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

This paper, presented in three parts, discusses a computational methodology for form- nding of tension membrane structures (TMS), or fabric structures, used as roo ng forms. The term ‘form- nding’ describes a process of nding 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 e cient 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 lms). The numerical method adopted throughout is dynamic relaxation (DR) with kinetic damping. Part 1 describes a new form- nding approach, based on the Laplace–Young equation and cubic spline tting 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- nding approach, based on a constant tension eld and faceted (triangular mesh) representation of the minimal surface. It presents techniques for controlling mesh distortion and discusses e ects of mesh control on the accuracy and computational e ciency of 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 e ciency of each method, are presented