Multi-objective optimization for lightweight design of bi-directional functionally graded beams for maximum frequency and buckling load

Abstract

This paper proposes a multi-objective particle swarm optimization (MOPSO) algorithm integrated with a semi-analytical solution to solve optimization problems of lightweight design of bi-directional functionally graded (BDFG) beams for maximum fundamental frequency and/or maximum critical buckling load. The MOPSO algorithm is utilized to search for Pareto-optimal solutions that present the best trade-off optimal material distributions to achieve the required objectives. The proposed semi-analytical solution replaces the time-consuming numerical methods in computing the objective functions during the optimization process. The material volume fraction is described in both longitudinal and thickness directions by a 10-parameter trigonometric function. These parameters are taken as the design variables of the optimization problem. The effective material properties of the BDFG beam are estimated according to the Mori–Tanaka homogenization scheme. The governing variable-coefficients differential equations of BDFG beams under different boundary conditions are derived based on Euler-Bernoulli beam theory then solved by the proposed semi-analytical method for the fundamental frequency and buckling load. The accuracy, efficiency, and applicability of the proposed method are demonstrated through several multi-objective optimization problems. The predicted optimal results are compared with those of other methods to investigate the reliability of the proposed method. The optimization results show that the proposed 10-parameter function provides flexible material profiles and gives designers a powerful tool for optimal material distributions. More importantly, the obtained optimal material distribution is essential for the manufacture of BDFG beams.