Operator Splits and Multiscale Methods in Computational Dynamics
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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