Skip to main content

Scale-bridging with the extended/generalized finite element method for linear elastodynamics

Abstract

This paper presents an extended/generalized finite element method for bridging scales in linear elastodynamics in the absence of scale separation. More precisely, the GFEMgl framework is expanded to enable the numerical solution of multiscale problems through the automated construction of specially-tailored shape functions, thereby enabling high-fidelity finite element modeling on simple, fixed finite element meshes. This introduces time-dependencies in the shape functions in that they are subject to continuous adaptation with time. The temporal aspects of the formulation are investigated by considering the Newmark-\(\beta \) time integration scheme, and the efficacy of mass lumping strategies is explored in an explicit time-stepping scheme. This method is demonstrated on representative wave propagation examples as well as a dynamic fracture problem to assess its accuracy and flexibility.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

References

  1. 1.

    Ambrosio L, Tortorelli VM (1990) Approximation of functionals depending on jumps by elliptic functionals via \(\gamma \)-convergence. Commun Pure Appl Math 43(8):999–1036. https://doi.org/10.1002/cpa.3160430805

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Armando Duarte C, Kim DJ, Babuška I (2007) 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, Armando Duarte C (eds) Advances in meshfree techniques. Springer, Dordrecht, pp 1–26

    Google Scholar 

  3. 3.

    Asareh I, Song JH, Mullen RL, Qian Y (2020) A general mass lumping scheme for the variants of the extended finite element method. Int J Numer Methods Eng 121(10):2262–2284. https://doi.org/10.1002/nme.6308

    MathSciNet  Article  Google Scholar 

  4. 4.

    Babuška I, Melenk JM (1997) The partition of unity method. Int J Numer Methods Eng 40(4):727–758. https://doi.org/10.1002/(SICI)1097-0207(19970228)40:4≤727::AID-NME86≥3.0.CO;2-N

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bangerth W, Geiger M, Rannacher R (2010) Adaptive Galerkin finite element methods for the wave equation. Comput Methods Appl Math 10(1):3–48

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bažant ZP, Belytschko TB, Chang T (1984) Continuum theory for strain-softening. J Eng Mech 110(12):1666–1692. https://doi.org/10.1061/(ASCE)0733-9399(1984)110:12(1666)

    Article  Google Scholar 

  7. 7.

    Belytschko T, Mullen R (1978) Stability of explicit-implicit mesh partitions in time integration. Int J Numer Methods Eng 12(10):1575–1586. https://doi.org/10.1002/nme.1620121008

    Article  MATH  Google Scholar 

  8. 8.

    Belytschko T, Yen HJ, Mullen R (1979) Mixed methods for time integration. Comput Methods Appl Mech Eng 17–18:259–275. https://doi.org/10.1016/0045-7825(79)90022-7

    Article  MATH  Google Scholar 

  9. 9.

    Bourdin B, Francfort G, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826. https://doi.org/10.1016/S0022-5096(99)00028-9

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Braides A (1998) Approximation of free-discontinuity problems, vol 1694. Lecture notes in mathematics, Springer, Berlin

    Book  Google Scholar 

  11. 11.

    Chessa J, Belytschko T (2003) An extended finite element method for two-phase fluids. J Appl Mech 70(1):10–17. https://doi.org/10.1115/1.1526599

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Chessa J, Belytschko T (2004) Arbitrary discontinuities in space-time finite elements by level sets and x-fem. Int J Numer Methods Eng 61(15):2595–2614. https://doi.org/10.1002/nme.1155

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\(\alpha \) method. J Appl Mech 60(2):371–375. https://doi.org/10.1115/1.2900803

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Duarte C, Kim DJ (2008) Analysis and applications of a generalized finite element method with global-local enrichment functions. Comput Methods Appl Mech Eng 197(6):487–504. https://doi.org/10.1016/j.cma.2007.08.017

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Duarte C, Babuška I, Oden J (2000) Generalized finite element methods for three-dimensional structural mechanics problems. Comput Struct 77(2):215–232. https://doi.org/10.1016/S0045-7949(99)00211-4

    MathSciNet  Article  Google Scholar 

  16. 16.

    Elguedj T, Gravouil A, Maigre H (2009) An explicit dynamics extended finite element method. Part 1: mass lumping for arbitrary enrichment functions. Comput Methods Appl Mech Eng 198(30):2297–2317. https://doi.org/10.1016/j.cma.2009.02.019

  17. 17.

    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(3):253–304. https://doi.org/10.1002/nme.2914

  18. 18.

    Fries TP, Zilian A (2009) On time integration in the xfem. Int J Numer Methods Eng 79(1):69–93. https://doi.org/10.1002/nme.2558

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Geelen R (2020) Towards simulations of pervasive fracture across structural scales. PhD thesis, Duke University

  20. 20.

    Geelen R, Liu Y, Hu T, Tupek M, Dolbow J (2019) A phase-field formulation for dynamic cohesive fracture. Comput Methods Appl Mech Eng 348:680–711. https://doi.org/10.1016/j.cma.2019.01.026

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Geelen R, Plews J, Tupek M, Dolbow J (2020) An extended/generalized phase-field finite element method for crack growth with global-local enrichment. Int J Numer Methods Eng 121(11):2534–2557. https://doi.org/10.1002/nme.6318

    MathSciNet  Article  Google Scholar 

  22. 22.

