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'''Smoothed finite element methods''' ('''S-FEM''')<ref>Liu, G.R., 2010 ''Smoothed Finite Element Methods'', CRC Press, {{ISBN|978-1-4398-2027-8}}.</ref> are a particular class of [[Numerical analysis|numerical simulation algorithms]] for the simulation of physical phenomena. It was developed by combining [[Meshfree methods|meshfree methods]]<ref>Liu, G.R. 2nd edn: 2009 ''Mesh Free Methods'',  CRC Press.  978-1-4200-8209-9</ref> with the [[Finite element method|finite element method]]. S-FEM are applicable to [[Physics:Solid mechanics|solid mechanics]] as well as [[Physics:Fluid dynamics|fluid dynamics]] problems, although so far they have mainly been applied to the former.
==Description==
The essential idea in the S-FEM is to use a finite element mesh (in particular triangular mesh) to construct numerical models of good performance.  This is achieved by modifying the compatible strain field, or construct a strain field using only the displacements, hoping a Galerkin model using the modified/constructed strain field can deliver some good properties. Such a modification/construction can be performed within elements but more often beyond the elements (meshfree concepts): bring in the information from the neighboring elements. Naturally, the strain field has to satisfy certain conditions, and the standard Galerkin weak form needs to be modified accordingly to ensure the stability and convergence. A comprehensive review of S-FEM covering both methodology and applications can be found in<ref>W. Zeng, G.R. Liu. Smoothed finite element methods (S-FEM): An overview and recent developments. ''Archives of Computational Methods in Engineering'', 2016, doi: 10.1007/s11831-016-9202-3</ref> ("Smoothed Finite Element Methods (S-FEM): An Overview and Recent Developments").

==History==
The development of S-FEM started from the works on meshfree methods, where the so-called weakened weak (W2) formulation based on the G space theory<ref>G.R. Liu. A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory and Part II applications to solid mechanics problems. ''International Journal for Numerical Methods in Engineering'', 81: 1093-1126, 2010</ref> were developed. The W2 formulation offers possibilities for formulate various (uniformly) "soft" models that works well with triangular meshes. Because triangular mesh can be generated automatically, it becomes much easier in re-meshing and hence automation in modeling and simulation. In addition, W2 models can be made soft enough (in uniform fashion) to produce upper bound solutions (for force-driving problems). Together with stiff models (such as the fully compatible FEM models), one can conveniently bound the solution from both sides.  This allows easy error estimation for generally complicated problems, as long as a triangular mesh can be generated. Typical W2 models are the Smoothed Point Interpolation Methods (or S-PIM).<ref>Liu, G.R. 2nd edn: 2009 ''Mesh Free Methods'', CRC Press.  978-1-4200-8209-9</ref> The S-PIM can be node-based (known as NS-PIM or LC-PIM),<ref>Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li GY and Han X, A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems, ''[[International Journal of Computational Methods]]'', 2(4): 645-665, 2005.</ref> edge-based (ES-PIM),<ref>G.R. Liu, G.R. Zhang. Edge-based Smoothed Point Interpolation Methods. ''International Journal of Computational Methods'', 5(4): 621-646, 2008</ref> and cell-based (CS-PIM).<ref>G.R. Liu, G.R. Zhang. A normed G space and weakened weak (W2) formulation of a cell-based Smoothed Point Interpolation Method. ''International Journal of Computational Methods'', 6(1): 147-179, 2009</ref> The NS-PIM was developed using the so-called SCNI technique.<ref>Chen, J. S., Wu, C. T., Yoon, S. and You, Y. (2001). A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Meth. Eng. 50: 435–466.</ref> It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free.<ref>G. R. Liu and G. Y. Zhang. Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM). ''International Journal for Numerical Methods in Engineering'', 74: 1128-1161, 2008.</ref> The ES-PIM is found superior in accuracy, and CS-PIM behaves in between the NS-PIM and ES-PIM. Moreover, W2 formulations allow the use of polynomial and radial basis functions in the creation of shape functions (it accommodates the discontinuous displacement functions, as long as it is in G1 space), which opens further rooms for future developments.{{Citation needed|date=March 2023}}

