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Asymptotic analysis of dynamic problems for elastic media with clusters of small inclusions

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MWSW02 - Theory of wave scattering in complex and random media

In this talk, we discuss an asymptotic approach used to accurately predict the vibration response of elastic media containing clusters of many, small, closely interacting inclusions of arbitrary shape [1]. The approach is based on the so-called method of mesoscale approximations, previously developed for quasi-static problems [2]. The resulting approximations make use of model solutions to dynamic problems (i) posed in the medium when the inclusions are absent and (ii) involving individual inclusions. These model solutions are combined with appropriate weights that solve a linear algebraic system, which naturally appears in attempting to satisfy the boundary conditions to a high order of accuracy and contains information about the size and shape of each inclusion, as well as their interaction. The method presented yields uniformly accurate approximations for fields associated with scattering problems and eigenmodes of finite systems containing defect clusters and allows one to (i) obtain effective models for the cluster at low-frequencies and (ii) address scenarios involving dilute clusters. The theoretical results are accompanied by numerical illustrations that demonstrate the effectiveness of the approach. References: [1] Nieves, M.J. and Movchan, A.B. (2022). Asymptotic analysis of in-plane dynamic problems for elastic media with rigid clusters of small inclusions. Phil. Trans. R. Soc. A.380: 20210392. doi: [2] Maz’ya V.G., Movchan, A.B., Nieves, M.J. (2013): Green’s Kernels and Meso-Scale Approximations in Perforated Domains, Lecture Notes in Mathematics, pp. XVII , 258, Springer Cham. doi:

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