Finding Minimal and Non-Minimal Surfaces through the Natural Force Density Method

Abstract

This paper discusses the Natural Force Density Method, an extension of the well known Force Density Method for the shape finding of continuous membrane structures, which preserves the linearity of the original method, overcoming the need for regular meshes. The method is capable of providing viable membrane configurations, comprising the membrane shape and its associates stress field in a single iteration. Besides, if the NFDM is applied iteratively, it is capable of converging to a configuration under a uniform and isotropic plane stress field. This means that a minimal surface for a membrane can be achieved through a succession of viable configurations, in such a way that the process can be stopped at any iteration, and the result assumed as good. The NFDM can also be employed to the shape finding of non-minimal surfaces. In such cases, however, there is no guarantee that a prescribed, non-isotropic stress field can be achieved through iterations. The paper presents several examples of application of the NFDM to the shape finding of minimal and nonminimal membrane surfaces.