Stretched grid method
The first problem is a reliability of engineering analysis that strongly depends upon the quality of initial data generated at the pre-processing stage.
It is known that automatic element mesh generation techniques at this stage have become commonly used tools for the analysis of complex real-world models.
For instance smoothing (also referred to as mesh refinement) is one such method, which repositions nodal locations, so as to minimize element distortion.
The Stretched Grid Method (SGM) allows the obtaining of pseudo-regular meshes very easily and quickly in a one-step solution(see [3]).
Let one assume that there is an arbitrary triangle grid embedded into plane polygonal single-coherent contour and produced by an automeshing procedure (see fig.
Therefore, the SGM has all the positive values peculiar to Laplacian and other kinds of smoothing approaches but much easier and reliable because of integer-valued final matrices representation.
The best-known minimum surface sample is a soap film bounded by wire frame.
Usually to create a minimum surface, a fictitious constitutive law, which maintains a constant prestress, independent of any changes in strain, is used.
[4] The alternative approximated approach to the minimum surface problem solution is based on SGM.
The idea is to approximate a surface part embedded into 3D non-plane contour by an arbitrary triangle grid.
The numerical area of catenoidal surface determined by SGM is equal to 2,9967189 (exact value is 2.992).
Since the membrane in a tension structure possesses no flexural stiffness, its form or configuration depends upon initial prestressing and the loads to which it is subjected.
Thus, the load-bearing behaviour and the shape of the membrane cannot be separated and cannot be generally described by simple geometric models only.
The membrane shape, the loads on the structure and the internal stresses interact in a non-linear manner to satisfy the equilibrium equations.
The preliminary design of tension structures involves the determination of an initial configuration referred to as form finding.
In addition to satisfying the equilibrium conditions, the initial configuration must accommodate both architectural (aesthetics) and structural (strength and stability) requirements.
Further, the requirements of space and clearance should be met, the membrane principal stresses must be tensile to avoid wrinkling, and the radii of the double-curved surface should be small enough to resist out-of-plane loads and to insure structural stability (work [5]).
Several variations on form finding approaches based on FEM have been developed to assist engineers in the design of tension fabric structures.
The physical meaning of SGM consists in convergence of the energy of an arbitrary grid structure embedded into rigid (or elastic) 3D contour to minimum that is equivalent to minimum sum distances between arbitrary pairs of grid nodes.
It allows the minimum surface energy problem solution substituting for finding grid structure sum energy minimum finding that provides much more plain final algebraic equation system than the usual FEM formulation.
We may obtain the following expression for such SGM formulation where Once a satisfactory shape has been found, a cutting pattern may be generated.
It is essential for a cutting pattern generation method to minimize possible approximation and to produce reliable plane cloth data.
The objective is to develop the shapes described by these data, as close as possible to the ideal doubly curved strips.
The corresponding cutting pattern at the second step can be found by simply taking each cloth strip and unfolding it on a planar area.
In the case of the ideal doubly curved membrane surface the subsurface cannot be simply unfolded and they must be flattened.
In the case of single-curved surface that can be unfolded precisely equi-areal mapping allows one to obtain a cutting pattern for fabric structure without any distortions.
Remembering that the first stage of form finding is based on triangular mesh of a surface and using the method of weighted residuals for the description of isometric and equi-areal mapping of the minimum surface onto a plane area we may write the following function which is defined by the sum of integrals along segments of curved triangles where Considering further weight ratios
into approximate finite sum that is a combination of linear distances between nodes of the surface grid and write the basic condition of equi-areal surface mapping as a minimum of following non-linear function where The initial and final lengths of segment number
As a rule, the plane mapping allows to obtain a pattern with linear dimensions 1–2% less than corresponding spatial lines of a final surface.
The typical sample of cut out — also called a cutout, a gore (segment), or a patch — is presented in Figs.
