Summary We present a modification of the multiscale finite element method (MsFEM) for modeling of heterogeneous viscoelastic materials and an enhancement of this method by the adaptive generation of both meshes, ie, a macroscale coarse one and a microscale fine one. The fine mesh refinements are performed independently within coarse elements adjusting the microscale discretization to the microstructure, whereas the coarse mesh adaptation optimizes the macroscale approximation. Besides the coupling of the hp‐adaptive finite element method with the MsFEM we propose a modification of the MsFEM to accommodate for the analysis of transient nonlinear problems. We illustrate the efficiency and accuracy of the new approach for a number of benchmark examples, including the modeling of functionally graded material, and demonstrate the potential of our improvement for upscaling nonperiodic and nonlinear composites.
In this paper, we present an enhanced framework for the synthetic asphalt concrete (AC) microstructure generation for the numerical analysis purposes. It is based on the Voronoi tessellation concept with some necessary extensions that allow for the reliable generation of the aggregate particles of the given size distribution. The synthetic microstructure generation allows for faster numerical modeling of the novel materials. It can partially replace the X-ray computed tomography approach, which is frequently used in such analysis. Our framework is a kind of compilation of the known techniques with the enhancements applied to expedite the microstructure modeling process. Therefore, the generated microstructure is used in the numerical upscaling to model the macroscale asphalt concrete properties. We restrict ourselves (in this paper only) to the 2D elastic computations. We also assume the perfect bonding between these two materials and the static load for the sake of simplicity. The upscaling is performed by the multiscale finite element method (MsFEM). A short recapitulation of the MsFEM foundations as well as the numerical test comparing the overkill mesh solution with the upscaled one is provided in the paper. The test results confirm that the whole presented methodology can serve as a fast and reliable tool for the tests on novel asphalt mixtures and other heterogeneous materials. It can reduce the cost of the design process substituting some of the laboratory experiments, giving the opportunity to test the developed constitutive models and expedite the numerical analysis itself.
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.
Computing the pseudoinverse of a matrix is an essential component of many computational methods. It arises in statistics, graphics, robotics, numerical modeling, and many more areas. Therefore, it is desirable to select reliable algorithms that can perform this operation efficiently and robustly. A demanding benchmark test for the pseudoinverse computation was introduced. The stiffness matrices for higher order approximation turned out to be such tough problems and therefore can serve as good benchmarks for algorithms of the pseudoinverse computation. It was found out that only one algorithm, out of five known from literature, enabled us to obtain acceptable results for the pseudoinverse of the proposed benchmark test.
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