The smoothed finite element method (S-FEM): A framework for the design of numerical models for desired solutions

doi:10.1007/s11709-019-0519-5

Frontiers of Structural and Civil Engineering

2019, Vol. 13

Issue (2): 456-477 https://doi.org/10.1007/s11709-019-0519-5

本期目录

The smoothed finite element method (S-FEM): A framework for the design of numerical models for desired solutions

Gui-Rong Liu(

)

Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati 45219, USA

全文: PDF(2888 KB) HTML

Abstract：

The smoothed finite element method (S-FEM) was originated by G R Liu by combining some meshfree techniques with the well-established standard finite element method (FEM). It has a family of models carefully designed with innovative types of smoothing domains. These models are found having a number of important and theoretically profound properties. This article first provides a concise and easy-to-follow presentation of key formulations used in the S-FEM. A number of important properties and unique features of S-FEM models are discussed in detail, including 1) theoretically proven softening effects; 2) upper-bound solutions; 3) accurate solutions and higher convergence rates; 4) insensitivity to mesh distortion; 5) Jacobian-free; 6) volumetric-locking-free; and most importantly 7) working well with triangular and tetrahedral meshes that can be automatically generated. The S-FEM is thus ideal for automation in computations and adaptive analyses, and hence has profound impact on AI-assisted modeling and simulation. Most importantly, one can now purposely design an S-FEM model to obtain solutions with special properties as wish, meaning that S-FEM offers a framework for design numerical models with desired properties. This novel concept of numerical model on-demand may drastically change the landscape of modeling and simulation. Future directions of research are also provided.

Key words： computational method finite element method smoothed finite element method strain smoothing technique smoothing domain weakened weak form solid mechanics softening effect upper bound solution

收稿日期: 2018-10-15 出版日期: 2019-03-12

Corresponding Author(s): Gui-Rong Liu

引用本文:

. [J]. Frontiers of Structural and Civil Engineering, 2019, 13(2): 456-477.
Gui-Rong Liu. The smoothed finite element method (S-FEM): A framework for the design of numerical models for desired solutions. Front. Struct. Civ. Eng., 2019, 13(2): 456-477.

链接本文:

https://academic.hep.com.cn/fsce/CN/10.1007/s11709-019-0519-5
https://academic.hep.com.cn/fsce/CN/Y2019/V13/I2/456

Fig.1

Fig.2

Fig.3

Fig.4

Fig.5

Fig.6

Type*	method for creation and number of SD’s ( $N s$ )	S-FEM models	dimension of problem; properties
Cell-based SD (CSD)	SD’s or smoothing cells (SC’s) are divided from and located within the elements ( $N s = ∑ i = 1 N e n s c i$ , $n s c i = 1, 2, 3, 4, ...$ )	CS-FEM nCS-FEM	1D, 2D, 3D softer; high accuracy; insensitive to mesh distortion
Edge-based SD (ESD)	SD’s are created based on edges by connecting portions of the surrounding elements sharing the associated edge ( $N s = N e d g e$ )	ES-FEM	2D, 3D softer; very high accuracy, less insensitive to mesh distortion
Node-based SD (NSD)	SD’s are created based on nodes by connecting portions of the surrounding elements sharing the associated node ( $N s = N n o d e$ )	NS-FEM	1D, 2D, 3D soft; upper bound, very insensitive to mesh distortion, volumetric locking free
Face-based SD (FSD)	SD’s are created based on faces by connecting portions of the surrounding elements sharing the associated face ( $N s = N f a c e$ )	FS-FEM	3D softer; very high accuracy, less insensitive to mesh distortion

Tab.1

Fig.7

Fig.8

dimension of the problem	minimum number of smoothing domains
1D	$N s min ?$ = n_t
2D	$N s min ?$ = 2n_t /3
3D	$N s min ?$ = 3n_t /6= n_t /2

