Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

Chechkin, Gregory A.; Mel'nyk, Taras A.

doi:10.1002/mma.2785

Cited by 15 publications

(12 citation statements)

References 20 publications

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“…Such successful applications of thick-junction constructions have stimulated active learning BVP's in thick junctions with more complex structures: thick junctions with the thin junction's body [2,3,4], thick multi-level junctions [8,9,18], thick cascade junctions [5,17], where new qualitative results were obtained. Specifically, it was shown that processes in thick multi-level junctions behave as a many-phase system and thick cascade junctions have new kind of eigenvibrations.…”

mentioning

confidence: 99%

Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure

Mel'nyk¹

2015

Journal of Mathematical Analysis and Applications

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We consider a semi-linear parabolic problem in a model plane thick fractal junction Ω ε , which is the union of a domain Ω 0 and a lot of joined thin trees situated ε-periodically along some interval on the boundary of Ω 0 . The trees have finite number of branching levels. The following nonlinear Robin boundary conditionε is given on the boundaries of the branches from the i-th branching layer; α i and β i are real parameters. The asymptotic analysis of this problem is made as ε → 0, i.e., when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the corresponding homogenized problem is found and the existence and uniqueness of its solution in an anizotropic Sobolev space of multi-sheeted functions is proved. We construct the asymptotic approximation for the solution v ε and prove the corresponding asymptotic estimate in the space C [0, T ]; L 2 (Ω ε ) ∩ L 2 0, T ; H 1 (Ω ε ) , which shows the influence of the parameters {α i } and {β i } on the asymptotic behavior of the solution.

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confidence: 99%

Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure

Mel'nyk¹

2015

Journal of Mathematical Analysis and Applications

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“…Let us verify the solvability condition (27) for the problem (24) at any fixed k ∈ N, k ≥ 2 . Taking into account the third relation in problems (9), (11) and (13), the equality (27) can be re-written as follows:…”

Section: Limit Problem and Problems For {ω K }mentioning

confidence: 99%

Asymptotic Approximations of the Solution to a Boundary Value Problem in a Thin Aneurysm Type Domain

Klevtsovskiy

Mel'nyk

2017

J Math Sci

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A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin 3D aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter O(ε). A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the parameter ε → 0.The asymptotic expansion consists of a regular part that is located inside of each cylinder, a boundary-layer part near the base of each cylinder, and an inner part discovered in a neighborhood of the aneurysm. Terms of the inner part of the asymptotics are special solutions of boundary-value problems in an unbounded domain with different outlets at infinity. It turns out that they have polynomial growth at infinity. By matching these parts, we derive the limit problem (ε = 0) in the corresponding graph and a recurrence procedure to determine all terms of the asymptotic expansion.Energetic and uniform pointwise estimates are proved. These estimates allow us to observe the impact of the aneurysm.

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“…Homogenization of oscillating boundaries with fixed amplitude is widely studied and we refer to the following main papers: [1], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [15], [21], [22], [23], [24], [26], [27], [28], [29], [30], [31], [32], [41] [42], [45], [46], [47], [48], and [53].…”

Section: Introductionmentioning

confidence: 99%

A Two-Dimensional Electrostatic Model of Interdigitated Comb Drive in Longitudinal Mode

Gaudiello¹,

Lenczner²

2020

SIAM J. Appl. Math.

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A periodic homogenization model of the electrostatic equation is constructed for a comb drive with a large number of fingers and whose mode of operation is in-plane and longitudinal. The model is obtained in the case where the distance between the rotor and the stator is of an order ε α , α ≥ 2, where ε denotes the period of distribution of the fingers. The model derivation uses the two-scale convergence technique. Strong convergences are also established. This allows us to find, after a proper scaling, the limit of the electrostatic force applied to the rotor in the longitudinal direction.

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Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

Cited by 15 publications

References 20 publications

Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure

Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure

Asymptotic Approximations of the Solution to a Boundary Value Problem in a Thin Aneurysm Type Domain

A Two-Dimensional Electrostatic Model of Interdigitated Comb Drive in Longitudinal Mode

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