This paper studies the discretization of a homogenization and dimension reduction model for the elastic deformation of microstructured thin plates proposed by Hornung, Neukamm, and Vel\v{c}i\'c in 2014. Thereby, a nonlinear bending energy is based on a homogenized quadratic form which acts on the second fundamental form associated with the elastic deformation. Convergence is proven for a multi-affine finite element discretization of the involved three-dimensional microscopic cell problems and a discrete Kirchhoff triangle discretization of the two-dimensional isometry-constrained macroscopic problem. Finally, the convergence properties are numerically verified in selected test cases and qualitatively compared with deformation experiments for microstructured sheets of paper.

翻译：本文研究了Hornung、Neukamm和Velčić于2014年提出的微结构薄板弹性变形均质化与降维模型的离散化方法。该模型基于一个作用于弹性变形相关第二基本形式的均质化二次型，构建了非线性弯曲能量。研究证明了所涉及三维微观胞元问题的多仿射有限元离散化，以及二维等距约束宏观问题的离散Kirchhoff三角形离散化的收敛性。最后，通过选定测试案例对收敛特性进行了数值验证，并与微结构纸张的变形实验进行了定性对比。