Non-intrusive Coupling: Recent Advances and Scalable Nonlinear Domain Decomposition

  • Mickaël Duval
  • Jean-Charles Passieux
  • Michel Salaün
  • Stéphane Guinard
Original Paper

Abstract

This paper provides a detailed review of the global/local non-intrusive coupling algorithm. Such method allows to alter a global finite element model, without actually modifying its corresponding numerical operator. We also look into improvements of the initial algorithm (Quasi-Newton and dynamic relaxation), and provide comparisons based on several relevant test cases. Innovative examples and advanced applications of the non-intrusive coupling algorithm are provided, granting a handy framework for both researchers and engineers willing to make use of such process. Finally, a novel nonlinear domain decomposition method is derived from the global/local non-intrusive coupling strategy, without the need to use a parallel code or software. Such method being intended to large scale analysis, we show its scalability. Jointly, an efficient high level Message Passing Interface coupling framework is also proposed, granting an universal and flexible way for easy software coupling. A sample code is also given.

Supplementary material

11831_2014_9132_MOESM1_ESM.zip (46 kb)
ESM 1 (ZIP 46 kb)

References

  1. 1.
    Agence Nationale de la Recherche (2014) Icare project. URL http://www.institut-clement-ader.org/icare/
  2. 2.
    Agouzal A, Thomas JM (1995) Une méthode d’éléments finis hybrides en décomposition de domaines. ESAIM Math Model Num Anal 29:749–764MATHMathSciNetGoogle Scholar
  3. 3.
    Akgün MA, Garcelon JH, Haftka RT (2001) Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas. Int J Numer Meth Eng 50(7):1587–1606MATHCrossRefGoogle Scholar
  4. 4.
    Amdouni S, Moakher M, Renard Y (2014) A local projection stabilization of fictitious domain method for elliptic boundary value problems. Appl Numer Math 76:60–75MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Barrière L (2014) Stratégies de calcul intensif pour la simulation du post-flambement local des grandes structures composites raidies aéronautiques. PhD Thesis, INSA de ToulouseGoogle Scholar
  6. 6.
    Barrière L, Marguet S, Castanié B, Cresta P, Passieux JC (2013) An adaptive model reduction strategy for post-buckling analysis of stiffened structures. Thin Wall Struct 73:81–93CrossRefGoogle Scholar
  7. 7.
    Becker R, Hansbo P, Stenberg R (2003) A finite element method for domain decomposition with non-matching grids. ESAIM Math Model Numer Anal 37(02):209–225MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Belgacem FB (1999) The mortar finite element method with Lagrange multipliers. Numer Math 84(2):173–197MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Ben Dhia H, Jamond O (2010) On the use of XFEM within the Arlequin framework for the simulation of crack propagation. Comput Methods Appl Mech Eng 199(21):1403–1414MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Ben Dhia H, Rateau G (2005) The Arlequin method as a flexible engineering design tool. Int J Numer Meth Eng 62(11):1442–1462MATHCrossRefGoogle Scholar
  11. 11.
    Ben Dhia H, Elkhodja N, Roux FX (2008) Multimodeling of multi-alterated structures in the Arlequin framework: solution with a domain-decomposition solver. Eur J Comput Mech 17(5–7):969–980MATHGoogle Scholar
  12. 12.
    Bernardi C, Maday Y, Patera AT (1994) A new nonconforming approach to domain decomposition: the Mortar element method. Nonlinear partial differential equations and their applications, Collège de France Seminar XI, H Brezis and JL Lions (Eds) pp 13–51Google Scholar
  13. 13.
