A numerical form-finding method for the minimal surface of membrane structures

This paper proposes a convenient numerical form-finding method for designing the minimal surface, or the equally tensioned surface of membrane structures with specified arbitrary boundaries. Area minimization problems are formulated as a distributed-parameter shape optimization problem. The internal volume or the perimeter is added as a constraint according to the structure type such as a pneumatic or a suspension membrane. It is assumed that the membrane is varied in the out-of-plane and/or the in-plane direction to the surface. The shape sensitivity function for each problem is derived using the material derivative method. The minimal surface is determined without shape parameterization by the free-form optimization method, a gradient method in the Hilbert space, where the shape is varied by the traction force in proportion to the sensitivity function under the Robin boundary condition. The calculated results show the effectiveness and practical utility of the proposed method for optimal form-finding of membrane structures.