A conformable-inspired two-parameter regularized Hermite curve family for airfoil geometry modeling
Year: 2026
Type: article
Abstract: We introduce a local two-parameter cubic Hermite-type interpolation family motivated by the conformable identity \(T_\alpha f(t)=t^{1-\alpha }f'(t)\) and the power-law change of variable \(s=t^\alpha \) . For differentiable functions, the conformable operator is local and, up to the factor \(\alpha \) , corresponds to an ordinary derivative with respect to s ; therefore, the proposed construction is not interpreted as a nonlocal fractional model. The parameters \(\alpha \in (0,1]\) and \(\varepsilon >0\) act, respectively, as a power-law tangent-scaling parameter and a displaced endpoint-constraint/asymmetry parameter. The construction yields explicit basis functions that interpolate the endpoint control points, preserve affine invariance, and satisfy a regularized conformable tangent-matching condition under the reparametrization \(s=t^\alpha \) . The displaced tangent constraint imposed at \(t=\varepsilon \) , however, does not generally guarantee classical \(C^1\) or geometric \(G^1\) continuity at the actual knot. Accordingly, knotwise continuity is assessed through diagnostics based on jumps in the classical derivative, tangent angle, and curvature. The limiting behavior is characterized by distinguishing fixed-parameter and coupled limits. Classical cubic Hermite recovery is obtained under the coupled condition \((1-\alpha (\varepsilon ))\log (1/\varepsilon )\rightarrow 0\) as \(\varepsilon \rightarrow 0^+\) . In contrast, for fixed \(0<\alpha <1\) , the initial tangent basis diverges as \(\varepsilon \rightarrow 0^+\) unless the associated tangent datum is suitably renormalized; the corresponding coefficient system also becomes ill-conditioned in this regime. Thus, an unconditional recovery of the classical Hermite structure as \(\varepsilon \rightarrow 0^+\) is not claimed. The regularization-induced loss of symmetry is quantified and interpreted as a bias-type shape effect. Moreover, for fixed admissible parameters, a uniform first-order consistency estimate is established, including the first mesh interval, without asserting an unqualified \(h^{2\alpha }\) convergence rate. Based on this theoretical formulation, we develop a comprehensive computational framework to evaluate the proposed model in airfoil geometry reconstruction. Although the conformable Hermite formulation introduces additional local shape flexibility, experimental observations indicate that it does not consistently outperform classical Hermite interpolation when used without parameter tuning. To exploit this flexibility effectively, the proposed interpolation family is coupled with the Improved Starfish Optimization Algorithm (ISFOA), which adaptively identifies suitable values of \((\alpha ,\varepsilon )\) for each airfoil geometry. The proposed framework is evaluated on 11 different airfoil profiles and assessed through both an initial comparison with Cubic Spline, Classical Hermite, and B’ezier interpolation and an extended equal-control-point-budget comparison including shape-parameter Hermite, TCB/Kochanek–Bartels, rational cubic Hermite, trigonometric Hermite, cubic B-spline, CST, PARSEC, and Natural Cubic Spline constructions. In the initial restricted comparison, the optimization-assisted conformable Hermite framework achieves lower reconstruction errors than Cubic Spline, Classical Hermite, and B’ezier interpolation under the adopted experimental protocol. In particular, it attains an average RMSE of 0.270314, compared with 0.420562 for Cubic Spline, 0.421578 for Bézier interpolation, and 0.423351 for Classical Hermite interpolation. However, the broader equal-budget comparison shows that several established shape-representation methods achieve substantially lower reconstruction errors, indicating that the proposed framework should not be interpreted as universally superior in reconstruction accuracy. In addition, the proposed framework retains the local controllability of classical Hermite interpolation while providing an efficient shape representation with fewer control points. Quantitative knotwise diagnostics confirm that classical \(C^1\) and geometric \(G^1\) continuity are not universally guaranteed, while conditioning analysis shows that the optimized parameter pairs remain in moderately conditioned regions of the investigated parameter domain. Complementary symmetric Hausdorff and Chamfer distances are additionally used to quantify worst-case and average bidirectional geometric discrepancies without requiring one-to-one point correspondence. A controlled mesh-refinement experiment further yields an observed convergence order approaching unity, consistent with the theoretically established first-order consistency. The numerical results are interpreted as a characterization of the capabilities and limitations of adaptive parameter identification within the proposed local shape family, rather than as evidence of universal reconstruction superiority, nonlocal fractional effects, or aerodynamic equivalence. The ISFOA stage therefore serves exclusively as a parameter-selection mechanism, whereas the primary scientific contribution is the formulation and theoretical analysis of the regularized conformable Hermite-type interpolation family.
Source: Scientific Reports
Institution: Fırat University
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,690
Award: G1274786054
OpenAlex ID: W7223681858