Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-16T01:21:45.576Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  21 March 2016

M. Labibzadeh
M. Labibzadeh*
Affiliation:
Civil Engineering DepartmentFaculty of EngineeringShahid Chamran UniversityAhvaz, Iran
*
*Corresponding author ([email protected])
Get access

Abstract

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.

Keywords

VoronoiMeshlessLeast squaresCrack
Type
Research Article
Information
Journal of Mechanics , Volume 32 , Issue 3 , June 2016 , pp. 267 - 276
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Belytschko, T. and Black, T., “Elastic Crack Growth in Finite Elements with Minimal Remeshing,” International Journal of Numerical Methods in Engineering, 45, pp. 601–620 (1999).Google Scholar
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. 131–150 (1999).Google Scholar
3.Motamedi, D. and Mohammadi, S., “Fracture Analysis of Composites by Time Independent Moving-Crack Orthotropic XFEM,” International Journal of Mechanical Science, 54, pp. 20–37 (2011).Google Scholar
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. 1419–1443 (2010).Google Scholar
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. 1042–1065 (2011).CrossRefGoogle Scholar
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. 1017–1048 (2011).CrossRefGoogle Scholar
7.Belytschko, T., Lu, Y. Y. and Gu, L., “Crack Propagation by Element-Free Galerkin Methods,” Engineering Fracture Mechanics, 51, pp. 295–315 (1995).Google Scholar
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. 3–47 (1996).Google Scholar
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. 1483–1504 (1997).Google Scholar
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. 1906–1927 (2011).Google Scholar
11.Lucy, L. B., “A Numerical Approach to the Testing of the Fission Hypothesis,” Astron Journal, 82, pp. 1013–1024 (1977).Google Scholar
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. 375–389 (1977).Google Scholar
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. 531–546 (2007).CrossRefGoogle Scholar
14.Das, R. and Cleary, P. W., “Simulating Brittle Fracture of Rocks Using Smoothed Particle Hydrodynamics,” AIP Conference Proceeding, 1138 (2009).CrossRefGoogle Scholar
15.Liu, W. K., Jun, S. and Zhang, Y. F., “Reproducing Kernel Particle Methods,” International Journal of Numerical Methods, 20, pp. 1081–1106 (1995).CrossRefGoogle Scholar
16.Atluri, S. and Zhu, T., “A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics,” Computational Mechanics, 22, pp. 117–127 (1998).Google Scholar
17.Atluri, S. and Zhu, T. L., “The Meshless Local Petrov-Galerkin (MLPG) Approach for Solving Problems in Elasto-Statics,” Computational Mechanics, 25, pp. 169–179 (2000).Google Scholar
18.Lin, H. and Atluri, S., “Meshless Local Petrov-Galerkin (MLPG) Method for Convection Diffusion Problems,” Computational Model of Engineering Science, 1, pp. 45–60 (2000).Google Scholar
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. 65–81 (2011).Google Scholar
20.Liszka, T. J., Duarte, C. and Tworzydlo, W. W., “Hp-Meshless Cloud Method,” Computational Methods in Applied Mechanical Engineering, 139, pp. 263–288 (1996).Google Scholar
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. 1–16 (2012).Google Scholar
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. 225–235 (1996).Google Scholar
23.Fleming, M., “Element-Free Galerkin Method for Fatigue and Quasi-Static Fracture,” Ph.D. Dissertation, North-Western University, U.S. (1997).Google Scholar
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. 9–18 (2009).Google Scholar
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. 894–902 (2012).Google Scholar
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. 1315–1325 (2011).Google Scholar
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. 385–393 (2012).Google Scholar