Skip to main content
SpringerLink
Log in

Space–time enriched finite elements for elastodynamic wave propagation

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

This article investigates a generalised finite-element method for time-dependent elastic wave propagation, based on plane-wave enrichments of the approximation space. The enrichment in both space and time allows good approximation of oscillatory solutions even on coarse mesh grids and for large time steps. The proposed method is based on a discontinuous Galerkin discretisation in time and conforming finite elements in space. Numerical experiments study the stability and accuracy and confirm significant reductions of the computational effort required to achieve engineering accuracy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

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
Fig. 14

Similar content being viewed by others

Data availability

No data sets are used and the results are from solving the numerical method. Extensions to the code are being developed before making it publically available.

References

  1. Melenk JM, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139(1):289–314

    Article  MathSciNet  Google Scholar 

  2. Babuska I, Melenk JM (1997) The partition of unity method. Int J Numer Meth Eng 40(4):727–758

    Article  MathSciNet  Google Scholar 

  3. Laghrouche O, Bettess P, Astley RJ (2002) Modelling of short wave diffraction problems using approximating systems of plane waves. Int J Numer Meth Eng 54(10):1501–1533

    Article  Google Scholar 

  4. Laghrouche O, Bettess P, Perrey-Debain E, Trevelyan J (2005) Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed. Comput Methods Appl Mech Eng 194(2):367–381

    Article  MathSciNet  Google Scholar 

  5. Perrey-Debain E, Trevelyan J, Bettess P (2005) On wave boundary elements for radiation and scattering problems with piecewise constant impedance. IEEE Trans Antennas Propag 53(2):876–879

    Article  MathSciNet  Google Scholar 

  6. El Kacimi A, Laghrouche O (2009) Numerical modelling of elastic wave scattering in frequency domain by the partition of unity finite element method. Int J Numer Meth Eng 77(12):1646–1669

    Article  MathSciNet  Google Scholar 

  7. Mahmood MS, Laghrouche O, Trevelyan J, El Kacimi A (2017) Implementation and computational aspects of a 3d elastic wave modelling by pufem. Appl Math Model 49:568–586

    Article  MathSciNet  Google Scholar 

  8. Drolia M, Mohamed MS, Laghrouche O, Seaid M, Trevelyan J (2017) Enriched finite elements for initial-value problem of transverse electromagnetic waves in time domain. Comput Struct 182:354–367

    Article  Google Scholar 

  9. Drolia M, Mohamed M, Laghrouche O, Seaid M, El-Kacimi A (2020) Explicit time integration with lumped mass matrix for enriched finite elements solution of time domain wave problems. Appl Math Model 77:1273–1293

    Article  MathSciNet  Google Scholar 

  10. Ham S, Bathe KJ (2012) A finite element method enriched for wave propagation problems. Comput Struct 94–95:1–12

    Article  Google Scholar 

  11. Iqbal M, Gimperlein H, Mohamed MS, Laghrouche O (2017) An a posteriori error estimate for the generalized finite element method for transient heat diffusion problems. Int J Numer Meth Eng 110(12):1103–1118

    Article  MathSciNet  Google Scholar 

  12. Barucq H, Calandra H, Diaz J, Shishenina E (2020) Space-time Trefftz-DG approximation for elasto-acoustics. Appl Anal 99:747–760

    Article  MathSciNet  Google Scholar 

  13. Gimperlein H, Stark D (2019) Algorithmic aspects of enriched time domain boundary element methods. Eng Anal Boundary Elem 100:118–124

    Article  MathSciNet  Google Scholar 

  14. Antonietti PF, Mazzieri I, Migliorini F (2020) A space-time discontinuous galerkin method for the elastic wave equation. J Comput Phys 419:109685. https://doi.org/10.1016/j.jcp.2020.109685

    Article  MathSciNet  Google Scholar 

  15. Yang Y, Chirputkar S, Alpert DN, Eason T, Spottswood S, Qian D (2012) Enriched space-time finite element method: a new paradigm for multiscaling from elastodynamics to molecular dynamics. Int J Numer Meth Eng 92(2):115–140

