# On meso-scale approximations for vibrations of membranes with lower-dimensional clusters of inertial inclusions

@article{Mazya2020OnMA, title={On meso-scale approximations for vibrations of membranes with lower-dimensional clusters of inertial inclusions}, author={Vladimir Maz'ya and Alexander B. Movchan and Michael J. Nieves}, journal={ArXiv}, year={2020}, volume={abs/2002.02810} }

In this paper we consider formal asymptotic algorithms for a class of meso-scale approximations for problems of vibration of elastic membranes, which contain clusters of small inertial inclusions distributed along contours of pre-defined smooth shapes. Effective transmission conditions have been identified for inertial structured interfaces, and approximations to solutions of eigenvalue problems have been derived for domains containing lower-dimensional clusters of inclusions.

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## References

SHOWING 1-10 OF 16 REFERENCES

Eigenvalue Problem in a Solid with Many Inclusions: Asymptotic Analysis

- MathematicsMultiscale Model. Simul.
- 2017

We construct the asymptotic approximation to the first eigenvalue and corresponding eigensolution of Laplace's operator inside a domain containing a cloud of small rigid inclusions. The separation of…

Green's Kernels and Meso-Scale Approximations in Perforated Domains

- Mathematics
- 2013

Systematic step-by-step approach to asymptotic algorithms that enables the reader to develop an insight to compound asymptotic approximations Presents a novel, well-explained method of meso-scale…

Mesoscale Asymptotic Approximations to Solutions of Mixed Boundary Value Problems in Perforated Domains

- MathematicsMultiscale Model. Simul.
- 2011

We describe a method of asymptotic approximations to solutions of mixed boundary value problems for the Laplacian in a three-dimensional domain with many perforations of arbitrary shape, with the…

Asymptotic treatment of perforated domains without homogenization

- Mathematics
- 2009

As a main result of the paper, we construct and justify an asymptotic approximation of Green's function in a domain with many small inclusions. Periodicity ofthe array ofinclusions is not required.…

Green's kernels for transmission problems in bodies with small inclusions

- Mathematics
- 2010

The uniform asymptotic approximation of Green's kernel for the transmission problem of antiplane shear is obtained for domains with small inclusions. The remainder estimates are provided. Numerical…

The Green's Function for the Two-Dimensional Helmholtz Equation in Periodic Domains

- Mathematics
- 1998

Analytical techniques are described for transforming the Green's function for the two-dimensional Helmholtz equation in periodic domains from the slowly convergent representation as a series of…

Homogenization of Partial Differential Equations

- Mathematics
- 2005

* Preface * Introduction * The Dirichlet Boundary Value Problem in Strongly Perforated Domains with Fine-Grained Boundary * The Dirichlet Boundary Value Problem in Strongly Perforated Domains with…

Periodic Integral and Pseudodifferential Equations with Numerical Approximation

- Mathematics
- 2001

1 Preliminaries.- 2 Single Layer and Double Layer Potentials.- 3 Solution of Boundary Value Problems by Integral Equations.- 4 Singular Integral Equations.- 5 Boundary Integral Operators in Periodic…

Bloch Waves in an Arbitrary Two-Dimensional Lattice of Subwavelength Dirichlet Scatterers

- MathematicsSIAM J. Appl. Math.
- 2017

The method presented, that simplifies and expands on Krynkin & McIver [Waves Random Complex, 19 347 2009], could be applied in the future to study more sophisticated designs entailing resonant subwavelength elements distributed over a lattice with periodicity on the order of the operating wavelength.

Semi-Infinite Arrays of Isotropic Point Scatterers. A Unified Approach

- MathematicsSIAM J. Appl. Math.
- 2004

This work solves the two-dimensional problem of acoustic scattering by a semi-infinite array of identical isotropic point scatterers and confirms that a number of phenomena reported for specific geometries are in fact present in the general case.