ON OPTIMUM DESIGN OF FRAME STRUCTURES

Didier Henrion,Jan Zeman,Martin Kružík,Marek Tyburec

doi:10.14311/app.2020.26.0117

Abstract

Optimization of frame structures is formulated as a non-convex optimization problem, which is currently solved to local optimality. In this contribution, we investigate four optimization approaches: (i) general non-linear optimization, (ii) optimality criteria method, (iii) non-linear semidefinite programming, and (iv) polynomial optimization. We show that polynomial optimization solves the frame structure optimization to global optimality by building the (moment-sums-of-squares) hierarchy of convex linear semidefinite programming problems, and it also provides guaranteed lower and upper bounds on optimal design. Finally, we solve three sample optimization problems and conclude that the local optimization approaches may indeed converge to local optima, without any solution quality measure, or even to infeasible points. These issues are readily overcome by using polynomial optimization, which exhibits a finite convergence, at the prize of higher computational demands.

Highlights

Designing economical, efficient, and sustainable structures represents a major challenge of the contemporary society
We develop four different methods in Section 2: (i) general non-linear optimization solved by the interior-point method of fmincon, (ii) the first-order optimality criteria (OC) method [1, 5], (iii) reformulation of the problem into a non-linear semidefinite program (NSDP) solved by PenLab [6], and (iv) a suitably modified polynomial optmization (PO) problem (iii) solved globally using polynomial optimization methods [7] and the Mosek [8] optimizer
We show that the latter PO approach generates guaranteed lower and upper bounds on the solution, providing a means of assessing the design quality

Summary

Introduction

Efficient, and sustainable structures represents a major challenge of the contemporary society. We develop four different methods in Section 2: (i) general non-linear optimization solved by the interior-point method of fmincon, (ii) the first-order optimality criteria (OC) method [1, 5], (iii) reformulation of the problem into a non-linear semidefinite program (NSDP) solved by PenLab [6], and (iv) a suitably modified polynomial optmization (PO) problem (iii) solved globally using polynomial optimization methods [7] and the Mosek [8] optimizer We show that the latter PO approach generates guaranteed lower and upper bounds on the solution, providing a means of assessing the design quality.

Solution techniques to frame optimization

Optimality criteria

Polynomial optimization

Solution process

Recognizing global optimality

Cantilever beam

I-shaped girder with self-weight

Conclusions