Computational Mechanics

, Volume 57, Issue 2, pp 211–235 | Cite as

Asynchronous space–time algorithm based on a domain decomposition method for structural dynamics problems on non-matching meshes

  • Waad Subber
  • Karel MatoušEmail author
Original Paper


Large-scale practical engineering problems featuring localized phenomena often benefit from local control of mesh and time resolutions to efficiently capture the spatial and temporal scales of interest. To this end, we propose an asynchronous space–time algorithm based on a domain decomposition method for structural dynamics problems on non-matching meshes. The three-field algorithm is based on the dual-primal like domain decomposition approach utilizing the localized Lagrange multipliers along the space and time common-refinement-based interface. The proposed algorithm is parallel in nature and well suited for a heterogeneous computing environment. Moreover, two-levels of parallelism are embedded in this novel scheme. For linear dynamical problems, the algorithm is unconditionally stable, shows an optimal order of convergence with respect to space and time discretizations as well as ensures conservation of mass, momentum and energy across the non-matching grid interfaces. The method of manufactured solutions is used to verify the implementation, and an engineering application is considered, where a sandwich plate is impacted by a projectile.


Asynchronous time integration Multi-time-step methods  Local Lagrange multipliers Domain decomposition methods  Non-matching grids High-performance computing 



This work has been supported by the Department of Energy, National Nuclear Security Administration, under Award No. DE-NA0002377.