    Article  MathSciNet  Google Scholar 

  16. Destuynder P, Hervella-Nieto L, López-Pérez PM, Orellana J, Prieto A (2022) A modal-based partition of unity finite element method for elastic wave propagation problems in layered media. Comput Struct 265:106759

    Article  Google Scholar 

  17. Iqbal M, Gimperlein H, Laghrouche O, Alam K, Shadi Mohamed M, Abid M (2020) A residual a posteriori error estimate for partition of unity finite elements for three-dimensional transient heat diffusion problems using multiple global enrichment functions. Int J Numer Meth Eng 121(12):2727–2746

    Article  MathSciNet  Google Scholar 

  18. Iqbal M, Alam K, Gimperlein H, Laghrouche O, Mohamed MS (2020) Effect of enrichment functions on gfem solutions of time-dependent conduction heat transfer problems. Appl Math Model 85:89–106

    Article  MathSciNet  Google Scholar 

  19. Iqbal M, Stark D, Gimperlein H, Mohamed MS, Laghrouche O (2020) Local adaptive q-enrichments and generalized finite elements for transient heat diffusion problems. Comput Methods Appl Mech Eng 372:113359

    Article  MathSciNet  Google Scholar 

  20. Moiola A, Perugia I (2018) A space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation. Numer Math 138(2):389–435

    Article  MathSciNet  Google Scholar 

  21. Imbert-Gérard L-M, Moiola A, Stocker P (2023) A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients. Math Comput 92(341):1211–1249

    Article  MathSciNet  Google Scholar 

  22. MATLAB: Version 9.12.0 R2022b. The MathWorks Inc., Natick, Massachusetts (2022)

  23. Quaine, K.: Space-time enriched finite elements for acoustic and elastodynamic problems. PhD Thesis, Heriot-Watt University

  24. Laghrouche O, Bettes P (2000) Short wave modelling using special finite elements. J Comput Acoust 08(01):189–210

    Article  MathSciNet  Google Scholar 

  25. El Kacimi A, Laghrouche O (2011) Wavelet based ilu preconditioners for the numerical solution by pufem of high frequency elastic wave scattering. J Comput Phys 230(8):3119–3134

    Article  MathSciNet  Google Scholar 

  26. Richardson CL, Hegemann J, Sifakis E, Hellrung J, Teran JM (2011) An xfem method for modeling geometrically elaborate crack propagation in brittle materials. Int J Numer Meth Eng 88(10):1042–1065

    Article  MathSciNet  Google Scholar 

  27. Sheng X, Liu Y, Zhou X (2016) The response of a high-speed train wheel to a harmonic wheel-rail force. In: Journal of Physics: Conference Series, vol 744, p 012145. IOP Publishing

  28. Gilvey B, Trevelyan J, Hattori G (2020) Singular enrichment functions for helmholtz scattering at corner locations using the boundary element method. Int J Numer Meth Eng 121(3):519–533

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Omar Laghrouche for the fruitful discussions on the topics of this article.

Author information

Authors and Affiliations

  1. Engineering Mathematics, Leopold-Franzens-Universität Innsbruck, Technikerstraße 13, 6020, Innsbruck, Austria

    Kieran Quaine & Heiko Gimperlein

  2. Maxwell Institute for Mathematical Sciences and School of Mathematics, Edinburgh University, James Clerk Maxwell Building, Edinburgh, EH9 3FD, UK

    Kieran Quaine

  3. FEN Research GmbH, Technikerstraße 1/3, 6020, Innsbruck, Austria

    Kieran Quaine

  4. Department of Mathematical, Physical and Computer Sciences, University of Parma, Parco Area delle Scienze, 53/A, 43124, Parma, Italy

    Heiko Gimperlein

Authors
  1. Kieran Quaine
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Heiko Gimperlein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kieran Quaine.

Ethics declarations

Conflict of interest

The authors would like to thank The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (Grant EP/L016508/01), Heriot-Watt University, University of Edinburgh and AWE (Contract No. PO30408299) for their sponsorship. The authors have no further competing interests to declare that are relevant to the content of this article.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Quaine, K., Gimperlein, H. Space–time enriched finite elements for elastodynamic wave propagation. Engineering with Computers 39, 4077–4091 (2023). https://doi.org/10.1007/s00366-023-01874-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-023-01874-z

Keywords

Search

Navigation