Abstract
In this paper, we present a new approach to model the steady-state heat transfer in heterogeneous materials. The multiscale finite element method (MsFEM) is improved and used to solve this problem. MsFEM is a fast and flexible method for upscaling. Its numerical efficiency is based on the natural parallelization of the main computations and their further simplifications due to the numerical nature of the problem. The approach does not require the distinct separation of scales, which makes its applicability to the numerical modeling of the composites very broad. Our novelty relies on modifications to the standard higher-order shape functions, which are then applied to the steady-state heat transfer problem. To the best of our knowledge, MsFEM (based on the special shape function assessment) has not been previously used for an approximation order higher than p = 2, with the hierarchical shape functions applied and non-periodic domains, in this problem. Some numerical results are presented and compared with the standard direct finite-element solutions. The first test shows the performance of higher-order MsFEM for the asphalt concrete sample which is subject to heating. The second test is the challenging problem of metal foam analysis. The thermal conductivity of air and aluminum differ by several orders of magnitude, which is typically very difficult for the upscaling methods. A very good agreement between our upscaled and reference results was observed, together with a significant reduction in the number of degrees of freedom. The error analysis and the p-convergence of the method are also presented. The latter is studied in terms of both the number of degrees of freedom and the computational time.
Highlights
Numerical modeling of the heterogeneous materials is a very active research field [1,2,3,4,5,6,7].This is due to the fact that such materials are widely used in many important industry branches, e.g., civil engineering [1,2,6,7], automotive engineering [3,4], aerospace engineering [8] and many others
We proposed a higher-order multiscale finite element method (MsFEM) for the steady-state heat-transfer problem
We demonstrated the convergence of this method with respect to both the number of degrees of freedom and the computational time
Summary
Numerical modeling of the heterogeneous materials is a very active research field [1,2,3,4,5,6,7]. This is due to the fact that such materials are widely used in many important industry branches, e.g., civil engineering [1,2,6,7], automotive engineering [3,4], aerospace engineering [8] and many others. We make use of the higher-order, finite-element method. In our previous studies [14], we presented the application of the automatic hp-adaptivity [15,16] coupled with the multiscale finite-element method
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