Relating Graph Thickness to Planar Layers and Bend Complexity

The thickness of a graph $G=(V,E)$ with $n$ vertices is the minimum number of planar subgraphs of $G$ whose union is $G$. A polyline drawing of $G$ in $\mathbb{R}^2$ is a drawing $\Gamma$ of $G$, where each vertex is mapped to a point and each edge is mapped to a polygonal chain. Bend and layer complexities are two important aesthetics of such a drawing. The bend complexity of $\Gamma$ is the maximum number of bends per edge in $\Gamma$, and the layer complexity of $\Gamma$ is the minimum integer $r$ such that the set of polygonal chains in $\Gamma$ can be partitioned into $r$ disjoint sets, where each set corresponds to a planar polyline drawing. Let $G$ be a graph of thickness $t$. By F\'{a}ry's theorem, if $t=1$, then $G$ can be drawn on a single layer with bend complexity $0$. A few extensions to higher thickness are known, e.g., if $t=2$ (resp., $t>2$), then $G$ can be drawn on $t$ layers with bend complexity 2 (resp., $3n+O(1)$). However, allowing a higher number of layers may reduce the bend complexity, e.g., complete graphs require $\Theta(n)$ layers to be drawn using 0 bends per edge. In this paper we present an elegant extension of F\'{a}ry's theorem to draw graphs of thickness $t>2$. We first prove that thickness-$t$ graphs can be drawn on $t$ layers with $2.25n+O(1)$ bends per edge. We then develop another technique to draw thickness-$t$ graphs on $t$ layers with bend complexity, i.e., $O(\sqrt{2}^{t} \cdot n^{1-(1/\beta)})$, where $\beta = 2^{\lceil (t-2)/2 \rceil }$. Previously, the bend complexity was not known to be sublinear for $t>2$. Finally, we show that graphs with linear arboricity $k$ can be drawn on $k$ layers with bend complexity $\frac{3(k-1)n}{(4k-2)}$.Comment: A preliminary version appeared at the 43rd International Colloquium on Automata, Languages and Programming (ICALP 2016

Edge Partitions of Optimal 22-plane and 33-plane Graphs

A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for $k=1$, we focus on optimal $2$-plane and $3$-plane graphs, which are $2$-plane and $3$-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple optimal $2$-plane graph into a $1$-plane graph and a forest, while (ii) an edge partition formed by a $1$-plane graph and two plane forests always exists and can be computed in linear time. (iii) We describe efficient algorithms to partition the edges of a simple optimal $2$-plane graph into a $1$-plane graph and a plane graph with maximum vertex degree $12$, or with maximum vertex degree $8$ if the optimal $2$-plane graph is such that its crossing-free edges form a graph with no separating triangles. (iv) We exhibit an infinite family of simple optimal $2$-plane graphs such that in any edge partition composed of a $1$-plane graph and a plane graph, the plane graph has maximum vertex degree at least $6$ and the $1$-plane graph has maximum vertex degree at least $12$. (v) We show that every optimal $3$-plane graph whose crossing-free edges form a biconnected graph can be decomposed, in linear time, into a $2$-plane graph and two plane forests

The Impact of Inter-Bed Cohesion on Fold-Related Fracture Development, Stillwell Anticline, West Texas (USA)

The interpretation of fracture networks in contractional folds is challenging due to the range of factors that control fracture formation. We use outcrop-based analysis of fractures in plan-view pavements and in a 9-bed cross-sectional exposure to evaluate the fracture system within Cretaceous limestone layers of a Laramide fold in west Texas, the Stillwell anticline. Opening-mode extension fractures (veins) at high angles to bedding dominate the fracture population, although shear fractures and faults cut bedding at low angles within the forelimb. Analysis of extension fractures reveals NW-striking axial parallel and NE-striking axial-perpendicular fracture sets interpreted to have formed during contractional folding, a third N-striking fracture set formed during subsequent Basin and Range extension, and a fourth ESE-striking fracture formed due to unloading during exhumation. Fracture fill textures suggest that many fracture apertures increased during exhumation. The relative abundances of the four fracture sets and the intensity of each set vary from bed to bed in cross section. Because beds display no significant differences in mechanical strength and there is no correlation between bed thickness and fracture intensity, we attribute this bed-to-bed variability to differences in cohesion between beds. Bed decoupling, when combined with low extensional

Development of a Fabrication Technique for Soft Planar Inflatable Composites

Soft robotics is a rapidly growing field in robotics that combines aspects of biologically inspired characteristics to unorthodox methods capable of conforming and/or adapting to unknown tasks or environments that would otherwise be improbable or complex with conventional robotic technologies. The field of soft robotics has grown rapidly over the past decade with increasing popularity and relevance to real-world applications. However, the means of fabricating these soft, compliant and intricate robots still poses a fundamental challenge, due to the liberal use of soft materials that are difficult to manipulate in their original state such as elastomers and fabric. These material properties rely on informal design approaches and bespoke fabrication methods to build soft systems. As such, there are a limited variety of fabrication techniques used to develop soft robots which hinders the scalability of robots and the time to manufacture, thus limiting their development. This research focuses towards developing a novel fabrication method for constructing soft planar inflatable composites. The fundamental method is based on a sub-set of additive manufacturing known as composite layering. The approach is designed from a planar manner and takes layers of elastomeric materials, embedded strain-limiting and mask layers. These components are then built up through a layer-by-layer fabrication method with the use of a bespoke film applicator set-up. This enables the fabrication of millimetre-scale soft inflatable composites with complex integrated masks and/or strain-limiting layers. These inflatable composites can then be cut into a desired shape via laser cutting or ablation. A design approach was also developed to expand the functionality of these inflatable composites through modelling and simulation via finite element analysis. Proof of concept prototypes were designed and fabricated to enable pneumatic driven actuation in the form of bending soft actuators, adjustable stiffness sensor, and planar shape change. This technique highlights the feasibility of the fabrication method and the value of its use in creating multi-material composite soft actuators which are thin, compact, flexible, and stretchable and can be applicable towards real-world application

