Smoothed finite element method

Short description: Class of numerical simulation algorithms

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[2] 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.

## Contents

* 1 Description
* 2 History
* 3 List of S-FEM models
* 4 Applications
* 5 Basic Formulation of S-FEM
* 5.1 Weak Formulation
* 5.2 S-FEM Formulation
* 5.3 Challenges
* 6 B-spline S-FEM (BFSEM)
* 7 See also
* 8 References
* 9 External links

## 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[3] ("Smoothed Finite Element Methods (S-FEM): An Overview and Recent Developments").

## History

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.[4]

## List of S-FEM models

* Node-based Smoothed FEM (NS-FEM)[5]
* Edge-based Smoothed FEM (ES-FEM)[6]
* Face-based Smoothed FEM (FS-FEM)[7]
* Cell-based Smoothed FEM (CS-FEM)[8][9][10]
* Node/Edge-based Smoothed FEM (NS/ES-FEM)[11][12]
* Alpha FEM method (Alpha FEM)[13][14]
* Beta FEM method (Beta FEM)[15][16]

## Applications

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

1. Mechanics for solid structures and piezoelectrics;[17][18]
2. Fracture mechanics and crack propagation;[19][20][21][22]
3. Nonlinear and contact problems;[23][24]
4. Stochastic analysis;[25]
5. Heat transfer;[26][27]
6. Structural acoustics;[28][29][30]
7. Adaptive analysis;[31][11]
8. Limited analysis;[32]
9. Crystal plasticity modeling.[33]

## 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.

S-FEM involves discretizing the domain Ω using finite element meshes, which can be global or local. The global mesh represents the entire domain, while the local mesh is used to discretize regions requiring high resolution within the global domain. The local domain is assumed to be included in the global domain (ΩL​⊆ΩG).

### Weak Formulation

The weak form of the problem is derived by multiplying the equation by suitable test functions and integrating over the domain. In SFEM, the weak form is expressed as follows: Given f and g, find u∈U such that for all w∈V,

aΩ​(w,u)=LΩ​(w)

where aΩ is a bilinear form, and LΩ is a linear functional.

### S-FEM Formulation

In S-FEM, the trial solution u and test functions w are defined separately for the global (ΩG) and local (ΩL) domains. The trial solution spaces UG, UL and test function spaces VG, VL are defined accordingly. The weak form in the S-FEM formulation becomes:

aΩ′​(w,u)=LΩ′​(w)

where aΩ′​(⋅,⋅) and LΩ′​(⋅) are modified bilinear forms and linear functionals, respectively, to accommodate the S-FEM approach.

### Challenges

One of the primary challenges of S-FEM is the difficulty in exact integration of the submatrices representing the relationship between global and local meshes (KGL and KLG). Additionally, the matrix K can become singular, posing numerical challenges in solving the resulting linear algebraic equations.

These challenges and potential solutions are discussed in detail in the literature, aiming to improve the efficiency and accuracy of S-FEM for various applications.[34]

## B-spline S-FEM (BFSEM)

S-FEM can reasonably model an analytical domain by superimposing meshes with different spatial resolutions, it has intrinsic advantages of local high accuracy, low computation time, and simple meshing procedure. However, it has disadvantages such as accuracy of numerical integration and matrix singularity. Although several additional techniques have been proposed to mitigate these limitations, they are computationally expensive or ad-hoc, and detract from the method's strengths. These issues can be address by incorporating cubic B-spline functions with C squared continuity across element boundaries as the global basis function. To avoid matrix singularity, applying different basis functions to different meshes. In a recent study Lagrange basis functions were used as local basis functions. With this method the numerical integration can be calculated with sufficient accuracy without any additional techniques used in conventional S-FEM. Furthermore, the proposed method avoids matrix singularity and is superior to conventional methods in terms of convergence for solving linear equations. Therefore, the proposed method has the potential to reduce computation time while maintaining a comparable accuracy to conventional S-FEM.[34]

## See also

* Finite element method
* Meshfree methods
* Weakened weak form
* Loubignac iteration

## 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. ↑ 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
4. ↑ 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
5. ↑ 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.
6. ↑ 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.
7. ↑ 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
8. ↑ Liu GR, Dai KY, Nguyen-Thoi T (2007) A smoothed finite element method for mechanics problems. Computational Mechanics; 39: 859-877
9. ↑ 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.
10. ↑ 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.
11. ↑ 11.0 11.1 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
12. ↑ 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
13. ↑ 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
14. ↑ 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
15. ↑ 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
16. ↑ 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
17. ↑ 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
18. ↑ 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
19. ↑ 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
20. ↑ 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
21. ↑ 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
22. ↑ 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
23. ↑ 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
24. ↑ 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
25. ↑ 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
26. ↑ 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
27. ↑ 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
28. ↑ 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
29. ↑ 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
30. ↑ 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
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. ↑ 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
33. ↑ 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
34. ↑ 34.0 34.1 Magome, Nozomi; Morita, Naoki; Kaneko, Shigeki; Mitsume, Naoto (January 2024). "Higher-continuity s-version of finite element method with B-spline functions". Journal of Computational Physics 497. doi:10.1016/j.jcp.2023.112593. ISSN 0021-9991. Bibcode: 2024JCoPh.49712593M.

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