The hole-filling method and multiscale algorithm for the heat transfer performance of periodic porous materials (Q730587)

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 \(\nu _{\delta }=\{y\in Y\setminus \omega \), \(\mathrm{dist}(y,\partial \omega )\leq \delta \}\), \(\nu _{0}=\{y\in Y\setminus \omega \), \(\mathrm{dist}(y,\partial \omega )\geq \delta \}\), \(\phi _{ij}^{\delta }\in C^{\infty }(\nu _{\delta })\) and \( \delta ^{1/2}\leq \left\| \phi _{ij}^{\delta }\right\| \leq M\) and \( \delta _{ij}\) is Kronecker's symbol. Thus doing, the authors fill the holes with compliant material. The first main result gives estimates on the difference between the solutions of the above problems in terms of \(\varepsilon \) and \(\delta \), under regularity hypotheses on \( T_{\varepsilon }\). For the proof, they draw explicit computations from the two equations. The further main results build correctors for the first problem which are based on expressions of the solution to the homogenized problem and their spatial partial derivatives. The paper ends with the presentation and discussion of numerical simulations which are obtained for the first problem using the finite element method and in the case of spherical holes.