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Numerical simulation of holes and inclusions using adaptive polygonal finite element method

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Abstract

An adaptive polygonal finite element method using the techniques of cut-cell and quadtree refinement is presented for modeling holes and inclusions in 2-D solids. A mesh template is used to ensure the high-quality refined elements generated in quadtree refinement. By coupling the level set method, the polygonal computational mesh is directly extracted from the mesh template in every adaptive cycle. An error estimator based on recovery stress is devoted for adaptive purpose, which allows the mesh where it is needed is further refining. This method allows to model arbitrary shape holes and inclusions in arbitrary-geometry 2-D solid using the initial mesh of few rectangular elements, which considerably simplifies construction of the finite element model. And one curved boundary can be accurately represented though several steps of refinement. Numerical examples are solved and the obtained results are compared with reference solutions to show the simplicity and efficiency of the proposed method.

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References

  1. T. Rabczuk, P. Areias and T. Belytschko, A simplified mesh-free method for shear bands with cohesive surfaces, International J. for Numerical Methods in Engineering, 69 (5) (2007) 993–1021.

    Article  MATH  Google Scholar 

  2. M. Fleming, Y. Chu, B. Moran, T. Belytschko, Y. Lu and L. Gu, Enriched element-free galerkin methods for crack tip fields, International J. for Numerical Methods in Engineering, 40 (8) (1997) 1483–1504.

    Article  MathSciNet  Google Scholar 

  3. L. Dong and S. N. Atluri, T-trefftz voronoi cell finite elements with elastic/rigid inclusions or voids for micromechanical analysis of composite and porous materials, Computer Modeling in Engineering & Sciences (CMES), 83 (2) (2012) 183–219.

    MathSciNet  MATH  Google Scholar 

  4. S. Li and S. Ghosh, Extended voronoi cell finite element model for multiple cohesive crack propagation in brittle materials, International J. for Numerical Methods in Engineering, 65 (7) (2006) 1028–1067.

    Article  MATH  Google Scholar 

  5. S. Ghosh, K. Lee and S. Moorthy, Multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method, International J. of Solids and Structures, 32 (1) (1995) 27–62.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Réthoré, A. Gravouil and A. Combescure, An energyconserving scheme for dynamic crack growth using the extended finite element method, International J. for Numerical Methods in Engineering, 63 (5) (2005) 631–659.

    Article  MathSciNet  MATH  Google Scholar 

  7. N. Sukumar, D. L. Chopp, N. Moës and T. Belytschko, Modeling holes and inclusions by level sets in the extended finite-element method, Computer Methods in Applied Mechanics and Engineering, 190 (46) (2001) 6183–6200.

    Article  MathSciNet  MATH  Google Scholar 

  8. N. Sukumar and T. Belytschko, Arbitrary branched and intersecting cracks with the extended finite element method, International J. for Numerical Methods in Engineering, 48 (2000) 1741–1760.

    Article  MATH  Google Scholar 

  9. T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, International J. for Numerical Methods in Engineering, 45 (5) (1999) 601–620.

    Article  MATH  Google Scholar 

  10. S. H. Lee, J. H. Song, Y. C. Yoon, G. Zi and T. Belytschko, Combined extended and superimposed finite element method for cracks, International J. for Numerical Methods in Engineering, 59 (8) (2004) 1119–1136.

    Article  MATH  Google Scholar 

  11. A. Khoei, M. Anahid and K. Shahim, An extended arbitrary lagrangian-eulerian finite element method for large deformation of solid mechanics, Finite Elements in Analysis and Design, 44 (6) (2008) 401–416.

    Article  MathSciNet  Google Scholar 

  12. A. Khoei and S. T. Mousavi, Modeling of large deformation-large sliding contact via the penalty x-fem technique, Computational Materials Science, 48 (3) (2010) 471–480.

    Article  Google Scholar 

  13. A. Khoei, R. Yasbolaghi and S. Biabanaki, A polygonal finite element method for modeling crack propagation with minimum remeshing, International J. of Fracture, 194 (2) (2015) 123–148.

    Article  Google Scholar 

  14. S. Biabanaki, A. Khoei and P. Wriggers, Polygonal finite element methods for contact-impact problems on nonconformal meshes, Computer Methods in Applied Mechanics and Engineering, 269 (2014) 198–221.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Biabanaki and A. Khoei, A polygonal finite element method for modeling arbitrary interfaces in large deformation problems, Computational Mechanics, 50 (1) (2012) 19–33.

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Chi, C. Talischi, O. Lopez-Pamies and G. H Paulino, Polygonal finite elements for finite elasticity, International J. for Numerical Methods in Engineering, 101 (4) (2015) 305–328.

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Chi, O. Lopez-Pamies and G. H. Paulino, A variational formulation with rigid-body constraints for finite elasticity: Theory, finite element implementation, and applications, Computational Mechanics, 57 (2) (2016) 325–338.

    Article  MathSciNet  MATH  Google Scholar 

  18. E. L. Wachspress, A rational finite element basis, Academic Press (1975).

    MATH  Google Scholar 

  19. E. A. Malsch and G. Dasgupta, Interpolations for temperature distributions: a method for all non-concave polygons, International J. of Solids and Structures, 41 (8) (2004) 2165–2188.

    Article  MathSciNet  MATH  Google Scholar 

  20. E. A. Malsch, J. J. Lin and G. Dasgupta, Smooth twodimensional interpolations: a recipe for all polygons, J. of Graphics, GPU, and Game Tools, 10 (2) (2005) 27–39.

