On the Matlab S-FEM code for 2D problems using

T3 and Q4 elements

These codes were developed by Liu, Nguyen and workers. The detailed

theoretical background, formulation and implementation procedure are given

in the book:

Liu, G. R. and Nguyen Thoi Trung, Smoothed Finite Element Method, CRC

press, Boca Raton, USA, 2010.

Important note: The authors decided to release the source codes free of charge with the hope that the S-

FEM technique can be applied to more problems and can be further developed into even more powerful

methods. The authors are not be able to provide any services or user-guides, but appreciate very much

your feedback on errors and suggestions, so that we can improve these codes/methods in the future. If the

idea, method, and any part of these codes are used in anyway, the users are required to cite the book and

the following related original papers of the authors:

• Liu, G. R., The Finite element method – a practical course, Elsevier (BH), UK. 2003.

• Liu, G. R. and Nguyen Thoi Trung, Smoothed Finite Element Method, CRC press, Boca Raton,

USA, 2010.

• Liu GR, Dai KY, Nguyen-Thoi T (2007) A smoothed finite element method for mechanics

problems. Computational Mechanics; 39: 859-877.

• 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.

• 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.

• Nguyen-Thoi T, Liu GR, Dai KY, Lam KY (2007) Selective Smoothed Finite Element

Method. Tsinghua Science and Technology; 12(5): 497-508.

• Liu GR, Nguyen-Thoi T, Dai KY, Lam KY (2007) Theoretical aspects of the smoothed finite

element method (SFEM). International Journal for Numerical Methods in Engineering; 71: 902-

930.

• Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, Dai KY, Lam KY (2009) On the essence and the

evaluation of the shape functions for the smoothed finite element method (SFEM) (Letter to

Editor). International Journal for Numerical Methods in Engineering; 77: 1863-1869.

• 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.

• Nguyen-Thoi T, Liu GR, Nguyen-Xuan H (2009) Additional properties of the node-based

smoothed finite element method (NS-FEM) for solid mechanics problems. International Journal

of Computational Methods, 6(4): 633-666.

• Nguyen-Thoi T, Liu GR, Nguyen-Xuan H, Nguyen-Tran C (2010) Adaptive analysis using the

node-based smoothed finite element method (NS-FEM). International Journal for Numerical

Methods in Biomedical Engineering; in press, doi: 10.1002/cnm.1291.

• 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.

• Nguyen-Xuan H, Liu GR, Thai-Hoang C, Nguyen-Thoi T (2009) An edge-based

smoothed finite element method with stabilized discrete shear gap technique for analysis

of Reissner-Mindlin plates. Computer Methods in Applied Mechanics and Engineering;

199: 471-489.

• Nguyen-Xuan H, Liu GR, Nguyen-Thoi T, Nguyen-Tran C (2009) An edge-based smoothed

finite element method for analysis of two-dimensional piezoelectric structures, Smart Materials

and Structures 18 (6). no. 065015.

• Nguyen-Thoi T, Liu GR, Vu-Do HC, Nguyen-Xuan H (2009) An edge-based smoothed finite

element method (ES-FEM) for visco-elastoplastic analyses of 2D solids using triangular mesh.

Computational Mechanics; 45: 23-44.

• Nguyen-Thoi T, Liu GR, Nguyen-Xuan H (2010) An n-sided polygonal edge-based smoothed

finite element method (nES-FEM) for solid mechanics. International Journal for Numerical

Methods in Biomedical Engineering; doi:10.1002/cnm.1375.

• 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.

• Nguyen-Thoi T, Liu GR, Vu-Do HC, Nguyen-Xuan H (2009) A face-based smoothed finite

element method (FS-FEM) for visco-elastoplastic analyses of 3D solids using tetrahedral mesh.

Computer Methods in Applied Mechanics and Engineering; 198: 3479-3498.

• 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.

• Liu GR, Nguyen-Thoi T, Lam KY (2008) A novel Alpha Finite Element Method (αFEM) for

exact solution to mechanics problems using triangular and tetrahedral elements. Computer

Methods in Applied Mechanics and Engineering; 197: 3883-3897.

• 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.