Tab.2

Fig.9

point	N₁	N₂	N₃	N₄	description
1	1.0	0	0	0	field node
2	0	1.0	0	0	field node
3	0	0	1.0	0	field node
4	0	0	0	1.0	field node
5	1/2	1/2	0	0	side midpoint
6	0	1/2	1/2	0	side midpoint
7	0	0	1/2	1/2	side midpoint
8	1/2	0	0	1/2	side midpoint
9	1/4	1/4	1/4	1/4	intersection of two bi-medians
g1	3/4	1/4	0	0	Gauss point (mid-segment point of $Γ k, p s$ )
g2	3/8	3/8	1/8	1/8	Gauss point (mid-segment point of $Γ k, p s$ )
g3	3/8	1/8	1/8	3/8	Gauss point (mid-segment point of $Γ k, p s$ )
g4	3/4	0	0	1/4	Gauss point (mid-segment point of $Γ k, p s$ )
g5	1/4	3/4	0	0	Gauss point (mid-segment point of $Γ k, p s$ )
g6	0	3/4	1/4	0	Gauss point (mid-segment point of $Γ k, p s$ )
g7	1/8	3/8	3/8	1/8	Gauss point (mid-segment point of $Γ k, p s$ )
g8	0	1/4	3/4	0	Gauss point (mid-segment point of $Γ k, p s$ )
g9	0	0	3/4	1/4	Gauss point (mid-segment point of $Γ k, p s$ )
g10	1/8	1/8	3/8	3/8	Gauss point (mid-segment point of $Γ k, p s$ )
g11	0	0	1/4	3/4	Gauss point (mid-segment point of $Γ k, p s$ )
g12	1/4	0	0	3/4	Gauss point (mid-segment point of $Γ k, p s$ )

Tab.3

point	N_1’	N_2’	N_3’	N_4’	N_5’	N_6’	description
1’	1.0	0	0	0	0	0	field node
2’	0	1.0	0	0	0	0	field node
3’	0	0	1.0	0	0	0	field node
4’	0	0	0	1.0	0	0	field node
5’	0	0	0	0	1.0	0	field node
6’	0	0	0	0	0	1.0	field node
O	1/6	1/6	1/6	1/6	1/6	1/6	centroid point
g1’	7/12	1/12	1/12	1/12	1/12	1/12	Gauss point (mid-segment point of $Γ k, p s$ )
g2’	1/2	1/2	0	0	0	0	Gauss point (mid-segment point of $Γ k, p s$ )
g3’	1/12	7/12	1/12	1/12	1/12	1/12	Gauss point (mid-segment point of $Γ k, p s$ )
g4’	0	1/2	1/2	0	0	0	Gauss point (mid-segment point of $Γ k, p s$ )
g5’	1/12	1/12	7/12	1/12	1/12	1/12	Gauss point (mid-segment point of $Γ k, p s$ )
g6’	0	0	1/2	1/2	0	0	Gauss point (mid-segment point of $Γ k, p s$ )
g7’	1/12	1/12	1/12	7/12	1/12	1/12	Gauss point (mid-segment point of $Γ k, p s$ )
g8’	0	0	0	1/2	1/2	0	Gauss point (mid-segment point of $Γ k, p s$ )
g9’	1/12	1/12	1/12	1/12	7/12	1/12	Gauss point (mid-segment point of $Γ k, p s$ )
g10’	0	0	0	0	1/2	1/2	Gauss point (mid-segment point of $Γ k, p s$ )
g11’	1/12	1/12	1/12	1/12	1/12	7/12	Gauss point (mid-segment point of $Γ k, p s$ )
g12’	1/2	0	0	0	0	1/2	Gauss point (mid-segment point of $Γ k, p s$ )

