标题： 非线性弹性应变限制Cosserat模型的广义多尺度有限元方法 显示英文标题

标题： Generalized multiscale finite element method for a nonlinear elastic strain-limiting Cosserat model

摘要： 对于非线性Cosserat弹性，本文考虑多尺度方法。 特别是，我们探索广义多尺度有限元方法（GMsFEM）来求解具有应变限制特性的各向同性Cosserat问题（即使在高应力下也能确保线性化应变有界）。 这种应变限制的Cosserat模型可以在固体和生物纤维中找到潜在应用。 然而，具有自然旋转自由度、非线性本构关系、高对比度和非均质性的Cosserat介质可能在解中产生具有挑战性的多尺度特征，因此通过多尺度方法进行升尺度是必要的。 因此，在处理非线性时，我们在这种情况下利用离线和基于残差的在线（自适应或均匀）GMsFEM。 通过各种二维实验（针对具有小和大应变限制参数的穿孔、复合和随机非均质介质），我们的数值结果展示了这些方法的收敛性、效率和鲁棒性。 此外，这些结果表明，这些方法提供了良好的准确性，在线GMsFEM比离线方法给出更精确的解，而在线自适应策略与均匀策略具有相似的精度，但自由度更少。 显示英文摘要

摘要： For nonlinear Cosserat elasticity, we consider multiscale methods in this paper. In particular, we explore the generalized multiscale finite element method (GMsFEM) to solve an isotropic Cosserat problem with strain-limiting property (ensuring bounded linearized strains even under high stresses). Such strain-limiting Cosserat model can find potential applications in solids and biological fibers. However, Cosserat media with naturally rotational degrees of freedom, nonlinear constitutive relations, high contrast, and heterogeneities may produce challenging multiscale characteristics in the solution, and upscaling by multiscale methods is necessary. Therefore, we utilize the offline and residual-based online (adaptive or uniform) GMsFEM in this context while handling the nonlinearity by Picard iteration. Through various two-dimensional experiments (for perforated, composite, and stochastically heterogeneous media with small and big strain-limiting parameters), our numerical results show the approaches' convergence, efficiency, and robustness. In addition, these results demonstrate that such approaches provide good accuracy, the online GMsFEM gives more accurate solutions than the offline one, and the online adaptive strategy has similar accuracy to the uniform one but with fewer degrees of freedom.