Mean curvature flow for generating discrete surfaces with piecewise constant mean curvatures

Piecewise constant mean curvature (P-CMC) surfaces are generated using the mean curvature flow (MCF). As an extension of the known fact that a CMC surface is the stationary point of an energy functional, a P-CMC surface can be obtained as the stationary point of an energy functional of multiple patch surfaces and auxiliary surfaces between them. A new formulation is presented for the MCF as the negative gradient flow of the energy functional for multiple patch continuous surfaces, which are further discretized so as to determine the change in the vertex positions of triangular meshes on the surface as well as along the internal boundaries between patches. Numerical examples show that multiple patch surfaces approximately reach the specified mean curvatures through the proposed method, which can diversify the options for the shape design using CMC surfaces

Discrete Gaussian Curvature Flow for Piecewise Constant Gaussian Curvature Surface

A method is presented for generating a discrete piecewise constant Gaussian curvature (CGC) surface. An energy functional is first formulated so that its stationary point is the linear Weingarten (LW) surface, which has a property such that the weighted sum of mean and Gaussian curvatures is constant. The CGC surface is obtained using the gradient derived from the first variation of a special type of the energy functional of the LW surface and updating the surface shape based on the Gaussian curvature flow. A filtering method is incorporated to prevent oscillation and divergence due to unstable property of the discretized Gaussian curvature flow. Two techniques are proposed to generate a discrete piecewise CGC surface with preassigned internal boundaries. The step length of Gaussian curvature flow is adjusted by introducing a line search algorithm to minimize the energy functional. The effectiveness of the proposed method is demonstrated through numerical examples of generating various shapes of CGC surfaces

五重塔の構造のモデル化に関する研究 : スネークダンスをする構造モデルについて

The five-story pagodas in Japan date back 1400 years or more. Despite the occurrence of many earthquakes during this period, there is no record of these pagodas being destroyed by earthquakes. Therefore, it is believed that these pagodas are earthquake resistant. However, the reason underlying this resistance has not been clarified yet. Of the various theories, that have been put forth, the most plausible explanation is that one offered by the snake-dance theory. According to this theory, the rocking movements of these pagodas during earthquakes, which resemble a snake dance, protects them from destruction. The pagodas are subjected to few horizontal vibrations during earthquakes. However, a structural model that can recreate these rocking vibrations has yet been created. While we attempted to create such a structural model on a laboratory scale, the aim of our study is that one of describing the structural process during the earthquake shaking, and explaining the results we obtain by writing reports

Differential geometric formulation of hanging membranes: Shell membrane theory and variational principle

In this paper, we present differential geometric formulation of hanging membrane forms based on the equilibrium equations in the shell membrane theory and the geometric variational problems. We also present the equations of hanging membranes in isothermic coordinates which will be an effective representation for numerical analysis in our forthcoming paper