标题： 用于四阶奇异摄动的异构多尺度方法 显示英文标题

标题： Heterogeneous multiscale methods for fourth-order singular perturbations

摘要： 我们开发了一种数值均质化方法，用于在非均匀多尺度方法框架下的四阶奇异摄动问题。 这些问题来源于非均匀应变梯度弹性理论和结构材料的弹性模型。 我们建立了适用于一般介质的均质解的误差估计，并推导了细尺度为$\varepsilon$的局部周期介质的显式收敛性。 对于尺寸为$\delta=\mathbb{N}\varepsilon$的单元问题，由于高阶算子的主导作用，经典的共振误差$\mathcal{O}(\varepsilon/\delta)$可以被消除。 尽管边界层效应出现，一般的边界条件下离散误差不一定恶化。 数值模拟验证了这些理论结果。 显示英文摘要

摘要： We develop a numerical homogenization method for fourth-order singular perturbation problems within the framework of heterogeneous multiscale method. These problems arise from heterogeneous strain gradient elasticity and elasticity models for architectured materials. We establish an error estimate for the homogenized solution applicable to general media and derive an explicit convergence for the locally periodic media with the fine-scale $\varepsilon$. For cell problems of size $\delta=\mathbb{N}\varepsilon$, the classical resonance error $\mathcal{O}(\varepsilon/\delta)$ can be eliminated due to the dominance of the higher-order operator. Despite the occurrence of boundary layer effects, discretization errors do not necessarily deteriorate for general boundary conditions. Numerical simulations corroborate these theoretical findings.