基于局部搜索算法的自然邻接点方法

NATURAL NEIGHBOUR METHOD BASED ON THE ALGORITHM OFLOCAL SEARCH^1

自然邻接点方法(NNM)采用自然邻接点形函数进行插值,其插值形函数具有严格定义,且与有限元形函数一样形式简洁、性能优良,因而避免了EFG法里难以准确施加位移边界条件和材料不连续条件等诸多主要困难.但是从形式上看自然邻接点方法仍然属于有网格的方法,其研究和应用受到了较大的限制.为了克服这个缺点,对于任意给定的数值积分点,提出了一种基于局部搜索自然邻接点的寻找算法对NNM进行改进.改进后的NNM与无单元伽辽金法(EFG)的插值和求解过程类似,兼具有EFG的真正无网格特性及NNM的便于处理边界和材料不连续条件等优点.所得计算结果表明,改进后的NNM的计算精度和计算时间与NNM相当,是一种比较理想的数值求解方法.

The natural neighbour method (or natural element method), which is based on the natural neighbour interpolation, is a method between meshless and mesh. The discrete model of the domain Ω in natural neighbour method(NNM) consists of a set of distinct nodes, and a polygonal description of the boundary. The whole displacement interpolations are constructed with respect to the nature neighbour nodes and Voronoi tessellation of the gived point. The natural neighbours of the gived point have been definitely defined. The properties of the natural neigbour interpolation are excellent. For instance, the conditions of linear consistency, partition of unitity, positivity, and delta properties are all satisfied in natural neigbour interpolation. The disadvantages in element-free Galerkin method(EFG), such as, the difficulties of imposition of essential boundary and treatment of material discontinuity, the complex algorithm of matrix inverse in the computation of Moving Least Squares(MLS) shape function, the uncertain choice of the weight functions can be avoided in NNM. But, NNM is usually regarded as a mesh-based method beacause the delaunay triangulations from the whole solution domain are still needed for neighbour-search. In stead of searching for the natural neighbors from delauny triangulation of the whole domain, an algorithm quantifies the natural neighbour nodes of the given point based on the locally delaunay triangles is proposed for the improvement of the NNM. Similar to the EFG method, the procedure of interpolation and construction in the improved NNM is meshiess. As a result, the improved NNM can possesses both the excellent properties of the natural neigbour interpolation and advantages of the EFG method. Numerical results show that the excellent agreement with exact solution is obtained in this method. Convergence studies in the numerical examples also show that the present method possesses an excellent rate of convergence for both the displacement and strain energy.