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.

A combined method of graph embedding (GE) and reinforcement learning (RL) is developed for discrete cross-section optimization of planar steel frames, in which the section size of each member is selected from a prescribed list of standard sections. The RL agent aims to minimize the total structural volume under various practical constraints. GE is a method for extracting features from data with irregular connectivity. While most of the existing GE methods aim at extracting node features, an improved GE formulation is developed for extracting features of edges associated with members in this study. Owing to the proposed GE operations, the agent is capable of grasping the structural property of columns and beams considering their connectivity in a frame with an arbitrary size as feature vectors of the same size. Using the feature vectors, the agent is trained to estimate the accurate return associated with each action and to take proper actions on which members to reduce or increase their size using an RL algorithm. The applicability of the proposed method is versatile because various frames different in the numbers of nodes and members can be used for both training and application phases. In the numerical examples, the trained agents outperform a particle swarm optimization method as a benchmark in terms of both computational cost and design quality for cross-sectional design changes; the agents successfully assign reasonable cross-sections considering the geometry, connectivity, and support and load conditions of the frames.

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.