二维弹性力学问题的光滑无网格伽辽金法

计算效率低的问题长期阻碍着无网格伽辽金法(element-free Galerkin method, EFGM) 的深入发展. 为了提高EFGM 的计算速度, 本文提出一种求解二维弹性力学问题的光滑无网格伽辽金法. 该方法在问题域内采用滑动最小二乘法(moving least square, MLS)近似、在域边界上采用线性插值建立位移场函数; 基于广义梯度光滑算子得到两层嵌套光滑三角形背景网格上的光滑应变, 根据广义光滑伽辽金弱形式建立系统离散方程. 两层嵌套光滑三角形网格是由三角形背景网格本身以及四个等面积三角形子网格组成. 为了提高方法的精度, 由Richardson外推法确定两层光滑网格上的最优光滑应变. 几个数值算例验证了该方法的精度和计算效率. 数值结果表明, 随着光滑积分网格数目的增加, 光滑无网格伽辽金法的计算精度逐步接近EFGM 的, 但计算效率要远远高于EFGM的. 另外, 光滑无网格伽辽金法的边界条件可以像有限元那样直接施加. 从计算精度和效率综合考虑, 光滑无网格伽辽金法比EFGM具有更好的数值表现, 具有十分广阔的发展空间. 

A SMOOTHED MESHFREE GALERKIN METHOD FOR 2D ELASTICITY PROBLEM1)

Despite clear general progress with element-free Galerkin method (EFGM), its low computational efficiency becomes a technical issue in the simulation of realistic problems. To improve the efficiency of EFGM, a smoothed meshfree Galerkin method is presented for the 2D elasticity problem. In the method presented, displacement fields are constructed using the moving least square (MLS) approximation and strains are smoothed over two-level nesting smoothed triangular cells based on the generalized gradient smoothing technique. Then, the generalized smoothed Galerkin (GS-Galerkin) weak form is used to create the discretized system equations. Each two-level nesting smoothed triangular cells include the triangular background cell itself and four equal-area triangular sub-cells, respectively. According to the Richardson extrapolation method, an optimal combination of the two-level smoothed strains can be obtained. Since the present method uses the linear interpolation on the boundary of problem domain, the boundary conditions including the essential and natural boundary conditions can directly impose as that in FEM. Several examples, including the cantilever beam, infinite plate with a circle hole, infinite plate with a central crack and the twin-arched tunnel, are investigated to demonstrate the accuracy and efficiency of the present method. The numerical results show that with more smoothing sub-cells by using in the smoothed meshfree Galerkin method, higher numerical accuracy can be obtained. In addition, the present method is higher efficient than EFGM. As a consequence, the smoothed meshfree Galerkin method with two-level nesting smoothed triangular cells significantly outperforms the EFGM and is very successful and attractive numerical method for solving the elasticity problems.