The S-FEM is largely the linear version of S-PIM, but with most of the properties of the S-PIM and much simpler. It has also variations of NS-FEM, ES-FEM and CS-FEM. The major property of S-PIM can be found also in S-FEM.<ref>Zhang ZQ, Liu GR, Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods, ''International Journal for Numerical Methods in Engineering''   Vol. 84   Issue: 2, 149-178, 2010</ref>

==List of S-FEM models==

* Node-based Smoothed FEM (NS-FEM)<ref>Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, Lam KY (2009) A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Computers and Structures; 87: 14-26.</ref>
* Edge-based Smoothed FEM (ES-FEM)<ref>Liu GR, Nguyen-Thoi T, Lam KY (2009) An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses in solids. Journal of Sound and Vibration; 320: 1100-1130.</ref>  
* Face-based Smoothed FEM (FS-FEM)<ref>Nguyen-Thoi T, Liu GR, Lam KY, GY Zhang (2009) A Face-based Smoothed Finite Element Method (FS-FEM) for 3D linear and nonlinear solid mechanics problems using 4-node tetrahedral elements. ''International Journal for Numerical Methods in Engineering''; 78: 324-353</ref>
* Cell-based Smoothed FEM (CS-FEM)<ref>Liu GR, Dai KY, Nguyen-Thoi T (2007) A smoothed finite element method for mechanics problems. Computational Mechanics; 39: 859-877</ref><ref>Dai KY, Liu GR (2007) Free and forced vibration analysis using the smoothed finite element method (SFEM). Journal of Sound and Vibration; 301: 803-820.</ref><ref>Dai KY, Liu GR, Nguyen-Thoi T (2007) An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics. Finite Elements in Analysis and Design; 43: 847－860.</ref> 
* Node/Edge-based Smoothed FEM (NS/ES-FEM)<ref name="auto">Li Y, Liu GR, Zhang GY, An adaptive NS/ES-FEM approach for 2D contact problems using triangular elements, ''Finite Elements in Analysis and Design''   Vol.47   Issue: 3, 256-275, 2011</ref><ref>Jiang C, Zhang ZQ, Liu GR, Han X, Zeng W, An edge-based/node-based selective smoothed finite element method using tetrahedrons for cardiovascular tissues, ''Engineering Analysis with Boundary Elements''   Vol.59, 62-77, 2015</ref> 
* Alpha FEM method (Alpha FEM)<ref>Liu GR, Nguyen-Thoi T, Lam KY (2009) A novel FEM by scaling the gradient of strains with factor α (αFEM). Computational Mechanics; 43: 369-391</ref><ref>Liu GR, Nguyen-Xuan H, Nguyen-Thoi T, Xu X (2009) A novel weak form and a superconvergent alpha finite element method (SαFEM) for mechanics problems using triangular meshes. Journal of Computational Physics; 228: 4055-4087</ref>
* Beta FEM method (Beta FEM)<ref>Zeng W, Liu GR, Li D, Dong XW (2016) A smoothing technique based beta finite element method (βFEM) for crystal plasticity modeling. Computers and Structures; 162: 48-67</ref><ref>Zeng W, Liu GR, Jiang C, Nguyen-Thoi T, Jiang Y (2016) A generalized beta finite element method with coupled smoothing techniques for solid mechanics. Engineering Analysis with Boundary Elements; 73: 103-119</ref>

==Applications==

S-FEM has been applied to solve the following physical problems:

#Mechanics for solid structures and piezoelectrics;<ref>Cui XY, Liu GR, Li GY, et al. A thin plate formulation without rotation DOFs based on the radial point interpolation method and triangular cells, ''International Journal for Numerical Methods in Engineering''   Vol.85   Issue: 8 , 958-986, 2011</ref><ref>Liu GR, Nguyen-Xuan H, Nguyen-Thoi T, A theoretical study on the smoothed FEM (S-FEM) models: Properties, accuracy and convergence rates, ''International Journal for Numerical Methods in Engineering''   Vol. 84   Issue: 10, 1222-1256, 2010</ref>
#Fracture mechanics and crack propagation;<ref>Liu GR, Nourbakhshnia N, Zhang YW, A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems, ''Engineering Fracture Mechanics''   Vol.78   Issue: 6   Pages: 863-876, 2011</ref><ref>Liu GR, Chen L, Nguyen-Thoi T, et al. A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems, ''International Journal for Numerical Methods in Engineering''   Vol.83   Issue: 11, 1466-1497, 2010</ref><ref>Zeng W, Liu GR, Kitamura Y, Nguyen-Xuan H. "A three-dimensional ES-FEM for fracture mechanics problems in elastic solids", ''Engineering Fracture Mechanics''   Vol. 114, 127-150, 2013</ref><ref>Zeng W, Liu GR, Jiang C, Dong XW, Chen HD, Bao Y, Jiang Y. "An effective fracture analysis method based on the virtual crack closure-integral technique implemented in CS-FEM", ''Applied Mathematical Modelling''   Vol. 40,   Issue: 5-6, 3783-3800, 2016</ref>
#Nonlinear and contact problems;<ref>Zhang ZQ, Liu GR, An edge-based smoothed finite element method (ES-FEM) using 3-node triangular elements for 3D non-linear analysis of spatial membrane structures, ''International Journal for Numerical Methods in Engineering'',  Vol. 86   Issue: 2    135-154, 2011</ref><ref>Jiang C, Liu GR, Han X, Zhang ZQ, Zeng W, 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'',  Vol. 31   Issue: 1,1-25, 2015</ref>
#Stochastic analysis;<ref>Liu GR, Zeng W, Nguyen-Xuan H. Generalized stochastic cell-based smoothed finite element method (GS_CS-FEM) for solid mechanics, ''Finite Elements in Analysis and Design''   Vol.63, 51-61, 2013</ref>
#Heat transfer;<ref>Zhang ZB, Wu SC, Liu GR, et al. Nonlinear Transient Heat Transfer Problems using the Meshfree ES-PIM, ''International Journal of Nonlinear Sciences and Numerical Simulation''   Vol.11   Issue: 12, 1077-1091, 2010</ref><ref>Wu SC, Liu GR, Cui XY, et al. An edge-based smoothed point interpolation method (ES-PIM) for heat transfer analysis of rapid manufacturing system, ''International Journal of Heat and Mass Transfer'' Vol.53 Issue: 9-10, 1938-1950, 2010</ref>
#Structural acoustics;<ref>He ZC, Cheng AG, Zhang GY, et al. Dispersion error reduction for acoustic problems using the edge-based smoothed finite element method (ES-FEM), ''International Journal for Numerical Methods in Engineering''   Vol. 86   Issue: 11   Pages: 1322-1338, 2011</ref><ref>He ZC, Liu GR, Zhong ZH, et al. A coupled ES-FEM/BEM method for fluid-structure interaction problems, Engineering Analysis with Boundary Elements   Vol. 35   Issue: 1, 140-147, 2011</ref><ref>Zhang ZQ, Liu GR, Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods, ''International Journal for Numerical Methods in Engineering''   Vol.84   Issue: 2,149-178, 2010</ref>
#Adaptive analysis;<ref>Nguyen-Thoi T, Liu GR, Nguyen-Xuan H, et al. Adaptive analysis using the node-based smoothed finite element method (NS-FEM), ''International Journal for Numerical Methods in Biomedical Engineering''   Vol. 27   Issue: 2, 198-218, 2011</ref><ref name="auto"/>
#Limited analysis;<ref>Tran TN, Liu GR, Nguyen-Xuan H, et al. An edge-based smoothed finite element method for primal-dual shakedown analysis of structures, ''International Journal for Numerical Methods in Engineering''   Vol.82   Issue: 7, 917-938, 2010</ref>
#Crystal plasticity modeling.<ref>Zeng W, Larsen JM, Liu GR. Smoothing technique based crystal plasticity finite element modeling of crystalline materials, ''International Journal of Plasticity''   Vol.65, 250-268, 2015</ref>

==See also==
* [[Finite element method]]
* [[Meshfree methods]]
* [[Weakened weak form]]
* [[Loubignac iteration]]
==References==
{{reflist}}

==External links==
* [https://link.springer.com/article/10.1007/s11831-016-9202-3/fulltext.html]

{{Numerical PDE}}

[[Category:Continuum mechanics]]
[[Category:Finite element method]]
[[Category:Numerical differential equations]]
[[Category:Partial differential equations]]
[[Category:Structural analysis]]

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