    Bernardi C, Maday Y, Rapetti F (2005) Basics and some applications of the mortar element method. GAMM-Mitteilungen 28(2):97–123MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Bettinotti O, Allix O, Malherbe B (2014) A coupling strategy for adaptive local refinement in space and time with a fixed global model in explicit dynamics. Comput Mech 53(4):561–574MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Bjorstad PE, Widlund OB (1986) Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J Num Anal 23(6):1097–1120MathSciNetCrossRefGoogle Scholar
  16. 16.
    Brancherie D, Dambrine M, Vial G, Villon P (2008) Effect of surface defects on structure failure: a two-scale approach. Eur J Comput Mech 17(5–7):613–624MATHGoogle Scholar
  17. 17.
    Brezzi F, Marini LD (2005) The three-field formulation for elasticity problems. GAMM-Mitteilungen 28(1):124–153MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Chahine E, Laborde P, Renard Y (2008) Spider-XFEM, an extended finite element variant for partially unknown crack-tip displacement. Eur J Comput Mech 17(5–7):625–636MATHGoogle Scholar
  19. 19.
    Chahine E, Laborde P, Renard Y (2009) A reduced basis enrichment for the eXtended finite element method. Math Model Nat Phenom 4(01):88–105MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Chantrait T, Rannou J, Gravouil A (2014) Low intrusive coupling of implicit and explicit time integration schemes for structural dynamics: application to low energy impacts on composite structures. Finite Elem Anal Des 86:23–33MathSciNetCrossRefGoogle Scholar
  21. 21.
    Chevreuil M, Nouy A, Safatly E (2013) A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties. Comput Methods Appl Mech Eng 255:255–274MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Conn AR, Gould NIM, Toint PL (1991) Convergence of quasi-newton matrices generated by the symmetric rank one update. Math Program 50(1–3):177–195MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Cresta P, Allix O, Rey C, Guinard S (2007) Nonlinear localization strategies for domain decomposition methods: application to post-buckling analyses. Comput Methods Appl Mech Eng 196(8):1436–1446MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Daghia F, Ladevèze P (2012) A micro-meso computational strategy for the prediction of the damage and failure of laminates. Compos Struct 94(12):3644–3653CrossRefGoogle Scholar
  25. 25.
    Daridon L, Dureisseix D, Garcia S, Pagano S (2011) Changement d’échelles et zoom structural. In: 10e colloque national en calcul des structures, Giens, FranceGoogle Scholar
  26. 26.
    Dohrmann C (2003) A preconditioner for substructuring based on constrained energy minimization. SIAM J Sci Comput 25(1):246–258MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Duarte CA, Kim DJ (2008) Analysis and applications of a generalized finite element method with global-local enrichment functions. Comput Methods Appl Mech Eng 197(6–8):487–504MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Farhat C, Roux FX (1991) A method of finite element tearing and interconnecting and its parallel solution algorithm. Int J Numer Meth Eng 32(6):1205–1227MATHCrossRefGoogle Scholar
  29. 29.
    Farhat C, Mandel J, Roux FX (1994) Optimal convergence properties of the FETI domain decomposition method. Comput Methods Appl Mech Eng 115(3–4):365–385MathSciNetCrossRefGoogle Scholar
  30. 30.
    Farhat C, Lesoinne M, LeTallec P, Pierson K, Rixen D (2001) FETI-DP: a dual-primal unified FETI method - Part I: A faster alternative to the two-level FETI method. Int J Numer Meth Eng 50(7):1523–1544MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Fritz A, Hüeber S, Wohlmuth B (2004) A comparison of mortar and Nitsche techniques for linear elasticity. Calcolo 41(3):115–137MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Gander MJ (2008) Schwarz methods over the course of time. Electron Trans Numer Anal 31:228–255MATHMathSciNetGoogle Scholar
