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A single point integration rule for numerical manifold method without locking and hourglass issues

数值流形方法无自锁和沙漏问题的一种单点积分策略

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Abstract

Due to the salient feature of cutting operation, the numerical manifold method (NMM) can deal with an any-shaped problem domain by the simplest regular grid. However, this usually creates many irregularly shaped lower-order manifold elements. As a result, the NMM not only needs lots of integration points, but also encounters severe locking issues on nearly incompressible or bending-dominated conditions. This study shows a robust single-point integration rule to handle the above issue in the two-dimensional NMM. The essential idea is to separate the virtual work of an element in terms of moments to the center, so that a zero-order main term and higher-order stabilizing terms are obtained. Further, the volumetric locking and the shearing locking are avoided by modifications to the spherical part and shearing part of the stabilizing terms, and hourglass deformation is overcome since stabilizing terms are always non-zero. Consequently, in addition to fewer integration points, the rule improves accuracy since it is free from locking or hourglass issues. Numerical examples verify the robustness and accuracy improvement of the new rule.

摘要

凭借切割运算, 数值流形方法(NMM)可通过最简单的规则网格处理任意形状的问题域. 然而, 这通常会生成很多低阶的、形状 不规则的单元. 因此NMM不但需要大量积分点, 而且在几乎不可压缩和弯曲变形占优时常常存在自锁现象. 为解决上述问题, 本研究 将建立一个数值稳定的单点积分策略. 通过单元的矩, 该策略将单元虚功分解为零阶主项和高阶稳定项, 进一步通过修正高阶稳定项 中的球应变部分和剪切应变部分, 达到克服体积自锁和剪切自锁的目的, 并用始终非零的稳定项克服沙漏变形. 除了积分点更少外, 新 方法不存在体积自锁、剪切自锁和沙漏模式, 因而也具有更高精度. 文中数值算例验证了新规则的鲁棒性和精度提升.

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References

  1. O. C. Zienkiewicz, R. L. Taylor, and D. D. Fox, The Finite Element Method for Solid and Structural Mechanics (Elsevier, Singapore, 2015).

    MATH  Google Scholar 

  2. T. Belytschko, Y. Y. Lu, and L. Gu, Element-free Galerkin methods, Int. J. Numer. Meth. Eng. 37, 229 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  3. N. Sukumar, D. L. Chopp, N. Moës, and T. Belytschko, Modeling holes and inclusions by level sets in the extended finite-element method, Comput. Methods Appl. Mech. Eng. 190, 6183 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Shi, in Modeling rock joints and blocks by manifold method: Proceedings of the 33th US Symposium on Rock Mechanics (USRMS), Santa Fe, 1992.

  5. G. Shi, Manifold method of material analysis, Army Research Office Research Triangle Park NC (1992).

  6. J. A. Cottrell, T. J. Hughes, and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA (John Wiley & Sons, 2009).

  7. Q. Hu, D. Baroli, and S. Rao, Isogeometric analysis of multi-patch solid-shells in large deformation, Acta Mech. Sin. 37, 844 (2021).

    Article  MathSciNet  Google Scholar 

  8. G. Ma, X. An, and L. He, The numerical manifold method: A review, Int. J. Comput. Methods 07, 1 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Zheng, and D. Xu, New strategies for some issues of numerical manifold method in simulation of crack propagation, Int. J. Numer. Meth. Eng. 97, 986 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  10. N. Zhang, X. Li, and X. Lin, A frictional spring and cohesive contact model for accurate simulation of contact forces in numerical manifold method, Int. J. Numer. Methods Eng. 121, 2369 (2020).

    Article  MathSciNet  Google Scholar 

  11. N. Zhang, H. Zheng, X. Li, and W. Wu, On hp refinements of independent cover numerical manifold method–some strategies and observations, Sci. China Tech. Sci. 66 (2023), doi: https://doi.org/10.1007/s11431-022-2221-5.

  12. Y. Yang, G. Sun, H. Zheng, and Y. Qi, Investigation of the sequential excavation of a soil-rock-mixture slope using the numerical manifold method, Eng. Geol. 256, 93 (2019).