Tab.4

Fig.10

Fig.11

Fig.12

Fig.13

Fig.14

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81	NNourbakhshnia, G R Liu. A quasi-static crack growth simulation based on the singular ES-FEM. International Journal for Numerical Methods in Engineering, 2011, 88(5): 473–492 https://doi.org/10.1002/nme.3186
82	LChen, G R Liu, Y Jiang, KZeng, JZhang. A singular edge-based smoothed finite element method (ES-FEM) for crack analyses in anisotropic media. Engineering Fracture Mechanics, 2011, 78(1): 85–109 https://doi.org/10.1016/j.engfracmech.2010.09.018
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84	LChen, G R Liu, K Zeng, JZhang. A novel variable power singular element in G space with strain smoothing for bi-material fracture analyses. Engineering Analysis with Boundary Elements, 2011, 35(12): 1303–1317 https://doi.org/10.1016/j.enganabound.2011.06.007
85	LChen, G R Liu, K Zeng. A combined extended and edge-based smoothed finite element method (ES-XFEM) for fracture analysis of 2D elasticity. International Journal of Computational Methods, 2011, 8(4): 773–786 https://doi.org/10.1142/S0219876211002812
86	NNourbakhshnia, G R Liu. Fatigue analysis using the singular ES-FEM. International Journal of Fatigue, 2012, 40: 105–111 https://doi.org/10.1016/j.ijfatigue.2011.12.018
87	HNguyen-Xuan, G R Liu, N Nourbakhshnia, LChen. A novel singular ES-FEM for crack growth simulation. Engineering Fracture Mechanics, 2012, 84: 41–66 https://doi.org/10.1016/j.engfracmech.2012.01.001
88	PLiu, T Q Bui, C Zhang, T TYu, G RLiu, M VGolub. The singular edge-based smoothed finite element method for stationary dynamic crack problems in 2D elastic solids. Computer Methods in Applied Mechanics and Engineering, 2012, 233–236: 68–80 https://doi.org/10.1016/j.cma.2012.04.008
89	YJiang, T E Tay, LChen, X S.Sun An edge-based smoothed XFEM for fracture in composite materials. International Journal of Fracture, 2013, 179(1–2):179–199
90	HNguyen-Xuan, G R Liu, S Bordas, SNatarajan, TRabczuk. An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order. Computer Methods in Applied Mechanics and Engineering, 2013, 253: 252–273 https://doi.org/10.1016/j.cma.2012.07.017
91	NVu-Bac, H Nguyen-Xuan, LChen, C KLee, GZi, X Zhuang, G RLiu, TRabczuk. A phantom-node method with edge-based strain smoothing for linear elastic fracture mechanics. Journal of Applied Mathematics, 2013, 2013: 1 https://doi.org/10.1155/2013/978026
92	G RLiu, L Chen, MLi. S-FEM for fracture problems, theory, formulation and application. International Journal of Computational Methods, 2014, 11(03): 1343003 https://doi.org/10.1142/S0219876213430032
93	P NJiki, J U Agber. Damage evaluation in gap tubular truss ‘K’ bridge joints using SFEM. Journal of Constructional Steel Research, 2014, 93: 135–142 https://doi.org/10.1016/j.jcsr.2013.10.010
94	YJiang, T E Tay, L Chen, Y WZhang. Extended finite element method coupled with face-based strain smoothing technique for three-dimensional fracture problems. International Journal for Numerical Methods in Engineering, 2015, 102(13): 1894–1916 https://doi.org/10.1002/nme.4878
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96	HChen, Q Wang, G RLiu, YWang, J Sun. Simulation of thermoelastic crack problems using singular edge-based smoothed finite element method. International Journal of Mechanical Sciences, 2016, 115–116: 123–134 https://doi.org/10.1016/j.ijmecsci.2016.06.012
97	LWu, P Liu, CShi, ZZhang, T QBui, DJiao. Edge-based smoothed extended finite element method for dynamic fracture analysis. Applied Mathematical Modelling, 2016, 40(19–20): 8564–8579 https://doi.org/10.1016/j.apm.2016.05.027
98	G RLiu, W Zeng, HNguyen-Xuan. Generalized stochastic cell-based smoothed finite element method (GS_CS-FEM) for solid mechanics. Finite Elements in Analysis and Design, 2013, 63: 51–61 https://doi.org/10.1016/j.finel.2012.08.007
