Fine Properties and Applications of Thin-Sheet Folding

在研究项目的第一个资助期，已经讨论了薄板折叠过程的分析理解和可靠的数值近似。特别地，从三维超弹性材料描述的预损伤板导出了二维弯曲-折叠模型，建立了不连续Galerkin方法，建立了相应的小挠度界面问题的准最优误差估计，并确定了折叠角与折线曲率量关系的精确表征。进一步的结果正在进行中的研究中。在第二个资助期，将讨论具体的分析问题，如导致其他界面条件的比例关系的最佳性、对分段光滑折叠线系统的理解以及折叠角不连续性的表征。此外，还将分析通过加速过程和基于后验误差估计的自适应局部网格加密来提高基本数值格式的效率。利用分析结果和新的计算方法，将研究与应用相关的问题，包括不同折叠结构的杠杆效应，基于奇异扑动效应的新机构的开发，以及双稳器件开关过程中不稳定临界构型的确定。

In the first funding period of the research project, the analytical understanding and the reliable numerical approximation of thin-sheet folding processes have been addressed. In particular, a two-dimensional bending-folding model has been derived from a three-dimensional hyper-elastic material description of a pre-damaged plate, a discontinuous Galerkin method has been devised, quasi-optimal error estimates for a corresponding small deflection interface problem have been established, and a precise characterization of the relation of folding angles and curvature quantities along creases has been identified. Further results are subject of ongoing research. In a second funding period specific analytical questions such as the optimality of scaling relations leading to other interface conditions, the understanding of piecewise smooth folding line systems, and the characterization of folding angle discontinuities will be addressed. Besides this, the improvement of the efficiency of basic numerical schemes via acceleration procedures and adaptive local mesh refinement based on a posteriori error estimates will be analyzed. Using the analytical results and new computational methodology, application-related questions including the leverage effect of different folding constructions, the development of new mechanisms based on singular flapping effects, and the determination of unstable critical configurations arising in switching processes of bistable devices will be investigated.