Geometric optimization principles for multi-constraint structural configuration design in advanced engineering applications

Abstract

Multi-constraint structural configuration design in aerospace, mechanical, and civil engineering leads to non-convex and high-dimensional optimization problems, in which stress constraint and bounds of displacements compete in terms of buckling stability and minimization of weight with rigorous nonlinear coupling, and standard tools based on the gradient are frequently trapped at local minima and fail near constraints intersections of singularity. The paper describes a geometric optimization scheme based on the mapping of the nonconvex structural design domain into the transformed coordinate system through posynomial geometric programming, which transforms the governing nonlinear structural equations into a dual convex form, which allows one to formulate the conditions of Karush-Kuhn-Tucker necessity and to prove global optimality. The proof of the existence and uniqueness of the transformed optimum is done under the standard posynomial regularity, and a special algorithm combining geometric mapping for coordinates, interior-point method for the solution of convex dual problems, Pareto handling of vector-optimization and inverse mapping back to the dimension of physical variables is designed and described with formal bounds for its complexity and convergence rate. The numerical results concerning three benchmark problems refer to a multi-member truss frame under stress and stability constraints, a thin-walled aerospace component, and a multi-material system to coupled thermal-mechanical loading and show that proposed methodology allows reducing the optimization time by 61-74% and the number of iterations by 55-68% as compared with classical interior-point method, genetic algorithm, and method of moving asymptotes, having shown the maximum amount of constraint violation of the value equal to 0.021 against 0.084-0.238 for the classic procedures. Furthermore, the proposed approach provides the dominating positions along the Pareto front in the trade-off weighing of compliance over the whole variety of weight settings and has linear scalability from 200 to 500000 degrees of freedom, in contrast to the super-linear scalability of the baseline methods. These results demonstrate that geometric transformation provides a mathematically rigorous and computationally efficient route to globally convergent multi-constraint structural configuration optimization.