離散的な曲面の幾何学的不変量を用いた膜構造の形状設計

Abstract

The target surfaces of the frame supported membrane structures are generated by using the Gaussian and mean curvatures. The surface is discretized into a triangular mesh, and the Gaussian and mean curvatures are defined at the interior vertices of the surface based on the formulations of discrete differential geometry. The minimal surface with zero mean curvature, which is equivalent to the uniform stressed surface, is often used as the target surface because uniform stress distribution is desirable. However, while the membrane structure is generated by connecting planar membrane sheets; i.e. cutting patterns, the minimal surface cannot be developed to a plane without out-of-plane deformation. In this respect, the developable surface with zero Gaussian curvature may be desirable; however, it cannot realize uniform stress distribution. Therefore, in this study, the curved surface with a geometric property intermediate between those of developable and minimal surfaces is generated and used as the target surface of a membrane structure. The Gaussian curvature at an inner vertex is defined using the angle defect, which is the difference between 2□ and the sum of the angles between edges connecting to the vertex. The mean curvature is defined using a cotangent formula for the mean curvature vector at the vertices. The developable and minimal surfaces are generated by minimizing the sum of the squares of Gaussian and mean curvatures, respectively. The positions of the vertices on the outer boundary of the surface are fixed, and the optimization problem is solved by using the z-coordinates of the interior vertices as variables. In addition, the intermediate surface can be obtained by solving a multi-objective optimization problem that minimizes the sum of squares of both Gaussian and mean curvatures. The constraint approach is applied to obtain a Pareto solution; i.e., the sum of squares of Gaussian curvature is minimized under the upper bound constraint on the sum of squares of the mean curvature. After obtaining the target surface, it is flattened by minimizing the sum of squares of differences between the edge lengths of on the surface in three-dimensional space and those in its development diagram on a plane. The cutting pattern is obtained by shrinking the obtained development diagram according to the target stress. Then, the equilibrium shape of the membrane structure is obtained by installing the obtained cutting patterns to the frame. The equilibrium state is achieved by solving the optimization problem that minimizes the strain energy of the membrane elements. The effectiveness of the proposed method is demonstrated in the examples of curved surfaces with cylindrical boundary shapes of various heights. The membrane structure is constructed with four cutting patterns. The intermediate surfaces obtained by the proposed method have shown to have intermediate properties between the minimal and developable surfaces with respect to Gaussian curvature, mean curvature, and surface area in every height. The intermediate surfaces have realized the most preferable distributions of stresses in two directions while the developable and minimal surfaces have realized the uniform stresses in only one direction. The average values of the stresses in two directions are also closest to the target stresses when the intermediate surface is used. Therefore, when the intermediate surface is used as the target surface of the membrane structure, the preferable equilibrium state can be obtained easily without optimization of the cutting pattern.