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Second-order asymptotic algorithm for heat conduction problems of periodic composite materials in curvilinear coordinates. (English) Zbl 1381.80003
Summary: A new second-order two-scale (SOTS) asymptotic analysis method is presented for the heat conduction problems concerning composite materials with periodic configuration under the coordinate transformation. The heat conduction problems are solved on the transformed regular domain with quasi-periodic structure in the general curvilinear coordinate system. By the asymptotic expansion, the cell problems, effective material coefficients and homogenized heat conduction problems are obtained successively. The main characteristic of the approximate model is that each cell problem defined on the microscopic cell domain is associated with the macroscopic coordinate. The error estimation of the asymptotic analysis method is established on some regularity hypothesis. Some common coordinate transformations are discussed and the reduced SOTS solutions are presented. Especially by considering the general one-dimensional problem, the explicit expressions of the SOTS solutions are derived and stronger error estimation is presented. Finally, the corresponding finite element algorithms are presented and numerical results are analyzed. The numerical errors presented agree well with the theoretical prediction, which demonstrate the effectiveness of the second-order asymptotic analysis method. By the coordinate transformation, the asymptotic analysis method can be extended to more general domain with periodic microscopic structures.

MSC:
 80A20 Heat and mass transfer, heat flow (MSC2010) 80M35 Asymptotic analysis for problems in thermodynamics and heat transfer 35Q79 PDEs in connection with classical thermodynamics and heat transfer 74F05 Thermal effects in solid mechanics 80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer
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