The hole-filling method and multiscale algorithm for the heat transfer performance of periodic porous materials
type: Item
instance of: scholarly article
title: The hole-filling method and multiscale algorithm for the heat transfer performance of periodic porous materials
zbMATH Open document ID: 1358.80003
author: Zhiqiang Yang
author: Yi Sun
author: Zihao Yang
author: Junzhi Cui
published in: Journal of Computational and Applied Mathematics
publication date: 2016-12-28T00:00:00Z
review text: The authors start with the parabolic heat problem posed in a periodically perforated domain \(\Omega ^{\varepsilon }\) and written as \[ \frac{\partial T_{\varepsilon }}{\partial t}(x,t)-\frac{\partial }{\partial x_{i}} (k_{ij}^{\varepsilon }(x)\frac{\partial T_{\varepsilon }}{\partial x_{j}} (x,t))=f(x,t). \] \(\Omega \) is a bounded Lipschitz and convex domain of \( \mathbb{R}^{3}\), \(\omega \) is a smooth unbounded domain of \(\mathbb{R}^{3}\) with 1-periodicity and \(\Omega ^{\varepsilon }=\Omega \cap \varepsilon \omega \). The boundary of \(\Omega \) is the union of \(\Gamma _{1}\), where homogeneous Dirichlet boundary conditions are imposed, and of \(\Gamma _{2}\), where Neumann boundary conditions \(\nu _{i}k_{ij}^{\varepsilon }\frac{ \partial T_{\varepsilon }}{\partial x_{j}}=\overline{q}\) are imposed. Homogeneous Neumann boundary conditions are imposed on the boundary of the holes \( \varepsilon \omega \). The coefficients \(k_{ij}^{\varepsilon }\) are defined through \(k_{ij}^{\varepsilon }(x)=k_{ij}(x/\varepsilon )\), where the \(k_{ij}\) are bounded and satisfy coercivity and continuity conditions. The authors also consider the problem \(\frac{\partial T_{\varepsilon }^{\ast }}{\partial t} (x,t)-\frac{\partial }{\partial x_{i}}(k_{ij}^{\ast }(\frac{x}{\varepsilon }) \frac{\partial T_{\varepsilon }^{\ast }}{\partial x_{j}}(x,t))=\eta _{\varepsilon }(x)f(x,t)\), where \(\eta _{\varepsilon }(x)=\eta (x/\varepsilon )=1\) if \(x\in \Omega ^{\varepsilon }\) and \(\eta _{\varepsilon }(x)=0\) if \(x\in \Omega \setminus \overline{\Omega }^{\varepsilon }\) and \( k_{ij}^{\ast }(y)=k_{ij}(y)\) if \(y\in Y^{\ast }=Y\cap \omega \), \( k_{ij}^{\ast }(y)=\phi _{ij}^{\delta }(y)\) if \(y\in \nu _{\delta }\) and \( k_{ij}^{\ast }(y)=\delta ^{1/2}\delta _{ij}\) if \(y\in \nu _{0}\), where \(\nu _{\delta }=\{y\in Y\setminus \omega \), \(\mathrm{dist}(y,\partial \omega )\leq \delta \}\), \(\nu _{0}=\{y\in Y\setminus \omega \), \(\mathrm{dist}(y,\partial \omega )\geq \delt…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 80A20
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 80M40
zbMATH DE Number: 6668827
zbMATH Keywords: multiscale analysis
zbMATH Keywords: transient heat transfer problem
zbMATH Keywords: periodic porous materials
zbMATH Keywords: hole-filling method
zbMATH Keywords: homogenized problem
zbMATH Keywords: corrector
zbMATH Keywords: numerical resolution
zbMATH Keywords: finite element method
MaRDI profile type: Publication
full work available at URL: j.cam.2016.08.017
OpenAlex ID: W2511991067
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DOI: 10.1016/J.CAM.2016.08.017
description: scientific article; zbMATH DE number 6668827