Semi-algebraic optimization methods for structural configuration analysis and design improvement

Abstract

The structural design problem is often nonlinear, and it includes constraints, geometric relations, and many performance measures. In that way, traditional optimization techniques cannot be used effectively for improving structural design. The necessity of mathematically well-defined optimization methods has grown up because of the necessity of accurate analysis and design and verification of stability of the structure under various engineering situations. In this research, a new method of semi-algebraic optimization has been introduced for structural configuration analysis and optimization. The mathematical representation of engineering constraints in polynomial forms enables semi-algebraic optimization of the structures using lifted variables, semidefinite programming, and moment-sum-of-squares relaxations. Experimental optimization based on a five-bar truss structure proves the effectiveness of the suggested approach in creating physically valid and optimized structural designs. Scalability tests show that the dimension of the moment matrix grows from 19 × 19 for 10 bars and 8 degrees of freedom to 1,801 × 1,801 for 1,000 bars and 800 degrees of freedom. It is shown that the structural design formulated as compliance maximization problem does not have any gap between the relaxation bounds and the optimal value, while moment-SOS hierarchy allows achieving design improvement and its certification. The main point discussed in this section is that exploiting sparsity is necessary in order to be able to handle large structural systems.