Computational form-finding of tension membrane structures-Non-finite element approaches: Part 1. Use of cubic splines in finding minimal surface membranes

摘要

This paper, presented in three parts, discusses a computational methodology for form-finding of tension membrane structures (TMS), or fabric structures, used as roofing forms. The term 'form-finding' describes a process of finding the shape of a TMS under its initial tension. Such a shape is neither known a priori, nor can it be described by a simple mathematical function. The work is motivated by the need to provide an efficient numerical tool, which will allow a better integration of the design/analysis/manufacture of TMS. A particular category of structural forms is considered, known as minimal surface membranes (such as can be reproduced by soap films). The numerical method adopted throughout is dynamic relaxation (DR) with kinetic damping. part 1 describes a new form-finding approach, based on the Laplace-Young equation and cubic spline fitting to give a full, piecewise, analytical description of a minimal surface. The advantages arising from the approach, particularly with regard to manufacture of cutting patterns for a membrane, are highlighted. Part 2 describes an alternative and novel form-finding approach, based on a constant tension field and faceted (triangular mesh) representation of the minimal surface, It presents techniques for controlling the solution, as well as on the subsequent stages in design. Part 3 gives a comparison of the performance of the initial method (Part 1) and the faceted approximations (Part 2). Functional relations, which encapsulate the numerical efficiency of each method, are presented.