    Article  Google Scholar 

  13. Y. Yang, G. Sun, H. Zheng, and C. Yan, An improved numerical manifold method with multiple layers of mathematical cover systems for the stability analysis of soil-rock-mixture slopes, Eng. Geol. 264, 105373 (2020).

    Article  Google Scholar 

  14. H. Zheng, Y. Yang, and G. Shi, Reformulation of dynamic crack propagation using the numerical manifold method, Eng. Anal. Bound. Elem. 105, 279 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  15. G. W. Ma, X. M. An, H. H. Zhang, and L. X. Li, Modeling complex crack problems using the numerical manifold method, Int. J. Fract. 156, 21 (2009).

    Article  MATH  Google Scholar 

  16. M. Hu, Y. Wang, and J. Rutqvist, On continuous and discontinuous approaches for modeling groundwater flow in heterogeneous media using the numerical manifold method: Model development and comparison, Adv. Watern Resour. 80, 17 (2015).

    Article  Google Scholar 

  17. Y. Wang, M. Hu, Q. Zhou, and J. Rutqvist, A new second-order numerical manifold method model with an efficient scheme for analyzing free surface flow with inner drains, Appl. Math. Model. 40, 1427 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  18. Y. Ning, X. Liu, G. Kang, and Q. Lu, Simulations of crack development in brittle materials under dynamic loading using the numerical manifold method, Eng. Fract. Mech. 275, 108830 (2022).

    Article  Google Scholar 

  19. G. Kang, Y. Ning, P. Chen, S. Pang, and Y. Shao, Comprehensive simulations of rock fracturing with pre-existing cracks by the numerical manifold method, Acta Geotech. 17, 857 (2022).

    Article  Google Scholar 

  20. C. Pu, X. Yang, H. Zhao, Z. Chen, D. Xiao, C. Zhou, and B. Xue, Numerical study on crack propagation under explosive loads, Acta Mech. Sin. 38, 421376 (2022).

    Article  MathSciNet  Google Scholar 

  21. Q. Jiang, S. Deng, C. Zhou, and W. Lu, Modeling unconfined seepage flow using three-dimensional numerical manifold method, J. Hydrodyn. 22, 554 (2010).

    Article  Google Scholar 

  22. L. He, X. M. An, and Z. Y. Zhao, Development of contact algorithm for three-dimensional numerical manifold method, Int. J. Numer. Meth. Eng. 97, 423 (2014).

    Article  MATH  Google Scholar 

  23. Q. Zhang, Advances in three-dimensional block cutting analysis and its applications, Comput. Geotech. 63, 26 (2015).

    Article  Google Scholar 

  24. E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing, CutFEM: Discretizing geometry and partial differential equations, Int. J. Numer. Meth. Eng. 104, 472 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Lozinski, CutFEM without cutting the mesh cells: A new way to impose Dirichlet and Neumann boundary conditions on unfitted meshes, Comput. Methods Appl. Mech. Eng. 356, 75 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  26. N. Zhang, X. Li, H. Zheng, and L. L. Zhang, Some displacement boundary inaccuracies in numerical manifold method and treatments, J. Eng. Mech. 147, (2021).

  27. S. Z. Lin, and Z. Q. Xie, A new recursive formula for integration of polynomial over simplex, Appl. Math. Comput. 376, 125140 (2020).

    MathSciNet  MATH  Google Scholar 

  28. A. F. Bower, Applied Mechanics of Solids (CRC Press, Boca Raton, 2009).

    Book  Google Scholar 

  29. T. J. R. Hughes, Generalization of selective integration procedures to anisotropic and nonlinear media, Int. J. Numer. Meth. Eng. 15, 1413 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  30. E. A. de Souza Neto, D. Perić, M. Dutko, and D. R. J. Owen, Design of simple low order finite elements for large strain analysis of nearly incompressible solids, Int. J. Solids Struct. 33, 3277 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  31. W. Wu, and H. Zheng, Mixed multiscale three-node triangular elements for incompressible elasticity, Eng. Comput. 36, 2859 (2019).

