A parametrized reduced order modeling methodology for cracked two dimensional solids is presented, where the parameters correspond to geometric properties of the crack, such as location and size. The method follows the offline‐online paradigm, where in the offline, training phase, solutions are obtained for a set of parameter values, corresponding to specific crack configurations and a basis for a lower dimensional solution space is created. Then in the online phase, this basis is used to obtain solutions for configurations that do not lie in the training set. The use of the same basis for different crack geometries is rendered possible by defining a reference configuration and employing mesh morphing to map the reference to different target configurations. To enable the application to complex geometries, a mesh morphing technique is introduced, based on inverse distance weighting, which increases computational efficiency and allows for special treatment of boundaries. Applications in linear elastic fracture mechanics are considered, with the extended finite element method being used to represent discontinuous and asymptotic fields.

中文翻译：

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裂纹固体的参数化降阶建模

提出了一种用于裂纹二维固体的参数化降阶建模方法，其中参数对应于裂纹的几何特性，例如位置和大小。该方法遵循离线-在线范式，在离线-在线训练阶段，将获得一组参数值的解，该值对应于特定的裂纹配置，并为较低维的解空间创建基础。然后，在在线阶段，此基础用于获取不在训练集中的配置的解决方案。通过定义参考配置并采用网格变形将参考映射到不同的目标配置，可以针对不同的裂纹几何形状使用相同的基础。为了使应用程序适用于复杂的几何图形，引入了网格变形技术，基于距离反比加权，可提高计算效率并允许对边界进行特殊处理。考虑了线性弹性断裂力学中的应用，并使用扩展有限元方法来表示不连续和渐近场。