Quadrilateral meshes are widely used in numerical simulation, particularly for high-order methods, due to their tensor-product structure and favorable approximation properties. However, generating high-quality quadrilateral meshes on complex geometries remains challenging. In layout-based approaches, mesh generation relies on cross-fields, i.e. directional fields with fourfold rotational symmetry, used to control element orientation and domain partitioning. This work presents a robust method to construct cross-fields from prescribed singular points, enabling quadrilateral mesh generation with user-controlled topology. Unlike classical approaches, where singularities arise from the field solution, the proposed framework allows the number, type, and location of singular points to be imposed explicitly. The method follows a two-step procedure. A discrete vector field is first computed by solving a mixed finite element formulation of a linear elasticity-type system, enforcing divergence-free constraints away from singular regions. The system is discretized using lowest-order Raviart–Thomas elements, while singularities are introduced through local boundary flux constraints. An angle correction field is then obtained by solving a Laplace problem, ensuring boundary alignment while preserving the prescribed singular structure. The approach is detailed for planar domains, extended to non-simply connected and drilled geometries, and generalized to surface meshes using Gauss–Bonnet relations. The resulting cross-fields satisfy the required topological constraints and enable the construction of structured domain partitions and high-quality quadrilateral meshes.