Isotropic, size-controlled integrable frame fields in representation space: a simplified formulation for quadrilateral meshing

We present a novel, simplified representation space formulation for isotropic, size-controlled integrable frame fields intended for quadrilateral meshing, offering an efficient alternative to heavy fourth-order tensor formalism. Within our framework, local mesh size and frame orientation are embedded into a unified 3D representation, allowing for the analytical derivation of the exact Lie bracket. To prevent numerical lock-up on aggressively graded, non-uniform background meshes, we implement an adimensional unweighted face-based residuals minimization approach, coupled with an elastic-cone relaxation. Rather than relying on tedious manual parameter tuning, we introduce a pragmatic two-stage calibration strategy: a stochastic Monte Carlo routine first evaluates mathematical expectations of the discrete operators on randomized fields to balance their intrinsic numerical stiffness, which are then compounded with a topological activation factor. This balance strategy is embedded within an autonomous, homotopy-driven optimization loop that dynamically stretches the elastic membrane. This protocol grants the representation space variables the physical freedom to overcome local energy barriers, facilitating the organic nucleation of topological singularities before flattening the field onto the admissible cone manifold. Numerical results on complex configurations demonstrate that our approach effectively neutralizes grid-induced "resolution locking", providing a stable baseline ideally suited for subsequent quadrilateral mesh extraction stages.