99	X BHu, X Y Cui, H Feng, G YLi. Stochastic analysis using the generalized perturbation stable node-based smoothed finite element method. Engineering Analysis with Boundary Elements, 2016, 70: 40–55 https://doi.org/10.1016/j.enganabound.2016.06.002
100	Z QZhang, G R Liu. Temporal stabilization of the node-based smoothed finite element method and solution bound of linear elastostatics and vibration problems. Computational Mechanics, 2010, 46(2): 229–246 https://doi.org/10.1007/s00466-009-0420-5
101	Z QZhang, G R Liu. Upper and lower bounds for natural frequencies: a property of the smoothed finite element methods. International Journal for Numerical Methods in Engineering, 2010, 84(2): 149–178 https://doi.org/10.1002/nme.2889
102	LWang, D Han, G RLiu, XCui. Free vibration analysis of double-walled carbon nanotubes using the smoothed finite element method. International Journal of Computational Methods, 2011, 8(4): 879–890 https://doi.org/10.1142/S0219876211002873
103	ZHe, G Li, ZZhong, ACheng, GZhang, ELi. An improved modal analysis for three-dimensional problems using face-based smoothed finite element method. Acta Mechanica Solida Sinica, 2013, 26(2): 140–150 https://doi.org/10.1016/S0894-9166(13)60014-2
104	X YCui, G Y Li, G R Liu. An explicit smoothed finite element method (SFEM) for elastic dynamic problems. International Journal of Computational Methods, 2013, 10(1): 1340002 https://doi.org/10.1142/S0219876213400021
105	TNguyen-Thoi, P Phung-Van, TRabczuk, HNguyen-Xuan, CLe-Van. Free and forced vibration analysis using the n-sided polygonal cell-based smoothed finite element method (nCS-FEM). International Journal of Computational Methods, 2013, 10(01): 1340008 https://doi.org/10.1142/S0219876213400082
106	HFeng, X Y Cui, G Y Li, S Z Feng. A temporal stable node-based smoothed finite element method for three-dimensional elasticity problems. Computational Mechanics, 2014, 53(5): 859–876 https://doi.org/10.1007/s00466-013-0936-6
107	GYang, D Hu, GMa, DWan. A novel integration scheme for solution of consistent mass matrix in free and forced vibration analysis. Meccanica, 2016, 51(8): 1897–1911 https://doi.org/10.1007/s11012-015-0343-5
108	X YCui, X Hu, G YLi, G RLiu. A modified smoothed finite element method for static and free vibration analysis of solid mechanics. International Journal of Computational Methods, 2016, 13(6), 1650043 https://doi.org/10.1142/S0219876216500432
109	Z CHe, G R Liu, Z H Zhong, G Y Zhang, A G Cheng. Dispersion free analysis of acoustic problems using the alpha finite element method. Computational Mechanics, 2010, 46(6): 867–881 https://doi.org/10.1007/s00466-010-0516-y
110	Z CHe, G R Liu, Z H Zhong, G Y Zhang, A G Cheng. Coupled analysis of 3D structural–acoustic problems using the edge-based smoothed finite element method/finite element method. Finite Elements in Analysis and Design, 2010, 46(12): 1114–1121 https://doi.org/10.1016/j.finel.2010.08.003
111	L YYao, D J Yu, X Y Cui, X G Zang. Numerical treatment of acoustic problems with the smoothed finite element method. Applied Acoustics, 2010, 71(8): 743–753 https://doi.org/10.1016/j.apacoust.2010.03.006
112	Z CHe, A G Cheng, G Y Zhang, Z H Zhong, G R Liu. Dispersion error reduction for acoustic problems using the edge‐based smoothed finite element method (ES-FEM). International Journal for Numerical Methods in Engineering, 2011, 86(11): 1322–1338 https://doi.org/10.1002/nme.3100
113	Z CHe, G Y Li, Z H Zhong, A G Cheng, G Y Zhang, E Li, G RLiu. An ES-FEM for accurate analysis of 3D mid-frequency acoustics using tetrahedron mesh. Computers & Structures, 2012, 106–107: 125–134 https://doi.org/10.1016/j.compstruc.2012.04.014
114	WLi, Y Chai, MLei, G RLiu. Analysis of coupled structural-acoustic problems based on the smoothed finite element method (S-FEM). Engineering Analysis with Boundary Elements, 2014, 42: 84–91 https://doi.org/10.1016/j.enganabound.2013.08.009
115	ELi, Z C He, X Xu, G RLiu. Hybrid smoothed finite element method for acoustic problems. Computer Methods in Applied Mechanics and Engineering, 2015, 283: 664–688 https://doi.org/10.1016/j.cma.2014.09.021
