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A two-level nesting smoothed meshfree method for structural dynamic analysis. (English) Zbl 1481.65238

Summary: This paper presents a novel two-level nesting smoothed meshfree method (NSMM), which significantly improves the computational efficiency of the meshfree Galerkin methods without losing their accuracy, thus facilitates the employment of meshfree methods in applications where background integration cells would be prohibitively expensive. In the NSMM, the system stiffness matrix is calculated using the general smoothing strain technique over the two-level nesting smoothing sub-domains where fewer integration points are used and the costly derivative computation of meshfree shape functions is avoided. The accuracy, efficiency and stability of the present method are assessed by virtue of several numerical examples for problems involving free and forced vibration analysis of the linear elastic continua and dynamic crack response of elastic solid. The results reveal that the NSMM stands out and achieves better performance compared to other existing approaches in the literature.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
74H45 Vibrations in dynamical problems in solid mechanics

Software:

XFEM
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Full Text: DOI

References:

[1] Hughes, T., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987), Prentice-Hall: Prentice-Hall New Jersey · Zbl 0634.73056
[2] Liu, W. K.; Jun, S.; Li, S. F.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. Numer. Methods Eng., 38, 1655-1679 (1995) · Zbl 0840.73078
[3] Gu, Y. T.; Liu, G. R., A local point interpolation method for static and dynamic analysis of thin beams, Comput. Methods Appl. Mech. Eng., 190, 32, 5515-5528 (2001) · Zbl 1059.74060
[4] Liu, G. R.; Gu, Y. T., A local radial point interpolation method (LRPIM) for free vibration of analyses of 2-D solids, J. Sound Vib., 246, 1, 29-46 (2001)
[5] Gu, Y. T.; Liu, G. R., A meshless local Petrov-Galerkin (MLPG) method for free and forced vibration analyses for solids, Comput. Mech., 27, 188-198 (2001) · Zbl 1162.74498
[6] Liu, I.; Chua, L. P.; Ghista, D. N., Element-free Galerkin method for static and dynamic analysis of spatial shell structures, J. Sound Vib., 295, 1-2, 388-406 (2006)
[7] Bui, T. Q.; Nguyen, M. N.; Zhang, C., A moving Kriging interpolation-based element-free Galerkin method for structural dynamic analysis, Comput. Methods Appl. Mech. Eng., 200, 13-16, 1354-1366 (2011) · Zbl 1228.74110
[8] Zhang, B. R.; Rajendran, S., FE-Meshfree’ QUAD4 element for free-vibration analysis, Comput. Methods Appl. Mech. Eng., 197, 3595-3604 (2008) · Zbl 1194.74489
[9] Belytschko, T.; Lu, Y.; Gu, L.; Tabbara, M., Element-free Galerkin methods for static and dynamic fracture, Int. J. Solids Struct., 32, 17, 2547-2570 (1995) · Zbl 0918.73268
[10] Krysl, P.; Belytschko, T., The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks, Int. J. Numer. Methods Eng., 44, 767-800 (1999) · Zbl 0953.74078
[11] Schwer, L.; Oerlach, C.; Belytschko, T., Element-free Galerkin simulations of concrete failure in dynamic uniaxial tension test, J. Eng. Mech., 126, 443-454 (2000)
[12] Belytschko, T.; Organ, D.; Gerlach, C., Element- free Galerkin methods for dynamic fracture in concrete, Comput. Methods Appl. Mech. Eng., 187, 385-399 (2000) · Zbl 0962.74077
[13] Rabczuk, T.; Bordas, S.; Zi, G., A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics, Comput. Mech., 40, 3, 473-495 (2007) · Zbl 1161.74054
[14] Belytschko, T.; Organ, D.; Gerlach, C., Element- free Galerkin methods for dynamic fracture in concrete, Comput. Methods Appl. Mech. Eng., 187, 3-4, 385-399 (2000) · Zbl 0962.74077
