Isogeometric analysis (IGA), which was introduced by T.J.R. Hughes et al. [1], integrates computer-aided design and analysis by employing spline/NURBS bases for both geometry and field approximation, enabling exact geometries and higher-order continuity. It has demonstrated notable advantagesin accuracy, adaptivity, and the efficient treatment of complex geometries across mechanics and wave propagation. The scaled boundary finite element method (SBFEM) is a semi-analytical, boundary-basedtechnique that introduces a scaling center and polar-like coordinates, discretizes only the boundary (circumferential direction) and solves analytically in the radial direction, leading to an ODE eigenproblem [2]. This yields a dimensional reduction and mitigates meshing burdens by working primarily on theboundary. Building on this, the idea of enhancing IGA with the Scaled Boundary Finite Element Method (SBFEM) was introduced by Natarajan et al. [3], extending the isogeometric framework to scaled boundary discretizations. Given a polar parameterization obtained by uniformly scaling the domain boundary, the method couples a NURBS-based discretization in the circumferential direction with analytical solutions in the radial direction. We extend this framework to a broader class of computational domains, termed weakly star-shaped spline elements. These elements are defined by polar parameterizations whose “rays” are general spline curves rather than straight lines in the radial direction, offering greater geometric flexibility and more geometric degrees of freedom. We construct such elements by applying conformal mappings to scaling-based parameterizations and show that the IGA-based SBFEM enhancement carries over naturally to thisgeneralized setting. The talk will discuss the construction of domains via suitable choices of conformal maps, and it will detail the discretization and solution procedures, illustrated through representative computational examples.