Journal of Computational Methods in Engineering

Journal of Computational Methods in Engineering

A novel technique for local mesh refinement in the finite element method based on the alternative shape functions

Document Type : Original Article

Authors
Nasrin Kheirkhah Barzaki
Alireza Sadeghirad
Department of Civil and Environmental Engineering, Amirkabir University of Technology
10.47176/jcme.43.2.1031
Abstract
Unlike triangular/tetrahedral elements used in the finite element method for two/three-dimensional problems, local refinement of meshes composed of quadrilateral/hexahedral elements while maintaining the compatibility is challenging, and often results in severe distortion of the elements. A well-known and widely used approach to address this issue is the local mesh refinement based on transitional elements with hanging nodes. The key point in this method is enforcing the displacement continuity at the transitional element boundaries in the presence of hanging nodes. Transitional elements introduced in the literature employ various formulations depending on their placement within the mesh, and are also constrained by a maximum number of hanging nodes. along 
the element boundary. Therefore, their implementation for a general case is quite complicated. This paper presents a novel transitional element based on alternative shape functions, which offers a unified formulation for different placements of transitional elements in the mesh, applicable to any number of hanging nodes. Additionally, an analytical proof is provided to demonstrate the continuity and partition of unity properties in the proposed method, used in local mesh refinement. Finally, numerical examples in two and three dimensions are simulated to compare the accuracy and convergence of the proposed method against the existing methods in the literature.
Keywords
Finite element method
Local mesh refinement
Hanging nodes
Transition elements
Alternative shape functions
Subjects
  1. Zienkiewicz, O. C., and Taylor, R. L., “The Finite Element Method For Solid And Structural Mechanics,” Elsevier, 2005.
  2. Aksoylu B., Bond, S., and Holst, M., “An Odyssey Into Local Refinement And Multilevel Preconditioning Iii: Implementation And Numerical Experiments”, SIAM Journal on Scientific Computing, Vol. 25, No. 2, pp. 478-498, 2003.
  3. Aksoylu, B., and Holst, M., “Optimality of Multilevel Preconditioners for Local Mesh Refinement in Three Dimensions”, SIAM Journal on Numerical Analysis, 44, No. 3, pp. 1005–1025, 2006.
  4. Chen, H., Hoppe, R. H. W., and Xu, X., “Uniform Convergence Of Local Multigrid Methods For The Time-Harmonic Maxwell Equation”, ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 47, No. 1, pp. 125–147, 2013.
  5. Hannukainen, A., Korotov, S., and Křížek, M., “On Global and Local Mesh Refinements by a Generalized Conforming Bisection Algorithm,” Journal of computational and applied mathematics, Vol. 235, No. 2, pp. 419–436, 2010.
  6. Fries, T. P., Byfut, A., Alizada, A., Cheng, K. W., and Schröder, A., “Hanging Nodes and XFEM”, International Journal for Numerical Methods in Engineering, Vol. 86, No. 4–5, pp. 404–430, 2011.
  7. Demkowicz, L., Oden, J. T., Rachowicz, W., and Hardy, O., “Toward a Universal h-p Adaptive Finite Element Strategy, Part 1. Constrained Approximation and Data Structure”, Computer Methods in Applied Mechanics and Engineering, Vol. 77, No. 1–2, pp. 79–112, 1989.
  8. Ainsworth, M., and Senior, B., “Aspects of an Adaptive Hp-Finite Element Method: Adaptive Strategy, Conforming Approximation And Efficient Solvers”, Computer Methods in Applied Mechanics and Engineering, Vol. 150, No. 1–4, pp. 65–87, 1997.
