结合三角形、四边形等实体单元的位移插值函数和一维轴力杆单元的刚度矩阵推导出了一种用于锚杆支护数值模拟的单元列式，可以在地下结构有限元计算中的任何开挖步自由地加设或拆除锚杆，而且不必在划分初始计算网格时事先考虑或预留锚杆的结点位置，较大地简化了有限元的前处理或网格划分过程。

基于数值流形方法的四节点四边形流形单元在物理覆盖上使用高阶覆盖位移函数能够得到高精度的数值结果，且可以在求解区域的不同地方混合使用各阶覆盖函数来提高求解效率，具有编程和前后处理简单等优点，弥补了有限元法的不足。计算结果表明，数值解与理论解吻合。

基于Laplace 插值函数提出了一种类似于无单元伽辽金法（EFG）的无网格方法—无网格自然邻接点法（MNNM）。该方法克服了自然单元法（NEM）需要全域三角形网格以及EFG法难以准确施加位移边界条件和材料不连续条件、形函数的计算复杂、权函数的选择困难等缺点，适合于考虑多种材料、多步施工过程等复杂岩土工程的自动数值模拟。文中详细讨论了这种MNNM 的分析过程和基本理论，给出了其在杆、梁、节理单元和材料不连续面等方面的处理办法，并用一些标准算例和实际的地下工程算例对本文方法的效率、精度和可靠性进行了验证。

本文采用规则的矩形作为流形方法的数学覆盖网格来模拟裂纹试件中张裂纹的扩展过程，并在裂纹尖端区域的物理覆盖上采用高阶覆盖位移函数计算得到裂纹尖端的应力强度因子，与仅在物理覆盖上采用常位移函数相比，在相同计算网格的情况下，本文方法得到的应力强度因子精度有了较大幅度的提高。数值算例及实验结果的比较表明了文中数值方法的有效性。

A local basis algorithm for searching natural neighbours in Natural Element Method (NEM) is presented for solving the elasticity problems in this paper. Comparison with the global sweep algorithm used in natural element method or Natural Neighbour Method (NNM) for searching natural neighbours, the proposed algorithm is more expedient and convenient in the constructions and computation of natural neighbour interpolations. In the proposed NEM based on local search, the Laplace (non-sibson) interpolations are constructed with respect to the natural neighbour nodes of the given point which have been locally defined. The shape functions from the Laplace approximations have the delta function property and the Laplace interpolants are strictly linear between adjacent nodes, which facilitate imposition of essential boundary conditions and treatment of material discontinuity with ease as it is in the conventional finite element method. The Laplace interpolants derived from the local algorithm and the global algorithm in NEM are identical because of the uniqueness of the Voronoi diagram. Numerical results and convergence studies also show that the present NEM based on local search algorithm possesses the same accuracy and rate of convergence as they are in previous NEM.