Multi-Objective Optimization of the Dynamic of a Bridge Pillar and a Truss Structure Subjected to Random Load by a New Hybridized Method

Abstract

In this paper two linear dynamic mechanical problems involving random Gaussian loading are considered for their design through multi-objective optimization problems involving the mean and the standard deviation. It is also presented a new hybridization, called Representation Formula Nelder-Mead, RFNM, for solving global optimization problems. This new hybridization, motivated by the Pincus representation formula hybridized with Nelder-Mead Algorithm, is proposed to solve the two multi-objective optimization problems. The Pincus representation formula gives a good approximation of the global minimum and once one is near the minimum the Nelder-Mead algorithm converges to it rather quickly. The Pincus representation is obtained generating through an uniform distribution a sequence of points and computing an approximation of the minimum; this is done a certain number of times until convergence is achieved.
The first problem is a design of a pillar geometry with respect to a compressive random load process.
The second problem is the design of a truss structure with respect to a vertical random load process for several frequency bands. In both problems, the load is characterized by an ergodic, stationary, Gaussian random process. To generate the Pareto front, the Normal Boundary Intersection (NBI) method is used to produce a series of constrained single-objective optimizations. The penalty method is introduced to deal with the new constrains introduced by NBI and in order to put the problems in the framework of the RFNM algorithm used in the single-objective optimizations. The second problem, depending on the frequency band of excitation, can have as Pareto curve a single point, a standard Pareto curve, or a discontinuous Pareto curve. Hence, it is showing through this simple example that difficult situations can occur for designing the mechanical systems when considering their dynamic responses due to random loads, even for this simple situation. But the strategy proposed here have shown its ability to give valuable results, able to help designers to choose for the best compromise between the mean and the standard deviation for this kind of problems.