  1. 1.
    Abedi R, Hawker MA, Haber RB, Matouš K (2010) An adaptive spacetime discontinuous Galerkin method for cohesive models of elastodynamic fracture. Int J Numer Methods Eng 81(10):1207–1241zbMATHGoogle Scholar
  2. 2.
    Arbogast T, Pencheva G, Wheeler MF, Yotov I (2007) A multiscale mortar mixed finite element method. Multiscale Model Simul 6(1):319–346CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    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):209–225CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Belytschko T, Mullen R (1978) Mesh partitions of explicit-implicit time integration. In: Bath K, Oden J, Wunderlich W (eds) Formulations and computational algorithms in finite element analysis. MIT Press, Cambridge, pp 673–690Google Scholar
  5. 5.
    Belytschko T, Mullen R (1978) Stability of explicit-implicit mesh partitions in time integration. Int J Numer Methods Eng 12(10):1575–1586CrossRefzbMATHGoogle Scholar
  6. 6.
    Belytschko T, Yen HJ, Mullen R (1979) Mixed methods for time integration. Comput Methods Appl Mech Eng 17:259–275CrossRefGoogle Scholar
  7. 7.
    Beneš M, Matouš K (2010) Asynchronous multi-domain variational integrators for nonlinear hyperelastic solids. Comput Methods Appl Mech Eng 199(2932):1992–2013zbMATHGoogle Scholar
  8. 8.
    Brezzi F, Marini LD (2005) The three-field formulation for elasticity problems. GAMM-Mitteilungen 28(2):124–153CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Brun M, Gravouil A, Combescure A, Limam A (2015) Two feti-based heterogeneous time step coupling methods for newmark and \(\alpha \)-schemes derived from the energy method. Comput Methods Appl Mech Eng 283:130–176CrossRefMathSciNetGoogle Scholar
  10. 10.
    Combescure A (2002) G.A.: a numerical scheme to couple subdomains with different time-steps for predominantly linear transient analysis. Comput Methods Appl Mech Eng 191:1129–1157CrossRefzbMATHGoogle Scholar
  11. 11.
    De Boer A, Van Zuijlen A, Bijl H (2007) Review of coupling methods for non-matching meshes. Comput Methods Appl Mech Eng 196(8):1515–1525CrossRefzbMATHGoogle Scholar
  12. 12.
    Falcone M, Ferretti R (1998) Convergence analysis for a class of high-order semi-lagrangian advection schemes. SIAM J Numer Anal 35(3):909–940CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Farhat C, Chen PS, Mandel J (1995) A scalable lagrange multiplier based domain decomposition method for time-dependent problems. Int J Numer Methods Eng 38(22):3831–3853CrossRefzbMATHGoogle Scholar
  14. 14.
    Farhat C, Crivelli L, Géradin M (1995) Implicit time integration of a class of constrained hybrid formulationspart i: Spectral stability theory. Computer methods in applied mechanics and engineering 125(1):71–107CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Farhat C, Crivelli L, Roux FX (1994) A transient FETI methodology for large-scale parallel implicit computations in structural mechanics. Int J Numer Methods Eng 37(11):1945–1975CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Farhat C, Lesoinne M, Le Tallec P (1998) Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity. Comput Methods Appl Mech Eng 157(1):95–114CrossRefzbMATHGoogle Scholar
  17. 17.
    Farhat C, Roux FX (1991) A method of finite element tearing and interconnecting and its parallel solution algorithm. Int J Numer Methods Eng 32(6):1205–1227CrossRefzbMATHGoogle Scholar
  18. 18.
    Faucher V, Combescure A (2003) A time and space mortar method for coupling linear modal subdomains and non-linear subdomains in explicit structural dynamics. Comput Methods Appl Mech Eng 192(5):509–533CrossRefzbMATHGoogle Scholar
  19. 19.
    Gander MJ, Vandewalle S (2007) Analysis of the parareal timeparallel timeintegration method. SIAM J Sci Comput 29(2):556–578CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Gates M, Matouš K, Heath MT (2008) Asynchronous multi-domain variational integrators for non-linear problems. Int J Numer Methods Eng 76(9):1353–1378CrossRefzbMATHGoogle Scholar
  21. 21.
    Gravouil A, Combescure A (2001) Multi-time-step explicit-implicit method for non-linear structural dynamics. Int J Numer Methods Eng 50(1):199–225CrossRefzbMATHGoogle Scholar
  22. 22.
    Gravouil A, Combescure A, Brun M. (2014) Heterogeneous asynchronous time integrators for computational structural dynamics. Int J Numer Methods EngGoogle Scholar
  23. 23.
    Hauret P, Le Tallec P (2007) A discontinuous stabilized mortar method for general 3d elastic problems. Comput Methods Appl Mech Eng 196(49):4881–4900CrossRefzbMATHGoogle Scholar