Flexible Object Manipulation

Flexible objects are a challenge to manipulate. Their motions are hard to predict, and the high number of degrees of freedom makes sensing, control, and planning difficult. Additionally, they have more complex friction and contact issues than rigid bodies, and they may stretch and compress. In this thesis, I explore two major types of flexible materials: cloth and string. For rigid bodies, one of the most basic problems in manipulation is the development of immobilizing grasps. The same problem exists for flexible objects. I have shown that a simple polygonal piece of cloth can be fully immobilized by grasping all convex vertices and no more than one third of the concave vertices. I also explored simple manipulation methods that make use of gravity to reduce the number of fingers necessary for grasping. I have built a system for folding a T-shirt using a 4 DOF arm and a fixed-length iron bar which simulates two fingers. The main goal with string manipulation has been to tie knots without the use of any sensing. I have developed single-piece fixtures capable of tying knots in fishing line, solder, and wire, along with a more complex track-based system for autonomously tying a knot in steel wire. I have also developed a series of different fixtures that use compressed air to tie knots in string. Additionally, I have designed four-piece fixtures, which demonstrate a way to fully enclose a knot during the insertion process, while guaranteeing that extraction will always succeed

Ultrathin silicon wafer bonding physics and applications

Ultrathin silicon wafer bonding is an emerging process that simplifies device fabrication, reduces manufacturing costs, increases yield, and allows the realization of novel devices. Ultrathin silicon wafers are between 3 and 200 microns thick with all the same properties of the thicker silicon wafers (greater than 300 microns) normally used by the semiconductor electronics industry. Wafer bonding is one technique by which multiple layers are formed. In this thesis, the history and practice of wafer bonding is described and applied to the manufacture of microelectomechanical systems (MEMS) devices with layer thickness on the scale of microns. Handling and processing problems specific to ultrathin silicon wafers and their bonding are addressed and solved. A model that predicts the conformal nature of these flexible silicon wafers and its impact on bonding is developed in terms of a relatively new description of surface quality, the Power Spectral Density (PSD). A process for reducing surface roughness of silicon is elucidated and a model of this process is described. A method of detecting particle contamination in chemical baths and other processes using wafer bonding is detailed. A final section highlights some recent work that has used ultrathin silicon wafer bonding to fabricate MEMS devices that have reduced existing design complexity and made possible novel, and otherwise difficult to produce, sensors. A new fabrication process that can reduce the required time for proof-of-principle devices using ultrathin silicon wafers is also described

Numerical simulation of fracture of a nano-paper coated e-glass/polyester composite with thermal damage

Aerospace research for next-generation travel increasingly focuses on the use of advanced composites to reduce weight and cost while retaining strength. One subset of materials with great potential is based on the combination of resin matrix and glass-fiber reinforcement. This research explores the application of a candidate nanopaper coating with a given composite. Prior research applied a set of given heat fluxes to the top surface of the composite for a set of given periods of time, and subsequently performed a 3-point flexural test to determine the elastic modulus for both the coated and uncoated composite for all of the combinations of heat flux and time. A finite element (FE) model is developed using the ANSYS general purpose finite element analysis (FEA) software that models the degradation in strength/stiffness properties based on heating condition and with the goal of predicting cracking using the element death feature in ANSYS. This thesis describes the prior research suggesting both the need for and novelty of this model, and the procedures used to form the model. The loading conditions of the 3-point flexural test are replicated, and four measures of accuracy are developed based on the force versus displacement curve of the test and the FE model. It is envisioned that continuum-level models developed as a part of these research be applied for design of next-generation space components These measurements are used to verify the FE model, and this model is then employed to extrapolate beyond the context of experimental conditions

Design of research oriented cylinder head for a heavy duty engine

The swirl flow is considered beneficial to enhance air-fuel mixing in CI engines during compression stroke as the piston reaches TDC, and helps in faster burn during combustion phase. Contrary to that, with the advancement of highly pressurized fuel injection systems and optimized types of IC engines like dual fuel engine demands for low swirl or preferably no swirl intake configuration is prevailing. The swirl structure induces by intake port differs from other flow structures like tumble; swirl not only survives during compression stroke but also throughout the expansion stroke. Therefore, swirl influences spray evolution and evaporation process during combustion and affect heat release due to the crumbling of the large-scale structure into small scale by adding more turbulence. This thesis work is aimed at designing, performing steady-state CFD analysis, and exercising additive manufacturing technique for a new single-cylinder research-oriented cylinder head with no induced swirl flow. The study also incorporates inclusive evaluation of flow structures produced by existing model of cylinder head through computational fluid analysis by employing Star-CCM+ software and experimental validation; then conjunction with that inquisition a new directed port model is devised. In addition, new ports position is designed, analyzed, and final model is selected based on admissible results. The new exhaust ports, cooling channels, and the main body of the cylinder head with appropriate thickness values are also designed. Additive manufacturing is a customized fabrication process to produce cost-effective products. AM has completely revolutionized current manufacturing techniques with a diverse selection of methods for different materials. Selective laser sintering is one of the powders based AM techniques with a range of available materials as polymers and metals used to contrive good quality densely structured light parts with flexible, interlocking and functional features. Therefore, SLS technique is adopted for new cylinder head manufacturing for later experimentally check of swirl flow