We develop a fully discrete numerical method for the singular perturbation homogenization problems in the framework of heterogeneous multiscale method, such problems arise from the heterogeneous strain gradient elasticity. We prove the stability and the convergence rate of the proposed method with general boundary conditions. The theoretical results predict that the resonance error may behave differently at different scales, and the discrete error may not degenerate by boundary layer effects. The numerical results validate the theoretical predictions. This is a joint work with Yulei Liao.