基于最佳质量网格的薄板问题的非协调流形方法

Incompatible manifold method for thin plate problems based on the best quality mesh

对于大部分非协调板单元,使用规则网格能得到很好的效果。但是,当网格不规则时,非协调元的数值特性将变得很差,甚至收敛性得不到保证。为解决网格依赖性问题,许多专家学者提出了改造单元,如拟协调元法和广义协调元法,这些方法能解决收敛性问题,但是数值实践证明没有一种单元能在所有情况下都具有良好的数值特性。考虑到流形方法采用两套完全独立的覆盖系统,可以用规则的数学网格来作为数学覆盖进行插值,取得最佳的插值效果,单元收敛性便能得到保证。再结合适用于流形方法的变分提法,建立起流形方法处理非规则物理边界非协调板单元的一般格式。以ACM薄板单元为例,与ANSYS、拟协调元法和广义协调元法进行了对比,证明本文方法在处理具有曲线边界的薄板弯曲问题时具有收敛快和精度高等优势。

For most incompatible plate elements, good effects can be achieved if regular meshes are used. But if the mesh is irregular, the numerical properties will become worse, and even the convergence cannot be guaranteed. In order to solve mesh dependence, many transformation elements are raised by many experts, in which the quasi-conforming elements and the generalized conforming elements can be used to solve convergence. But it has been proved by numerical practice that no good numerical character can be obtained by one element in any situation. Considering the two completely independent covering systems are used in the numerical manifold method （NMM）,we can always use the best mesh as the mathematical cover for interpolation. In this way, the best interpolation precision can be achieved and the convergence is accordingly reached. With the variational formulation of Kirchhoff ＇s plate problems fitted to NMM,an unified scheme is proposed for NMM to deal with irregular boundaries of domains. By taking the ACM plate element as an example, finally, comparisons among the proposed scheme, ANSYS, the quasi-conforming elements and the generalized conforming elements in the literature are made, indicating that the proposed scheme is advantageous in treating thin plate bending problems where the plate has a curve boundary.