数值流形法中基于适合分析T样条的局部网格加密算法

Local mesh refinement algorithm based on analysis-suitable T-spline in numerical manifold method

数值流形方法中一般采用有限元网格或规则网格作为其数学覆盖系统,而规则的网格突出的优点是不需要适应求解域的边界和各种不连续面。采用规则的矩形网格作为数值流形方法中的数学网格,并借助适合分析的T样条实现了数值流形方法中的局部加密。适合分析的T样条定义在一个限制的T网格上,其基函数具有线性无关、单位分解、局部加密等许多重要性质,使得其非常适合用于工程设计及分析。当对一个适合分析的T网格加密后,所产生的新的网格往往不再是适合分析的T网格。基于此,提出了一种简单的数学网格加密算法,该算法能保证局部加密后的数学网格仍然是适合分析的。算例结果表明:在应力集中区和裂纹尖端等应力梯度较大区域,该算法均具有较强的适用性。

Generally,a finite element mesh or a regular mesh is used as the mathematical covering system in the numerical manifold method(NMM).The advantage of the regular mesh is that it has no requirement to conform to the boundary of the solution domain and various discontinuities.In this paper,the regular rectangular mesh was used as the mathematical mesh in NMM,and the analysis-suitable T-spline was introduced into NMM to realize the local refinement.As the analysis-suitable T-spline was defined over a mildly restricted T-mesh,it presents many important mathematical properties,such as linear independence,partition of unity and highly localized refinement capability.However,after an analysis-suitable T-mesh was locally refined,the generated new T-spline was not analysis-suitable.Therefore,a simple local refinement algorithm was developed in this paper to make sure that the refined mathematical mesh was still analysis-suitable.Moreover,the results from numerical examples show that the algorithm has strong applicability in large-stress gradient areas such as stress concentration area and crack tips.