A New Description of Nuclear Rotational Motion in terms of Intrinsic Pair Mode

A new method describing nuclear rotational motion microscopically is proposed. We extract the rotational Hamiltonian by introducing the intrinsic pair modes which commute with the rotational mode. Thereby the rotational mode is not treated as zero energy mode in contrast with the conventional RPA formalism so that we circumvent the difficulty related with infrared divergence. The wave function is constructed by angular momentum projection on each intrinsic state. Without numerical integration for projection we calculate the matrix elements analytically under a certain approximation. The numerical calculations are carried out to illustrate the applicability of our method and they show that our method works well.Comment: 14pages,1figur

A locally minimal, but not globally minimal bridge position of a knot

We give a locally minimal, but not globally minimal bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.Comment: 27 pages, 12 figures, v3: minor corrections throughout the pape

Non-minimal bridge positions of torus knots are stabilized

We show that any non-minimal bridge decomposition of a torus knot is stabilized and that $n$-bridge decompositions of a torus knot are unique for any integer $n$. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphere in two essential loops.Comment: 11 pages, 4 figure

FDMopt: Force density method for optimal geometry and topology of trusses

This paper presents a new efficient tool for simultaneous optimization of topology and geometry of truss structures. Force density method is applied to formulate optimization problem to minimize compliance under constraint on total structural volume, and objective and constraint functions are expressed as explicit functions of force density only. This method does not need constraints on nodal locations to avoid coalescent nodes, and enables to generate optimal solutions with a variety in topology and geometry. Furthermore, for the purpose of controlling optimal shapes, tensor product Bézier surface is introduced as a design surface. The optimization problem is solved using sensitivity coefficients and the optimizer is compiled as a component compatible with Grasshopper, an algorithmic modeling plug-in for Rhinoceros, which is a popular 3D modeling software. Efficiency and accuracy of the proposed method are demonstrated through two numerical examples of semi-cylindrical and semi-spherical models

Reinforcement Learning and Graph Embedding for Binary Truss Topology Optimization Under Stress and Displacement Constraints

This paper addresses a combined method of reinforcement learning and graph embedding for binary topology optimization of trusses to minimize total structural volume under stress and displacement constraints. Although conventional deep learning methods owe their success to a convolutional neural network that is capable of capturing higher level latent information from pixels, the convolution is difficult to apply to discrete structures due to their irregular connectivity. Instead, a method based on graph embedding is proposed here to extract the features of bar members. This way, all the members have a feature vector with the same size representing their neighbor information such as connectivity and force flows from the loaded nodes to the supports. The features are used to implement reinforcement learning where an action taker called agent is trained to sequentially eliminate unnecessary members from Level-1 ground structure, where all neighboring nodes are connected by members. The trained agent is capable of finding sub-optimal solutions at a low computational cost, and it is reusable to other trusses with different geometry, topology, and boundary conditions

Graph-based reinforcement learning for discrete cross-section optimization of planar steel frames

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