[15] Rabczuk, T.; Belytschko, T., Cracking particles: a simplified meshfree method for arbitrary evolving cracks, Int. J. Numer. Methods Eng., 61, 13, 2316-2343 (2004) · Zbl 1075.74703
[16] Rabczuk, T.; Belytschko, T.; Xiao, S. P., Stable particle methods based on Lagrangian kernels, Comput. Methods Appl. Mech. Eng., 193, 12-14, 1035-1063 (2004) · Zbl 1060.74672
[17] Rabczuk, T.; Belytschko, T., A three-dimensional large deformation meshfree method for arbitrary evolving cracks, Comput. Methods Appl. Mech. Eng., 196, 2777-2799 (2007) · Zbl 1128.74051
[18] Rabczuk, T.; Zi, G.; Bordas, S.; Nguyen-Xuan, H., A simple and robust three dimensional cracking particle method without enrichment, Comput. Methods Appl. Mech. Eng., 199, 2437-2455 (2010) · Zbl 1231.74493
[19] Rabczuk, T.; Bordas, S.; Zi, G., On three-dimensional modeling of crack growth using partition of unity methods, Comput. Struct., 88, 23-24, 1391-1411 (2010)
[20] Bui, T. Q.; Nguyen, N. T.; Lich, L. V.; Nguyen, M. N.; Truong, T. T., Analysis of transient dynamic fracture parameters of cracked functionally graded composites by improved meshfree methods, Theoret. Appl. Fract. Mech., 96, 642-657 (2018)
[21] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. Methods Appl. Mech. Eng., 139, 3-47 (1996) · Zbl 0891.73075
[22] Nguyen, V. P.; Rabczuk, T.; Bordas, S.; Duflot, M., Meshless Methods: a review and computer implementation aspects, Math. Comput. Simul., 79, 763-813 (2008) · Zbl 1152.74055
[23] Liu, G. R., A generalized gradient smoothing technique and smoothed bilinear form for Galerkin formulation of a wide class of computational methods, Int. J. Comput. Method, 5, 02, 199-236 (2008) · Zbl 1222.74044
[24] Chen, J. S.; Wu, C. T.; Yoon, S.; You, Y., A stabilized conforming nodal integration for Galerkin meshfree methods, Int. J. Numer. Methods Eng., 50, 435-466 (2001) · Zbl 1011.74081
[25] Liu, G. R.; Zhang, G. Y., A normed G space and weaked weak \((W^2)\) formulation of a cell-based smoothed point interpolation method, Int. J. Numer. Methods Eng., 6, 1, 147-179 (2009) · Zbl 1264.74285
[26] Liu, G. R.; Nguyen, T. T., Smoothed Finite Element Method (2010), CRC Press: CRC Press Boca Raton, USA
[27] Dai, K. Y.; Liu, G. R., Free and forced vibration analysis using the smoothed finite element method (SFEM), J. Sound Vib., 301, 803-820 (2007)
[28] Li, Y.; Liu, G. R., A novel node-based smoothed finite element method with linear strain fields for static, free and forced vibration analyses of solids, Appl. Math. Comput., 352, 30-58 (2019) · Zbl 1428.74211
[29] 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 in solids, J. Sound Vib., 320, 1100-1130 (2009)
[30] Nguyen-Thoi, T.; Liu, G. R.; Lam, K. Y.; Zhang, G. Y., A face-based smoothed finite element method (FS-FEM) for 3D linear and nonlinear solid mechanics problems using 4-node tetrahedral elements, Int. J. Numer. Methods Eng., 78, 324-353 (2009) · Zbl 1183.74299
[31] Nguyen-Thanh, N.; Rabczuk, T.; Nguyen-Xuan, H., An alternative alpha finite element method (AαFEM) for free and forced structural vibration using triangular meshes, J. Comput. Appl. Math., 233, 2112-2135 (2010) · Zbl 1423.74910
[32] Liu, G. R.; Zhang, G. Y., Smoothed Point Interpolation Methods: G Space Theory and Weakened Weak Forms (2013), World Scientific: World Scientific New Jersey · Zbl 1278.65002
[33] Tootoonchi, A.; Khoshghalb, A.; Liu, G. R.; Khalili, N., A cell-based smoothed point interpolation method for flow-deformation analysis of saturated porous media, Comput. Geotech., 75, 159-173 (2016)
[34] Nguyen-Thoi, T.; Liu, G. R.; Nguyen-Xuan, H.; Nguyen-Tran, C., Adaptive analysis using the node-based smoothed finite element method (NS-FEM), Int. J. Numer. Methods Eng., 27, 198-218 (2011) · Zbl 1370.74144
[35] Liu, G. R.; Zhang, G. Y., Edge-based smoothed point interpolation methods, Int. J. Comput. Method, 5, 04, 621-646 (2008) · Zbl 1264.74284
