As the convergence, good numerical accuracy and high computing efficiency of nonconforming elements cannot be achieved simultaneously using the finite element method (FEM) or the current numerical manifold method (NMM), the first-order NMM was developed to analyze the bending of thin plates. The first-order Taylor expansion was selected to construct the local displacement function, which endowed the generalized degrees of freedom with physical meanings and decreased the rank deficiency. Additionally, the new relations between the global and local rotation functions in the first-order approximation were derived by adopting two sets of rotation functions, $\left\{{𝜃}_{xi},{𝜃}_{yi}\right\}$ and $\left\{{𝜃}_x^i,{𝜃}_y^i\right\}$. Regular meshes were selected to improve the convergence performance. With the penalized formulation fitted to the NMM for Kirchhoff’s thin plate problems, a unified scheme was proposed to deal with irregular and regular boundaries of the domain. The typical examples indicated that the numerical solutions achieved using the first-order NMM rapidly converged to the analytical solutions, and the accuracy of such numerical solutions was vastly superior to that achieved using the FEM and the zero-order NMM.

由于使用有限元法（FEM）或当前的数值流形方法（NMM）无法同时实现不合格元素的收敛，良好的数值精度和较高的计算效率，因此开发了一阶NMM来分析薄板的弯曲。选择一阶泰勒展开来构造局部位移函数，该局部位移函数赋予了广义的自由度以物理意义，并降低了秩不足。此外，通过采用两组旋转函数，得出了一阶近似中全局旋转函数和局部旋转函数之间的新关系，$\left\{{𝜃}_{X\mathrm{一世}}，{𝜃}_{ÿ\mathrm{一世}}\right\}$ 和 $\left\{{𝜃}_X^{\mathrm{一世}}，{𝜃}_{ÿ}^{\mathrm{一世}}\right\}$。选择常规网格以提高收敛性能。针对Kirchhoff的薄板问题，采用适合NMM的惩罚公式，提出了一个统一的方案来处理域的不规则和规则边界。典型示例表明，使用一阶NMM所获得的数值解迅速收敛到了解析解，并且这种数值解的精度大大优于使用FEM和零阶NMM所实现的数值解。