A comparative analysis of force-density based form-finding software to minimal surface equation#

Abstract

The shape of membrane and cable-net structures is usually modeled by geometry of minimal surfaces and its approximations. After experimenting with physical models during 1960s, computational methods were developed to find the initial geometry of tensile structures. Among the early numerical methods applied on formfinding of tensile structures was the finite difference method. An algorithm based on central finite differences combined with a nonlinear iterative process for finding the minimal surface over given stiff boundary conditions is developed and implemented in Mathematica®. The explicit 2-variable formulation as a second order quasi-linear partial differential equation with boundary condition, arising from the Euler-Lagrange areaminimizing condition, has been used for obtaining the soap film geometry. The forcedensity method, developed in 1970s by Linkwitz and Schek for the roof of the Olympia Stadium in Munich, found its implementation in commercial software EASY®, made by Technet GmbH, Germany. The commercial software used generates a surface which corresponds to the solution of the Laplace's equation. The form finding results obtained by these two methods are compared on some typical examples: the asymmetric hyperboloid membrane, the Concus’ sine arc example and a saddle-like structure example.