  9. Byfut, A., and Schröder, A., “Hp-Adaptive Extended Finite Element Method”, International Journal for Numerical Methods in Engineering, Vol. 89, No. 11, pp. 1392–1418, 2012.
  10. Kůs, P., “Automatic hp-Adaptivity on Meshes with Arbitrary-Level Hanging Nodes in 3D,” D. Dissertation, Charles University, Prague, 2011.
  11. Šolín, P., Červený, J., and Doležel, I., “Arbitrary-Level Hanging Nodes And Automatic Adaptivity in the hp-FEM,” Mathematics and Computers in Simulation, Vol. 77, No. 1, pp. 117–132, 2008.
  12. Jeong G. E., Song, Y. U., Youn, S. K., and Park, K. C., “A New Approach for Nonmatching Interface Construction by the Method Of Localized Lagrange Multipliers”, Computer Methods in Applied Mechanics and Engineering, Vol. 361, p. 112728, 2020.
  13. Muixí, A., Fernández-Méndez, S., and Rodríguez-Ferran, A., “Adaptive Refinement for Phase-Field Models of Brittle Fracture Based on Nitsche’s Method”, Computational mechanics, Vol. 66, No. 1, pp. 69–85, 2020.
  14. Gupta, A. K., “A Finite Element for Transition From A Fine to A Coarse Grid”, International Journal for Numerical Methods in Engineering, 12, No. 1, pp. 35-45, 1978.
  15. Tabarraei, A., and Sukumar, N., “Adaptive Computations on Conforming Quadtree Meshes”, Finite Elements in Analysis and Design, 2005, Vol. 41, No. 7–8, pp. 686–702.
  16. Baitsch, M., and Hartmann, D., “Piecewise Polynomial Shape Functions for Hp-Finite Element Methods”, Computer Methods in Applied Mechanics and Engineering, Vol. 198, No. 13–14, pp. 1126–1137, 2009.
  17. McDill, J. M., Goldak, J. A., Oddy, A. S., and Bibby, M. J., “Isoparametric Quadrilaterals and Hexahedrons for Mesh-Grading Algorithms”, Communications in applied numerical methods, Vol. 3, No. 2, pp. 155–163, 1987.
  18. Morton, D. J., Tyler, J. M., and Dorroh, J. R., “A New 3D Finite Element for Adaptive h-Refinement in 1-Irregular Meshes”, International Journal for Numerical Methods in Engineering, Vol. 38, No. 23, pp. 3989–4008, 1995.
  19. Lim, J. H., Sohn, D., and Im, S., “Variable-Node Element Families for Mesh Connection and Adaptive Mesh Computation”, Structural Engineering and Mechanics, Vol. 43, No. 3, pp. 349–370, 2012.
  20. Ling, D., Bu, L., Tu, F., Yang, Q., and Chen, Y., “A Finite Element Method with Mesh-Separation-Based Approximation Technique and Its Application in Modeling Crack Propagation with Adaptive Mesh Refinement”, International Journal for Numerical Methods in Engineering, Vol. 99, No. 7, pp. 487–521, 2014.
  21. Sadeghirad, A., Brannon, R. M., and Burghardt, J., “A Convected Particle Domain Interpolation Technique to Extend Applicability of the Material Point Method for Problems Involving Massive Deformations”, International Journal for Numerical Methods in Engineering, Vol. 86, No. 12, pp. 1435–1456, 2011.
  22. Sadeghirad, A., Brannon, R. M., and Guilkey, J. E., “Second-Order Convected Particle Domain Interpolation (CPDI2) with Enrichment for Weak Discontinuities at Material Interfaces”, International Journal for Numerical Methods in Engineering, Vol. 95, No. 11, pp. 928–952, 2013.
  23. MATLAB Release 2019, MathWorks, Inc., Natick, Massachusetts, United States.
  24. Tecplot 360 EX, Tecplot, Inc., Bellevue, WA, United States.
  25. Timoshenko, S., and J. Goodier, “Theory of Elasticity”, McGraw-Hill, 1969.
  26. Gronwall, T. H., “Elastic Stresses in an Infinite Solid with a Spherical Cavity”, Annals of Mathematics, Vol. 19, No. 4, p. 295, 1918.

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