116	Z CHe, G Y Li, G R Liu, A G Cheng, E Li. Numerical investigation of ES-FEM with various mass re-distribution for acoustic problems. Applied Acoustics, 2015, 89: 222–233 https://doi.org/10.1016/j.apacoust.2014.09.017
117	FWu, G R Liu, G Y Li, A G Cheng, Z C He, Z H Hu. A novel hybrid FS‐FEM/SEA for the analysis of vibro-acoustic problems. International Journal for Numerical Methods in Engineering, 2015, 102(12): 1815–1829 https://doi.org/10.1002/nme.4871
118	ZHe, G Li, GZhang, G RLiu, YGu, E Li. Acoustic analysis using a mass-redistributed smoothed finite element method with quadrilateral mesh. Engineering Computation, 2015, 32(8): 2292–2317 https://doi.org/10.1108/EC-10-2014-0219
119	Z CHe, E Li, G YLi, FWu, G R Liu, X Nie. Acoustic simulation using α-FEM with a general approach for reducing dispersion error. Engineering Analysis with Boundary Elements, 2015, 61: 241–253 https://doi.org/10.1016/j.enganabound.2015.07.018
120	GWang, X Y Cui, H Feng, G YLi. A stable node-based smoothed finite element method for acoustic problems. Computer Methods in Applied Mechanics and Engineering, 2015, 297: 348–370 https://doi.org/10.1016/j.cma.2015.09.005
121	GWang, X Y Cui, Z M Liang, G Y Li. A coupled smoothed finite element method (S-FEM) for structural-acoustic analysis of shells. Engineering Analysis with Boundary Elements, 2015, 61: 207–217 https://doi.org/10.1016/j.enganabound.2015.07.017
122	YChai, W Li, ZGong, TLi. Hybrid smoothed finite element method for two-dimensional underwater acoustic scattering problems. Ocean Engineering, 2016, 116: 129–141 https://doi.org/10.1016/j.oceaneng.2016.02.034
123	YChai, W Li, ZGong, TLi. Hybrid smoothed finite element method for two dimensional acoustic radiation problems. Appl Acoust., 2016, 103: 90–101
124	FWu, Z C He, G R Liu, G Y Li, A G Cheng. A novel hybrid ES-FE-SEA for mid-frequency prediction of Transmission losses in complex acoustic systems. Applied Acoustics, 2016, 111: 198–204 https://doi.org/10.1016/j.apacoust.2016.04.011
125	VKumar, R Metha. Impact simulations using smoothed finite element method. International Journal of Computational Methods, 2013, 10(4): 1350012 https://doi.org/10.1142/S0219876213500126
126	TNguyen-Thoi, G R Liu, H Nguyen-Xuan, CNguyen-Tran. Adaptive analysis using the node-based smoothed finite element method (NS-FEM). International Journal for Numerical Methods in Biomedical Engineering, 2011, 27(2): 198–218 https://doi.org/10.1002/cnm.1291
127	HNguyen-Xuan, C T Wu, G R Liu. An adaptive selective ES-FEM for plastic collapse analysis. European Journal of Mechanics-A/Solids, 2016, 58: 278–290 https://doi.org/10.1016/j.euromechsol.2016.02.005
128	M JKazemzadeh-Parsi, FDaneshmand. Solution of geometric inverse heat conduction problems by smoothed fixed grid finite element method. Finite Elements in Analysis and Design, 2009, 45(10): 599–611 https://doi.org/10.1016/j.finel.2009.03.008
129	ELi, G R Liu, V Tan. Simulation of hyperthermia treatment using the edge-based smoothed finite-element method. Numerical Heat Transfer, 2010, 57(11): 822–847 https://doi.org/10.1080/10407782.2010.489483
130	ELi, G R Liu, V Tan, Z CHe. An efficient algorithm for phase change problem in tumor treatment using αFEM. International Journal of Thermal Sciences, 2010, 49(10): 1954–1967 https://doi.org/10.1016/j.ijthermalsci.2010.06.003
131	VKumar. Smoothed finite element methods for thermo-mechanical impact problems. International Journal of Computational Methods, 2013, 10(1): 1340010 https://doi.org/10.1142/S0219876213400100
132	B YXue, S C Wu, W H Zhang, G R Liu. A smoothed FEM (S-FEM) for heat transfer problems. International Journal of Computational Methods, 2013, 10(1): 1340001 https://doi.org/10.1142/S021987621340001X
133	S ZFeng, X Y Cui, G Y Li. Analysis of transient thermo-elastic problems using edge-based smoothed finite element method. International Journal of Thermal Sciences, 2013, 65: 127–135 https://doi.org/10.1016/j.ijthermalsci.2012.10.007
