Generalization of unfolding operator for highly oscillating smooth boundary domains and homogenization

Aiyappan, S.; Nandakumaran, A. K.; Prakash, Ravi

doi:10.1007/s00526-018-1354-6

Cited by 27 publications

(29 citation statements)

References 33 publications

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“…Similarly, we bound the adjoint state, that is ∃ C >0 such that

false‖ {\overset{v}{true}}_{ε} false‖ ⩽ C

. Then it follows from theorem 4.1 in Aiyappan et al that up to a subsequence ∃ u 0 , v 0 ∈ W (Ω) such that

\begin{matrix} right & left \tilde{ū_{ε}^{+}} ⇀ h (x_{3}) u_{0}^{+}, \tilde{{\bar{v}}_{ε}^{+}} ⇀ h (x_{3}) v_{0}^{+} weakly in L^{2} ({normalΩ}^{+}), & right \\ right & left \tilde{\frac{\partial ū_{ε}^{+}}{\partial x_{3}}} ⇀ h (x_{3}) \frac{\partial u_{0}^{+}}{\partial x_{3}}, \tilde{\frac{\partial {\bar{v}}_{ε}^{+}}{\partial x_{3}}} ⇀ h (x_{3}) \frac{\partial v_{0}^{+}}{\partial x_{2}} weakly in L^{2} ({normalΩ}^{+}), \\ right & left \tilde{\frac{\partial ū_{ε}^{+}}{\partial x_{i}}} ⇀ 0, \tilde{\frac{\partial {\bar{v}}_{ε}^{+}}{\partial x_{i}}} ⇀ 0 weakly in L^{2} ({normalΩ}^{+}) for i = 1, 2, \\ right & left ū_{ε}^{-} ⇀ u_{0}^{-}, {\bar{v}}_{ε}^{-} ⇀ v_{0}^{-} weakly in H^{1} \end{matrix}

…”

Section: Main Results and Their Proofsmentioning

confidence: 99%

“…Aiyappan and Sardar have studied a biharmonic boundary optimal control problem. For more literature on homogenization of optimal control problems, one can look into previous works and the references therein. For general periodic homogenization theory, we refer to other studies and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we consider Neumann controls acting on the oscillating part of the boundary which are smooth. Since we are considering smooth oscillating domains, we can workout all the details to circular oscillating domains, which we believe has far reaching applications, using unfolding operators in polar coordinates . We obtain the optimality system at every ε stage, and then we have analyzed the asymptotic behavior of this optimality system as ε →0.…”

Section: Introductionmentioning

confidence: 99%

“…Although plenty of research articles on oscillating domain have appeared in the last 10 to 15 years, most of them consider pillar type oscillating domains and some generalization of the same. But recently, Aiyappan et al have considered an oscillatory domain with smooth oscillatory part. There was a novel approach by these authors in defining new unfolding operators which they have used to study a homogenization problem, in fact, a nonlinear problem.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Asymptotic analysis of a boundary optimal control problem on a general branched structure

Aiyappan

Nandakumaran

Sufian

2019

Math Methods in App Sciences

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We consider an optimal control problem posed on a domain with a highly oscillating smooth boundary where the controls are applied on the oscillating part of the boundary. There are many results on domains with oscillating boundaries where the oscillations are pillar-type (non-smooth) while the literature on smooth oscillating boundary is very few. In this article, we use appropriate scaling on the controls acting on the oscillating boundary leading to different limit control problems, namely, boundary optimal control and interior optimal control problem. In the last part of the article, we visualize the domains as a branched structure, and we introduce unfolding operators to get contributions from each level at every branch. KEYWORDS asymptotic analysis, homogenization, optimal control, oscillating boundary domain, unfolding operator MSC CLASSIFICATION 80M35; 80M40; 49J20; 35B27 Math Meth Appl Sci. 2019;42:6407-6434.wileyonlinelibrary.com/journal/mma

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“…Similarly, we bound the adjoint state, that is ∃ C >0 such that

false‖ {\overset{v}{true}}_{ε} false‖ ⩽ C

. Then it follows from theorem 4.1 in Aiyappan et al that up to a subsequence ∃ u 0 , v 0 ∈ W (Ω) such that

\begin{matrix} right & left \tilde{ū_{ε}^{+}} ⇀ h (x_{3}) u_{0}^{+}, \tilde{{\bar{v}}_{ε}^{+}} ⇀ h (x_{3}) v_{0}^{+} weakly in L^{2} ({normalΩ}^{+}), & right \\ right & left \tilde{\frac{\partial ū_{ε}^{+}}{\partial x_{3}}} ⇀ h (x_{3}) \frac{\partial u_{0}^{+}}{\partial x_{3}}, \tilde{\frac{\partial {\bar{v}}_{ε}^{+}}{\partial x_{3}}} ⇀ h (x_{3}) \frac{\partial v_{0}^{+}}{\partial x_{2}} weakly in L^{2} ({normalΩ}^{+}), \\ right & left \tilde{\frac{\partial ū_{ε}^{+}}{\partial x_{i}}} ⇀ 0, \tilde{\frac{\partial {\bar{v}}_{ε}^{+}}{\partial x_{i}}} ⇀ 0 weakly in L^{2} ({normalΩ}^{+}) for i = 1, 2, \\ right & left ū_{ε}^{-} ⇀ u_{0}^{-}, {\bar{v}}_{ε}^{-} ⇀ v_{0}^{-} weakly in H^{1} \end{matrix}

…”

Section: Main Results and Their Proofsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic analysis of a boundary optimal control problem on a general branched structure

Aiyappan

Nandakumaran

Sufian

2019

Math Methods in App Sciences

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View full text Add to dashboard Cite

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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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Homogenization of the heat equation in a noncylindrical domain with randomly oscillating boundary

Nandakumaran

Sankar

2022

Math Methods in App Sciences

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Award Number: No.F.4-2/2006 (BSR)/MA/19-20/0060In this article, we study the homogenization of heat equations in a domain with randomly oscillating boundary parts. The random oscillating boundary is time-dependent and confined by a stationary random field. Here, we follow a new homogenization technique that deals with the evolving domains, which covers many applications. We obtain the asymptotic limit as 𝜀 → 0 in the reference configuration, in which the heat equation becomes a parabolic equation with random oscillating coefficients in the reference domain. To the best of our knowledge, this is the first result of the homogenization of problems on the random evolving boundary domain. One of the major contributions is the corrector result which we establish in this article.

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Generalization of unfolding operator for highly oscillating smooth boundary domains and homogenization

Cited by 27 publications

References 33 publications

Asymptotic analysis of a boundary optimal control problem on a general branched structure

Asymptotic analysis of a boundary optimal control problem on a general branched structure

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

Homogenization of the heat equation in a noncylindrical domain with randomly oscillating boundary

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