SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2121-2142, October 2024.
Abstract. 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 Velčić [Calc. Var. Partial Differential Equations, 51 (2014), pp. 677–699]. 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 proved 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.

中文翻译：

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均质板模型的两尺度有限元近似


《SIAM 数值分析杂志》，第 62 卷，第 5 期，第 2121-2142 页，2024 年 10 月。

抽象的。本文研究了 Hornung、Neukamm 和 Velčić 提出的微结构薄板弹性变形均匀化和降维模型的离散化 [Calc.变种。偏微分方程，51 (2014)，第 677–699 页]。由此，非线性弯曲能基于均匀二次形式，其作用于与弹性变形相关的第二基本形式。证明了所涉及的三维微观细胞问题的多重仿射有限​​元离散化和二维等距约束宏观问题的离散基尔霍夫三角形离散化的收敛性。最后，在选定的测试案例中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比较。