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The heterogeneous multiscale finite element method for the homogenization of linear elastic solids and a comparison with the FE~2 method

机译:线性弹性固体均质化的异质多尺度有限元方法及与FE〜2方法的比较

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AbstractThe Heterogeneous Multiscale Finite Element Method (FE-HMM) is a two-scale FEM based on asymptotic homogenization for solving multiscale partial differential equations. It was introduced in [W. E, B. Engquist, Commun. Math. Sci. 1 (2003) 87–132]. The objective of the present work is an FE-HMM formulation for the homogenization of linear elastic solids in a geometrical linear frame, and doing so, of a vector-valued field problem. A key ingredient of FE-HMM is that macrostiffness is estimated by stiffness sampling on heterogeneous microdomains in terms of a modified quadrature formula, which implies an equivalence of energy densities of the microscale with the macroscale. Beyond this coincidence with the Hill–Mandel condition, which is the cornerstone of the FE2method, we elaborate a conceptual comparison with the latter method. After developing an algorithmic framework we (i) assess the existing a priori convergence estimates for the micro- and macro-errors in various norms, (ii) verify optimal strategies in uniform micro–macromesh refinements based on the estimates, (iii) analyze superconvergence properties of FE-HMM, and (iv) compare FE-HMM with FE2by numerical results.HighlightsAspects of FE-HMM implementation for the vector-valued field problem of linear elasticity.Explanation of FE-HMM by direct links to asymptotic homogenization.Unified conceptual and numerical comparison of FE-HMM with FE2.Assessment and verification of a priori error estimates and of superconvergence.Assessment of optimal uniform micro-/macromesh refinement strategies.
机译: 摘要 异构多尺度有限元方法(FE-HMM)是基于渐近均匀化的两尺度FEM,用于求解多尺度偏微分方程。它是在[W. E. B. Engquist,Commun。数学。科学1(2003)87–132]。本工作的目的是一种FE-HMM公式,用于使几何线性框架中的线性弹性固体均质化,从而实现矢量值场问题。 FE-HMM的一个关键因素是,通过修改正交公式,通过对异质微域的刚度采样来估计宏观刚度,这意味着微观尺度的能量密度与宏观尺度相等。除了Hill-Mandel条件之外,这是FE 2 < / mml:math>方法,我们将与后一种方法进行概念比较。建立算法框架后,我们(i)评估各种规范中微误差和宏观误差的现有先验收敛估计,(ii)根据估计值验证统一的微宏改进的最优策略,(iii)分析超收敛FE-HMM的特性,以及(iv)将FE-HMM与FE 2 通过数值结果。 突出显示 FE的方面-HMM实现线性弹性的矢量值场问题。 通过直接链接到渐近同质化来解释FE-HMM。 FE-HMM与FE 2 先验误差估计和超收敛性的评估和验证。 评估最佳的均匀微/微距提纯策略。

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