Upward Three-Dimensional Grid Drawings of Graphs

A \emph{three-dimensional grid drawing} of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. We prove that every $n$-vertex graph with bounded degeneracy has a three-dimensional grid drawing with $O(n^{3/2})$ volume. This is the broadest class of graphs admiting such drawings. A three-dimensional grid drawing of a directed graph is \emph{upward} if every arc points up in the z-direction. We prove that every directed acyclic graph has an upward three-dimensional grid drawing with $(n^3)$ volume, which is tight for the complete dag. The previous best upper bound was $O(n^4)$. Our main result is that every $c$-colourable directed acyclic graph ($c$ constant) has an upward three-dimensional grid drawing with $O(n^2)$ volume. This result matches the bound in the undirected case, and improves the best known bound from $O(n^3)$ for many classes of directed acyclic graphs, including planar, series parallel, and outerplanar

Visibility Representations of Boxes in 2.5 Dimensions

We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane $z=0$ and edges are unobstructed lines of sight parallel to the $x$- or $y$-axis. We prove that: $(i)$ Every complete bipartite graph admits a 2.5D-BR; $(ii)$ The complete graph $K_n$ admits a 2.5D-BR if and only if $n \leq 19$; $(iii)$ Every graph with pathwidth at most $7$ admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an $n$-vertex graph that admits a 2.5D-GBR has at most $4n - 6 \sqrt{n}$ edges and this bound is tight. Finally, we prove that deciding whether a given graph $G$ admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR $\Gamma$ is the set of bottom faces of the boxes in $\Gamma$.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Effect of thermal exposure, forming, and welding on high-temperature, dispersion-strengthened aluminum alloy: Al-8Fe-1V-2Si

The feasibility of applying conventional hot forming and welding methods to high temperature aluminum alloy, Al-8Fe-1V-2Si (FVS812), for structural applications and the effect of thermal exposure on mechanical properties were determined. FVS812 (AA8009) sheet exhibited good hot forming and resistance welding characteristics. It was brake formed to 90 deg bends (0.5T bend radius) at temperatures greater than or equal to 390 C (730 F), indicating the feasibility of fabricating basic shapes, such as angles and zees. Hot forming of simple contoured-flanged parts was demonstrated. Resistance spot welds with good static and fatigue strength at room and elevated temperatures were readily produced. Extended vacuum degassing during billet fabrication reduced porosity in fusion and resistance welds. However, electron beam welding was not possible because of extreme degassing during welding, and gas-tungsten-arc welds were not acceptable because of severely degraded mechanical properties. The FVS812 alloy exhibited excellent high temperature strength stability after thermal exposures up to 315 C (600 F) for 1000 h. Extended billet degassing appeared to generally improve tensile ductility, fatigue strength, and notch toughness. But the effects of billet degassing and thermal exposure on properties need to be further clarified. The manufacture of zee-stiffened, riveted, and resistance-spot-welded compression panels was demonstrated

Pixel and Voxel Representations of Graphs

We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for $k$-outerplanar graphs with $n$ vertices, $\Theta(kn)$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, $\Theta(n^2)$ voxels are always sufficient and sometimes necessary for any $n$-vertex graph. We improve this bound to $\Theta(n\cdot \tau)$ for graphs of treewidth $\tau$ and to $O((g+1)^2n\log^2n)$ for graphs of genus $g$. In particular, planar graphs admit representations with $O(n\log^2n)$ voxels

Design of a simulated cruise scene visual attachment. Volume 3 - Assembly and detail drawings

Cruise scene visual attachment system assembly and detail design drawings for manned spacecraft or aircraft flight simulator

Optimization of Formula SAE Electric Vehicle Frame with Finite Element Analysis

Optimization of Formula SAE Electric Vehicle Frame with Finite Element Analysi

A novel approach towards a lubricant-free deep drawing process via macro-structured tools

In today’s industry, the sustainable use of raw materials and the development of new green technology in mass production, with the aim of saving resources, energy and production costs, is a significant challenge. Deep drawing as a widely used industrial sheet metal forming process for the production of automotive parts belongs to one of the most energy-efficient production techniques. However, one disadvantage of deep drawing regarding the realisation of green technology is the use of lubricants in this process. Therefore, a novel approach for modifying the conventional deep drawing process to achieve a lubricant-free deep drawing process is introduced within this thesis. In order to decrease the amount of frictional force for a given friction coefficient, the integral of the contact pressure over the contact area has to be reduced. To achieve that, the flange area of the tool is macro-structured, which has only line contacts. To avoid the wrinkling, the geometrical moment of inertia of the sheet should be increased by the alternating bending mechanism of the material in the flange area through immersing the blankholder slightly into the drawing die