## A variational framework for the strain-smoothed element method.(English)Zbl 07351734

Summary: This paper is devoted to a rigorous mathematical foundation for the convergence properties of the strain-smoothed element (SSE) method. The SSE method has demonstrated improved convergence behaviors compared to other strain smoothing methods through various numerical examples; however, there has been no theoretical evidence for the convergence behavior. A unique feature of the SSE method is the construction of smoothed strain fields within elements by fully unifying the strains of adjacent elements. Owing to this feature, convergence analysis is required, which is different from other existing strain smoothing methods. In this paper, we first propose a novel mixed variational principle wherein the SSE method can be interpreted as a Galerkin approximation of that. The proposed variational principle is a generalization of the well-known Hu-Washizu variational principle; thus, various existing strain smoothing methods can be expressed in terms of the proposed variational principle. With a unified view of the SSE method and other existing methods through the proposed variational principle, we analyze the convergence behavior of the SSE method and explain the reason for the improved performance compared to other methods. We also present numerical experiments that support our theoretical results.

### MSC:

 65-XX Numerical analysis 74-XX Mechanics of deformable solids

JuSFEM
Full Text:

### References:

 [1] Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2000), Dover Publications: Dover Publications Mineola, New York · Zbl 1191.74002 [2] Bathe, K. J., Finite Element Procedures (1996), Prentice Hall [3] Liu, G. R.; Nguyen-Thoi, T., Smoothed Finite Element Methods (2010), CRC Press: CRC Press New York [4] Hughes, T. J.R., Generalization of selective integration procedures to anisotropic and nonlinear media, Int. J. Numer. Methods Eng., 15, 1413-1418 (1980) · Zbl 0437.73053 [5] Simo, J. C.; Taylor, R. L.; Pister, K. S., Variational and projection methods for the volume constraint in finite deformation elasto-plasticity, Comput. Methods Appl. Mech. Eng., 51, 177-208 (1985) · Zbl 0554.73036 [6] Belytschko, T.; Bachrach, W. E., Efficient implementation of quadrilaterals with high coarse-mesh accuracy, Comput. Methods Appl. Mech. Eng., 54, 279-301 (1986) · Zbl 0579.73075 [7] Wilson, E. L.; Ibrahimbegovic, A., Use of incompatible displacement modes for the calculation of element stiffnesses or stresses, Finite Elem. Anal. Des., 7, 229-241 (1990) [8] Ibrahimbegovic, A.; Wilson, E. L., A modified method of incompatible modes, Commun. Appl. Numer. Methods, 7, 187-194 (1991) · Zbl 0723.73097 [9] Simo, J. C.; Hughes, T. J.R., On the variational foundations of assumed strain methods, J. Appl. Mech., 53, 51-54 (1986) · Zbl 0592.73019 [10] Melenk, J. M.; Babuška, I., The partition of unity finite element method: basic theory and applications, Comput. Methods Appl. Mech. Eng., 139, 289-314 (1996) · Zbl 0881.65099 [11] Babuška, I.; Melenk, J. M., The partition of unity method, Int. J. Numer. Methods Eng., 40, 727-758 (1997) · Zbl 0949.65117 [12] Strouboulis, T.; Babuška, I.; Copps, K., The design and analysis of the Generalized Finite Element Method, Comput. Methods Appl. Mech. Eng., 181, 43-69 (2000) · Zbl 0983.65127 [13] Belytschko, T.; Black, T., Elastic crack growth in finite elements with minimal remeshing, Int. J. Numer. Methods Eng., 45, 601-620 (1999) · Zbl 0943.74061 [14] Moës, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, Int. J. Numer. Methods Eng., 46, 131-150 (1999) · Zbl 0955.74066 [15] Chen, J. S.; Wu, C. T.; Yoon, S.; You, Y., A stabilized conforming nodal integration for Galerkin mesh-free methods, Int. J. Numer. Methods Eng., 50, 435-466 (2001) · Zbl 1011.74081 [16] Liu, G. R., A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods, Int. J. Comput. Methods, 5, 199-236 (2008) · Zbl 1222.74044 [17] Liu, G. R.; Zhang, G. Y.; Dai, K. Y.; Wang, Y. Y.; Zhong, Z. H.; Li, G. Y.; Han, X., A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems, Int. J. Comput. Methods, 2, 645-665 (2005) · Zbl 1137.74303 [18] Liu, G. R.; Jiang, Y.; Chen, L.; Zhang, G. Y.; Zhang, Y. W., A singular cell-based smoothed radial point interpolation method for fracture problems, Comput. Struct., 89, 1378-1396 (2011) [19] Li, Y.; Liu, G. R.; Yue, J. H., A novel node-based smoothed radial point interpolation method for 2D and 3D solid mechanics problems, Comput. Struct., 196, 157-172 (2018) [20] Li, Y.; Liu, G. R., An element-free smoothed radial point interpolation method (EFS-RPIM) for 2D and 3D solid mechanics problems, Comput. Math. Appl., 77, 441-465 (2019) · Zbl 1442.65406 [21] You, X.; Chai, Y.; Li, W., Edged-based smoothed point interpolation method for acoustic radiation with perfectly matched layer, Comput. Math. Appl., 80, 1596-1618 (2020) · Zbl 1452.65387 [22] Liu, G. R.; Dai, K. Y.; Nguyen, T. T., A smoothed finite element method for mechanics problems, Comput. Mech., 39, 859-877 (2007) · Zbl 1169.74047 [23] Liu, G. R.; Nguyen-Thoi, T.; Nguyen-Xuan, H.; Lam, K. Y., A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems, Comput. Struct., 87, 14-26 (2009) [24] Liu, G. R.; Nguyen-Thoi, T.; Lam, K. Y., An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids, J. Sound Vib., 320, 1100-1130 (2009) [25] Nguyen-Thoi, T.; Vu-Do, H. C.; Rabczuk, T.; Nguyen-Xuan, H., A node-based smoothed finite element method (NS-FEM) for upper bound solution to visco-elastoplastic analyses of solids using triangular and tetrahedral meshes, Comput. Methods Appl. Mech. Eng., 199, 3005-3027 (2010) · Zbl 1231.74432 [26] Vu-Bac, N.; Nguyen-Xuan, H.; Chen, L.; Lee, C. K.; Zi, G.; Zhuang, X.; Liu, G. R.; Rabczuk, T., A phantom-node method with edge-based strain smoothing for linear elastic fracture mechanics, J. Appl. Math., 2013 (2013) · Zbl 1271.74418 [27] Natarajan, S.; Bordas, S. P.A.; Ooi, E. T., Virtual and smoothed finite elements: a connection and its application to polygonal/polyhedral finite element methods, Int. J. Numer. Methods Eng., 104, 1173-1199 (2015) · Zbl 1352.65531 [28] Lee, C. K.; Angela Mihai, L.; Hale, J. S.; Kerfriden, P.; Bordas, S. P.A., Strain smoothing for compressible and nearly-incompressible finite elasticity, Comput. Struct., 182, 540-555 (2017) [29] Francis, A.; Ortiz-Bernardin, A.; Bordas, S. P.A.; Natarajan, S., Linear smoothed polygonal and polyhedral finite elements, Int. J. Numer. Methods Eng., 109, 1263-1288 (2017) [30] Nguyen-Hoang, S.; Phung-Van, P.; Natarajan, S.; Kim, H. G., A combined scheme of edge-based and node-based smoothed finite element methods for Reissner-Mindlin flat shells, Eng. Comput., 32, 267-284 (2016) [31] Kim, J.; Lee, C.; Kim, H. G.; Im, S., The surrounding cell method based on the S-FEM for analysis of FSI problems dealing with an immersed solid, Comput. Methods Appl. Mech. Eng., 341, 658-694 (2018) · Zbl 1440.74409 [32] Sohn, D.; Han, J.; Cho, Y. S.; Im, S., A finite element scheme with