パラメトリック曲面として記述された極小曲面の形態創生手法

Abstract

The membrane architectural structures are constituted by flexible members which do not resist bending such as film materials and cable materials, and their mechanical stability greatly depends on the shape of the initial surface. In order to design a membrane structure having mechanical stability which does not cause wrinkles or slackness, it is common to adopt an isotonic surface as the initial design surface of the membrane structure. Some previous studies proposed methods of obtaining a surface with minimal surface area under appropriate subsidiary conditions by utilizing the fact that the isotonic surface is the same as the minimal surface. However, in any of these studies, since the surface is approximated by triangular elements and the sum of the triangular elements area is minimized, the error from the true minimal surface greatly depends on the division method of triangular elements. If the triangulation is made finer, a minimal surface with higher precision can be obtained, but the computational cost increases and the obtained minimal surface becomes a collection of enormous nodal information. The amount of data becomes enormous and it is difficult to smoothly convert or deliver the obtained solution as 3D graphic data in the situation of using CAD/CAE. In contrast, if we use a parametric surface, it is possible to explicitly calculate the surface area in parametric expression. Also, the minimal surface is known as a surface in which the mean curvature is equal to 0 at any point on the surface. By measuring the mean curvature, it is possible to quantitatively evaluate the error from the true minimal surface. In this research, to create the minimal surface which can explicitly express numerical functions with a small data volume, we solve the three optimization problems as follows: 1. Minimization of the sum of areas of triangular elements of triangulated surfaces (Previous method). 2. Minimizing total surface area of the parametric surface (Proposed method). 3. Minimizing the square of mean curvature of the parametric surface (Proposed method). The results obtained in this study are summarized as follows: · Method 1 requires high computational cost in order to get a solution with sufficient accuracy. · In the case of methods 2 and 3, if the weights of the parametric surface are set as design variables in addition to the vertical control point coordinates, the objective function becomes highly nonlinear. As a result, the computational cost becomes high and the obtained solution has a greater chance of being a non excellent local optimal solution. So the design variables should be the vertical control point coordinates only. · If the vertical control point coordinates are defined as the design variables, methods 2 and 3 can drastically decrease the computational cost as compared with method 1. · Methods 2 and 3 are suitable for CAD/CAE use because of their low quantity of the shape information. It is confirmed that methods 2 and 3 are very effective methods for finding minimal surface.