TheoryFormfinding and Optimization
of Membranes and Minimal Surfaces
Formfinding with the Updated Reference Strategy (URS)
Copyright by
Kai-Uwe Bletzinger
Lehrstuhl für Statik
Technische Universität München
1 Introduction
Membrane structures are very attractive alternatives to span large distances. They are very light, elegant, and effective [1-3]. The material is optimally used since the structures are subjected only to membrane tension stresses. The art of form finding means to find the optimal deflected and finally visual shape due to a given stress distribution acting on the deformed structure. This fact should be clearly pointed out since it is different compared to conventional structural optimization where the undeformed shape is optimized with respect to displacements due to given load cases.
The problem is very closely related to the determination of minimal surfaces. Since centuries, they are one of the oldest toys of mathematicians. There exists an enormous amount of knowledge of how to describe and to generate them [4]. From the mechanical point of view minimal surfaces are determined by a isotropic stress field that can be experimentally simulated by the soap film analogy. For the engineering application “form finding of tensile membrane structures” the later approach is most promising since it is directly formulated with respect to stresses which restrict and dominate the final design because of strength resistance and the anisotropic material behavior in the directions of weft and warp [5]. Furthermore, an unlimited variation of shapes is generated by prescribing anisotropic stress distributions.
Inspired e.g. by the pioneer work of Frei Otto [6] many technical procedures and algorithms had been developed which many of them are based on dynamic relaxation [7,8]. Others, like the force density method are based on special discretization and linearization techniques. Originally, it was developed for the form finding of cable structures [9-11]. Recently the method was extended to triangular membrane elements [12]. The existence of all the different methods is explained by the mathematical problem which arises by solving an inverse mechanical problem. It is defined by the prescribed stress distribution as the driving degree of freedom in the design process. This is inverse to standard mechanics where stresses are the structural response to the deformation of material. As a consequence the related numerical solution methods are faced to mathematical singularities which are overcome by several techniques, e.g. by a modified Newton-Raphson iteration [13-15]. The updated reference strategy which will be presented in the following is a further alternative [16-18]. It is consistently derived from continuum mechanics of elastic bodies with respect to large deflections and small strains. The numerical solution follows standard finite element discretization procedures which means that the method can be applied to any triangular or quadrilateral finite element formulation. The above mentioned singularities are regularized by a homotopy mapping [19] which is based on approximations for Cauchy and Piola-Kirchhoff stresses. It will be shown that this approach leads to the force density method if it is applied to one dimensional elastic bodies, e.g. cables. Vice versa the updated reference strategy is the consistent generalization of the force density method applicable for any structure. It can also be used as an approximation technique in structural shape optimization [18].
2 Minimal surfaces, variational approach
3 Differential geometry of surfaces
4 The governing equation of form finding
5 Discretization of the governing equations
6 Linearization
7 Linearization
8 Surface pressure load
9 Extension to structural weight minimization
10 Examples
11 References
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