Stochastic phase-field modeling of brittle fracture: computing multiple crack patterns and their probabilities
Stochastic phase-field modeling of brittle fracture: computing multiple crack patterns and their probabilities
复制标题
DOI：
10.1016/j.cma.2020.113353
复制
发表时间：
2020-05
期刊：
影响因子：
--
通讯作者：
T. Gerasimov;U. Römer;J. Vondrejc;H. Matthies;L. Lorenzis
中科院分区：
文献类型：
--
作者：
T. Gerasimov;U. Römer;J. Vondrejc;H. Matthies;L. Lorenzis
在脆性断裂的变分相场模拟中，要极小化的泛函不是凸的，因此泛函的必要平稳性条件可能允许多个解。在实际计算中获得的解通常是若干局部极小值中的一个。偶尔会记录到数值或物理参数的小扰动引起的多解的证据，但文献中没有明确研究。在这项工作中，我们专注于这个问题，并主张一个范式的转变，远离一个特定的解决方案，对所有可能的解决方案（局部极小）的同时描述的搜索，沿着其发生的概率。最近的方法倡导测度值的解决方案（杨措施，以及他们的推广到统计解决方案）和他们的数值近似在流体力学的启发，我们提出了随机松弛的变分脆性断裂问题，通过随机扰动的功能。我们引入了随机解的概念，其主要优点是可以捕获底层域中裂纹相场的点到点相关性。这些随机解由随机场或随机变量表示，其值在经典的确定性解空间中。在数值实验中，我们使用一个简单的蒙特卡罗方法来计算这种随机解的近似。计算的最终结果不是单一的裂纹模式，而是几种可能的裂纹模式及其概率。随机解框架，使用不断变化的随机场允许额外的有趣的可能性条件的概率进一步裂纹路径上的中间裂纹图案。
In variational phase-field modeling of brittle fracture, the functional to be minimized is not convex, so that the necessary stationarity conditions of the functional may admit multiple solutions. The solution obtained in an actual computation is typically one out of several local minimizers. Evidence of multiple solutions induced by small perturbations of numerical or physical parameters was occasionally recorded but not explicitly investigated in the literature. In this work, we focus on this issue and advocate a paradigm shift, away from the search for one particular solution towards the simultaneous description of all possible solutions (local minimizers), along with the probabilities of their occurrence. Inspired by recent approaches advocating measure-valued solutions (Young measures as well as their generalization to statistical solutions) and their numerical approximations in fluid mechanics, we propose the stochastic relaxation of the variational brittle fracture problem through random perturbations of the functional. We introduce the concept ofstochastic solution, with the main advantage that point-to-point correlations of the crack phase fields in the underlying domain can be captured. These stochastic solutions are represented by random fields or random variables with values in the classical deterministic solution spaces. In the numerical experiments, we use a simple Monte Carlo approach to compute approximations to such stochastic solutions. The final result of the computation is not a single crack pattern, but rather several possible crack patterns and their probabilities. The stochastic solution framework using evolving random fields allows additionally the interesting possibility of conditioning the probabilities of further crack paths on intermediate crack patterns.