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Computational Mechanics

, Volume 51, Issue 5, pp 743–764 | Cite as

Efficient analysis of transient heat transfer problems exhibiting sharp thermal gradients

  • P. O’Hara
  • C. A. Duarte
  • T. Eason
  • J. Garzon
Original Paper

Abstract

In this paper, heat transfer problems with sharp spatial gradients are analyzed using the Generalized Finite Element Method with global-local enrichment functions (GFEM gl). With this approach, scale-bridging enrichment functions are generated on the fly, providing specially-tailored enrichment functions for the problem to be analyzed with no a-priori knowledge of the exact solution. In this work, a decomposition of the linear system of equations is formulated for both steady-state and transient heat transfer problems, allowing for a much more computationally efficient analysis of the problems of interest. With this algorithm, only a small portion of the global system of equations, i.e., the hierarchically added enrichments, need to be re-computed for each loading configuration or time-step. Numerical studies confirm that the condensation scheme does not impact the solution quality, while allowing for more computationally efficient simulations when large problems are considered. We also extend the GFEM gl to allow for the use of hexahedral elements in the global domain, while still using tetrahedral elements in the local domain, to allow for automatic localized mesh refinement without the use of constrained approximations. Simulations are run with the use of linear and quadratic hexahedral and tetrahedral elements in the global domain. Convergence studies indicate that the use of a different partition of unity (PoU) in the global (hexahedral elements) and local (tetrahedral elements) domains does not adversely impact the solution quality.

Keywords

Generalized finite elements Global-local finite elements Multi scale methods hp-Methods Transient analysis 

