and Analysis of Elastic Plates and Shells, Second Edition, CRC Press, 2006. URL: https://www. taylorfrancis.com/books/mono/10.1201/9780849384165/theory-analysis-elastic-plates-shells-reddy . doi: https:// doi.org/10.1201/9780849384165 . [9] Y. Leng, L. Svolos, D. Adak, I. Boureima, G. Manzini, H. Mourad, J. Plohr, Y. Leng, L. Svolos, D. Adak, I. Boureima, G. Manzini, H. Mourad, J. Plohr, A guide to the design of the virtual element methods for second- and fourth-order partial di ff erential equations, Mathematics in Engineering 2023 6:1 5 (2023) 1–22. URL: http://www.aimspress.com/article/doi/10.3934/mine.2023100 . doi: 10.3934/MINE. 2023100 . [10] B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, A. Russo, Equivalent projectors for virtual element methods, Computers & Mathematics with Applications 66 (2013) 376–391. URL: https://www.sciencedirect.com/science/article/pii/S0898122113003179?ref=pdf_ download&fr=RR-2&rr=97219a062c3b4177 . doi: 10.1016/J.CAMWA.2013.05.015 .