    Gupta V, Kim DJ, Duarte CA (2012) Analysis and improvements of global-local enrichments for the generalized finite element method. Comput Methods Appl Mech Eng 245–246:47–62. https://doi.org/10.1016/j.cma.2012.06.021

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Hilber HM, Hughes TJR, Taylor RL (1977) Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq Eng Struct Dyn 5(3):283–292. https://doi.org/10.1002/eqe.4290050306

    Article  Google Scholar 

  24. 24.

    Hughes TJ (2012) The finite element method: linear static and dynamic finite element analysis. Courier Corporation, North Chelmsford

    Google Scholar 

  25. 25.

    Ji H, Chopp D, Dolbow JE (2002) A hybrid extended finite element/level set method for modeling phase transformations. Int J Numer Methods Eng 54(8):1209–1233. https://doi.org/10.1002/nme.468

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Kim DJ, 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): https://doi.org/10.1115/1.2936240

    Article  Google Scholar 

  27. 27.

    Kim DJ, 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. https://doi.org/10.1002/nme.2690

    Article  MATH  Google Scholar 

  28. 28.

    Marigo JJ, Maurini C, Pham K (2016) An overview of the modelling of fracture by gradient damage models. Meccanica 51(12):3107–3128. https://doi.org/10.1007/s11012-016-0538-4

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Matouš K, Geers MG, Kouznetsova VG, Gillman A (2017) A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials. J Comput Phys 330:192–220. https://doi.org/10.1016/j.jcp.2016.10.070

    MathSciNet  Article  Google Scholar 

  30. 30.

    Melenk J, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139(1):289–314. https://doi.org/10.1016/S0045-7825(96)01087-0

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Menouillard T, Réthoré J, Combescure A, Bung H (2006) Efficient explicit time stepping for the extended finite element method (x-fem). Int J Numer Methods Eng 68(9):911–939. https://doi.org/10.1002/nme.1718

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Menouillard T, Réthoré J, Moës N, Combescure A, Bung H (2008) Mass lumping strategies for x-fem explicit dynamics: application to crack propagation. Int J Numer Methods Eng 74(3):447–474. https://doi.org/10.1002/nme.2180

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Merle R, Dolbow J (2002) Solving thermal and phase change problems with the extended finite element method. Comput Mech 28(5):339–350

    Article  Google Scholar 

  34. 34.

    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150. https://doi.org/10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    O’Hara P, Duarte CA, Eason T (2011) Transient analysis of sharp thermal gradients using coarse finite element meshes. Comput Methods Appl Mech Eng 200(5):812–829. https://doi.org/10.1016/j.cma.2010.10.005

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    O’Hara P, Duarte CA, Eason T, Garzon J (2013) Efficient analysis of transient heat transfer problems exhibiting sharp thermal gradients. Comput Mech 51(5):743–764

    MathSciNet  Article  Google Scholar 

  37. 37.

    Plews JA, Duarte CA (2016) A two-scale generalized finite element approach for modeling localized thermoplasticity. Int J Numer Methods Eng 108(10):1123–1158. https://doi.org/10.1002/nme.5241

    MathSciNet  Article  Google Scholar 

  38. 38.

    Schweitzer MA (2013) Variational mass lumping in the partition of unity method. SIAM J Sci Comput 35(2):A1073–A1097. https://doi.org/10.1137/120895561

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Schweitzer MA, Wu S (2015) Numerical integration of on-the-fly-computed enrichment functions in the pum. In: Griebel M, Schweitzer MA (eds) Meshfree methods for partial differential equations VII. Springer, Cham, pp 247–267

    Chapter  Google Scholar 

  40. 40.

    Shadi Mohamed M, Seaid M, Trevelyan J, Laghrouche O (2013) A partition of unity fem for time-dependent diffusion problems using multiple enrichment functions. Int J Numer Methods Eng 93(3):245–265. https://doi.org/10.1002/nme.4383

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    van der Meer FP, Al-Khoury R, Sluys LJ (2009) Time-dependent shape functions for modeling highly transient geothermal systems. Int J Numer Methods Eng 77(2):240–260. https://doi.org/10.1002/nme.2414

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

Rudy Geelen and John Dolbow would like to thank Professor Carlos Armando Duarte for several helpful conversations on GFEMgl. This work was performed under a research grant from Sandia National Laboratories, to Duke University. That support is gratefully acknowledged. The work was also partially supported by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under Contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

Author information

Affiliations

Authors

Corresponding author

Correspondence to John Dolbow.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Displacement-difference form of Newmark’s equations

Displacement-difference form of Newmark’s equations

In this appendix it is shown how the displacement-difference form of Newmark’s algorithm, (12), can be derived (see also Exercise 4 in Section 9.3 of [24]).

