Smoothed finite element methods (S-FEM)Liu, G.R., 2010 Smoothed Finite Element Methods, CRC Press, . are a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methodsLiu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9 with the finite element method. S-FEM are applicable to solid mechanics as well as 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 inW. 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 ("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 theoryG.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 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).Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9 The S-PIM can be node-based (known as NS-PIM or LC-PIM),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. edge-based (ES-PIM),G.R. Liu, G.R. Zhang. Edge-based Smoothed Point Interpolation Methods. International Journal of Computational Methods, 5(4): 621-646, 2008 and cell-based (CS-PIM).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 The NS-PIM was developed using the so-called SCNI technique.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. It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free.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. 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.
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.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
List of S-FEM models
Node-based Smoothed FEM (NS-FEM)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. Edge-based Smoothed FEM (ES-FEM)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. Face-based Smoothed FEM (FS-FEM)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 Cell-based Smoothed FEM (CS-FEM)Liu GR, Dai KY, Nguyen-Thoi T (2007) A smoothed finite element method for mechanics problems. Computational Mechanics; 39: 859-877Dai KY, Liu GR (2007) Free and forced vibration analysis using the smoothed finite element method (SFEM). Journal of Sound and Vibration; 301: 803-820.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. Node/Edge-based Smoothed FEM (NS/ES-FEM)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, 2011Jiang 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 Alpha FEM method (Alpha FEM)Liu GR, Nguyen-Thoi T, Lam KY (2009) A novel FEM by scaling the gradient of strains with factor α (αFEM). Computational Mechanics; 43: 369-391Liu 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 Beta FEM method (Beta FEM)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: 48-67Zeng 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
Applications
S-FEM has been applied to solve the following physical problems:
Mechanics for solid structures and piezoelectrics;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, 2011Liu 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 Fracture mechanics and crack propagation;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, 2011Liu 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, 2010Zeng 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, 2013Zeng 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 Nonlinear and contact problems;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, 2011Jiang 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 Stochastic analysis;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 Heat transfer;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, 2010Wu 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 Structural acoustics;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, 2011He 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, 2011Zhang 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 Adaptive analysis;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 Limited analysis;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 Crystal plasticity modeling.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
Basic Formulation of S-FEM The fundamental problem addressed by SFEM is typically the solution of Poisson's equation with Dirichlet boundary conditions, given as follows:
Δu+f=0 in Ω, u=g on ΓD
where Ω is the domain and Γ is its boundary, consisting of ΓD=Γ. Here, u: Ω→R is the trial solution, f: Ω→R is a given function, and g represents Dirichlet boundary conditions.