# Smoothed finite element method

Smoothed Finite Element methods (S-FEM) [1] are a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methods with the finite element method. S-FEM are applicable to solid mechanics as well as fluid dynamics, but so far mainly applied for solid mechanics problems.

## 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 weakform needs to be modified accordingly to ensure the stability and convergence.

## History

The development of S-FEM started from the works on meshfree methods,[2] where the so-called weakened weak (W2) formulation based on the G space theory [3] 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).[4] The S-PIM can be node-based (known as NS-PIM or LC-PIM),[5] edge-based (ES-PIM),[6] and cell-based (CS-PIM).[7] The NS-PIM was developed using the so-called SCNI technique.[8] It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free.[9] 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.[10]

## Applications

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

1) Mechanics for solids, structures and piezoelectrics;[20][21]

2) Fracture mechanics and crack propagation;[22][23][24]

3) Heat transfer;[25][26]

4) Structural acoustics;[27][28][29]

5) Nonlinear and contact problems;[30]

7) Phase change problem;[33]

8) Limited analysis.[34]

## References

1. ^ Liu, G.R., 2010 Smoothed Finite Element Methods, CRC Press, ISBN 978-1-4398-2027-8.
2. ^ Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9
3. ^ 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
4. ^ Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9
5. ^ 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.
6. ^ G.R. Liu, G.R. Zhang. Edge-based Smoothed Point Interpolation Methods. International Journal of Computational Methods, 5(4): 621-646, 2008
7. ^ 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
8. ^ 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.
9. ^ 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.
10. ^ 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
11. ^ 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.
12. ^ 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.
13. ^ 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
14. ^ Liu GR, Dai KY, Nguyen-Thoi T (2007) A smoothed finite element method for mechanics problems. Computational Mechanics; 39: 859-877
15. ^ 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.
16. ^ 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.
17. ^ 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
18. ^ 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
19. ^ 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
20. ^ 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
21. ^ 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
22. ^ 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
23. ^ 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
24. ^ Liu GR, Nourbakhshnia N, Chen L, et al. "A Novel General Formulation for Singular Stress Field Using the Es-Fem Method for the Analysis of Mixed-Mode Cracks", International Journal OF Computational Methods Vol. 7 Issue: 1, 191-214, 2010
25. ^ 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
26. ^ 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
27. ^ 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
28. ^ 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
29. ^ 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
30. ^ 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
31. ^ 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
32. ^ 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
33. ^ Li E, Liu GR, Tan V, et al.An efficient algorithm for phase change problem in tumor treatment using alpha FEM, International Journal of Thermal Sciences Vol.49 Issue: 10, 1954-1967, 2010
34. ^ 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
35. ^ Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9