Development of a Lattice Boltzmann Method for the Simulation of Dynamic Crack Propagation in Brittle Materials

该项目的目标是开发一种数值方法，用于模拟脆性（即线弹性）材料中的裂纹扩展，该方法还考虑了周围材料的惯性。为此，将使用所谓的格子玻尔兹曼方法（LBM）。通常，LBM用于模拟流体流动。LBM的思想是通过计算分布函数的演化来模拟连续介质的宏观行为，分布函数在微观水平上代表流体的行为。分布函数在离散格点处定义，并且在一个离散时间步长中沿预定义的空间格点沿着传输到相邻格点。在随后的碰撞步骤中，分布函数被重新计算依赖于所有其他分布函数在所考虑的晶格点，材料参数和lattice quantities. To模拟结构问题与LBM，它是必要的扩展已经存在的LBM简化变形假设的固体（即反平面剪切变形）。这意味着现有的方法需要修改，以便能够模拟平面应变和平面应力问题。随后，将通过正确地对裂纹处的边界条件进行建模来实现静止（即，不移动）裂纹。这不是一个微不足道的任务，因为它旨在使该方法对可能的裂纹模式设置很少的限制。最后，基于断裂力学准则的裂纹扩展也将在LBM中实现。为此，将从文献中选择合适的方法并使其适应LBM。在这一点上，开发的LBM将能够逼真地模拟脆性材料中的动态裂纹扩展。这意味着，在模拟中可以观察到裂纹速度的特征上限和动态裂纹分支等动态断裂的特征。上述方法将在允许模拟各种用户定义的动态线弹性问题以及线弹性材料中的动态裂纹扩展问题的软件中实现。与其他方法相比，LBM固有的显式和可并行化特性将被利用，以显着减少运行时间。此外，该软件将包括用户自定义扩展的接口，并将为今后扩展到三维提供框架。

The goal of this project is to develop a numerical method for the simulation of crack propagation in brittle (i.e. linear elastic) materials, which also considers the inertia of the surrounding material. For this purpose a so-called Lattice Boltzmann Method (LBM) will be used. Typically, the LBM is used to simulate fluid flows. The idea of the LBM is to model the macroscopic behavior of a continuum by computing the evolution of distribution functions, which represent the behavior of the fluid on a microscopic level. The distribution functions are defined at discrete lattice points and are transported along a predefined spatial lattice to the neighbor lattice points in one discrete time step. In the subsequent collision step, distribution functions are recomputed in dependence of all other distribution functions at the considered lattice point, material parameters and lattice quantities.In order to simulate structural problems with the LBM, it is necessary to extend an already existing LBM for simplified deformation assumptions of a solid (i.e. antiplane shear deformation). This means that the existing method needs to be modified in order to be able to model plane strain and plane stress problems. Subsequently, stationary (i.e. non-moving) cracks will be implemented by modeling the boundary conditions at the crack correctly. This is not a trivial task since it is intended that the method sets few limitations on possible crack patterns. Eventually, crack growth based on a criterion of fracture mechanics will also be implemented in the LBM. To this end, a suited approach will be chosen from the literature and adapted to the LBM. At this point the developed LBM will be able to realistically model dynamic crack growth in brittle materials. This implies that features of dynamic fracture such as characteristic upper limits of the crack speed and dynamic crack branching can be observed in simulations.The method described above will be implemented in a software that allows the simulation of various user-defined dynamic linear elastic problems as well as problems of dynamic crack growth in linear elastic materials. The inherent explicit and parallelizable character of the LBM will be exploited in order to significantly decrease the runtime compared to other methods. Furthermore, the software will include interfaces for user-defined extensions and will provide the framework for a future extension to three-dimensions.