Optimum shape design of thin-walled cross sections using a parameter-free optimization method

Abstract

In this paper, a parameter-free method is presented for optimizing the cross-sectional shape of thin-walled structures, which are often demanded at the early stage in structural designs. The thin-walled cross section is minimized subject to constraints of sectional properties including torsion constant, moment of inertia of area, centroid and shear center of the cross section. The problem is formulated as a distributed shape optimization problem, and the shape gradient function is derived using the Lagrange multipliers and the material derivative method. The H1 gradient method, which was proposed as a gradient method in a Hilbert space, is applied to determine the smooth optimal shape. The constraint conditions are satisfied using a linearised constraint equation. The validity of this parameter-free method is verified through several design examples for obtaining the optimal shape of a thin-walled cross section under the constraints of sectional properties.