Robustness Analysis and Important Element Evaluation Method of Truss Structures

Robustness and vulnerability are important evaluation criteria for structural safety to improve progressive collapse resistance. Modeling rapid evaluation on important elements in the conceptual analysis stage is an essential and efficient way to ensure the robustness of a design. This paper proposes an evaluation method for element importance by using structural strain energy, which can reflect the influence of geometric topology and external load distribution on the structural safety status simultaneously. Three simple planar trusses are chosen to clarify the proposed method. Moreover, the influences of geometric topology, boundary condition, stiffness distribution and load distribution on the element importance are investigated. This evaluation method also extends to the two spatial truss structures, regular quadrangular pyramid grid structures and chessboard-shaped pyramid grid structures. The distribution characteristics of important elements are obtained. This work defines the element importance distribution law of typical structures. It provides a method for rapid robustness analysis of general truss structures, which can be effectively integrated into the existing analysis platform.

The minimum deformation retract on the wormhole spacetime

In this paper, we investigate the topology of a wormhole from the viewpoint of the theory of retracting and var-folding. We deduce the equatorial geodesics on the line element of the wormhole and discuss the minimum deformation retract related to this space. Using the Lagrangian equations we find that there is a type of minimum retraction of a wormhole with associated topology. We also extend the result to the [Formula: see text]-dimensional wormhole and show that the end limit of var-folding is 0-dimensional wormhole and obtain the relation between limit var-folding and limit retraction. We find a new application in geometric topology and astrophysics.

Automated Layout Generation Methods for 2D Spatial Packing

Development of a computationally-tractable design method for combined multi-physics optimization of packing and routing problems, at a relevant scale, within compact packaging volumes, will offer benefits across several engineering domains. But for performing multi-physics packing and routing optimization, the generation of spatially feasible initial layouts is essential. Three new and computationally efficient methods are demonstrated in this article to produce automatically interference-free 2D geometric layouts. First, a novel 2D force-directed layout method (FDLM) is proposed that implicitly ensures noninterference between components and/or the interconnect network by utilizing spring force theory without using explicit geometric constraints. Second, the A* algorithm, a well-established 2D shortest path algorithm (SPA), has been modified significantly to perform efficient routing of complex interconnect systems. Third, a new geometric topology (GT) enumeration algorithm is presented that produces all unique interconnect routing configurations for given multi-component system architecture. These layout generation methods are then compared with respect to average computational efficiencies and average success rates in attaining feasible layouts for a restricted class of topologies, including evaluation of how the methods scale to problems with an increased number of components. Limitations and future work items for each method are discussed. These methods are presented as an important step toward solution strategies that are compatible with the currently unmet challenges of real-world 2D and 3D combined packing and routing problems, including efficient navigation of the space of discrete options for interconnect geometric topology, as well as scaling to more complex problems.

Virtually spinning hyperbolic manifolds

We give a new proof of a result of Sullivan [Hyperbolic geometry and homeomorphisms, in Geometric topology (ed. J. C. Cantrell), pp. 543–555 (Academic Press, New York, 1979)] establishing that all finite volume hyperbolic n-manifolds have a finite cover admitting a spin structure. In addition, in all dimensions greater than or equal to 5, we give the first examples of finite-volume hyperbolic n-manifolds that do not admit a spin structure.

Cross-sections of unknotted ribbon disks and algebraic curves

We resolve parts (A) and (B) of Problem 1.100 from Kirby’s list [Problems in low-dimensional topology, in Geometric topology, AMS/IP Studies in Advanced Mathematics, vol. 2 (American Mathematical Society, Providence, RI, 1997), 35–473] by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontrivial cross-sections.