Advanced Finite Element Modelling of 3D Crack Propagation by a Phase Field Approach

该项目的主要目标是建立一个健壮和有效的裂缝相场模型的数值实现。虽然优先方案1748的第一阶段的重点是一些建模问题，如塑性和退化函数，但第二阶段将用于利用所谓的指数形函数的数值效率。在初步一维分析的基础上，建立了指数型的形函数。这些特殊的形函数满足有限元形函数(Kronecker Delta性质、单位分解)的标准要求。类似于扩展有限元方法中裂纹尖端位移的平方根性质，指数形函数将相场模型的正则化长度尺度和演化方程一维解析解的指数性质结合到离散化中，从而获得更好的逼近性质。因此，与交叉有限元方法不同，不需要进行节点浓缩。由于在形函数中引入了相场长度标度，并且需要指数元的适当取向，这种新的近似需要特殊的数值处理和分析。到目前为止，由于缺乏对不断发展的裂纹场的适应性，该方法的适用性仅限于具有先验已知裂纹路径的相当简单的二维问题。为了推广这一方法，针对二维和三维问题，开发了指数形函数的自适应积分和定向格式。此外，借助于新发展的自适应指数形函数，还考虑了动态断裂问题。

The main aim of the project is to establish a robust and efficient numerical implementation of a fracture phase field model. While in the first period of the Priority Programme 1748 the focus was on some modelling issues, such as plasticity and the degradation function, the second phase will be used to exploit the numerical efficiency of so called exponential shape functions. Motivated by preliminary 1d analyses shape functions of exponential character are developed. These special shape functions satisfy standard requirements for finite element shape functions (Kronecker delta property, partition of unity). Similar to the incorporation of the square root behaviour of the displacements at the crack tip in the extended finite element method (XFEM), exponential shape functions incorporate the regularization length scale of the phase field model and the exponential character of the 1d analytic solution of the evolution equation into the discretization resulting in a superior approximation behaviour. In contrast to XFEM approaches, no nodal enrichment is required therefore. Due to the incorporation of the phase field length scale into the shape functions and the need for a proper orientation of the exponential elements, this new type of approximation requires special numerical treatment and analysis. So far the applicability of the approach is limited to rather simple 2d problems with a priori known crack paths due to a lack of adaptivity with respect to the evolving fracture field. In order to generalize the approach, adaptive integration and orientation schemes for the exponential shape functions are developed for 2d and 3d problems. In addition dynamic fracture problems are considered with help of the newly developed adaptive exponential shape functions.