We consider the elastic scattering problem by multiple disjoint arcs or \emph{cracks} in two spatial dimensions. A key aspect of our approach lies in the parametric description of each arc's shape, which is controlled by a potentially high-dimensional, possibly countably infinite, set of parameters. We are interested in the efficient approximation of the parameter-to-solution map employing model order reduction techniques, specifically the reduced basis method. Initially, we utilize boundary potentials to transform the boundary value problem, originally posed in an unbounded domain, into a system of boundary integral equations set on the parametrically defined open arcs. Our aim is to construct a rapid surrogate for solving this problem. To achieve this, we adopt the two-phase paradigm of the reduced basis method. In the offline phase, we compute solutions for this problem under the assumption of complete decoupling among arcs for various shapes. Leveraging these high-fidelity solutions and Proper Orthogonal Decomposition (POD), we construct a reduced-order basis tailored to the single arc problem. Subsequently, in the online phase, when computing solutions for the multiple arc problem with a new parametric input, we utilize the aforementioned basis for each individual arc. To expedite the offline phase, we employ a modified version of the Empirical Interpolation Method (EIM) to compute a precise and cost-effective affine representation of the interaction terms between arcs. Finally, we present a series of numerical experiments demonstrating the advantages of our proposed method in terms of both accuracy and computational efficiency.

翻译：我们考虑二维空间中由多个不相交弧线或“裂纹”构成的弹性散射问题。该方法的关键在于每条弧线形状的参数化描述，该描述由一组可能为高维（甚至可数无限维）的参数控制。我们感兴趣的是利用模型降阶技术，特别是降阶基底法，高效逼近参数到解的映射。首先，利用边界势将原本定义在无界域中的边值问题转化为定义在参数化开放弧线上的边界积分方程组系统。我们的目标是构建一个快速替代模型来解决此问题。为此，我们采用降阶基底法的两阶段范式。在离线阶段，我们假设各弧线形状完全解耦的情况下计算该问题的解。利用这些高保真解和本征正交分解（POD），我们为单个弧线问题构建降阶基底。随后，在在线阶段，当计算新参数输入下的多弧线问题时，我们对每条弧线分别使用上述基底。为加速离线阶段，我们采用经验插值法（EIM）的改进版本来计算弧线间相互作用项的精确且经济高效的仿射表示。最后，通过一系列数值实验展示了所提方法在精度和计算效率方面的优势。