Simultaneous optimization of GPL distribution and GPL volume fraction for frequency-constrained design of AFG-GPL curved beams using MSS-IGP

This paper presents a novel optimization methodology to determine optimal graphene platelet (GPL) distribution patterns and GPL volume fraction (GVF) in axially functionally graded GPL-reinforced (AFG-GPLs) composite curved beams. Existing studies are limited to predefined distribution patterns, restricting GPL-efficient structural design. To overcome this, a Multi-Scale Seeded Iterative Genetic Programming (MSS-IGP) algorithm coupled with Isogeometric Analysis (IGA) is proposed to autonomously generate optimal GPL distribution functions that minimize overall GVF while satisfying frequency constraints. An IGA model based on the Third-Order Shear Deformation Theory (TSDT) evaluates the free vibration characteristics of parabolic and semi-elliptical AFG-GPLs curved beams under various boundary conditions and height-to-span ratios. The MSS-IGP evolves populations of mathematical functions to identify the minimum GVF required for a target fundamental frequency. Results demonstrate that the MSS-IGP achieves the target frequency with consistently lower GVFs than conventional predefined patterns, reaching as low as 0.5815%. Furthermore, the optimal GPL distributions predominantly exhibit zero GVF at the curved beam center, and a distinctive re-elevation phenomenon emerges near simply supported boundaries at larger H/L ratios. These findings provide practical guidelines for GPL-efficient design in AFG-GPLs curved beams and demonstrate the potential of the MSS-IGP for the optimal design of advanced composite structures.