A Non-parametric Form-Finding Method for Designing Membrane Structure

In this paper, we present a non-parametric form-finding method for designing the minimal surface, or the uniformly tensioned surface of membrane structures with arbitrary specified boundaries. The area minimization problems are formulated as distributed-parameter shape optimization problems, and solved numerically. The internal volume or the perimeter is added as the constraints according to the type of a structure such as pneumatic or suspended membranes. It is assumed that the membrane is varied in the out-of-plane and/or the in-plane direction to the surface. The shape gradient function for each problem is derived using the material derivative method. The minimal surface is numerically determined without the shape parameterization by the free-form optimization method, a gradient method in a Hilbert space, where the shape is varied by the traction force in proportion to the sensitivity function. The calculated results show the effectiveness of the proposed method for finding the optimal form of membrane structures.