Master's Dissertation
DOI
10.11606/D.3.2008.tde-07112008-092620
Document
Author
Full name
Diogo Carlos Bernardes de Souza
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2008
Supervisor
Committee
Pauletti, Ruy Marcelo de Oliveira (President)
Almeida Neto, Edgard Sant'Anna de
Arcaro, Vinicius Fernando
Title in Portuguese
Sobre a busca de superfícies minimais e seu emprego nas estruturas de membrana.
Keywords in Portuguese
Estruturas de membranas
Otimização não linear
Superfícies mínimas
Abstract in Portuguese
Title in English
On finding minimal surfaces and their application to membrane structures.
Keywords in English
Form finding
Membranes
Minimal surfaces
Numerical methods
Theory of membranes
Abstract in English
This dissertation presents a historical review on the theoretical developments on minimal surfaces, highlighting the important analogy between the problems of finding minimal area surfaces and finding membrane surfaces with homogeneous and isotropic stress fields. Some geometric concepts of the parametric surfaces are placed, on the basis of differential geometry, in order to do the differential equilibrium of membranes and to achieve its equilibrium equations. Moreover, a purely geometric methodology for the determination of minimal surfaces is presented, based on the minimization of the area functional, which is computed by the simple addition of a finite number of triangular facet areas in which the surface is divided. It discusses the mathematical formulation of the problem as well as some results obtained with the algorithms implemented in MATLAB and others obtained with the aid of MATLAB optimization routines. Finally, some examples and a convergence test are produced, comparing their analytical and numerical results. The final geometry of one of examples is verified by means of the soap film analogy, with a nonlinear equilibrium analysis through Ansys. The solutions have been sufficiently satisfactory, resulting forms very close to the analytical ones and with small areas relative errors. Convergence test also confirm that the method lead to numerical solutions as close to the analytical one as required, as long as the triangular facets mesh is refined. Therefore, the presented procedures can be used in structural membranes form finding.