Skip to main content Accessibility help

Voronoi Based Discrete Least Squares Meshless Method for Assessment of Stress Field in Elastic Cracked Domains

  • M. Labibzadeh (a1)


A new approach in meshless methods has been introduced for stress assessment around a crack in two-dimensional elastic solids. This method with the name VDLSM (Voronoi Based Discrete Least Squares Meshless) is a pure meshless method which does not implement nodal mesh for trial and test functions. Rather, it uses a collection of the scattered nodal points and implements the discrete least squares approach to discretize the strong form of the governing differential equations on the domain of interest. This can reduce considerably the pre-processing cost of the analysis. Meshless methods generally are faced with some difficulty to accommodate the stress analysis in the vicinity of sharply concave surfaces such as cracks. Some techniques have been used to fix that problem such as visibility, transparency and diffraction, but these methods require some additional back corrective analyses for those parts of the domain located near the crack, which lengthen the time consumed for the solution and, moreover, do not provide the desired accuracy for the unknowns in these regions. VDLSM as a new, straightforward and easy applicable method has been suggested here for overcoming such deficiency using the algorithm of the Voronoi tessellation for constructing the Moving Least Squares (MLS) shape functions.


Corresponding author

*Corresponding author (


Hide All
1.Belytschko, T. and Black, T., “Elastic Crack Growth in Finite Elements with Minimal Remeshing,” International Journal of Numerical Methods in Engineering, 45, pp. 601620 (1999).
2.Moes, N., Dolbow, J. and Belytschko, T., “A Finite Element Method for Crack Growth Without Remeshing,” International Journal of Numerical Methods in Engineering, 46, pp. 131150 (1999).
3.Motamedi, D. and Mohammadi, S., “Fracture Analysis of Composites by Time Independent Moving-Crack Orthotropic XFEM,” International Journal of Mechanical Science, 54, pp. 2037 (2011).
4.Rabczuk, T., Bordas, S. and Zi, G., “On Three-Dimensional Modelling of Crack Growth Using Partition of Unity Methods,” Computers and Structures, 88, pp. 14191443 (2010).
5.Richardson, C., Hegemann, J., Sifakis, E., Hellrung, J. and Teran, J., “An XFEM Method for Modeling Geometrically Elaborate Crack Propagation,” International Journal of Numerical Methods in Engineering, 88, pp. 10421065 (2011).
6.Azadi, H. and Khoei, A. R., “Numerical Simulation of Multiple Crack Growth in Brittle Materials with Adaptive Remeshing,” International Journal of Numerical Methods in Engineering, 85, pp. 10171048 (2011).
7.Belytschko, T., Lu, Y. Y. and Gu, L., “Crack Propagation by Element-Free Galerkin Methods,” Engineering Fracture Mechanics, 51, pp. 295315 (1995).
8.Belytschko, T., Krongauz, Y., Organ, D., Fleming, M. and Krysl, P., “Meshless Methods: An Overview and Recent Developments,” Computer Methods in Applied Mechanical Engineering, 139, pp. 347 (1996).
9.Fleming, M., Chu, Y. A., Moran, B. and Belytschko, T., “Enriched Element-Free Galerkin Methods for Crack Tip Fields,” International Journal of Numerical Methods in Engineering, 40, pp. 14831504 (1997).
10.Ghorashi, S. S., Mohammadi, S. and Sabbagh-Yazdi, S. R., “Orthotropic Enriched Element Free Galerkin Method for Fracture Analysis of Composites,” Engineering Fracture Mechanics, 78, pp. 19061927 (2011).
11.Lucy, L. B., “A Numerical Approach to the Testing of the Fission Hypothesis,” Astron Journal, 82, pp. 10131024 (1977).
12.Gingold, R. A. and Monaghan, J. J., “Smoothed Particle Hydrodynamics-Theory and Application to Non-Spherical Stars,” Monthly Notices of the Royal Astronomical Society, 181, pp. 375389 (1977).
13.Batra, R. C. and Zhang, G. M., “Search Algorithm, and Simulation of Elastodynamic Crack Propagation by Modified Smoothed Particle Hydrodynamics (MSPH) Method,” Computational Mechanics, 40, pp. 531546 (2007).
14.Das, R. and Cleary, P. W., “Simulating Brittle Fracture of Rocks Using Smoothed Particle Hydrodynamics,” AIP Conference Proceeding, 1138 (2009).
15.Liu, W. K., Jun, S. and Zhang, Y. F., “Reproducing Kernel Particle Methods,” International Journal of Numerical Methods, 20, pp. 10811106 (1995).
16.Atluri, S. and Zhu, T., “A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics,” Computational Mechanics, 22, pp. 117127 (1998).
17.Atluri, S. and Zhu, T. L., “The Meshless Local Petrov-Galerkin (MLPG) Approach for Solving Problems in Elasto-Statics,” Computational Mechanics, 25, pp. 169179 (2000).
18.Lin, H. and Atluri, S., “Meshless Local Petrov-Galerkin (MLPG) Method for Convection Diffusion Problems,” Computational Model of Engineering Science, 1, pp. 4560 (2000).
19.RezaeiMojdehi, A., Darvizeh, A. and Basti, A., “Three Dimensional Static and Dynamic Analysis of Thick Plates by the Meshless Local Petrov-Galerkin (MLPG) Method Under Different Loading Conditions,” Computational Mathematics in Civil Engineering, 2, pp. 6581 (2011).
20.Liszka, T. J., Duarte, C. and Tworzydlo, W. W., “Hp-Meshless Cloud Method,” Computational Methods in Applied Mechanical Engineering, 139, pp. 263288 (1996).
21.Pirali, H., Djavanroodi, F. and Haghpanahi, M., “Combined Visibility and Surrounding Triangles Method for Simulation of Crack Discontinuities in Meshless Methods,” Journal of Applied Mathematics, 2012, pp. 116 (2012).
22.Organ, D., Fleming, M., Terry, T. and Belytschko, T., “Continuous Meshless Approximation for Non-Convex Bodies by Diffraction and Transparency,” Computational Mechanics, 18, pp. 225235 (1996).
23.Fleming, M., “Element-Free Galerkin Method for Fatigue and Quasi-Static Fracture,” Ph.D. Dissertation, North-Western University, U.S. (1997).
24.Naisipour, M., Afshar, M. H., Hassani, B. and Firoozjaee, A. R., “Collocation Discrete Least Square (CDLS) Method for Elasticity Problems,” International Journal of Civil Engineering, 7, pp. 918 (2009).
25.Amani, J., Afshar, M. H. and Naisipour, M., “Mixed Discrete Least Squares Meshless Method for Planar Elasticity Problems Using Regular and Irregular Nodal Distributions,” Engineering Analysis with Boundary Elements, 36, pp. 894902 (2012).
26.Afshar, M. H., Naisipour, M. and Amani, J., “Node Moving Adaptive Refinement Strategy for Planar Elasticity Problems Using Discrete Least Squares Meshless Method,” Finite Element Analysis and Design, 47, pp. 13151325 (2011).
27.Afshar, M. H., Amani, J. and Naisipour, M., “A Node Enrichment Adaptive Refinement by Discrete Least Squares Meshless Method for Solution of Elasticity Problems,” Engineering Analysis with Boundary Elements, 36, pp. 385393 (2012).


Related content

Powered by UNSILO

Voronoi Based Discrete Least Squares Meshless Method for Assessment of Stress Field in Elastic Cracked Domains

  • M. Labibzadeh (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.