Operator Splits and Multiscale Methods in Computational Dynamics

  • Harm AskesEmail author
  • Dario De Domenico
  • Mingxiu Xu
  • Inna M. Gitman
  • Terry Bennett
  • Elias C. Aifantis
Part of the Mathematics of Planet Earth book series (MPE, volume 6)


Gradient enriched continua are an elegant and versatile class of material models that are able to simulate a variety of physical phenomena, ranging from singularity-free descriptions of crack tips and dislocations, via size dependent mechanical response, to dispersive wave propagation. However, the increased order of the governing partial differential equations has historically complicated analytical and numerical solution methods. Inspired by the work of Ru and Aifantis (Acta Mech. 101(1-4), 59–68 (1993)), this contribution focusses on operator split methods that allow to reduce the order of the governing equations. It will be shown that this order reduction leads to multiscale reformulations of the original equations in which the macrolevel unknowns are fully coupled to the microlevel unknowns. As a first example, gradient enriched equations of elastodynamics are considered with second order and fourth order microinertia terms. The second example concerns dynamic piezomagnetics with gradient enrichment of both the mechanical fields and the magnetic fields.


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HA, IMG and ECA gratefully acknowledge support of the EU RISE project FRAMED-734485. MX gratefully acknowledges financial support from the China Scholarship Council and the Fundamental Research Funds for the Central Universities (FRFBR-16-017A).


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Harm Askes
    • 1
    Email author
  • Dario De Domenico
    • 2
  • Mingxiu Xu
    • 3
  • Inna M. Gitman
    • 4
  • Terry Bennett
    • 5
  • Elias C. Aifantis
    • 6
  1. 1.Department of Civil and Structural EngineeringUniversity of SheffieldSheffieldUK
  2. 2.Department of EngineeringUniversity of MessinaMessinaItaly
  3. 3.Department of Applied MechanicsUniversity of Science and Technology BeijingBeijingChina
  4. 4.Department of Mechanical EngineeringUniversity of SheffieldSheffieldUK
  5. 5.School of Civil, Environmental and Mining EngineeringUniversity of AdelaideAdelaideAustralia
  6. 6.Laboratory of Mechanics and MaterialsAristotle University of ThessalonikiThessalonikiGreece

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