Phase-field computation of brittle fracture: robustness, efficiency, and characterisation of solution non-uniqueness

模拟和准确预测弹性固体中复杂的断裂过程，包括裂纹的萌生、扩展、分支和合并，是一个非常具有挑战性的工程研究课题。我们专注于脆性材料，其中断裂发生在显著的塑性(永久)变形之前。相场方法通过将连续的场(裂纹相场)与材料的状态(从完整到完全断裂)相关联，使得能够真实地模拟这种材料的破坏过程。材料软化的模型可能会导致数学上不唯一的解。它们的出现概率在理论上是由能量关系决定的，而在实际材料中则是由随机的不均匀性决定的。在确定性计算中，出现以下困难：尖锐的裂纹是由相场的急剧变化来模拟的，需要在数值离散中进行非常精细的网格划分，这可能是计算非常昂贵的。为了通过自适应网格来处理这一问题，需要开发高效可靠的误差估计器。由于损伤演化和断裂是不可逆的过程，解在时间上演化，计算上是离散的时间步长。时间步长的选择对求解过程和可能出现的不唯一性有很大影响。将开发误差指示器，以控制算法效率和稳健性方面的时间步长选择。在每个时间步长中，一个可能的非凸能量泛函必须最小化，以显示多个局部极小值，从而给数值算法带来问题。现有的分区和整体办法将在稳健性和效率方面得到进一步改进。为了捕捉解的非唯一性，将用随机场来描述它们出现的概率(由它们的能量关系决定)。这种非唯一性在计算上通过不同的网格细化、时间步进算法以及在每一步的最小化算法中出现。所有这些都将被适当的随机变量局部参数化，从而通过随机场捕捉所有可能解的演变。裂纹扩展过程的描述将空间上每个点和时间上每个时刻的位移场和相场建模为随机变量，然后该随机变量就是时空随机场。这种方法描述了材料状态的不同概率，即裂纹发展的概率和相关的位移。这种方法产生了大量描述解及其演化的参数。处理高维参数问题的数值稀疏技术将进一步发展，以适应相场方法。

Modelling and accurate prediction of complicated fracture processes in elastic solids including crack initiation, propagation, branching, and coalescence, is a very challenging topic of engineering interest. We focus on brittle materials where fracture occurs prior to significant plastic (permanent) deformation.The phase-field approach enables realistic simulations of such material failure processes by associating a continuous field (the crack phase-field) to the state of the material, ranging from intact to fully broken. Models of material softening may result in mathematically non-unique solutions. Their probability of appearance is in theory determined by energetic relations, and in the real material by random inhomogeneities.In deterministic computations the following difficulties appear: sharp cracks are modelled by a steep variation of the phase-field, necessitating very fine meshes in numerical discretisations which may be computationally very expensive. To deal with this via adaptive meshing, the development of efficient and reliable error estimators is required. As damage evolution and fracture are irreversible processes, the solution evolves in time, computationally in discrete time-steps. The time-step choice strongly influences the solution process and the possible appearance of non-uniqueness. Error indicators will be developed to control the time-step selection as regards algorithmic efficiency and robustness. In each time-step, a possibly non-convex energy functional has to be minimised displaying multiple local minima and hence causing problems for numerical algorithms. Existing partitioned and monolithic approaches will be further advanced as regards the robustness and efficiency. Particular attention will be paid to the capture of all local minima.To capture the non-unique nature of solutions, the probability of their appearance (determined by their energetic relations) will be described by random fields. This non-uniqueness computationally appears with different mesh-refinements, time-stepping algorithms, and within the minimisation algorithm in each step. All of these will be parameterised locally by appropriate random variables, thus capturing the evolution of all possible solutions via random fields.The description of the crack propagation process will model both the displacement and phase-field in each point in space and each instant in time as a random variable, which then is a spatio-temporal random field. This describes the different probabilities of the state of the material, i.e. probability of crack development and the associated displacement.Such an approach generates a large number of parameters which describe the solutions and their evolution. Numerically sparse techniques for dealing with high-dimensional parametric problems will be further developed to suit the phase-field approach.