Minimum weight shape design for the natural vibration problem of plate and shell structures

In this paper we present a solution to a shape optimization problem involving plate and shell structures subject to natural vibration. The volume is chosen as the response to be minimized under a specified eigenvalue constraint with mode tracking. The designed boundaries are assumed to be movable in the in-plane direction so as to maintain the initial curvatures. The surfaces are discretized by plane elements based on the Mindlin-Reissner plate theory. A non-parametric or a distributed shape optimization problem is formulated and the shape gradient function is theoretically derived using the material derivative method and the Lagrange multiplier method. The traction method, a shape optimization method developed by the authors, is applied to obtain the optimal shape in this problem. The validity of the numerical solution for minimizing the weight of the plate and shell structures is verified through application to fundamental design problems and an actual automotive suspension component.