Genetic-Algorithm Optimization of Dynamic Efficiency in Bidirectional Porous Functionally Graded Beams

This study develops an analytical–evolutionary framework for optimizing the dynamic efficiency of bidirectional porous functionally graded beams. Touratier’s higher-order shear deformation theory is coupled with a real-coded genetic algorithm. The material-gradation indices in the thickness and width directions, the porosity coefficient, and the cross-sectional aspect ratio are treated as four coupled design variables. Dynamic efficiency is defined as the first modal frequency per unit mass, J = f1/m (Hz/kg), with f1 evaluated at β1 = π/L for the simply supported finite beam. Uniform and non-uniform porosity laws are examined under three admissible design domains. After correcting and consistently implementing the modified rule of mixtures, the restricted-domain efficiencies are 117.819 and 96.766 Hz/kg for uniform and non-uniform porosity, respectively. Extending the material-gradation bounds increases them to 920.823 and 328.627 Hz/kg, while extension of the geometric domain gives 2302.058 and 821.568 Hz/kg. Thirty independent GA runs for each case yield 100% success under a 0.5% tolerance. Deterministic corner and one-at-a-time sampled checks confirm the observed boundary-directed trends within the investigated boxes. The results are mathematical optima for the stated objective and constraints, not production-ready designs.