    Article  Google Scholar 

  32. E. L. Wilson, R. L. Taylor, W. P. Doherty, and J. Ghaboussi, Incompatible displacement models, Numer. Comput. Meth. Struct. Mech. 43 (1973).

  33. G. R. Liu, K. Y. Dai, and T. T. Nguyen, A smoothed finite element method for mechanics problems, Comput. Mech. 39, 859 (2007).

    Article  MATH  Google Scholar 

  34. G. R. Liu, and T. T. Nguyen, Smoothed Finite Element Methods (CRC Press, Boca Raton, 2016), p. 170.

    Book  Google Scholar 

  35. D. P. Flanagan, and T. Belytschko, A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, Int. J. Numer. Meth. Eng. 17, 679 (1981).

    Article  MATH  Google Scholar 

  36. T. Belytschko, and L. P. Bindeman, Assumed strain stabilization of the eight node hexahedral element, Comput. Methods Appl. Mech. Eng. 105, 225 (1993).

    Article  MATH  Google Scholar 

  37. J. F. Caseiro, R. A. F. Valente, A. Reali, J. Kiendl, F. Auricchio, and R. J. Alves de Sousa, On the Assumed Natural Strain method to alleviate locking in solid-shell NURBS-based finite elements, Comput. Mech. 53, 1341 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  38. M. A. Puso, J. S. Chen, E. Zywicz, and W. Elmer, Meshfree and finite element nodal integration methods, Int. J. Numer. Meth. Eng. 74, 416 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  39. T. Nguyen-thoi, G. R. Liu, and H. Nguyen-xuan, Additional properties of the node-based smoothed finite element method (NS-FEM) for solid mechanics problems, Int. J. Comput. Methods 06, 633 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  40. C. R. Dohrmann, M. W. Heinstein, J. Jung, S. W. Key, and W. R. Witkowski, Node-based uniform strain elements for three-node triangular and four-node tetrahedral meshes, Int. J. Numer. Meth. Eng. 47, 1549 (2000).

    Article  MATH  Google Scholar 

  41. H. D. Su, Z. Fu, and Z. Q. Xie, Numerical computations based on cover meshes with arbitrary shapes and on exactly geometric boundaries (In Chinese), J. Yangtze River Sci. Res. Inst. 37, 167 (2020).

    Google Scholar 

  42. J. R. H. Thomas, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (Prentice-Hall, Inc., Cham, 2000).

    MATH  Google Scholar 

  43. Y. J. Cheng, Y. Li, L. Tao, P. Joli, and Z. Q. Feng, An adaptive smoothed particle hydrodynamics for metal cutting simulation, Acta Mech. Sin. 38, 422126 (2022).

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52130905, 52079002, and 12202024). The authors would like to thank Prof. Xu Li for his kind encouragement.

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Authors and Affiliations

  1. School of Civil Engineering, Qinghai University, Qinghai, 810016, China

    Ning Zhang  (张宁)

  2. Key Laboratory of Urban Security and Disaster Engineering, Ministry of Education, Beijing University of Technology, Beijing, 100124, China

    Ning Zhang  (张宁), Hong Zheng  (郑宏), Liang Yang  (杨亮) & Wenan Wu  (武文安)

  3. Agiletech Engineering Consultants Co., Ltd., Beijing, 100037, China

    Yichen Wang  (王熠琛)

  4. National Engineering Lab for Green and Safe Construction Technology in Urban Rail Transit, Beijing, 100037, China

    Yichen Wang  (王熠琛)

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  1. Ning Zhang  (张宁)
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Contributions

Ning Zhang conceived the idea, provided analyses, and write the original draft. Hong Zheng provided the funding and contributed significantly to the analysis and revision of the paper. The remaining authors contributed a lot to refining the ideas, carrying out additional analyses, and finalizing this paper.

Corresponding author

Correspondence to Hong Zheng  (郑宏).

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Zhang, N., Zheng, H., Yang, L. et al. A single point integration rule for numerical manifold method without locking and hourglass issues. Acta Mech. Sin. 39, 422318 (2023). https://doi.org/10.1007/s10409-023-22318-x

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