Frequency-constrained topology optimization of an airfoil with variable supports using super-Gaussian function parameterization

Abstract This paper introduces a procedure that optimizes the topology of an airfoil structure and its support locations subject to mass-fraction, natural frequency constraints, and aerodynamic loading. We use a node-based density formulation with a density filter and solid isotropic material penalization (SIMP) interpolation. This ensures a continuous density field and provides a smooth transition between solid and void element states. A super-Gaussian function projects circular supports onto the airfoil’s finite element mesh, applying a soft-min filter to the distance function, thus preserving differentiability. By imposing a Kreisselmeier–Steinhauser (KS) aggregation constraint on the fundamental natural frequency, we address possible eigenvalue crossover issues known to affect frequency-centered structural problems without needing direct eigenmode tracking. In addition, loads obtained from a 2D aerodynamic analysis are interpolated and applied to the boundary nodes of the structural mesh using 1D linear shape functions. We provide new insight into enforcing a “skin” layer in which elements on the outer boundary of the airfoil structure maintain some requisite density. This ensures that the design can withstand the distributed aerodynamic loads and alleviates numerical performance issues that could potentially hinder convergence of the optimizer. This is the first procedure to co‑optimize topology, movable supports, and a graded skin while respecting a minimum eigenfrequency; the scheme remains fully differentiable and requires no mode‑tracking. We present several numerical examples that showcase the benefits of simultaneous consideration of mass, frequency, and aerodynamic constraints in designing airfoil structures with variable support locations.