In this paper, we view homogenization within the framework of variational multiscale methods. The standard (primal) variational format lends itself naturally to the choice of Dirichlet boundary conditions on the Representative Volume Element (RVE). However, how to impose flux boundary conditions, treated as Neumann conditions in the standard variational format, is less obvious. Therefore, in this paper we propose and investigate a novel mixed variational setting, where the fluxes are treated as additional primary fields, in order to provide the natural variational environment for such flux boundary conditions. This mixed dual formulation allows for a conforming implementation of (lower bound) flux boundary conditions in the framework of discretization-based homogenization. To focus on essential features, a very simple problem is studied: the classical stationary linear heat equation. Furthermore, we consider the standard context of model-based homogenization (without loss of generality), since we are only concerned with the RVE problem and merely assume that the relevant macroscale fields are properly prolonged. Numerical results from the primary and the mixed dual variational formats are compared and their convergence properties for mesh finite element (FE) refinement and RVE size are assessed.

中文翻译：

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关于通量边界条件的变数一致计算均质化的主对偶形式

在本文中，我们认为变分多尺度方法框架内的均质化。标准（原始）变式格式自然适合于在“代表体积元素”（RVE）上选择Dirichlet边界条件。但是，如何强加通量边界条件（在标准变分格式中被视为诺伊曼条件）并不明显。因此，在本文中，我们提出并研究了一种新颖的混合变分设置，其中将通量视为附加的主场，以便为此类通量边界条件提供自然的变化环境。这种混合双重公式允许在基于离散化的均质化框架中以一致的方式实现（下限）磁通边界条件。为了关注基本功能，研究了一个非常简单的问题：经典平稳线性热方程。此外，我们考虑了基于模型的同质化的标准上下文（不失一般性），因为我们仅关注RVE问题，并且仅假设适当延长了相关宏尺度字段。比较了主要和混合对偶变分格式的数值结果，并评估了它们对网格有限元（FE）细化和RVE大小的收敛性。