含裂纹体的单位分解扩展无网格方法

不连续体的数值模拟尤其是动态裂纹的追踪问题一直是工程界研究的热点和难点问题。无网格方法仅仅需要结点信息,非常适合于求解这类问题。基于单位分解思想,在移动最小二乘近似函数(MLS)中根据裂纹面的不连续位移增加一个Heaviside函数,在裂尖则增加四个扩展函数描述渐进裂纹位移场;应用Galerkin方法推导了平衡方程的离散线性方程,并给出了求解裂纹问题应力强度因子的计算公式。与其他类型的扩展无网格相比,在裂尖处近似函数不需要使用可视准则,很容易生成r1/2奇异;另一个优势是影响域并没有因为裂纹的存在而改变,不会降低方程的稀疏性,求解效率较高。数值算例表明,该方法能方便有效地模拟不连续问题,具有十分广阔的应用空间。

Enriched meshless method based on partition of unity for a body with cracks

Numerical simulation of discontinuities, especially for solving the dynamic cracks, is a hot and difficult problem. The meshless method based only on nodes is very suitable for solving those typical questions. A new method for crack approximation is introduced. Based on the idea of partition of unity, one Heaviside function accounting for the displacement discontinuity along the crack faces and four enriched functions near the crack tip in order to capture the singularity of the asymptotic crack tip displacement fields are enriched in the function of moving least square(MLS) approximation.Then the linear discrete system of equilibrium differential equations is derived by using the Galerkin method and the interaction integral method to evaluate the stress intensity factors is presented. Compared with other styles of enriched meshless, one advantage of this methods is that the visibility criterion is not to be used around the crack tip.Therefore, r^1/2singularity is reproduced very well. The second advantage is that the domain of influence of the nodes not changed by the passage of the crack, holding the equations of sparsity, leading to rather high computations. Numerical simulations illustrate that the approach can effectively model the discontinuities, and it has practical merits.