Heterogeneous Multiscale Methods for Fourth-Order Singular Perturbations

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 $δ=\mathbb{N}\varepsilon$, the classical resonance error $\mathcal{O}(\varepsilon/δ)$ 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.