Continuum limit of discrete Sommerfeld problems on square lattice

Sharma, Basant Lal

doi:10.1007/s12046-017-0636-6

Cited by 14 publications

(13 citation statements)

References 50 publications

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“…Last but not the least, there remains an issue of the continuum limit. For the considered case of positive imaginary part of ω, it is left as an exercise (one possibility involves the tools that are used in [54]) to prove that the low-frequency limit (i.e. with b → 0 but fixed ω and Nb; recall ω = bω) coincides with that of the well-known solution [27,28] provided the separation between the semi-infinite defects N scales naturally as 1/b.…”

Section: Discussionmentioning

confidence: 99%

Discrete scattering by a pair of parallel defects

Sharma

Maurya

2019

Phil. Trans. R. Soc. A.

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Scattering of a time-harmonic anti-plane shear wave due to either a pair of crack tips or a pair of rigid constraint tips on square lattice is considered. The two problems correspond to the so-called zero-offset case of scattering due to a pair of identical Sommerfeld screens. The peculiar structural symmetry allows the reduction of coupled equations to two scalar Wiener–Hopf equations and a total of four geometrically reduced problems on lattice half-plane. Exact solution of each problem for incidence from the bulk lattice, as well as from an associated lattice waveguide, is constructed. A suitable superposition of the four expressions is used to construct the solution of the main problem. The discrete paradigm involving the wave mode incident from the waveguide is relevant for modern applications where an investigation of mechanisms of electronic and thermal transport at nanoscale remains an interesting problem. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.

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Section: Discussionmentioning

confidence: 99%

Discrete scattering by a pair of parallel defects

Sharma

Maurya

2019

Phil. Trans. R. Soc. A.

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“…As the wavelength 2π/k 1 becomes large compared to b, the so called continuum limit is obtained, which can also be perceived as a low frequency approximation in the case of assumed square lattice model. The continuum limit of the discrete Helmholtz equation is the continuous Helmholtz equation [13,43]. From the perspective of the continuum model, the solutions obtained by [4,5] can be seen to be approximated via a lattice formulation and the solution of discrete scattering due to the two staggered cracks.…”

Section: Low Frequency Approximationmentioning

confidence: 99%

“…In the context of the well known solution of the two parallel, staggered plates problem [4,5] in the continuum model, using the numerical solution of the discrete scattering problem, it has been found that, as the frequency approaches zero, the solution coincides with that of the the continuum model, but this is expected [43].…”

Section: Introductionmentioning

confidence: 99%

Scattering by two staggered semi-infinite cracks on square lattice: an application of asymptotic Wiener–Hopf factorization

Maurya

Sharma

2019

Z. Angew. Math. Phys.

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Scattering of time-harmonic plane wave by two parallel semi-infinite rows, but with staggered edges, is considered on square lattice. The condition imposed on the semi-infinite rows is a discrete analogue of Neumann boundary condition. A physical interpretation assuming an out-of-plane displacement for the particles arranged in the form of a square lattice and interacting with nearest-neighbours, associates the scattering problem to lattice wave scattering due to the presence of two staggered but parallel crack tips. The discrete scattering problem is reduced to the study of a pair of Wiener-Hopf equation on an annulus in complex plane, using Fourier transforms. Due to the offset between the crack edges, the Wiener-Hopf kernel, a 2 × 2 matrix, is not amenable to factorization in a desirable form and an asymptotic method is adapted. Further, an approximation in the far field is carried out using the stationary phase method. A graphical comparison between the far-field approximation based on asymptotic Wiener-Hopf method and that obtained by a numerical solution is provided. Also included is a graphical illustration of the low frequency approximation, where it has been found that the numerical solution of the scattering problem coincides with the well known formidable solution in the continuum framework.

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“…A sketch of the proof of Theorem 4.4 is presented in this paragraph. The detailed statements and rigorous arguments appear in [42], a part of which is already included in the foregoing text; in particular the definitions and statements after (4.13). Suppose that P : H + 1 2 → H + 1 2 (R) is chosen as the suitably defined prolongation operator, which acts in a direction opposite to R .…”

Section: Accepted Manuscriptmentioning

confidence: 99%

Near-Tip Field for Diffraction on Square Lattice by Crack

Sharma¹

2015

SIAM J. Appl. Math.

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The displacement field near a tip of a finite crack, due to the diffraction of a wave on square lattice is studied. The finite section method, in the theory of Toeplitz operators on 2 , is invoked as the semi-infinite crack diffraction problem is shown equivalent to the inversion of a Toeplitz operator, a truncation of which appears in the finite crack diffraction problem for the same incident wave. The existence and uniqueness of the solution in 2 for the semi-infinite crack problem is established by an application of the well known Krein conditions. Continuum limit of the semiinfinite crack diffraction problem is established in a discrete Sobolev space; a graphical illustration of convergence in the relevant Sobolev norm is also included. A low frequency asymptotic approximation of the normalized shear force, in 'vertical' bonds ahead of the crack tip, recovers the classical crack tip singularity. Displacement of particles in the vicinity of the crack tip, a closed form expression of which is provided, is compared, graphically, with that obtained by a numerical solution of the diffraction problem on a finite grid. Graphical results are also included to demonstrate that the normalized shear force in 'horizontal' bonds, along the crack face, approaches the corresponding stress component in the continuum model at sufficiently low frequency. Numerical solutions indicate that the crack opening displacement of a semi-infinite crack approximates that of a finite crack of sufficiently large size, and at sufficiently high frequency of incident wave, away from the neighborhood of the other crack tip, while it differs significantly for low frequencies, as a result of multiple scattering due to the two crack tips.

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Continuum limit of discrete Sommerfeld problems on square lattice

Cited by 14 publications

References 50 publications

Discrete scattering by a pair of parallel defects

Discrete scattering by a pair of parallel defects

Scattering by two staggered semi-infinite cracks on square lattice: an application of asymptotic Wiener–Hopf factorization

Near-Tip Field for Diffraction on Square Lattice by Crack

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