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Computational Mechanics

, Volume 54, Issue 5, pp 1357–1374 | Cite as

A simple and efficient preconditioning scheme for heaviside enriched XFEM

  • Christapher Lang
  • David Makhija
  • Alireza DoostanEmail author
  • Kurt Maute
Original Paper

Abstract

The extended finite element method (XFEM) is an approach for solving problems with non-smooth solutions, which arise from geometric features such as cracks, holes, and material inclusions. In the XFEM, the approximate solution is locally enriched to capture the discontinuities without requiring a mesh which conforms to the geometric features. One drawback of the XFEM is that an ill-conditioned system of equations results when the ratio of volumes on either side of the interface in an element is small. Such interface configurations are often unavoidable, in particular for moving interface problems on fixed meshes. In general, the ill-conditioning reduces the performance of iterative linear solvers and impedes the convergence of solvers for nonlinear problems. This paper studies the XFEM with a Heaviside enrichment strategy for solving problems with stationary and moving material interfaces. A generalized formulation of the XFEM is combined with the level set method to implicitly define the embedded interface geometry. In order to avoid the ill-conditioning, a simple and efficient scheme based on a geometric preconditioner and constraining degrees of freedom to zero for small intersections is proposed. The geometric preconditioner is computed from the nodal basis functions, and therefore may be constructed prior to building the system of equations. This feature and the low-cost of constructing the preconditioning matrix makes it well suited for nonlinear problems with fixed and moving interfaces. It is shown by numerical examples that the proposed preconditioning scheme performs well for discontinuous problems and \(C^0\)-continuous problems with both the stabilized Lagrange and Nitsche methods for enforcing the continuity constraint at the interface. Numerical examples are presented which compare the condition number and solution error with and without the proposed preconditioning scheme. The results suggest that the proposed preconditioning scheme leads to condition numbers similar to that of a body-fitted mesh using the traditional finite element method without loss of solution accuracy.

Keywords

Level set method Extended finite element method Heaviside enrichment Ill-condition  Preconditioner 

Notes

Acknowledgments

The first author acknowledges the support of the NASA Fundamental Aeronautics Program Fixed Wing Project, and the second and fourth authors acknowledges the support of the National Science Foundation under grants CMMI-0729520 and EFRI SEED-1038305. This material is based upon work of third author supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, under Award Number DE-SC0006402. The opinions and conclusions presented are those of the authors and do not necessarily reflect the views of the sponsoring organizations.