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References

  1. 1.
    Albertelli G, Crawfis RA (1997) Efficient subdivision of finite-element datasets into consistent tetrahedra. In: Proceedings of IEEE visualization ’97. IEEE Computer Society Press, Los Alamitos, pp 213–219Google Scholar
  2. 2.
    Arnold DN, Mukherjee A, Pouly L (2000) Locally adapted tetrahedral meshes using bisection. SIAM J Sci Comput 22(2): 431–448MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Babuska I (1973) The finite element method with penalty. Math Comput 27(122): 221–228MathSciNetzbMATHGoogle Scholar
  4. 4.
    Babuška I, Melenk JM (1997) The partition of unity finite element method. Int J Numer Methods Eng 40: 727–758zbMATHCrossRefGoogle Scholar
  5. 5.
    Babuška I, Caloz G, Osborn JE (1994) Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J Numer Anal 31(4): 945–981MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Babuška I, Ihlenburg F, Paik E, Sauter S (1995) A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Comput Methods Appl Mech Eng 128(3–4): 325–360zbMATHCrossRefGoogle Scholar
  7. 7.
    Bansch E (1991) Local mesh refinement in 2 and 3 dimensions. Impact Comput Sci Eng 3: 181–191MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ching HK, Chen JK (2006) Thermomechanical analysis of functionally graded composites under laser heating by the MLPG method. Comput Model Eng Sci 13(3): 199–217MathSciNetzbMATHGoogle Scholar
  9. 9.
    Combescure A, Gravouil A, Gregoire D, Rethore J (2008) X-FEM a good candidate for energy conservation in simulation of brittle crack propagation. Comput Methods Appl Mech Eng 197: 309–318zbMATHCrossRefGoogle Scholar
  10. 10.
    De S, Bathe KJ (2000) The method of finite spheres. Comput Mech 25: 329–345MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Demkowicz L, Oden JT, Rachowicz W, Hardy O (1989) Toward a universal h–p adaptive finite element strategy, Part 1. Constrained approximation and data structure. Comput Methods Appl Mech Eng 77: 79–112MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Devloo PR (1992) On the convergence rate of an adaptive grid pattern for resolving point singularities in elliptic problems. Commun Appl Numer Methods 8: 161–169MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Dompierre J, Labb P, Vallet M-G, Camarero R (1999) How to subdivide pyramids, prisms and hexahedra into tetrahedra. Rapport CERCA R99-78Google Scholar
  14. 14.
    Duarte CA (1996) The hp cloud method. PhD dissertation, The University of Texas at Austin, AustinGoogle Scholar
  15. 15.
    Duarte CA, Babuška I (2002) Mesh-independent directional p-enrichment using the generalized finite element method. Int J Numer Methods Eng 55(12):1477–1492. http://10.1002/nme.557 Google Scholar
  16. 16.
    Duarte CA, Kim D-J (2008) Analysis and applications of a generalized finite element method with global-local enrichment functions. Comput Methods Appl Mech Eng 197(6–8):487–504. http://10.1016/j.cma.2007.08.017
  17. 17.
    Duarte CAM, Oden JT (1995) Hp clouds—a meshless method to solve boundary-value problems. Technical Report 95-05, TICAM, The University of Texas at AustinGoogle Scholar
  18. 18.
    Duarte CAM, Oden JT (1996) An hp adaptive method using clouds. Comput Methods Appl Mech Eng 139: 237–262MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Duarte CAM, Oden JT (1996) Hp clouds—an hp meshless method. Numer Methods Partial Differ Equ 12: 673–705MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Duarte CA, Babuška I, Oden JT (2000) Generalized finite element methods for three dimensional structural mechanics problems. Comput Struct 77: 215–232CrossRefGoogle Scholar
  21. 21.
    Duarte CA, Hamzeh ON, Liszka TJ, Tworzydlo WW (2001) A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. Comput Methods Appl Mech Eng 190(15–17):2227–2262. http://10.1016/S0045-7825(00)00233-4 Google Scholar
  22. 22.
    Duarte CA, Kim D-J, Babuška I (2007) Chapter: A global-local approach for the construction of enrichment functions for the generalized fem and its application to three-dimensional cracks. In: Leitão VMA, Alves CJS, Duarte CA (eds) Advances in meshfree techniques, vol 5 of Computational methods in applied sciences. Springer, Dordrecht. ISBN 978-1-4020-6094-6Google Scholar
  23. 23.
    Fries TP, Zilian A (2009) On time integration in the xfem. Int J Numer Methods Eng 79: 69–93MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Fries TP, Byfut A, Alizada A, Cheng KW, Schroder A (2011) Hanging nodes and xfem. Int J Numer Methods Eng 86: 404–430MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Griebel M, Schweitzer MA (2000) A particle-partition of unity method for the solution of elliptic, parabolic and hyperbolic PDEs. SIAM J Sci Comput 22(3): 853–890MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Gupta AK (1978) A finite element for transition from a fine to a coarse grid. Int J Numer Methods Eng 12: 35–45zbMATHCrossRefGoogle Scholar
  27. 27.
    Khoei AR, Gharehbaghi SA (2009) Three-dimensional data transfer operators in large plasticity deformations using modified-spr technique. Appl Math Modell 33: 3269–3285zbMATHCrossRefGoogle Scholar
  28. 28.
    Kim D-J, Duarte CA, Pereira JP (2008) Analysis of interacting cracks using the generalized finite element method with global-local enrichment functions. J Appl Mech 75(5):051107. http://link.aip.org/link/?AMJ/75/051107/1
  29. 29.
    Kim D-J, Duarte CA, Sobh NA (2010) Parallel simulations of three-dimensional cracks using the generalized finite element method. Comput Mech. Accepted for publication. http://10.1007/s00466-010-0546-5
  30. 30.
    Kim D-J, Pereira JP, Duarte CA (2010) Analysis of three-dimensional fracture mechanics problems: a two-scale approach using coarse generalized FEM meshes. Int J Numer Methods Eng 81(3):335–365. http://10.1002/nme.2690 Google Scholar
  31. 31.
    Lee NS, Bathe KJ (1994) Error indicators and adaptive remeshing in large deformation finite element analysis. Finite Elem Anal Des 16: 99–139MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Melenk JM, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139: 289–314zbMATHCrossRefGoogle Scholar
  33. 33.
    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46: 131–150zbMATHCrossRefGoogle Scholar
  34. 34.
    Niekamp R, Stein E (2002) An object-oriented approach for parallel two- and three-dimensional adaptive finite element computations. Comput Struct 80: 317–328CrossRefGoogle Scholar
  35. 35.
    Oden JT, Duarte CA, Zienkiewicz OC (1998) A new cloud-based hp finite element method. Comput Methods Appl Mech Eng 153: 117–126MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    O’Hara P, Duarte CA, Eason T (2009) Generalized finite element analysis of three-dimensional heat transfer problems exhibiting sharp thermal gradients. Comput Methods Appl Mech Eng 198(21–26):1857–1871. http://10.1016/j.cma.2008.12.024 Google Scholar
  37. 37.
    O’Hara P, Duarte CA, Eason T (2010) Transient analysis of sharp thermal gradients using coarse finite element meshes. Comput Methods Appl Mech Eng 200(5–8):812–829. http://10.1016/j.cma.2010.10.005
  38. 38.
    O’Hara PJ (2007) Finite element analysis of three-dimensional heat transfer for problems involving sharp thermal gradients. Master’s thesis, University of Illinois at Urbana-ChampaignGoogle Scholar
  39. 39.
    O’Hara PJ (2010) A multi-scale generalized finite element method for sharp, transient thermal gradients. PhD thesis, University of Illinois at Urbana-ChampaignGoogle Scholar
  40. 40.
    Rethore J, Gravouil A, Combescure A (2004) A stable numerical scheme for the finite element simulation of dynamic crack propagation with remeshing. Comput Methods Appl Mech Eng 193: 4493–4510zbMATHCrossRefGoogle Scholar
  41. 41.
    Rethore J, Gravouil A, Combescure A (2005) An energy-conserving scheme for dynamic crack growth using the extended finite element method. Int J Numer Methods Eng 63: 631–659MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Simone A, Duarte CA, van der Giessen E (2006) A generalized finite element method for polycrystals with discontinuous grain boundaries. Int J Numer Methods Eng 67(8):1122–1145. http://10.1002/nme.1658 Google Scholar
  43. 43.
    Strouboulis T, Babuška I, Copps K (2000) The design and analysis of the generalized finite element mehtod. Comput Methods Appl Mech Eng 81(1–3): 43–69CrossRefGoogle Scholar
  44. 44.
    Strouboulis T, Copps K, Babuška I (2001) The generalized finite element method. Comput Methods Appl Mech Eng 190: 4081–4193zbMATHCrossRefGoogle Scholar
  45. 45.
    Swenson DV, Ingraffea AR (1988) Modeling mixed-mode dynamic crack propagation using finite elements: Theory and applications. Comput Mech 3: 381–397zbMATHCrossRefGoogle Scholar
  46. 46.
    Tradegard A, Nilsson F, Ostlund S (1998) Fem-remeshing technique applied to crack growth problems. Comput Methods Appl Mech Eng 160: 115–131CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • P. O’Hara
    • 1
  • C. A. Duarte
    • 2
  • T. Eason
    • 3
  • J. Garzon
    • 2
  1. 1.Universal Technology CorporationDaytonUSA
  2. 2.Department of Civil and Environmental Engineering, Newmark LaboratoryUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Air Force Research LaboratoryAir Vehicles DirectorateWPAFBUSA

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