发表机构
State Key Laboratory of Fluid Power & Mechatronic Systems, Zhejiang University; College of Mechanical Engineering, Guizhou University; Advanced Technology Institute, Zhejiang University（浙江大学流体动力与机电系统国家重点实验室; 贵州大学机械工程学院; 浙江大学先进技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出一种仅用位置信息的薄壳离散化方法，基于棋盘格边中点几何特性，引入特定坐标处理盲模态，实现二阶曲率收敛等性能，可处理多种薄壳问题。
AI 中文摘要
棋盘格边中点几何为每个空间四边形提供了精确的平面Varignon平行四边形，即使原始四边形发生扭曲，也能直接从节点位置恢复局部切向量和法向量。基于这一特性，我们开发了一种仅用位置信息的薄壳离散化方法，无需独立的导向向量、旋转或应变变量。连接后的边中点表面被用作物理中面：第一基本形式在平面B面上计算，而以W为中心的第二基本形式则由相邻B面法向量的变化构建，因此膜效应和弯曲效应共享同一几何载体。变分核分析表明，平滑的二阶一致性无法消除晶格尺度的盲模态。在约去原始棋盘格规范后，B度量具有一个物理膜盲方向，而对称W曲率具有两个曲率盲方向。我们引入了商最小膜相容坐标$X_M$和目标参考相关曲率坐标$X_W^{rel}$，分别一致地置于$O(t)$膜扇区和$O(t^3)$弯曲扇区。该公式包含显式中点商、完备的平坦盲模态分类、刚体运动客观性、参考态一致性，以及对齐广义圆柱的$O(h^2)$近等距近似结果。数值测试显示二阶曲率收敛、膜缺陷的针对性去除，以及$X_W^{rel}$的亚百分之一、随网格细化衰减的影响。线性和非线性壳基准进一步证明，在单一仅用位置信息的框架内，可实现平坦弯曲、曲壳膜-弯曲耦合、厚度敏感性、大旋转、非线性捏缩及局部椭圆化。
英文摘要
Checkerboard edge-midpoint geometry provides an exact planar Varignon parallelogram for every spatial quadrilateral, allowing a local tangent frame and normal to be recovered directly from nodal positions even when the raw quadrilateral is warped. Building on this property, we develop a position-only thin-shell discretization with no independent director, rotation, or strain variables. The connected edge-midpoint surface is taken as the physical midsurface: the first fundamental form is evaluated on planar B faces, while a W-centered second fundamental form is constructed from variations of neighboring B-face normals, so membrane and bending share the same geometric carrier. A variational-kernel analysis shows that smooth second-order consistency does not eliminate lattice-scale blind modes. After quotienting out the raw checkerboard gauge, the B metric has one physical membrane blind direction and the symmetric W curvature has two curvature blind directions. We introduce a quotient-minimal membrane compatibility coordinate $X_M$ and an objective reference-relative curvature coordinate $X_W^{rel}$, placed consistently in the $O(t)$ membrane and $O(t^3)$ bending sectors. The formulation admits an explicit midpoint quotient, complete flat blind-mode classification, rigid-motion objectivity, reference-state consistency, and an $O(h^2)$ near-isometry approximation result for aligned generalized cylinders. Numerical tests show second-order curvature convergence, targeted removal of the membrane defect, and a sub-percent, refinement-decaying influence of $X_W^{rel}$. Linear and nonlinear shell benchmarks further demonstrate flat bending, curved-shell membrane-bending coupling, thickness sensitivity, large rotation, nonlinear pinching, and localized ovalization within a single position-only framework.
Comments39 pages, 8 figures