    Article  Google Scholar 

  21. M. S. Floater, Mean value coordinates, Computer Aided Geometric Design, 20 (1) (2003) 19–27.

    Article  MathSciNet  MATH  Google Scholar 

  22. N. Sukumar and A. Tabarraei, Conforming polygonal finite elements, International J. for Numerical Methods in Engineering, 61 (12) (2004) 2045–2066.

    Article  MathSciNet  MATH  Google Scholar 

  23. X. H. Tang, S. C. Wu, C. Zheng and J. H. Zhang, A novel virtual node method for polygonal elements, Applied Mathematics and Mechanics, 30 (2009) 1233–1246.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Tabarraei and N. Sukumar, Adaptive computations on conforming quadtree meshes, Finite Elements in Analysis and Design, 41 (7–8) (2005) 686–702.

    Article  Google Scholar 

  25. A. Tabarraei and N. Sukumar, Adaptive computations using material forces and residual-based error estimators on quadtree meshes, Computer Methods in Applied Mechanics and Engineering, 196 (25) (2007) 2657–2680.

    Article  MathSciNet  MATH  Google Scholar 

  26. C. Talischi, G. H. Paulino, A. Pereira and I. F. M. Menezes, Polygonal finite elements for topology optimization: A unifying paradigm, International J. for Numerical Methods in Engineering, 82 (2010) 671–698.

    MATH  Google Scholar 

  27. H. C. Oh and B. C. Lee, Hp-adaptive finite element method for linear elasticity using higher-order virtual node method, J. of Mechanical Science and Technology, 29 (10) (2015) 4299–4312.

    Article  Google Scholar 

  28. H. G. Kim and D. Sohn, A new finite element approach for solving three-dimensional problems using trimmed hexahedral elements, International J. for Numerical Methods in Engineering, 102 (9) (2015) 1527–1553.

    Article  MathSciNet  MATH  Google Scholar 

  29. D. Sohn, J. Han, Y. S. Cho and S. Im, A finite element scheme with the aid of a new carving technique combined with smoothed integration, Computer Methods in Applied Mechanics and Engineering, 254 (2013) 42–60.

    Article  MathSciNet  MATH  Google Scholar 

  30. N. Sukumar and E. A. Malsch, Recent advances in the construction of polygonal finite element interpolants, Archives of Computational Methods in Engineering, 13 (1) (2006) 129–163.

    Article  MathSciNet  MATH  Google Scholar 

  31. G. Dasgupta, Integration within polygonal finite elements, Journal of Aerospace Engineering, 16 (1) (2003) 9–18.

    Article  Google Scholar 

  32. P. Lancaster and K. Salkauskas, Surfaces generated by moving least squares methods, Mathematics of Computation, 37 (155) (1981) 141–158.

    Article  MathSciNet  MATH  Google Scholar 

  33. O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineerng analysis, International Journal for Numerical Methods in Engineering, 24 (2) (1987) 337–357.

    Article  MathSciNet  MATH  Google Scholar 

  34. S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on hamilton-jacobi formulations, J. of Computational Physics, 79 (1) (1988) 12–49.

    Article  MathSciNet  MATH  Google Scholar 

  35. E. T. Ooi, H. Man, S. Natarajan and C. Song, Adaptation of quadtree meshes in the scaled boundary finite element method for crack propagation modelling, Engineering Fracture Mechanics, 144 (2015) 101–117.

    Article  Google Scholar 

  36. C. Talischi, G. H. Paulino, A. Pereira and I. F. Menezes, Polymesher: a general-purpose mesh generator for polygonal elements written in matlab, Structural and Multidisciplinary Optimization, 45 (3) (2012) 309–328.

    Article  MathSciNet  MATH  Google Scholar 

  37. P. O. Persson and G. Strang, A simple mesh generator in matlab, SIAM Review, 46 (2) (2004) 329–345.

    Article  MathSciNet  MATH  Google Scholar 

  38. H. Samet, The quadtree and related hierarchical data structures, ACM Computing Surveys (CSUR), 16 (2) (1984) 187–260.

    Article  MathSciNet  Google Scholar 

  39. K. C. Chellamuthu and N. Ida, Algorithms and data structures for 2d and 3d adaptive finite element mesh refinement, Finite Elements in Analysis and Design, 17 (3) (1994) 205–229.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mechanics and Materials, Hohai University, Nanjing, 210098, China

    Shengyong Ding, Guojian Shao & Hougai Shi

  2. College of Harbour, Coastal and Offshore Engineering, Hohai University, Nanjing, 210098, China

    Ang Li & Jingbo Su

Authors
  1. Shengyong Ding
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  2. Guojian Shao
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  3. Ang Li
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  4. Jingbo Su
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  5. Hougai Shi
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Correspondence to Guojian Shao.

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Recommended by Associate Editor Gang-Won Jang

Guojian Shao received the B.E. (1983) of Mechanics from Hohai University, China, and Ph.D. (1997) in Hydraulic Structure Engineering from Hohai University, China. He is currently a Professor at the Department of Engineering Mechanics in Hohai University. His areas of interest are in finite element method and multiscale problems.

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Ding, S., Shao, G., Li, A. et al. Numerical simulation of holes and inclusions using adaptive polygonal finite element method. J Mech Sci Technol 31, 4305–4317 (2017). https://doi.org/10.1007/s12206-017-0829-2

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  • DOI: https://doi.org/10.1007/s12206-017-0829-2

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