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

脆性断裂的相场计算：稳健性、效率和解非唯一性表征

基本信息

批准号：
328873018
负责人：
Professorin Dr. Laura De Lorenzis
金额：
--
依托单位：
Departement Maschinenbau und Verfahrenstechnik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2018
资助国家：
德国
起止时间：
2017-12-31 至 2022-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/328873018?language=en
关键词：
Phase field computation brittle fracture

项目摘要

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.

弹性固体中复杂断裂过程（包括裂纹萌生、扩展、分支和合并）的建模和准确预测是一个非常具有挑战性的工程课题。我们专注于在显着塑性（永久）变形之前发生断裂的脆性材料。相场方法通过将连续场（裂纹相场）与材料状态（从完整到完全断裂）相关联，可以对此类材料失效过程进行真实模拟。材料软化模型可能会产生数学上非唯一的解决方案。它们出现的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不均匀性决定的。在确定性计算中，会出现以下困难：通过相场的急剧变化来模拟尖锐裂纹，需要在数值离散中使用非常精细的网格，这可能在计算上非常昂贵。为了通过自适应网格划分来解决这个问题，需要开发高效可靠的误差估计器。由于损伤演变和断裂是不可逆的过程，因此解决方案会随着时间的推移而演变，在计算上以离散的时间步长进行。时间步长的选择强烈影响求解过程和可能出现的非唯一性。将开发误差指示器来控制算法效率和鲁棒性方面的时间步选择。在每个时间步长中，必须最小化可能的非凸能量泛函，以显示多个局部最小值，从而导致数值算法出现问题。现有的分区和整体方法将在稳健性和效率方面得到进一步提升。将特别注意捕获所有局部最小值。为了捕获解的非唯一性，它们出现的概率（由它们的能量关系确定）将由随机场描述。这种非唯一性在计算上表现为不同的网格细化、时间步进算法以及每个步骤中的最小化算法。所有这些都将通过适当的随机变量进行局部参数化，从而通过随机场捕获所有可能解的演化。裂纹扩展过程的描述将空间中每个点和时间上每个瞬间的位移和相场建模为随机变量，这是一个时空随机场。这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。这种方法生成大量描述解决方案及其演变的参数。用于处理高维参数问题的数值稀疏技术将得到进一步发展，以适应相场方法。