Multiplying (10a), (10b) with the density and plugging in (1a) yields

$$\begin{aligned} \rho {\mathbf {u}}_{n+1}&= \rho {\mathbf {u}}_n + \varDelta t \rho \dot{{\mathbf {u}}}_n \nonumber \\&\qquad + \varDelta t^2 \left( \dfrac{1}{2} - \beta \right) \left( \nabla \cdot \varvec{\sigma }_n + {\mathbf {b}}_n \right) \nonumber \\&\qquad + \varDelta t^2 \beta \left( \nabla \cdot \varvec{\sigma }_{n+1} + {\mathbf {b}}_{n+1} \right) , \end{aligned}$$
(38a)
$$\begin{aligned} \rho \dot{{\mathbf {u}}}_{n+1}&= \rho \dot{{\mathbf {u}}}_n + \varDelta t (1-\gamma ) \left( \nabla \cdot \varvec{\sigma }_n + {\mathbf {b}}_n \right) \nonumber \\&\qquad + \varDelta t \gamma \left( \nabla \cdot \varvec{\sigma }_{n+1} + {\mathbf {b}}_{n+1} \right) . \end{aligned}$$
(38b)

Solving (38a) for \(\rho \dot{{\mathbf {u}}}_n\) and substituting into (38b) gives

$$\begin{aligned} \begin{aligned} \rho \dot{{\mathbf {u}}}_{n+1}&= \dfrac{1}{\varDelta t} \left( \rho {\mathbf {u}}_{n+1} - \rho {\mathbf {u}}_n \right) \\&\qquad - \varDelta t \left( \dfrac{1}{2} - \beta \right) \left( \nabla \cdot \varvec{\sigma }_n + {\mathbf {b}}_n \right) \\&\qquad - \varDelta t \beta \left( \nabla \cdot \varvec{\sigma }_{n+1} + {\mathbf {b}}_{n+1} \right) \\&\qquad + \varDelta t (1-\gamma ) \left( \nabla \cdot \varvec{\sigma }_n + {\mathbf {b}}_n \right) \\&\qquad + \varDelta t \gamma \left( \nabla \cdot \varvec{\sigma }_{n+1} + {\mathbf {b}}_{n+1} \right) . \end{aligned} \end{aligned}$$
(39)

Writing (39) at time step \(t_n\) and substituting into (38a), we obtain

$$\begin{aligned} \begin{aligned} \rho {\mathbf {u}}_{n+1}&= 2 \rho {\mathbf {u}}_{n} - \rho {\mathbf {u}}_{n-1} \\&\qquad - \varDelta t^2 \left( \dfrac{1}{2} - \beta \right) \left( \nabla \cdot \varvec{\sigma }_{n-1} + {\mathbf {b}}_{n-1} \right) \\&\qquad - \varDelta t^2 \beta \left( \nabla \cdot \varvec{\sigma }_{n} + {\mathbf {b}}_{n} \right) \\&\qquad + \varDelta t^2 (1-\gamma ) \left( \nabla \cdot \varvec{\sigma }_{n-1} + {\mathbf {b}}_{n-1} \right) \\&\qquad + \varDelta t^2 \gamma \left( \nabla \cdot \varvec{\sigma }_{n} + {\mathbf {b}}_{n} \right) \\&\qquad + \varDelta t^2 \left( \dfrac{1}{2} - \beta \right) \left( \nabla \cdot \varvec{\sigma }_n + {\mathbf {b}}_n \right) \\&\qquad + \varDelta t^2 \beta \left( \nabla \cdot \varvec{\sigma }_{n+1} + {\mathbf {b}}_{n+1} \right) . \end{aligned} \end{aligned}$$
(40)

Moving all the known terms to the right-hand side yields the desired equation in terms of displacement at time steps \(t_{n+1}, t_n\) and \(t_{n-1}\):

$$\begin{aligned} \begin{aligned} \rho {\mathbf {u}}_{n+1}&- \varDelta t^2 \beta \nabla \cdot \varvec{\sigma }_{n+1} = 2 \rho {\mathbf {u}}_{n} - \rho {\mathbf {u}}_{n-1} \\&\qquad + \varDelta t^2 \left( \dfrac{1}{2} + \beta - \gamma \right) \left( \nabla \cdot \varvec{\sigma }_{n-1} + {\mathbf {b}}_{n-1} \right) \\&\qquad + \varDelta t^2 \left( \dfrac{1}{2} -2\beta + \gamma \right) \left( \nabla \cdot \varvec{\sigma }_n + {\mathbf {b}}_n \right) \\&\qquad + \varDelta t^2 \beta {\mathbf {b}}_{n+1}, \end{aligned} \end{aligned}$$
(41)

in which all terms depending on unknown deformation state \({\mathbf {u}}_{n+1}\) are on the left-hand side of (41), and the remaining terms form the right-hand side.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Geelen, R., Plews, J. & Dolbow, J. Scale-bridging with the extended/generalized finite element method for linear elastodynamics. Comput Mech 68, 295–310 (2021). https://doi.org/10.1007/s00466-021-02032-2

Download citation

Keywords

  • Extended/generalized finite element methods
  • Scale-bridging
  • Transient dynamics
  • Multiscale problems
  • Time-dependent shape functions
  • Variational fracture modeling
  • Phase-field