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

This paper proposes a multi-objective particle swarm optimization (MOPSO) algorithm integrated with a semianalytical 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 variablecoefficients 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.