Accurate numerical modeling of fracture in solids is a challenging undertaking that often involves the use of computationally demanding modeling frameworks. Model order reduction techniques can be used to alleviate the computational effort associated with these models. However, the traditional offline-online reduction approach is unsuitable for complex fracture phenomena due to their excessively large parameter spaces. In this work, we present a reduction framework for fracture simulations that leaves out the offline training phase and instead adaptively constructs reduced solutions spaces online. We apply the framework to the thick level set (TLS) method, a novel approach for modeling fracture able to model crack initiation, propagation, branching, and merging. The analysis starts with a fully-solved load step, after which two consecutive reduction operations—the proper orthogonal decomposition and the empirical cubature method—are performed. Numerical features specific to the TLS method are used to define an adaptive domain decomposition scheme that allows for three levels of model reduction coexisting on the same finite element mesh. Special solutions are proposed that allow the framework to deal with enriched nodes and a dynamic number of integration points. We demonstrate and assess the performance of the framework with a number of numerical examples.

中文翻译：

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通过自适应模型降阶加速裂纹扩展模拟

对固体中的断裂进行精确的数值建模是一项具有挑战性的工作，通常涉及使用计算要求高的建模框架。模型降阶技术可用于减轻与这些模型相关的计算工作量。然而，传统的离线-在线还原方法由于参数空间过大，不适用于复杂的断裂现象。在这项工作中，我们提出了一种用于断裂模拟的简化框架，该框架省略了离线训练阶段，而是自适应地在线构建简化的解决方案空间。我们将该框架应用于厚水平集 (TLS) 方法，这是一种能够模拟裂纹萌生、扩展、分支和合并的断裂建模新方法。分析从完全求解的载荷步开始，之后执行两个连续的归约操作——适当的正交分解和经验培养方法。特定于 TLS 方法的数值特征用于定义自适应域分解方案，该方案允许在同一有限元网格上共存三个级别的模型简化。提出了特殊的解决方案，允许框架处理丰富的节点和动态数量的集成点。我们用许多数值例子来演示和评估框架的性能。提出了特殊的解决方案，允许框架处理丰富的节点和动态数量的集成点。我们用许多数值例子来演示和评估框架的性能。提出了特殊的解决方案，允许框架处理丰富的节点和动态数量的集成点。我们用许多数值例子来演示和评估框架的性能。