  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.
