A Numerical Form Finding Method for Minimal Surface of Membrane Structure

1. Abstract 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. The area minimization problems are formulated as a distributed-parameter shape optimization problem. The internal volume or the perimeter is added as the constraints according to the structure type such as a pneumatic or a suspension membrane. It is assumed that the membrane is varied in the normal and/or the tangential direction to the surface. The shape sensitivity function for the problems is derived using the material derivative method. The minimal surface is 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 under the Robin condition. The calculated results will show the effectiveness and practical utility of the proposed method for the optimal form-finding of membrane structures. 2.