通过能量公式耦合复变函数理论和有限元法求解裂纹扩展：共形映射方法和黎曼-希尔伯特问题的简化
Coupling of complex function theory and finite element method for crack propagation through energetic formulation: conformal mapping approach and reduction to a Riemann-Hilbert problem
摘要：
在本文中，我们提出了一个耦合的分析-数值方法来模拟裂纹扩展过程中的二维有界域的理论背景。解析-数值耦合方法的目的是利用复变函数理论的工具构造解析解，并在远离奇异点的区域将其与有限元解连续耦合，从而获得裂纹尖端附近的正确解行为。通过这种方式，裂纹扩展可以建模，而无需重新网格化。根据能量释放率，通过由势能和耗散能组成的总能量的最小化，可以计算出裂纹扩展的可能方向。在此基础上，利用复分析和共形映射技术，给出了含任意裂纹路径的圆形区域上的混合边值问题的解析解，将线弹性问题转化为单位圆盘上的全纯函数的黎曼-希尔伯特问题.利用裂纹尖端附近区域的解析解的优点，可以在短的计算时间内对各种裂纹扭结角和长度的总能量进行评估，从而导致计算最小化过程的潜在有效方式。为此，本文提出了一种新的耦合方法裂纹扩展建模的一般策略。此外，我们还讨论了这一战略的实际实现的道路上的障碍。
英文摘要：
In this paper we present a theoretical background of a coupled analytical-numerical approach to model a crack propagation process in two-dimensional bounded domains. The goal of the coupled analytical-numerical approach is to obtain the correct solution behaviour near the crack tip by help of the analytical solution constructed by using tools of the complex function theory and couple it continuously with the finite element solution in the region far from singularity. In this way, crack propagation could be modelled without using remeshing. Possible directions of crack growth can be calculated through the minimization of the total energy composed of the potential energy and the dissipated energy based on the energy release rate. Within this setting, an analytical solution of a mixed boundary value problem based on complex analysis and conformal mapping techniques is presented in a circular region containing an arbitrary crack path. More precisely, the linear elastic problem is transformed into a Riemann-Hilbert problem in the unit disk for holomorphic functions. Utilising advantages of the analytical solution in the region near the crack tip, the total energy could be evaluated within short computation times for various crack kink angles and lengths leading to a potentially efficient way of computing the minimization procedure. To this end, the paper presents a general strategy of the new coupled approach for crack propagation modelling. Additionally, we also discuss obstacles on the way of practical realisation of this strategy.
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