  24. 24.
    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 Methods Eng 55(8):973–1003CrossRefzbMATHGoogle Scholar
  25. 25.
    Hughes T (2000) The finite element method: linear static and dynamic finite element analysis., Dover civil and mechanical engineeringDover Publications, MineolaGoogle Scholar
  26. 26.
    Hughes T, Liu W (1978) Implicit-explicit finite elements in transient analysis: implementation and numerical examples. J Appl Mech 45:375–378CrossRefzbMATHGoogle Scholar
  27. 27.
    Hughes T, Liu W (1978) Implicit-explicit finite elements in transient analysis: stability theory. J Appl Mech 45(2):371–374CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Hutter K, Johnk K (2004) Continuum methods of physical modeling. Springer, New YorkCrossRefzbMATHGoogle Scholar
  29. 29.
    Jaiman RK, Jiao X, Geubelle PH, Loth E (2005) Assessment of conservative load transfer for fluid-solid interface with non-matching meshes. Int J Numer Methods Eng 64(15):2014–2038CrossRefzbMATHGoogle Scholar
  30. 30.
    Jiao X, Heath MT (2004) Common-refinement-based data transfer between non-matching meshes in multiphysics simulations. Int J Numer Methods Eng 61(14):2402–2427CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Jiao X, Heath MT (2004) Overlaying surface meshes, part I: algorithms. Int J Comput Geom Appl 14(06):379–402CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Jiao X, Heath MT (2004) Overlaying surface meshes, part II: Topology preservation and feature matching. Int J Comput Geom Appl 14(06):403–419CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Kane C, Marsden JE, Ortiz M, West M (2000) Variational integrators and the newmark algorithm for conservative and dissipative mechanical systems. Int J Numer Methods Eng 49(10):1295–1325Google Scholar
  34. 34.
    Karimi S, Nakshatrala K (2014) On multi-time-step monolithic coupling algorithms for elastodynamics. J Comput Phys 273:671–705CrossRefMathSciNetGoogle Scholar
  35. 35.
    Kim JS, Arronche L, Farrugia A, Muliana A, Saponara VL (2011) Multi-scale modeling of time-dependent response of smart sandwich constructions. Compos Struct 93(9):2196–2207CrossRefGoogle Scholar
  36. 36.
    Kruis J, Zeman J, Gruber P (2013) Model of imperfect interfaces in composite materials and its numerical solution by FETI method. In: Bank R, Holst M, Widlund O, Xu J (eds) Domain decomposition methods in science and engineering XX., Lecture notes incomputational science and engineeringSpringer, BerlinGoogle Scholar
  37. 37.
    Maday Y, Turinici G (2005) The parareal in time iterative solver: a further direction to parallel implementation. In: Barth T, Griebel M, Keyes D, Nieminen R, Roose D, Schlick T, Kornhuber R, Hoppe R, Priaux J, Pironneau O, Xu J (eds) Domain de-composition methods in science and engineering, vol 40., Lecture notes in computational scienceand engineeringSpringer, Berlin, pp 441–448CrossRefGoogle Scholar
  38. 38.
    Magouls F, Roux FX (2006) Lagrangian formulation of domain decomposition methods: a unified theory. Appl Math Model 30(7):593–615 (Parallel and vector processing in science and engineering)Google Scholar
  39. 39.
    Mahjoubi N, Gravouil A, Combescure A (2009) Coupling subdomains with heterogeneous time integrators and incompatible time steps. Comput Mech 44(6):825–843CrossRefMathSciNetzbMATHGoogle Scholar
  40. 40.
    Mandel J, Dohrmann CR (2003) Convergence of a balancing domain decomposition by constraints and energy minimization. Numer Linear Algebra Appl 10(7):639–659CrossRefMathSciNetzbMATHGoogle Scholar
  41. 41.
    Mathew T (2008) Domain decomposition methods for the numerical solution of partial differential equations, vol 61., Lecture notes in computational science and engineeringSpringer, BerlinzbMATHGoogle Scholar
  42. 42.
    Nakshatrala K, Prakash A, Hjelmstad K (2009) On dual Schur domain decomposition method for linear first-order transient problems. J Comput Phys 228(21):7957–7985CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    Neal MO, Belytschko T (1989) Explicit-explicit subcycling with non-integer time step ratios for structural dynamic systems. Comput Struct 31(6):871–880CrossRefMathSciNetzbMATHGoogle Scholar
  44. 44.
    Oberkampf WL, Trucano TG (2002) Verification and validation in computational fluid dynamics. Prog Aerosp Sci 38(3):209–272CrossRefGoogle Scholar
  45. 45.
    Park K, Felippa C, Rebel G (2002) A simple algorithm for localized construction of non-matching structural interfaces. Int J Numer Methods Eng 53(9):2117–2142CrossRefMathSciNetzbMATHGoogle Scholar
