自适应自然单元法研究——自适应细化

对h型自适应自然单元法的自适应细化方案进行了初步研究。在ZZ误差分析的基础上实现了节点的自动加密，使得随着自适应细化的进行，求解误差减小，而且误差分布趋于均匀。在数值分析中，主要有两种误差来源--插值误差和积分误差。随着节点的加密，Delaunay三角形的尺寸随之减小，三角形内的应力场趋于线性分布，那么插值误差和积分误差也都会随之减小。因而，h型自适应分析可以同时减小上述两种误差而达到不断提高求解精度的目的。由于自然单元法求解依赖于求解域内离散节点的Voronoi结构，建议的细化方案中新节点的引入只需局部调整Delaunay结构，算法的实现极为容易，程序实现简单、高效。研究表明，建议的自适应方案是可行的，自然单元法特别适合进行h型自适应分析。

Study of the adaptive natural element method — adaptive refinement

A preliminary study is made of the h-adaptive refinement scheme for the Natural Element Method(NEM). The solution error is estimated by using the ZZ error estimator which is based on the recovery stress filed acquired through the MLS/NNI stress recovery scheme according to the character of the NEM. Then new nodes are inserted automatically according to the error distribution information acquired. With the process of the adaptive analysis, the solution error reduced and the distribution of the error tend to be uniform. In numerical analysis, there is mainly two sources of error, namely the interpolation error and integration error, with the increase in nodes, the size of Delaunay triangle decreases and the stress tend to be linear, so the interpolation error and integration error decreases accordingly. In the process of adaptive analysis, solution errors can be reduced and the accuracy improved. The Voronoi tessellation is needed in the NEM analysis, the refinement scheme presented is just involved in the local adjustment of the Delaunay Tessellation. The implementation of adaptive process is effective and efficient. Numerical examples show that the adaptive scheme is successful and NEM is especially suitable for h-adaptive analysis.