• Liu GR, Chen L, Nguyen-Thoi T, Zeng K, G. Y. Zhang (2010) A novel singular node-based

smoothed finite element method (NS-FEM) for upper bound solutions of cracks. International

Journal for Numerical Methods in Engineering; doi: 10.1002/nme.2868.

• Liu GR, Nourbakhshnia N, Chen L, Zhang YW (2010) 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 7 (1), pp. 191-214.

• Liu GR, Nourbakhshnia N, Zhang YW (2010) A novel singular ES-FEM method for simulating

singular stress fields near the crack tips for linear fracture problems, Engineering Fracture

Mechanics, (in press).

• Wu SC, Liu GR, Zhang HO, Zhang GY (2009) A node-based smoothed point interpolation

method (NS-PIM) for thermoelastic problems with solution bounds. International Journal of Heat

and Mass Transfer; 52(5-6): 1464-1471.

• Wu SC, Liu GR, Zhang HO, Zhang GY (2008) A node-based smoothed point interpolation

method (NS-PIM) for three-dimensional thermoelastic problems. Numerical Heat Transfer, Part

A: Applications; 54(12): 1121-1147.

• Wu SC, Liu GR, Zhang HO, Xu X, Li ZR (2009) A node-based smoothed point interpolation

method (NS-PIM) for three-dimensional heat transfer problems. International Journal of Thermal

Sciences; 48(7): 1367-1376, 2009.

A brief overall view on the use of these codes is given as follows. Some guidance is given in the

commend lines of the source codes.

Main file : main_2D_problems.m

There are 3 major processes:

• Preprocess

• Solution

• Postprocess

1. Preprocessor 1.1. Choice of examples: there are two examples (with the analytical solutions)

1) cantilever (a rectangular cantilever loaded at the end)

2) platehole (infinite plate with a circular hole)

1.2. Choice of the methods: there are 7 methods including

1) FEM_T3 (FEM using 3-node triangular elements)

2) FEM_Q4 (FEM using 4-node quadrilateral elements)

3) CSFEM_Q4 (cell-based smoothed FEM using 4-node quadrilateral elements)

Input the number of smoothing cells per element (1, 2, 3, 4, 8, 16)

4) NSFEM_T3 (node-based smoothed FEM using 3-node triangular elements)

5) NSFEM_Q4 (node-based smoothed FEM using 4-node quadrilateral elements)

6) ESFEM_T3 (cell-based smoothed FEM using 3-node triangular elements)

7) Alpha_FEM_T3 (alpha FEM using 3-node triangular elements)

Input the value of parameter alpha [ ]0,1α ∈

1.3. Input the initial data of the problem: density of element mesh

1.4. Compute the necessary data from the initial data including: coordinates of nodes

(gcoord), element connectivity (ele_nods), boundary conditions (bcdof, bcval) and force vector (ff).

2. Solution procedure 2.1. Compute the stiffness matrix: K

For NSFEM_T3 and ESFEM_T3: there are two schemes (optional):

1st scheme: basing on the boundary of smoothing domains

NSFEM_T3: [K]=cal_K_NSFEM_T3_boun(nod_adjele,area_nod);

ESFEM_T3: [K]=cal_K_ESFEM_T3_boun(edge_data,area_edg);

2nd scheme: taking average of the adjacent elements of nodes or edges

NSFEM_T3: [K]=cal_K_NSFEM_T3_aver(nod_adjele,area_nod,area_T3);

ESFEM_T3: [K]=cal_K_ESFEM_T3_aver(edge_data,area_edg,area_T3);

For CSFEM_Q4 and ESFEM_T3: there is also the selective scheme used for nearly

incompressible materials

CSFEM_Q4: [K]=cal_K_CSFEM_Q4_selective(nSD);

ESFEM_T3: [K]=cal_K_ESFEM_T3_selective(edge_data,area_T3,area_edg);

2.2. Apply the boundary condition: [K,ff]=apply_bcdof(K,ff,bcdof,bcval);

2.3. Solve the system of equations to obtain the nodal displacement vector

 

3. Postprocess 3.1. Compute the strain energy of system (energy)

3.2. Compute error norms: in displacement (norm_disp), in energy (norm_E) and

recovery energy (norm_Ereco)

3.3. Output error norms and plot the results of displacement and stresses.