# zbMATH — the first resource for mathematics

A displacement smoothing induced strain gradient stabilization for the meshfree Galerkin nodal integration method. (English) Zbl 1329.74292
Summary: In this paper, we present a gradient-type stabilization formulation for the meshfree Galerkin nodal integration method in liner elastic analysis. The stabilization is introduced to the standard variational formulation through an enhanced strain induced by a decomposed smoothed displacement field using the first-order meshfree convex approximations. It leads to a penalization formulation containing a symmetric strain gradient stabilization term for the enhancement of coercivity in the direct nodal integration method. As a result, the stabilization parameter comes naturally from the enhanced strain field and provides the simplest means for effecting stabilization. This strain gradient stabilization formulation is also shown to pass the constant stress patch test if the SCNI scheme is applied to the non-stabilized terms. Several numerical benchmarks are examined to demonstrate the effectiveness and accuracy of the proposed stabilization method in linear elastic analysis.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
##### Keywords:
meshfree; nodal integration; stabilization
Full Text:
##### References:
 [1] Arroyo, M; Ortiz, M, Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods, Int J Numer Methods Eng, 65, 2167-2202, (2006) · Zbl 1146.74048 [2] Beissel, S; Belytschko, T, Nodal integration of the element-free Galerkin method, Comput Methods Appl Mech Eng, 139, 49-74, (1996) · Zbl 0918.73329 [3] Belytschko, T; Lu, YY; Gu, L, Element-free Galerkin methods, Int J Numer Methods Eng, 37, 229-256, (1994) · Zbl 0796.73077 [4] Bochev, PB; Gunzburger, MD, Finite element methods of least-squares type, SIAM Rev, 40, 789-837, (1998) · Zbl 0914.65108 [5] Brenner SC, Scott LR (2008) The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York · Zbl 1135.65042 [6] Chen, JS; Pan, C; Wu, CT; Liu, WK, Reproducing kernel particle methods for large deformation analysis of non-linear structures, Comput Methods Appl Mech Eng, 139, 195-227, (1996) · Zbl 0918.73330 [7] Chen, JS; Wu, CT; Belytschko, T, Regularization of material instabilities by meshfree approximations with intrinsic length scales, Int J Numer Methods Eng, 47, 1303-1322, (2000) · Zbl 0987.74079 [8] Chen, JS; Yoon, S; Wang, HP; Liu, WK, An improved reproducing kernel particle mthod for nearly incompressible finite elasticity, Comput Methods Appl Mech Eng, 181, 117-145, (2000) · Zbl 0973.74088 [9] Chen, JS; Wu, CT; 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 [10] Chen, JS; Yoon, S; Wu, CT, Nonlinear version of stabilized conforming nodal integration for Galerkin meshfree methods, Int J Numer Methods Eng, 53, 2587-2615, (2002) · Zbl 1098.74732 [11] Chen, JS; Hillman, M; Rüter, M, An arbitrary order variationally consistent integration method for Galerkin meshfree methods, Int J Numer Methods Eng, 95, 387-418, (2013) · Zbl 1352.65481 [12] Donning, B; Liu, WK, Meshless methods for shear deformable beams and plates, Comput Methods Appl Mech Eng, 152, 47-72, (1998) · Zbl 0959.74079 [13] Duan, Q; Li, X; Zhang, H; Belytschko, T, Second-order accurate derivatives and integration schemes for meshfree methods, Int J Numer Methods Eng, 92, 399-424, (2012) · Zbl 1352.65390 [14] Franca, LP; Hughes, TJR, Convergence analyses of Galerkin least-squares methods for symmetric advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations, Comput Methods Appl Mech Eng, 105, 285-298, (1993) · Zbl 0771.76037 [15] Guan, PC; Chi, SW; Chen, JS; Slawson, Roth MJ, Semi-Lagrangian reproducing kernel particle method for fragment-impact problems, Int J Impact Eng, 38, 1033-1047, (2011) [16] Günther, FC; Liu, WK, Implementation of boundary conditions for meshless methods, Comput Methods Appl Mech Eng, 163, 205-230, (1998) · Zbl 0963.76068 [17] Hao, S; Liu, WK, Moving particle finite element method with superconvergence: nodal integration formulation and applications, Comput Methods Appl Mech Eng, 195, 6059-6072, (2006) · Zbl 1120.74051 [18] Hillman, M; Chen, JS; Chi, SW, Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems, Comput Part Mech, 1, 245-256, (2014) [19] Hughes, TJR; Brooks, A; Gallagher, RH (ed.); Carey, GF (ed.); Oden, JT (ed.); Zienkiewicz, OC (ed.), A theoretical framework for petro-Galerkin methods with discontinuous weighing functions: applications to the streamline upwind procedure, No. 4, (1982), Chichester [20] Hughes, TJR; Franca, LP; Hulbert, GM, A new finite element formulation for computational fluid dynamics: VIII. the Galerkin/least-squares method for advective-diffusive equations, Comput Methods Appl Mech Eng, 73, 173-189, (1989) · Zbl 0697.76100 [21] Li S, Liu WK (2004) Meshfree particle method. Springer, Berlin [22] Libersky, LD; Petscheck, AG; Carney, TC; Hipp, JR; Allahadai, FA, High strength Lagrangian hydrodynamics, J Comput Phys, 109, 67-75, (1993) · Zbl 0791.76065 [23] Liu, GR; Zhang, GY; Wang, YY; Zhong, ZH; Li, GY; Han, X, A nodal integration technique for meshfree radial point interpolation method (NI-RPIM), Int J Solids Struct, 44, 3840-3860, (2007) · Zbl 1135.74050 [24] Liu GR (2010) Meshfree methods: moving beyond the finite element method. CRC Press, Florida · Zbl 1205.74003 [25] Liu, WK; Ong, JSJ; Uras, RA, Finite-element stabilization matrices—a unification approach, Comput Methods Appl Mech Eng, 53, 13-46, (1985) · Zbl 0553.73065 [26] Liu, WK; Jun, S; Zhang, YF, Reproducing kernel particle methods, Int J Numer Methods Fluids, 20, 1081-1106, (1995) · Zbl 0881.76072 [27] Park, CK; Wu, CT; Kan, CD, On the analysis of dispersion property and stable time step in meshfree method using generalized meshfree approximation, Finite Elem Anal Des, 47, 683-697, (2011) [28] Puso, MA; Chen, JS; Zywicz, W; Elmer, W, Meshfree and finite element nodal integration methods, Int J Numer Methods Eng, 74, 416-446, (2008) · Zbl 1159.74456 [29] Rabczuk, T; Gracie, R; Song, HJ; Belytschko, T, Immersed particle method for fluid-structure interaction, Int J Numer Methods Eng, 81, 48-71, (2010) · Zbl 1183.74367 [30] Simkins, DC; Li, S, Meshfree simulations of thermal-mechanical ductile fracture, Comput Mech, 38, 235-249, (2006) · Zbl 1162.74052 [31] Sukumar, N, Construction of polygonal interpolants: a maximum entropy approach, Int J Numer Methods Eng, 61, 2159-2181, (2004) · Zbl 1073.65505 [32] Thomson, LL; Pinsky, PM, A Galerkin least square finite element method for the two-dimensional Helmholtz equation, Int J Numer Methods Eng, 38, 371-397, (1995) · Zbl 0844.76060 [33] Timoshenko SP, Goodier JN (1970) Theory of elasticity. McGraw-Hill, New York · Zbl 0266.73008 [34] Vignjevic, R; Campbell, J; Libersky, LD, A treatment of zero-energy modes in the smoothed particle hydrodynamics method, Comput Methods Appl Mech Eng, 184, 67-85, (2000) · Zbl 0989.74079 [35] Wang, DD; Chen, JS, Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation, Comput Methods Appl Mech Eng, 193, 1065-1083, (2004) · Zbl 1060.74675 [36] Wang, HP; Wu, CT; Guo, Y; Botkin, M, A coupled meshfree/finite element method for automotive crashworthiness simulations, Int J Impact Eng, 36, 1210-1222, (2009) [37] Wu, CT; Park, CK; Chen, JS, A generalized approximation for the meshfree analysis of solids, Int J Numer Methods Eng, 85, 693-722, (2011) · Zbl 1217.74150 [38] Wu, CT; Hu, W; Chen, JS, A meshfree-enriched finite element method for compressible and nearly incompressible elasticity, Int J Numer Methods Eng, 90, 882-914, (2012) · Zbl 1242.74174 [39] Wu, CT; Koishi, M, Three-dimensional meshfree-enriched finite element formulation for micromechanical hyperelastic modeling of particulate rubber composites, Int J Numer Methods Eng, 91, 1137-1157, (2012) [40] Wu, CT; Guo, Y; Askari, E, Numerical modeling of composite solids using an immersed meshfree Galerkin method, Composites B, 45, 1397-1413, (2013) [41] Wu, CT; Hu, W; Liu, GR, Bubble-enhanced smoothed finite element formulation: a variational multi-scale approach for volume-constrained problems in two-dimensional linear elasticity, Int J Numer Methods Eng, 100, 374-398, (2014) · Zbl 1352.74451 [42] Wu CT, Guo Y, Hu W (2014) An introduction to the LS-DYNA smoothed particle Galerkin method for severe deformation and failure analysis in solids. In: 13th international LS-DYNA users conference, Detroit, MI, 8-10 June, pp 1-20 [43] Wu, YC; Wang, DD; Wu, CT, Three dimensional fragmentation simulation of concrete structures with a nodally regularized meshfree method, Theor Appl Fract Mech, 27, 89-99, (2014)
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.