134	S ZFeng, X Y Cui, G Y Li, H Feng, F XXu. Thermo-mechanical analysis of functionally graded cylindrical vessels using edge-based smoothed finite element method. International Journal of Pressure Vessels and Piping, 2013, 111–112: 302–309 https://doi.org/10.1016/j.ijpvp.2013.09.004
135	S ZFeng, X Y Cui, G Y Li. Transient thermal mechanical analyses using a face-based smoothed finite element method (FS-FEM). International Journal of Thermal Sciences, 2013, 74: 95–103 https://doi.org/10.1016/j.ijthermalsci.2013.07.002
136	ELi, Z C He, X Xu. An edge-based smoothed tetrahedron finite element method (ES-T-FEM) for thermomechanical problems. International Journal of Heat and Mass Transfer, 2013, 66: 723–732 https://doi.org/10.1016/j.ijheatmasstransfer.2013.07.063
137	SFeng, X Cui, GLi. Thermo-mechanical analyses of composite structures using face-based smoothed finite element method. International Journal of Applied Mechanics, 2014, 6(2): 1450020 https://doi.org/10.1142/S1758825114500203
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139	SFeng, X Cui, GLi. Thermo-mechanical analysis of composite pressure vessels using edge-based smoothed finite element method. International Journal of Computational Methods, 2014, 11(6): 1350089 https://doi.org/10.1142/S0219876213500898
140	X YCui, Z C Li, H Feng, S ZFeng. Steady and transient heat transfer analysis using a stable node-based smoothed finite element method. International Journal of Thermal Sciences, 2016, 110: 12–25 https://doi.org/10.1016/j.ijthermalsci.2016.06.027
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142	M SOlyaie, M R Razfar, E J Kansa. Reliability based topology optimization of a linear piezoelectric micromotor using the cell-based smoothed finite element method. Computer Modeling in Engineering & Sciences, 2011, 75(1): 43–87
143	M SOlyaie, M R Razfar, S Wang, E JKansa. Topology optimization of a linear piezoelectric micromotor using the smoothed finite element method. Computer Modeling in Engineering & Sciences, 2011, 82(1): 55–81
144	LChen, Y W Zhang, G R Liu, H Nguyen-Xuan, Z QZhang. A stabilized finite element method for certified solution with bounds in static and frequency analyses of piezoelectric structures. Computer Methods in Applied Mechanics and Engineering, 2012, 241–244: 65–81 https://doi.org/10.1016/j.cma.2012.05.018
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146	K S RAtia, A MHeikal, S S AObayya. Efficient smoothed finite element time domain analysis for photonic devices. Optics Express, 2015, 23(17): 22199–22213 https://doi.org/10.1364/OE.23.022199
147	Z CHe, G R Liu, Z H Zhong, G Y Zhang, A G Cheng. A coupled ES-FEM/BEM method for fluid-structure interaction problems. Engineering Analysis with Boundary Elements, 2011, 35(1): 140–147 https://doi.org/10.1016/j.enganabound.2010.05.003
148	Z QZhang, G R Liu, B C Khoo. Immersed smoothed finite element method for two dimensional fluid-structure interaction problems. International Journal for Numerical Methods in Engineering, 2012, 90(10): 1292–1320 https://doi.org/10.1002/nme.4299
149	JYao, G R Liu, D A Narmoneva, R B Hinton, Z Q Zhang. Immersed smoothed finite element method for fluid-structure interaction simulation of aortic valves. Computational Mechanics, 2012, 50(6): 789–804 https://doi.org/10.1007/s00466-012-0781-z
150	Z QZhang, G R Liu, B C Khoo. A three dimensional immersed smoothed finite element method (3D IS-FEM) for fluid-structure interaction problems. Computational Mechanics, 2013, 51(2): 129–150 https://doi.org/10.1007/s00466-012-0710-1
151	TNguyen-Thoi, P Phung-Van, TRabczuk, HNguyen-Xuan, CLe-Van. An application of the ES-FEM in solid domain for dynamic analysis of 2D fluid–solid interaction problems. International Journal of Computational Methods, 2013, 10(1): 1340003 https://doi.org/10.1142/S0219876213400033
152	SWang, B C Khoo, G R Liu, G X Xu, L Chen. Coupling GSM/ALE with ES-FEM-T3 for fluid-deformable structure interactions. Journal of Computational Physics, 2014, 276: 315–340 https://doi.org/10.1016/j.jcp.2014.07.016