References

  1. 1.
    Babuška I, Banerjee U (2012) Stable generalized finite element method (SGFEM). Comput Methods Appl Mech Eng 201–204:91–111CrossRefGoogle Scholar
  2. 2.
    Béchet E, Minnebo H, Moës N, Burgardt B (2005) Improved implementation and robustness study of the X-FEM for stress analysis around cracks. Int J Numer Methods Eng 64:1033–1056CrossRefzbMATHGoogle Scholar
  3. 3.
    Belytschko T, Parimi C, Moës N, Sukumar N, Usui S (2003) Structured extended finite element methods for solids defined by implicit surfaces. Int J Numer Methods Eng 56:609–635CrossRefzbMATHGoogle Scholar
  4. 4.
    Chessa J, Belytschko T (2003) An extended finite element method for two-phase fluids. J Appl Mech 70:10–17MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chessa J, Smolinski P, Belytschko T (2002) The extended finite element method (xfem) for solidification problems. Int J Numer Methods Eng 53:1959–1977MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Choi Y, Hulsen M, Meijer H (2012) Simulation of the flow of a viscoelastic fluid around a stationary cylinder using an extended finite element method. Comput Fluids 57:183–194MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ewing R, Iliev O, Lazarov R (2001) A modified finite volume approximation of second-order elliptic equations with discontinuous coefficients. SIAM J Sci Comput 23:1335–1351MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fernández-Méndez S, Huerta A (2004) Imposing essential boundary conditions in mesh-free methods. Comput Methods Appl Mech Eng 193:1257–1275CrossRefzbMATHGoogle Scholar
  9. 9.
    Fries TP (2008) A corrected X-FEM approximation without problems in blending elements. Int J Numer Methods Eng 75:503–532MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fries TP, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84:253–304MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gerstenberger A, Wall WA (2010) An embedded Dirichlet formulation for 3D continua. Int J Numer Methods Eng 82:537–563MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hansbo A, Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput Methods Appl Mech Eng 193:3523–3540MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hansbo P, Larson M, Zahedi S (2014) A cut finite element method for a stokes interface problem. Appl Numer Math 85:90–114Google Scholar
  14. 14.
    Juntunen M, Stenberg R (2008) Nitsche’s method for general boundary conditions. Math Comput 78:1353–1374MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kreissl S, Maute K (2012) Levelset based fluid topology optimization using the extended finite element method. Struct Multidiscip Optim 46:311–326MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lang C, Doostan A, Maute K (2013) Extended stochastic fem for diffusion problems with uncertain material interfaces. Comput Mech 51:1031–1049MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li S, Ghosh S (2007) Modeling interfacial debonding and matrix cracking in fiber reinforced composites by the extended Voronoi cell FEM. Finite Elem Anal Des 43:397–410CrossRefzbMATHGoogle Scholar
  18. 18.
    Makhija D, Maute K (2014) Numerical instabilities in level set topology optimization with the extended finite element method. Struct Multidiscip Optim 49(2):185–197Google Scholar
  19. 19.
    Mandel J, Brezina M (1996) Balancing domain decomposition for problems with large jumps in coefficients. Math Comput 65:1387–1401MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Menk A, Bordas S (2011) A robust preconditioning technique for the extended finite element method. Int J Numer Methods Eng 85:1609–1632MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150CrossRefzbMATHGoogle Scholar
  22. 22.
    Moës N, Cloirec M, Cartraud P, Remacle JF (2003) A computational approach to handle complex microstructure geometries. Comput Methods Appl Mech Eng 192:3163–3177CrossRefzbMATHGoogle Scholar
  23. 23.
    Osher S, Sethian J (1988) Fronts propogating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79:12–49MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Reusken A (2008) Analysis of an extended pressure finite element space for two-phase incompressible flows. Comput Vis Sci 11:293–305MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rüberg T, Cirak F (2012) Analysis of an extended pressure finite element space for two-phase incompressible flows. Comput Vis Sci 209–212:266–283Google Scholar
  26. 26.
    Saad Y, Schultz M (1986) GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7:856–869MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sauerland H, Fries TP (2013) The stable XFEM for two-phase flows. Comput Fluids 87:41–49MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sethian J (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Cambridge University Press, CambridgezbMATHGoogle Scholar
  29. 29.
    Soghrati S, Aragón A, Duarte C, Geubelle P (2010) An interface-enriched generalized finite element method for problems with discontinuous gradient fields. Int J Numer Methods Eng 00:1–19Google Scholar
  30. 30.
    Soghrati S, Thakre P, White S, Sottos N, Geubelle P (2012) Computational modeling and design of actively-cooled microvascular materials. Int J Heat Mass Transf 55:5309–5321CrossRefGoogle Scholar
  31. 31.
    Stenberg R (1995) On some techniques for approximating boundary conditions in the finite element method. J Comput Appl Math 63:139–148MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sukumar N, Chopp D, Moës N, Belytschko T (2001) Modeling holes and inclusions by level sets in the extended finite element method. Comput Methods Appl Mech Eng 190:6183–6200CrossRefzbMATHGoogle Scholar
  33. 33.
    Terada K, Asai M, Yamagishi M (2003) Finite cover method for linear and non-linear analyses of heterogeneous solids. Int J Numer Methods Eng 58:1321–1346CrossRefzbMATHGoogle Scholar
  34. 34.
    Tezduyar TE, Mittal S, Ray SE, Shih R (1992) Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput Methods Appl Mech Eng 95:221–242CrossRefzbMATHGoogle Scholar
  35. 35.
    Tran A, Yvonnet J, He QC, Toulemonde C, Sanahuja J (2010) A multiple level set approach to prevent numberical artefacts in complex microstructures with nearby inclusions within XFEM. Int J Numer Methods Eng 85:1436–1459CrossRefGoogle Scholar
  36. 36.
    Villanueva C, Maute K (2014) Density and level set-XFEM schemes for topology optimization of 3-d structures. Comput Mech 54(1):133–150Google Scholar
  37. 37.
    Wadbro E, Zahedi S, Kreiss G, Berggren M (2013) A uniformly well-conditioned, unfitted Nitsche method for interface problems. BIT Numer Math 53:791–820 Google Scholar
  38. 38.
    Zabaras N, Ganapathysubramanian B, Tan L (2006) Modelling dendritic solidification with melt convection using the extended finite element method. J Comput Phys 218:200–227MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Christapher Lang
    • 1
  • David Makhija
    • 2
  • Alireza Doostan
    • 2
    Email author
  • Kurt Maute
    • 2
  1. 1.Structural Mechanics and Concepts BranchNASA Langley Research CenterHamptonUSA
  2. 2.Aerospace Engineering SciencesUniversity of ColoradoBoulderUSA

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