Constrained dynamic optimization of bidirectional porous functionally graded beams using Touratier higher-order shear deformation theory

This study develops a constrained gradient-based optimization framework for the dynamic design of bidirectional porous functionally graded (FG) beams. Material properties are graded simultaneously along the thickness and width directions through independent power-law distributions, while even and uneven porosity patterns are considered. Touratier’s higher-order shear deformation theory (HSDT) is embedded directly in a constrained BFGS design loop so that the thickness power-law index ( n ), width power-law index ( k ), porosity coefficient ( α ), and geometric aspect ratio ( b / h ) enter the design formulation simultaneously. The principal novelty is not the use of HSDT or BFGS separately, but their integration into a feasibility-aware framework in which the HSDT stiffness and inertia operators are recomputed during the design iterations and bound constraints together with a modal-stress screening criterion are enforced. The scalar objective combines natural-frequency enhancement and structural-mass reduction. To address the reviewers’ concerns about objective weighting and numerical differentiation, the revision adds explicit a-posteriori sensitivity audits using the archived response data and distinguishes these reproducible post-processing results from historical run metadata that were not preserved unambiguously. The archived performance cases include a fundamental frequency as high as 162.8 Hz, while the archived parameter sweep reports a 15-20 % mass-reduction trend relative to its reference state. These values are presented as performance cases rather than as proof of a unique weight-independent global optimum. The parametric phase-velocity results further demonstrate the strong coupling between bidirectional gradation and porosity. The framework therefore provides an interpretable route for linking spatial material tailoring, porosity, geometry, structural feasibility, and dynamic performance, with clear extensions toward vibration-transmission and vibroacoustic design.