153	TNguyen-Thoi, P Phung-Van, SNguyen-Hoang, QLieu-Xuan (2014) A smoothed coupled NS/nES-FEM for dynamic analysis of 2D fluid-solid interaction problems.
154	THe. On a partitioned strong coupling algorithm for modeling fluid-structure interaction. International Journal of Applied Mechanics, 2015, 7(2): 1550021 https://doi.org/10.1142/S1758825115500210
155	THe. Semi-implicit coupling of CS-FEM and FEM for the interaction between a geometrically nonlinear solid and an incompressible fluid. International Journal of Computational Methods, 2015, 12(5): 1550025 https://doi.org/10.1142/S0219876215500255
156	Z QZhang, G R Liu. Solution bound and nearly exact solution to nonlinear solid mechanics problems based on the smoothed FEM concept. Engineering Analysis with Boundary Elements, 2014, 42: 99–114 https://doi.org/10.1016/j.enganabound.2014.02.003
157	CJiang, Z Q Zhang, X Han, G RLiu. Selective smoothed finite element methods for extremely large deformation of anisotropic incompressible bio-tissues. International Journal for Numerical Methods in Engineering, 2014, 99(8): 587–610 https://doi.org/10.1002/nme.4694
158	YOnishi, K Amaya. A locking-free selective smoothed finite element method using tetrahedral and triangular elements with adaptive mesh rezoning for large deformation problems. International Journal for Numerical Methods in Engineering, 2014, 99(5): 354–371 https://doi.org/10.1002/nme.4684
159	CJiang, G R Liu, X Han, Z QZhang, WZeng. A smoothed finite element method for analysis of anisotropic large deformation of passive rabbit ventricles in diastole. International Journal for Numerical Methods in Biomedical Engineering, 2015, 31(1): 1–25 https://doi.org/10.1002/cnm.2697
160	YOnishi, R Iida, KAmaya. F-bar aided edge-based smoothed finite element method using tetrahedral elements for finite deformation analysis of nearly incompressible solids. International Journal for Numerical Methods in Engineering, 2015, 109: 771–773 https://doi.org/10.1002/nme.5337
161	ELi, J Chen, ZZhang, JFang, G R Liu, Q Li. Smoothed finite element method for analysis of multi-layered systems‐Applications in biomaterials. Computers & Structures, 2016, 168: 16–29 https://doi.org/10.1016/j.compstruc.2016.02.003
162	ELi, W H Liao. An efficient finite element algorithm in elastography. International Journal of Applied Mechanics, 2016, 8(3): 1650037 https://doi.org/10.1142/S175882511650037X
163	R PNiu, G R Liu, J H Yue. Development of a software package of smoothed finite element method (S-FEM) for solid mechanics problems. International Journal of Computational Methods, 2018, 15(3): 1845004 https://doi.org/10.1142/S0219876218450044
164	CJiang, X Han, Z QZhang, G RLiu, G JGao. A locking-free face-based S-FEM via averaging nodal pressure using 4-nodes tetrahedrons for 3D explicit dynamics and quasi-statics. International Journal of Computational Methods, 2018, 15(6): 1850043 https://doi.org/10.1142/S0219876218500433
165	JYue, G R Liu, M Li, RNiu. A cell-based smoothed finite element method for multi-body contact analysis using linear complementarity formulation. International Journal of Solids and Structures, 2018, 141–142: 110–126 https://doi.org/10.1016/j.ijsolstr.2018.02.016
166	C FDu, D G Zhang, L Li, G RLiu. A node-based smoothed point interpolation method for dynamic analysis of rotating flexible beams. Chinese Journal of Theoretical and Applied Mechanics, 2018, 34(2): 409–420 https://doi.org/10.1007/s10409-017-0713-4
167	WZeng, G R Liu. Smoothed finite element methods (S-FEM): an overview and recent developments. Archives of Computational Methods in Engineering, 2018, 25(2): 397–435 https://doi.org/10.1007/s11831-016-9202-3
168	Y HLi, M Li, G RLiu. A novel alpha smoothed finite element method for ultra-accurate solution using quadrilateral elements. International Journal of Computational Methods, 2018, 15(3): 1845008 https://doi.org/10.1142/S0219876218450081
