Optimum Design of Plane Trusses Using Mathematical and Metaheuristic Algorithms on a Spreadsheet

Abstract

Mathematical optimization refers to the process of finding the values of variables that maximize or minimize a function. Structural optimization is the process of designing a structure in such a way as to minimize its weight or cost, while meeting a set of performance requirements, ensuring that it is robust, lightweight, and efficient. Two large categories of optimization algorithms are mathematical and metaheuristic algorithms. The ones of the first rely on mathematical principles, are deterministic and exact but may fail if the problem is too large or complex. The latter category, metaheuristics, represents algorithms that are used to find approximate solutions. They are high-level strategies that guide the search toward a good solution, rather than being a specific, deterministic algorithm. They are often used for problems where it is difficult or impractical to find the optimal solution using exact methods. Metaheuristics typically involve iteratively improving a solution through some type of search or exploration process. They make use of techniques from probability and statistics, such as randomization and stochastic optimization, to explore the search space and guide the search toward good solutions. Some examples include genetic algorithms, simulated annealing, differential evolution (DE), particle swarm optimization (PSO), and ant colony optimization. In this study, a mathematical optimizer and two metaheuristics (DE, PSO), are employed for the optimum structural design of plane truss structures aiming to minimize the weight of the structure under constraints on allowable displacements and stresses. A 10-bar plane truss is considered as the numerical example of the study. The constraints are checked by performing an analysis with matrix methods. All calculations are done on a spreadsheet. The results of the algorithms are compared to each other as well as to results from the literature in terms of convergence speed, number of function evaluations, and accuracy of the solution.