Non‐local dispersive model for wave propagation in heterogeneous media: one‐dimensional case

Fish, Jacob; Chen, Wen; Nagai, Gakuji

doi:10.1002/nme.423

Cited by 126 publications

(122 citation statements)

References 16 publications

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“…Existing methodologies (see for example [55,56,57]) are based on continuum (or quasi-continuum) enrichment. It remain to be seen if higher order enrichment functions (18 functions in 3D…”

Section: Conclusion and Future Research Directionsmentioning

confidence: 99%

Multiscale enrichment based on partition of unity

Fish¹,

Yuan²

2004

Numerical Meth Engineering

Self Cite

139

110

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A new Multiscale Enrichment method based on the Partition of Unity (MEPU) method is presented. It is a synthesis of mathematical homogenization theory and the Partition of Unity method. Its primary objective is to extend the range of applicability of mathematical homogenization theory to problems where scale separation may not be possible. MEPU is perfectly suited for enriching the coarse scale continuum descriptions (PDEs) with fine scale features and the quasi-continuum formulations with relevant atomistic data. Numerical results show that it provides a considerable improvement over classical mathematical homogenization theory and quasi-continuum formulations.

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“…Existing methodologies (see for example [55,56,57]) are based on continuum (or quasi-continuum) enrichment. It remain to be seen if higher order enrichment functions (18 functions in 3D…”

Section: Conclusion and Future Research Directionsmentioning

confidence: 99%

Multiscale enrichment based on partition of unity

Fish¹,

Yuan²

2004

Numerical Meth Engineering

Self Cite

139

110

View full text Add to dashboard Cite

show abstract

“…The interest in this question is not new, see [2] for dispersive limits in the case of large potentials and [3] in the case of high frequency initial data. Likewise, works such as [7,8,9,12] deal with the long time behavior of waves and give formal calculations. Nevertheless, it seems that a rigorous mathematical statement (or even an effective equation) is still missing.…”

Section: Introductionmentioning

confidence: 99%

Dispersive Effective Models for Waves in Heterogeneous Media

Lamacz

2011

Math. Models Methods Appl. Sci.

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We study the long time behavior of waves in a strongly heterogeneous medium, starting from the one-dimensional scalar wave equation with variable coefficients. We assume that the coefficients are periodic with period ε and ε > 0 is a small length parameter. Our main result is the rigorous derivation of two different dispersive models. The first is a fourth-order equation with constant coefficients including powers of ε. In the second model, the ε-dependence is completely avoided by considering a third-order linearized Korteweg-de-Vries equation. Our result is that both simplified models describe the long time behavior well. An essential tool in our analysis is an adaption operator which modifies smooth functions according to the periodic structure of the medium.

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“…Here, the finite element method will be used for the spatial discretisation, whereas the Newmark time integrator will be adopted to progress the solution in the time domain. The finite element equations of the irreducible form are well established and need not be revisited here -the interested reader is referred to [Fish et al 2002a;2002b;Askes and Aifantis 2011]. For the reducible form, we write u = N u d and σ = N σ s where u and σ are column vectors containing the relevant components of the displacements and Cauchy stresses, respectively.…”

Section: Finite Element Equationsmentioning

confidence: 99%

Reducible and irreducible forms of stabilised gradient elasticity in dynamics

Askes

Gitman

2017

Math. Mech. Compl. Sys.

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The continualisation of discrete particle models has been a popular tool to formulate higher-order gradient elasticity models. However, a straightforward continualisation leads to unstable continuum models. Padé approximations can be used to stabilise the model, but the resulting formulation depends on the particular equation that is transformed with the Padé approximation. In this contribution, we study two different stabilised gradient elasticity models; one is an irreducible form with displacement degrees of freedom only, and the other is a reducible form where the primary unknowns are not only displacements but also the Cauchy stresses -this turns out to be Eringen's theory of gradient elasticity. Although they are derived from the same discrete model, there are significant differences in variationally consistent boundary conditions and resulting finite element implementations, with implications for the capability (or otherwise) to suppress crack tip singularities.

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Non‐local dispersive model for wave propagation in heterogeneous media: one‐dimensional case

Cited by 126 publications

References 16 publications

Multiscale enrichment based on partition of unity

Multiscale enrichment based on partition of unity

Dispersive Effective Models for Waves in Heterogeneous Media

Reducible and irreducible forms of stabilised gradient elasticity in dynamics

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