Analysis of a linear 3D fluid–mesh–shell interaction problem

Čanić, Sunčica; Galić, Marija; Ljulj, Matko; Muha, Boris; Tambača, Josip; Wang, Yifan

doi:10.1007/s00033-019-1087-1

Cited by 10 publications

(14 citation statements)

References 44 publications

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“…[9,11,13,21,22,27,32,38] and references within. However, many biomedical devices (such as stents) are being developed with the view that vascular wall structures are composed of composite materials and not of single layer; see [19,20,24,25,39]. In short, some degree of physical realism is lost if the FSI PDE does not adequately describe arterial wall layers of composite type.…”

Section: Literaturementioning

confidence: 99%

“…In turn, with vector field m as specified in (30), applying (31) to the right hand side of (28), using (24), the Sobolev Trace Theorem and again Young's Inequality, we have the following Proposition:…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Applying this estimate to the right hand side of (38), along with (37), ( 22), (24), and the Sobolev Trace Theorem, we then have for |β| ≥ 1 (upon summing over j = 1, ..., K),…”

Section: An Appropriate Estimate For ∂Z ∂ν γSmentioning

confidence: 99%

“…the change of variable in(25), (47) and the regularity for D in(24), with respect to the thick layer wave solution components in(11), we have that for |β| ≥ 1,Ωs) = z − i β D u| Γs + w * 0 Ωs) = iβ z − i β D u| Γs + w * 0 | Γs − w * 0…”

mentioning

confidence: 99%

“…Completion of the Proof of Theorem 3Applying (44) to right hand side of (32) of Proposition 7 (and subsequently rescaling), we have for |β| ≥ 1,In turn, we take vector field m(x) in (29) of Proposition 6 to satisfy div( m) = 1. Afterwards, we estimate this relation by means of (45),(24), the Sobolev Trace Theorem and the Young's Inequality,|ab| ≤ ρa 2 + C ρ b 2 (1 > ρ > 0) to have Ωs |βz| 2 dΩ s ≤ (1 − ρ) ∇z 2 L 2 (Ωs) + β 2 z 2 L 2 (Ωs) + C ,ρ |β|…”

mentioning

confidence: 99%

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Rational Decay of A Multilayered Structure-Fluid PDE System

Avalos¹,

Geredeli²,

Muha³

2022

Preprint

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In this work, we consider a certain multilayered (thick layer) wave-(thin layer) wave-heat (fluid) interactive PDE system. Such coupled PDE systems have been used in the literature to describe the blood transport process in mammalian vascular systems. In particular, the deformations of the boundary interface (thin layer) are described via the two dimensional elastic equation. The present work constitutes an investigation of the extent of the stabilizing effects of the underlying fluid dissipation -across the boundary interface -upon both the thick and thin structural components. (All three PDE components evolve on their respective geometries.) In this regard, our main result is the derivation of uniform decay rates for classical solutions of this multilayered PDE model. To obtain these estimates, necessary a priori inequalities for certain static multilayered PDE models are generated here to ultimately allow an application of a wellknown resolvent criterion for rational decay.

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Section: Literaturementioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

“…Applying this estimate to the right hand side of (38), along with (37), ( 22), (24), and the Sobolev Trace Theorem, we then have for |β| ≥ 1 (upon summing over j = 1, ..., K),…”

Section: An Appropriate Estimate For ∂Z ∂ν γSmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Rational Decay of A Multilayered Structure-Fluid PDE System

Avalos¹,

Geredeli²,

Muha³

2022

Preprint

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3D structure–2D plate–1D rod interaction problem

Ljulj

Tambača

2023

Math Methods in App Sciences

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In this paper, we consider the equilibrium problem of interaction of three elastic bodies of different elastic properties. The main body is the unit cube. On top of it, there is a thin layer/cuboid of thickness 𝜀 of material whose stiffness is of order 1 𝜀 that in the middle contains another cuboid which is of width and thickness 𝜀 that is made of material with elasticity coefficients of order 1 𝜀 q for q > 0. We show that the family of solutions of linearized elasticity problems, when 𝜀 tends to zero, converges to a solution of a problem that is posed only on the unit cube with possibly additional elastic terms on the boundary related to the plate/rod energy of the thin elastic parts. It turns out that there are five different regimes related to different values of q (q ∈ ⟨0, 2⟩, {2}, ⟨2, 4⟩, {4}, ⟨4, ∞⟩) with different limit problems. We further formulate a model posed on the unit cube that has the same asymptotics when 𝜀 tends to zero as the full 3D problem posed on the union of the unit cube and thin cuboids. This model then can be used as the approximating model in all regimes.

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Fluid-Structure Interaction with Incompressible Fluids

Čanić

2020

Progress in Mathematical Fluid Dynamics

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Analysis of a linear 3D fluid–mesh–shell interaction problem

Cited by 10 publications

References 44 publications

Rational Decay of A Multilayered Structure-Fluid PDE System

Rational Decay of A Multilayered Structure-Fluid PDE System

3D structure–2D plate–1D rod interaction problem

Fluid-Structure Interaction with Incompressible Fluids

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