  33. 33.
    Gander MJ, Japhet C (2013) Algorithm 932: PANG: software for nonmatching grid projections in 2D and 3D with linear complexity. ACM Trans Math Softw 40(1):1–25MathSciNetCrossRefGoogle Scholar
  34. 34.
    Gendre L, Allix O, Gosselet P, Comte F (2009) Non-intrusive and exact global/local techniques for structural problems with local plasticity. Comput Mech 44(2):233–245MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Gendre L, Allix O, Gosselet P (2011) A two-scale approximation of the Schur complement and its use for non-intrusive coupling. Int J Numer Meth Eng 87(9):889–905MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Gerstenberger A, Tuminaro RS (2013) An algebraic multigrid approach to solve extended finite element method based fracture problems. Int J Numer Meth Eng 94(3):248–272MathSciNetCrossRefGoogle Scholar
  37. 37.
    Glowinski R, Le Tallec P (1990) Augmented lagrangian interpretation of the nonoverlapping Schwartz alternating method. SIAM, Philadelphia, pp 224–231Google Scholar
  38. 38.
    Gosselet P, Rey C (2006) Non-overlapping domain decomposition methods in structural mechanics. Archiv Comput Methods Eng 13(4):515–572MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Guguin G, Allix O, Gosselet P, Guinard S (2014) Nonintrusive coupling of 3D and 2D laminated composite models based on finite element 3D recovery. Int J Numer Meth Eng 98(5):324–343MathSciNetCrossRefGoogle Scholar
  40. 40.
    Guidault PA, Allix O, Champaney L, Cornuault C (2008) A multiscale extended finite element method for crack propagation. Comput Methods Appl Mech Eng 197(5):381–399MATHCrossRefGoogle Scholar
  41. 41.
    Gupta P, Pereira J, Kim DJ, Duarte C, Eason T (2012) Analysis of three-dimensional fracture mechanics problems: A non-intrusive approach using a generalized finite element method. Eng Fract Mech 90:41–64CrossRefGoogle Scholar
  42. 42.
    Hansbo A, Hansbo P (2002) An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput Methods Appl Mech Eng 191(47–48):5537–5552MATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Hansbo A, Hansbo P, Larson MG (2003) A finite element method on composite grids based on Nitsche’s method. ESAIM Math Model Numer Anal 37(03):495–514MATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Hautefeuille M, Annavarapu C, Dolbow JE (2012) Robust imposition of Dirichlet boundary conditions on embedded surfaces. Int J Numer Meth Eng 90(1):40–64MATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Herry B, Di Valentin L, Combescure A (2002) An approach to the connection between subdomains with non-matching meshes for transient mechanical analysis. Int J Numer Meth Eng 55(8):973–1003MATHCrossRefGoogle Scholar
  46. 46.
    Hirai I, Wang BP, Pilkey WD (1984) An efficient zooming method for finite element analysis. Int J Numer Meth Eng 20(9):1671–1683MATHCrossRefGoogle Scholar
  47. 47.
    Hughes TJR (1995) Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput Methods Appl Mech Eng 127(1–4):387–401MATHCrossRefGoogle Scholar
  48. 48.
    Ibrahimbegović A, Markovič D (2003) Strong coupling methods in multi-phase and multi-scale modeling of inelastic behavior of heterogeneous structures. Comput Methods Appl Mech Eng 192(28–30):3089–3107MATHCrossRefGoogle Scholar
  49. 49.
    Irons BM, Tuck RC (1969) A version of the Aitken accelerator for computer iteration. Int J Numer Meth Eng 1(3):275–277MATHCrossRefGoogle Scholar
  50. 50.