169	XRong, R Niu, GLiu. Stability analysis of smoothed finite element methods with explicit method for transient heat transfer problems. International Journal of Computational Methods, 2018, 15(3): 1845005 https://doi.org/10.1142/S0219876218450056
170	J FZhang, R P Niu, Y F Zhang, C Q Wang, M Li, G RLiu. Development of SFEM-Pre: a novel preprocessor for model creation for the smoothed finite element method. International Journal of Computational Methods, 2018, 15(1): 1845002 https://doi.org/10.1142/S0219876218450020
171	CJiang, Z Q Zhang, X Han, GLiu, TLin. A cell-based smoothed finite element method with semi-implicit CBS procedures for incompressible laminar viscous flows. International Journal for Numerical Methods in Fluids, 2018, 86(1): 20–45 https://doi.org/10.1002/fld.4406
172	FWu, W Zeng, L YYao, G RLiu. A generalized probabilistic edge-based smoothed finite element method for elastostatic analysis of Reissner-Mindlin plates. Applied Mathematical Modelling, 2018, 53: 333–352 https://doi.org/10.1016/j.apm.2017.09.005
173	TNguyen-Thoi, T Bui-Xuan, G RLiu, TVo-Duy. Static and free vibration analysis of stiffened flat shells by a cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using three-node triangular elements. International Journal of Computational Methods, 2018, 15(6): 1850056 https://doi.org/10.1142/S0219876218500561
174	SBhowmick, G R Liu. On singular ES-FEM for fracture analysis of solids with singular stress fields of arbitrary order. Engineering Analysis with Boundary Elements, 2018, 86: 64–81 https://doi.org/10.1016/j.enganabound.2017.10.013
175	CJiang, X Han, G RLiu, Z QZhang, GYang, G J Gao. Smoothed finite element methods (S-FEMs) with polynomial pressure projection (P3) for incompressible solids. Engineering Analysis with Boundary Elements, 2017, 84: 253–269 https://doi.org/10.1016/j.enganabound.2017.07.022
176	G RLiu, M Chen, MLi. Lower bound of vibration modes using the node-based smoothed finite element method (NS-FEM). International Journal of Computational Methods, 2017, 14(4): 1750036 https://doi.org/10.1142/S0219876217500360
177	C FDu, D G Zhang, G R Liu. A cell-based smoothed finite element method for free vibration analysis of a rotating plate. International Journal of Computational Methods, 2017, 14(5): 1840003 https://doi.org/10.1142/S0219876218400030
178	YChai, W Li, G RLiu, ZGong, T Li. A superconvergent alpha finite element method (SαFEM) for static and free vibration analysis of shell structures. Computers & Structures, 2017, 179: 27–47 https://doi.org/10.1016/j.compstruc.2016.10.021
179	YLi, J H Yue, R P Niu, G R Liu. Automatic mesh generation for 3D smoothed finite element method (S-FEM) based on the weaken-weak formulation. Advances in Engineering Software, 2016, 99: 111–120 https://doi.org/10.1016/j.advengsoft.2016.05.012
180	J HYue, M Li, G RLiu, R PNiu. Proofs of the stability and convergence of a weakened weak method using PIM shape functions. Computers & Mathematics with Applications, 2016, 72(4): 933–951 https://doi.org/10.1016/j.camwa.2016.06.002
181	MChen, M Li, G RLiu. Mathematical basis of g spaces. International Journal of Computational Methods, 2016, 13(4): 1641007 https://doi.org/10.1142/S0219876216410073
182	ATootoonchi, A Khoshghalb, G RLiu, NKhalili. A cell-based smoothed point interpolation method for flow-deformation analysis of saturated porous media. Computers and Geotechnics, 2016, 75: 159–173 https://doi.org/10.1016/j.compgeo.2016.01.027
183	G RLiu. On partitions of unity property of nodal shape functions: rigid-body-movement reproduction and mass conservation. International Journal of Computational Methods, 2016, 13(02): 1640003 https://doi.org/10.1142/S021987621640003X
184	Z CHe, G Y Zhang, L Deng, ELi, G RLiu. Topology optimization using node-based smoothed finite element method. International Journal of Applied Mechanics, 2015, 7(06): 1550085 https://doi.org/10.1142/S1758825115500854
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