基于覆盖细化的数值流形法及其在裂纹扩展中的应用

COVER BASED REFINEMENT IN THE NUMERICAL MANIFOLD METHOD AND ITS APPLICATION IN CRACK PROPAGATION

数值流形法是一种基于数学覆盖和物理覆盖的双重覆盖方法,权函数设于数学覆盖上,而位移函数设于物理覆盖上。该文提出了一种基于数值流形法的覆盖细化方法,用来解决二维裂纹扩展问题。对某一待细化流形单元,其加密点是预先确定的,并非所有流形单元均需细化,只有满足一定条件的裂纹前缘单元才有必要进行覆盖细化。所有的细化均基于数学覆盖,覆盖细化过程对于裂纹前缘单元的不同边界条件是不同的。在流形单元进行覆盖细化后,将会产生新的数学覆盖以及相应的物理覆盖和单元。同时,节理环路和接触信息也需要做相应的修正。选取三个常见裂纹扩展的算例进行比较分析,计算结果表明,该文的覆盖细化方法是可行的,相对于原始流形单元而言,裂纹尖端可以处于单元内部,覆盖细化方法获得更高的精度,而不显著地增加未知自由度。

The numerical manifold method （NMM） is a cover-based method with two different covers, namely mathematical cover and physical cover. The weight functions are defined on the mathematical covers and the unknown displacement coefficients are defined on the physical covers. A cover based refinement with the NMM to model two-dimensional crack propagation is described. The refined points of this method are pre-determined for a given element. Not all manifold elements are required to be refined, instead, only the frontal elements and relevant elements satisfied some conditions are required. The refinement is not based on the elements but based on the corresponding mathematical covers of those elements. The updated process is different for various boundary conditions of frontal elements and will be discussed. When a mathematical cover is updated, the corresponding physical covers and elements are updated. Furthermore, the loop of the considered domain is required to be updated as well. In order to demonstrate the utility of the proposed technique, three numerical examples are analyzed to validate and explain the present method. The results show that the refined method is more accurate, since the crack can propagate inside an element. Therefore, the refined method is suitable for improving the accuracy of considered problems without significantly increasing the degrees of freedom.