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 of stochastic 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.

中文翻译：

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脆性断裂的随机相场建模：计算多个裂纹模式及其概率

在脆性断裂变分相场建模中，要最小化的泛函不是凸的，因此泛函的必要平稳条件可能允许多个解。在实际计算中获得的解决方案通常是几个局部极小值之一。偶尔记录下由数值或物理参数的小扰动引起的多重解的证据，但没有在文献中明确研究。在这项工作中，我们关注这个问题并提倡范式转变，从寻找一个特定的解决方案转向同时描述所有可能的解决方案（局部最小化），以及它们发生的概率。受最近提倡测量值解决方案（杨氏测量及其对统计解决方案的泛化）及其在流体力学中的数值近似的方法的启发，我们提出了通过函数的随机扰动来随机松弛变分脆性断裂问题。我们引入了随机解的概念，其主要优点是可以捕获基础域中裂纹相场的点对点相关性。这些随机解由具有经典确定性解空间中的值的随机场或随机变量表示。在数值实验中，我们使用简单的蒙特卡罗方法来计算这种随机解的近似值。计算的最终结果不是单一的裂纹图案，而是几种可能的裂纹模式及其概率。使用演化随机场的随机求解框架还允许在中间裂纹模式上调节进一步裂纹路径的概率的有趣可能性。