the aid of a new carving technique combined with smoothed integration, Comput. Methods Appl. Mech. Eng., 254, 42-60 (2013) · Zbl 1297.74131 [33] Jin, S.; Sohn, D.; Im, S., Node-to-node scheme for three-dimensional contact mechanics using polyhedral type variable-node elements, Comput. Methods Appl. Mech. Eng., 304, 217-242 (2016) · Zbl 1425.74342 [34] Nguyen, T. K.; Nguyen, V. H.; Chau-Dinh, T.; Vo, T. P.; Nguyen-Xuan, H., Static and vibration analysis of isotropic and functionally graded sandwich plates using an edge-based MITC3 finite elements, Composites, Part B, Eng., 107, 162-173 (2016) [35] Chau-Dinh, T.; Nguyen-Duy, Q.; Nguyen-Xuan, H., Improvement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysis, Acta Mech., 228, 2141-2163 (2017) · Zbl 1369.74077 [36] Yuan, W. H.; Wang, B.; Zhang, W.; Jiang, Q.; Feng, X. T., Development of an explicit smoothed particle finite element method for geotechnical applications, Comput. Geotech., 106, 42-51 (2019) [37] Jin, Y. F.; Yuan, W. H.; Yin, Z. Y.; Cheng, Y. M., An edge-based strain smoothing particle finite element method for large deformation problems in geotechnical engineering, Int. J. Numer. Anal. Methods Geomech., 44, 923-941 (2020) [38] Huo, Z.; Mei, G.; Xu, N., juSFEM: a Julia-based open-source package of parallel Smoothed Finite Element Method (S-FEM) for elastic problems, Comput. Math. Appl., 81, 459-477 (2020) · Zbl 07288724 [39] Nguyen-Xuan, H.; Bordas, S.; Nguyen-Dang, H., Smooth finite element methods: convergence, accuracy and properties, Int. J. Numer. Methods Eng., 74, 175-208 (2008) · Zbl 1159.74435 [40] Liu, G. R.; Nguyen-Xuan, H.; Nguyen-Thoi, T., A theoretical study on the smoothed FEM (S-FEM) models: properties, accuracy and convergence rates, Int. J. Numer. Methods Eng., 84, 1222-1256 (2010) · Zbl 1202.74180 [41] Zeng, W.; Liu, G., Smoothed finite element methods (S-FEM): an overview and recent developments, Arch. Comput. Methods Eng., 25, 397-435 (2018) · Zbl 1398.65312 [42] Lee, C.; Lee, P. S., A new strain smoothing method for triangular and tetrahedral finite elements, Comput. Methods Appl. Mech. Eng., 341, 939-955 (2018) · Zbl 1440.74414 [43] Lee, C.; Lee, P. S., The strain-smoothed MITC3+ shell finite element, Comput. Struct., 223, Article 106096 pp. (2019) [44] Lee, C.; Kim, S.; Lee, P. S., The strain-smoothed 4-node quadrilateral finite element, Comput. Methods Appl. Mech. Eng., 373, Article 113481 pp. (2021) · Zbl 07337764 [45] Liu, G. R., On G space theory, Int. J. Comput. Methods, 6, 257-289 (2009) · Zbl 1264.74266 [46] Liu, G. R., A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: part I theory, Int. J. Numer. Methods Eng., 81, 1093-1126 (2010) · Zbl 1183.74358 [47] Chen, M.; Li, M.; Liu, G. R., Mathematical basis of G spaces, Int. J. Comput. Methods, 13, Article 1641007 pp. (2016) · Zbl 1359.46024 [48] Brenner, S.; Scott, R., The Mathematical Theory of Finite Element Methods (2008), Springer: Springer New York [49] Boffi, D.; Brezzi, F.; Fortin, M., Mixed Finite Element Methods and Applications (2013), Springer: Springer Heidelberg · Zbl 1277.65092 [50] Kim, S.; Lee, P. S., A new enriched 4-node 2D solid finite element free from the linear dependence problem, Comput. Struct., 202, 25-43 (2018) [51] Teschl, G., Mathematical Methods in Quantum Mechanics (2009), American Mathematical Society: American Mathematical Society Providence [52] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (2002), SIAM: SIAM Philadelphia
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.