  46. 46.
    Park K, Felippa CA (1998) A variational framework for solution method developments in structural mechanics. J Appl Mech 65(1):242–249CrossRefMathSciNetGoogle Scholar
  47. 47.
    Park K, Felippa CA (2000) A variational principle for the formulation of partitioned structural systems. Int J Numer Methods Eng 47(1–3):395–418CrossRefMathSciNetzbMATHGoogle Scholar
  48. 48.
    Park K, Felippa CA, Ohayon R (2001) Partitioned formulation of internal fluidstructure interaction problems by localized lagrange multipliers. Comput Methods Appl Mech Eng 190(2425):2989–3007 (Advances in computational methods for fluid–structure interaction)Google Scholar
  49. 49.
    Park KC, Felippa CA, Gumaste UA (2000) A localized version of the method of lagrange multipliers and its applications. Comput Mech 24(6):476–490CrossRefzbMATHGoogle Scholar
  50. 50.
    Park KC, Felippa CA, Rebel G (2002) A simple algorithm for localized construction of non-matching structural interfaces. Int J Numer Methods Eng 53(9):2117–2142CrossRefMathSciNetzbMATHGoogle Scholar
  51. 51.
    Prakash A, Hjelmstad KD (2004) A FETI-based multi-time-step coupling method for Newmark schemes in structural dynamics. Int J Numer Methods Eng 61(13):2183–2204Google Scholar
  52. 52.
    Prakash A, Taciroglu E, Hjelmstad KD (2014) Computationally efficient multi-time-step method for partitioned time integration of highly nonlinear structural dynamics. Comput Struct 133:51–63CrossRefGoogle Scholar
  53. 53.
    Quarteroni A, Valli A (1999) Domain decomposition methods for partial differential equations. Numerical mathematics and scientific computation. Oxford University Press, New YorkGoogle Scholar
  54. 54.
    Radu F, Pop IS, Knabner P (2004) Order of convergence estimates for an euler implicit, mixed finite element discretization of richards’ equation. SIAM J Numer Anal 42(4):1452–1478CrossRefMathSciNetzbMATHGoogle Scholar
  55. 55.
    Roache PJ (1998) Verification and validation in computational science and engineering. Hermosa, AlbuquerqueGoogle Scholar
  56. 56.
    Ross MR, Sprague MA, Felippa CA, Park K (2009) Treatment of acoustic fluid-structure interaction by localized lagrange multipliers and comparison to alternative interface-coupling methods. Comput Methods Appl Mech Eng 198(9):986–1005CrossRefzbMATHGoogle Scholar
  57. 57.
    Scovazzi G, Love E, Shashkov M (2008) Multi-scale Lagrangian shock hydrodynamics on Q1/P0 finite elements: theoretical framework and two-dimensional computations. Comput Methods Appl Mech Eng 197(912):1056–1079CrossRefMathSciNetzbMATHGoogle Scholar
  58. 58.
    Smith B, Bjorstad P, Gropp W (1996) Domain decomposition: parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, New YorkzbMATHGoogle Scholar
  59. 59.
    Smolinski P, Sleith S (1992) Explicit multi-time step methods for structural dynamics. In: New methods in transient analysis, pp. 1–4. ASME. PVP-Vol. 246 /AMD-Vol. 143Google Scholar
  60. 60.
    Smolinski P, Sleith S, Belytschko T (1996) Stability of an explicit multi-time step integration algorithm for linear structural dynamics equations. Comput Mech 18(3):236–244CrossRefMathSciNetzbMATHGoogle Scholar
  61. 61.
    Sousedik B, Sistek J, Mandel J (2013) Adaptive-multilevel BDDC and its parallel implementation. Computing 95(12):1087–1119CrossRefMathSciNetzbMATHGoogle Scholar
  62. 62.
    Souza F, Allen D, Kim YR (2008) Multiscale model for predicting damage evolution in composites due to impact loading. Compos Sci Technol 68(13):2624–2634CrossRefGoogle Scholar
  63. 63.
    Toselli A, Widlund O (2005) Domain decomposition methods: algorithms and theory, vol 34., Computational mathematicsSpringer, BerlinGoogle Scholar
  64. 64.
    Wiberg NE, Li X (1993) A post-processing technique and an a posteriori error estimate for the newmark method in dynamic analysis. Earthq Eng Struct Dyn 22(6):465–489CrossRefGoogle Scholar
  65. 65.
    Zeng LF, Wiberg NE, Li X, Xie Y (1992) Posteriori local error estimation and adaptive time-stepping for newmark integration in dynamic analysis. Earthq Eng Struct Dyn 21(7):555–571CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Center for Shock Wave-processing of Advanced Reactive Materials (C-SWARM)University of Notre DameNotre DameUSA
  2. 2.Department of Aerospace and Mechanical Engineering, Center for Shock Wave-processing of Advanced Reactive Materials (C-SWARM)University of Notre DameNotre DameUSA

Personalised recommendations