Multi-objective optimization of multi-directional functionally graded beams using an effective deep feedforward neural network-SMPSO algorithm

This paper proposes an intelligent multi-objective optimization approach using the deep feedforward neural network (DNN) integrated with the speed-constrained multi-objective particle swarm optimization (SMPSO) to give the so-called DNN-SMPSO algorithm for solving multi-objective optimization problems of two-dimensional functionally graded (2D-FG) beams under a static load and free vibration. In the proposed approach, a high accurate DNN integrated with an intelligent sampling technique is used as a surrogate model to replace time-consuming numerical models in predicting objectives and constraints during the optimization process. Meanwhile, the SMPSO algorithm is utilized to search a set of Pareto-optimal solutions which show the best trade-off solutions of the required objectives. The ceramic volume fraction values at control points defined by the isogeometric analysis (IGA) framework are taken into account as continuous design variables and input parameters of the DNN model while the objectives and constraints are considered as output signals. In order to avoid the overfitting phenomena and speed up the training process of the DNN model, the state-of-the-art dropout and mini-batch techniques are applied. Additionally, various activation functions, optimizers, and hyper-parameters such as number of hidden layers and hidden units of the DNN model are surveyed. The accuracy, efficiency, and applicability of the proposed method are illustrated through two different multi-objective optimization examples of the 2D-FG beams with various boundary conditions. Optimal results obtained by the DNN-SMPSO method are compared with those of other methods to investigate the reliability of the proposed method. The optimal material distribution of the 2D-FG beams is described by two-dimensional Non-Uniform Rational B-spline (2D-NURBS) basis functions. Through the obtained numerical results, the DNN-SMPSO shows its accuracy, effectiveness, and capability in solving multi-objective optimization problems of engineering structures, especially in aspect of saving the computational cost. In addition, the attained optimal material distribution is useful for the 2D-FG beam fabrication.