A solution to shape optimization problem of cross section in structural design

In this paper, we present a numerical solution to optimize a cross section shape of a solid structure, which is often demanded at the early stage in structural designs. The cross section area is minimized subject to the constraints of sectional properties including the torsional constant, the second moment of area, the center of figure and the boundary length. 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 traction method, which was proposed as a gradient method in the Hubert space, is applied to determine the smooth optimal shape. The constraint conditions are efficiently satisfied using the modified feasible direction method. The validity of this method is verified through several design examples for obtaining the optimal shape of the cross section under the constraints of sectional properties.