    Kelley CT, Sachs EW (1998) Local convergence of the symmetric rank-one iteration. Comput Optim Appl 9(1):43–63MATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Khalfan HF, Byrd RH, Schnabel RB (1993) A theoretical and experimental study of the symmetric rank-one update. SIAM J Optim 3(1):1–24MATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Khiyabani FM, Hassan MA, Leong WJ (2010) Convergence of symmetric rank-one method based on modified Quasi-Newton equation. J Math Res 2(3):97–102MATHCrossRefGoogle Scholar
  53. 53.
    Kim DJ, Duarte CA, Proenca SP (2012) A generalized finite element method with global-local enrichment functions for confined plasticity problems. Comput Mech 50(5):563–578MATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    Küttler U, Wall WA (2008) Fixed-point fluid-structure interaction solvers with dynamic relaxation. Comput Mech 43(1):61–72MATHCrossRefGoogle Scholar
  55. 55.
    Laborde P, Lozinski A (in progress) Numerical zoom for multi-scale and multi-model problemsGoogle Scholar
  56. 56.
    Ladeveze P (1985) Sur une famille d’algorithmes en mécanique des structures. Comptes-rendus des séances de l’Académie des sciences Série 2, Mécanique-physique, chimie, sciences de l’univers, sciences de la terre 300(2):41–44Google Scholar
  57. 57.
    Ladeveze P, Nouy A, Loiseau O (2002) A multiscale computational approach for contact problems. Comput Methods Appl Mech Eng 191(43):4869–4891MATHMathSciNetCrossRefGoogle Scholar
  58. 58.
    Ladevèze P, Dureisseix D (1999) Une nouvelle stratégie de calcul micro/macro en mécanique des structures. Comptes Rendus de l’Académie des Sciences - Series IIB - Mechanics-Physics-Astronomy 327(12):1237–1244MATHCrossRefGoogle Scholar
  59. 59.
    Ladevèze P, Loiseau O, Dureisseix D (2001) A micro-macro and parallel computational strategy for highly heterogeneous structures. Int J Numer Meth Eng 52(1–2):121–138CrossRefGoogle Scholar
  60. 60.
    Lions PL (1987) On the Schwarz method. In: Glowinski R, Golub GH, Meurant GA, Périaux J (eds). Domain decomposition methods for partial differential equations. Paris, FranceGoogle Scholar
  61. 61.
    Liu YJ, Sun Q, Fan XL (2014) A non-intrusive global/local algorithm with non-matching interface: derivation and numerical validation. Comput Methods Appl Mech Eng 277:81–103CrossRefGoogle Scholar
  62. 62.
    Mandel J (1993) Balancing domain decomposition. Commun Numer Methods Eng 9:233–241MATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    Mandel J, Dohrmann CR (2003) Convergence of a balancing domain decomposition by constraints and energy minimization. Numer Linear Algebra Appl 10(7):639–659MATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    Mao KM, Sun CT (1991) A refined global-local finite element analysis method. Int J Numer Meth Eng 32(1):29–43MATHCrossRefGoogle Scholar
  65. 65.
    Massing A, Larson MG, Logg A (2012) Efficient implementation of finite element methods on non-matching and overlapping meshes in 3D. arXiv preprintGoogle Scholar
  66. 66.
    Melenk J, Babuška I (1996) The partition of unity finite element method: Basic theory and applications. Comput Methods Appl Mech Eng 139(1–4):289–314MATHCrossRefGoogle Scholar
  67. 67.
    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Meth Eng 46(1):131–150MATHCrossRefGoogle Scholar
  68. 68.
    Nguyen VP, Kerfriden P, Claus S, Bordas SPA (2013) Nitsche’s method for mixed dimensional analysis: conforming and non-conforming continuum-beam and continuum-plate coupling. arXiv preprintGoogle Scholar
  69. 69.
    Nocedal J, Wright S (2006) Numerical optimization, 2nd edn. Springer, New YorkGoogle Scholar
  70. 70.
    Oden JT, Vemaganti K, Moës N (1999) Hierarchical modeling of heterogeneous solids. Comput Methods Appl Mech Eng 172(1–4):3–25MATHCrossRefGoogle Scholar
  71. 71.
    Park KC, Felippa CA (2000) A variational principle for the formulation of partitioned structural systems. Int J Numer Meth Eng 47(1–3):395–418MATHMathSciNetCrossRefGoogle Scholar