[36] Feng, S. Z.; Cui, X. Y.; Li, A. M., A face-based smoothed point interpolation methods (FS-PIM) for analysis of nonlinear heat conduction in multi-material bodies, Int. J. Therm. Sci., 100, 430-437 (2016)
[37] Liu, G. R., Smoothed finite element methods (S-FEM): an overview and recent developments, Arch Comput. Methods Eng., 25, 397-435 (2018) · Zbl 1398.65312
[38] Liu, G. R.; Gu, Y. T., A point interpolation method for two-dimensional solids, Int. J. Numer. Methods Eng., 50, 937-951 (2001) · Zbl 1050.74057
[39] Wang, J. G.; Liu, G. R., A point interpolation meshless method based on radial basis functions, Int. J. Numer. Methods Eng., 54, 1623-1648 (2002) · Zbl 1098.74741
[40] Chen, L.; Nguyen-xuan, H.; Nguyen_Thoi, T.; Zeng, K. Y.; Wu, S. C., Assessment of smoothed point interpolation methods for elastic mechanics, Int. J. Numer. Methods Eng., 26, 12, 1635-1655 (2010) · Zbl 1323.74080
[41] Bordas, P. A.S.; Rabczuk, T.; Nguyen-Xuan, H.; Vinh-Phu, N.; Natarajan, S.; Bog, T.; Quan, D. M.; Nguyen-Vinh, H., Strain smoothing in FEM and XFEM, Comput. Struct., 88, 1419-1443 (2010)
[42] Chen, L.; Rabczuk, T.; Bordas, S. P.A.; Liu, G. R.; Zeng, K. Y.; Kerfriden, P., Extended finite element method with edge-based strain smoothing (ESm-XFEM) for linear elastic crack growth, Comput. Method Appl. Mech. Eng., 209-212, 250-265 (2012) · Zbl 1243.74170
[43] Jiang, Y.; Tay, T. E.; Chen, L.; Sun, X. S., An edge-based smoothed XFEM for fracture in composite materials, Int. J. Fatigue, 179, 1-2, 179-199 (2013)
[44] Jiang, Y.; Tay, T. E.; Chen, L.; Zhang, Y. W., Extended finite element method coupled with face-based strain smoothing technique for three-dimensional fracture problems, Int. J. Numer. Methods Eng., 102, 13, 1894-1916 (2015) · Zbl 1352.74370
[45] Wu, L.; Liu, P.; Shi, C.; Zhang, Z.; Bui, T. Q.; Jiao, D., Edge-based smoothed extended finite element method for dynamic fracture analysis, Appl. Math. Model., 40, 19-20, 8564-8579 (2016) · Zbl 1471.74066
[46] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int. J. Numer. Methods Eng., 37, 229-256 (1994) · Zbl 0796.73077
[47] Fries, T. P.; Belytschko, T., Convergence and stabilization of stress-point integration in mesh-free and particle methods, Int. J. Numer. Methods Eng., 74, 1067-1087 (2008) · Zbl 1158.74525
[48] Brezinski, C.; Zaglia, M. R., Extrapolation Methods: Theory and Practice (1991), North-Holland · Zbl 0744.65004
[49] Duan, Q. L.; Li, X. K.; Zhang, H. W.; Belytschko, T., Second-order accurate derivatives and integration schemes for meshfree methods, Int. J. Numer. Methods Eng., 92, 399-424 (2012) · Zbl 1352.65390
[50] Ventura, G., On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method, Int. J. Numer. Methods Eng., 66, 5, 767-795 (2006) · Zbl 1110.74858
[51] Fleming, M.; Chu, Y. A.; Moran, B.; Belytschko, T., Enriched element-free Galerkin methods for crack tip fields, Int. J. Numer. Methods Eng., 40, 1483-1504 (1997)
[52] Timoshenko, S. P.; Goodier, J. N., Theory of Elasticity (1987), McGraw-Hill: McGraw-Hill New York · Zbl 0266.73008
[53] Li, H.; Wang, Q. X.; Li, H., A novel meshless approach-Local Kriging (LoKriging) method for structure dynamic analysis, Comput. Methods Appl. Mech. Eng., 193, 2599-2619 (2004) · Zbl 1067.74598
[54] Menouillard, T.; Belytschko, T., Dynamic fracture with meshfree-enriched XFEM, Acta Mech., 213, 53-69 (2010) · Zbl 1272.74604
[55] Zhuang, X.; Augarde, C.; Mathisen, K., Fracture modeling using meshless methods and level sets in 3D: framework and modeling, Int. J. Numer. Methods Eng., 92, 11, 969-998 (2012) · Zbl 1352.74312
[56] Ventura, G.; Xu, J. X.; Belytschko, T., A vector level set method and new discontinuity approximations for crack growth by EFG, Int. J. Numer. Methods Eng., 54, 6, 923-944 (2002) · Zbl 1034.74053
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