  72. 72.
    Passieux JC, Ladevèze P, Néron D (2010) A scalable time-space multiscale domain decomposition method: adaptive time scale separation. Comput Mech 46(4):621–633MATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    Passieux JC, Gravouil A, Réthoré J, Baietto MC (2011) Direct estimation of generalized stress intensity factors using a three-scale concurrent multigrid XFEM. Int J Numer Meth Eng 85(13):1648–1666MATHCrossRefGoogle Scholar
  74. 74.
    Passieux JC, Réthoré J, Gravouil A, Baietto MC (2013) Local/global non-intrusive crack propagation simulation using a multigrid XFEM solver. Comput Mech 52(6):1381–1393MATHCrossRefGoogle Scholar
  75. 75.
    Pebrel J, Rey C, Gosselet P (2008) A nonlinear dual domain decomposition method : application to structural problems with damage. Int J Multiscale Comput Eng 6(3):251–262CrossRefGoogle Scholar
  76. 76.
    Picasso M, Rappaz J, Rezzonico V (2008) Multiscale algorithm with patches of finite elements. Commun Numer Methods Eng 24(6):477–491MATHMathSciNetCrossRefGoogle Scholar
  77. 77.
    Pironneau OP, Lozinski A (2011) Numerical Zoom for localized Multiscales. Numer Methods Partial Differ Equ 27:197–207MATHMathSciNetCrossRefGoogle Scholar
  78. 78.
    Rannou J, Gravouil A, Baïetto-Dubourg MC (2009) A local multigrid XFEM strategy for 3D crack propagation. Int J Numer Meth Eng 77(4):581–600MATHCrossRefGoogle Scholar
  79. 79.
    Roux FX (2009) A FETI-2lm method for non-matching grids. In: Domain Decomposition Methods in Science and Engineering XVIII, no. 70 in Lecture Notes in Computational Science and Engineering, Springer, Berlin, pp 121–128Google Scholar
  80. 80.
    Réthoré J, Roux S, Hild F (2010) Hybrid analytical and extended finite element method (HAX-FEM): a new enrichment procedure for cracked solids. Int J Numer Meth Eng 81(3):269–285MATHCrossRefGoogle Scholar
  81. 81.
    Strouboulis T, Copps K, Babuška I (2001) The generalized finite element method. Comput Methods Appl Mech Eng 190(32–33):4081–4193Google Scholar
  82. 82.
    Whitcomb JD (1991) Iterative global/local finite element analysis. Comput Struct 40(4):1027–1031Google Scholar
  83. 83.
    Wohlmuth BI (2000) A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J Num Anal 38(3):989–1012MATHMathSciNetCrossRefGoogle Scholar
  84. 84.
    Wohlmuth BI (2003) A comparison of dual Lagrange multiplier spaces for mortar finite element discretizations. ESAIM Math Model Numer Anal 36(6):995–1012MathSciNetCrossRefGoogle Scholar
  85. 85.
    Wyart E, Duflot M, Coulon D, Martiny P, Pardoen T, Remacle JF, Lani F (2008) Substructuring FE-XFE approaches applied to three-dimensional crack propagation. J Comput Appl Math 215(2):626–638MATHMathSciNetCrossRefGoogle Scholar
  86. 86.
    Wyart E, Coulon D, Pardoen T, Remacle J, Lani F (2009) Application of the substructured finite element/extended finite element method (s-FE/XFE) to the analysis of cracks in aircraft thin walled structures. Eng Fract Mech 76(1):44–58CrossRefGoogle Scholar
  87. 87.
    Zohdi TI, Oden JT, Rodin GJ (1996) Hierarchical modeling of heterogeneous bodies. Comput Methods Appl Mech Eng 138(1–4):273–298MATHMathSciNetCrossRefGoogle Scholar
  88. 88.
    Électricité de France (2014) \(\text{ Code }\_\text{ aster }\). URL http://www.code-aster.org

Copyright information

© CIMNE, Barcelona, Spain 2014

Authors and Affiliations

  • Mickaël Duval
    • 1
  • Jean-Charles Passieux
    • 1
  • Michel Salaün
    • 1
  • Stéphane Guinard
    • 2
  1. 1. Institut Clément Ader (ICA), INSA de Toulouse, UPS, Mines Albi, ISAEUniversité de ToulouseToulouseFrance
  2. 2.Airbus Group InnovationsBlagnacFrance

Personalised recommendations