Compaction Algorithms for Non-Convex Polygons and Their Applications

Given a two-dimensional, non-overlapping layout of convex and non-convex polygons, compaction refers to a simultaneous motion of the polygons that generates a more densely packed layout. In industrial two-dimensional packing applications, compaction can improve the material utilization of already tightly packed layouts. Efficient algorithms for compacting a layout of non-convex polygons are not previously known. This dissertation offers the first systematic study of compaction of non-convex polygons. We start by formalizing the compaction problem as that of planning a motion that minimizes some linear objective function of the positions. Based on this formalization, we study the complexity of compaction and show it to be PSPACE-hard. The major contribution of this dissertation is a position-based optimization model that allows us to calculate directly new polygon positions that constitute a locally optimum solution of the objective via linear programming. This model yields the first practically efficient algorithm for translational compaction-compaction in which the polygons can only translate. This compaction algorithm runs in almost real time and improves the material utilization of production quality human-generated layouts from the apparel industry. Several algorithms are derived directly from the position-based optimization model to solve related problems arising from manual or automatic layout generation. In particular, the model yields an algorithm for separating overlapping polygons using a minimal amount of motion. This separation algorithm together with a database of human-generated markers can automatically generate markers that approach human performance. Additionally, we provide several extensions to the position-based optimization model. These extensions enables the model to handle small rotations, to offer the flexible control of the distances between polygons and to find optimal solution to two-dimensional packing of non-convex polygons. This dissertation also includes a compaction algorithm based on existing physical simulation approaches. Although our experimental results showed that it is not practical for compacting tightly packed layouts, this algorithm is of interest because it shows that the simulation can speed up significantly if we use geometrical constraints to replace physical constraints. It also reveals the inherent limitations of physical simulation algorithms in compacting tightly packed layouts. Most of the algorithms presented in this dissertation have been implemented on a SUN SparcStationTM and have been included in a software package licensed to a CAD company.Engineering and Applied Science

C6 Wheels

This document details the C6 Wheels project being undertaken for senior design. The objective is to design and manufacture carbon fiber reinforced polymer wheels for the Cal Poly Formula Society of Automotive Engineers (FSAE) team. The wheel shells will be used on FSAE’s competition vehicles. FSAE requested the wheels to improve the handling characteristics of their vehicles by reducing the unsprung and rotational mass. They have attempted carbon fiber wheels previously but have not yet run any on their vehicles. FSAE specifically proposed the design of carbon fiber shells with an aluminum center as opposed to full carbon fiber wheels on the recommendation of the 2018 attempt. C6 Wheels is responsible for designing the wheel shells—including interfacing with the aluminum centers, designing and manufacturing the mold tooling, and molding of the carbon fiber wheel shells—including any post machining. The aluminum centers are being designed and manufactured by the FSAE team

Auxetic power amplification mechanisms for low frequency vibration energy harvesting

Energy harvesting from locally available small amplitude vibrations can struggle to generate sufficient power for wireless sensor nodes, which thereby constrains their use for structural health monitoring. This work discusses a selection of two-dimensional auxetic substrate designs used to increase a piezoelectric harvester’s power output by 2.18-14.5 times by concentrating the ambient strain energy into the piezoelectric material. The harvesters were modelled and their auxetic designs optimised in COMSOL before empirical testing under sinusoidal or dynamic strain oscillations. The investigated auxetic designs included re-entrant honeycombs, rotating squares, triangles and hexagrams, and -hole structures; the most effective of which was found to be the honeycomb design, with a gain of 5.66 and a raw output of 570 μW at 10 Hz, 100 με. This work also compared PZT (Lead Zirconate Titanate), LN (Lithium Niobate), and MFC (Macro-Fibre Composite) as materials for the active piezoelectric layer. The former was found to be detrimentally brittle but delivered the greatest output, while the LN was stronger but with a significantly lower output. The MFC was more flexible, with only a modest reduction in output compared to PZT, and was found to be the most viable of these materials for future research. A crucial issue during the design stages was appropriately modelling the mechanical losses associated with the bonding between substrate and piezoelectric material; this adhesion was modelled using thin elastic layers (TELs) to emulate each sample by comparing to its output. The value of the stiffness constant per unit area in these TELs was found to be consistent for each sample across a range of input excitations. These kinds of energy harvesters open up many new avenues for wireless self-powered structural health monitoring sensor nodes in infrastructure, buildings, and vehicles, where the ambient vibration energy would otherwise be too diffuse to harvest from.Engineering and Physical Sciences Research Council (EPSRC

Composite Materials in Design Processes

The use of composite materials in the design process allows one to tailer a component’s mechanical properties, thus reducing its overall weight. On the one hand, the possible combinations of matrices, reinforcements, and technologies provides more options to the designer. On the other hand, it increases the fields that need to be investigated in order to obtain all the information requested for a safe design. This Applied Sciences Special Issue, “Composite Materials in Design Processes”, collects recent advances in the design methods for components made of composites and composite material properties at a laminate level or using a multi-scale approach

A Study in the Use of Maniplatives to Teach Topics in Differential and Integral Calculus

The use of mathematics manipulatives for the elementary grades is well-studied, but little research exists on their value for teaching calculus students. This project studied the role of physical manipulatives on student learning in two high school calculus classes. It explored the effect of two lessons taught with manipulatives, and compared two lessons on the same topic, one taught in the traditional way and the other incorporating the use of manipulatives. In evaluating the teaching method and process for the four lessons, quantitative measures involved statistical testing of mean pretest and posttest scores. Qualitative factors considered student feedback on a questionnaire, and the evaluation of the experience by the instructor. Overall, this research found that physical manipulatives improved student understanding and the students reported a positive experience with the visual and hands-on approach of the research study lessons. It is suggested that manipulatives be included among other good teaching practices in calculus, especially in classes taught at the regular and honors level

Characterizing the Realistic-ness of Word Problems in Secondary Mathematics Textbooks

Word problems are an integral part of any secondary mathematics curriculum and one purpose has been to prepare students for the real-world – for everyday events as well as workplace problem-solving. Prior literature suggests that word problems have not met this objective, in part, because the textbook problems do not mirror the kinds of problems commonly found in real life situations. In this dissertation, I investigate a sample of word problems from two contemporary non-traditional textbooks to uncover the aspects that may influence if and how the problems might be used in the classroom. I utilize a qualitative content analysis with a directed approach, using the literature to guide my initial codes and categories, and allowing other categories and subcategories to emerge during the analysis. I also conduct a numerical analysis of the data to reveal aspects which may be a common thread between the two books. These analyses allow me to answer the research question: Given that the two books chosen for this study have different approaches, what aspects of realistic-ness exist in the textbooks’ word problems that encourage students to use their real-world knowledge of the context of the problems? This study suggests that changes to the manner in which problems are presented can be beneficial to re-negotiating the didactical contract. Textbook word problems should be posed in a variety of ways, breaking from the tradition of the three-component structure. Additionally, secondary mathematics textbooks should use scaffolding throughout the curricula to afford students the opportunities to grapple with problems as they would in the real world. This study recommends a digital database to organize and update problems with a real-world context

Development and characterization of spiral additions in a ceramic matrix

A novel spiral architecture was formed using titanium diboride and silicon carbide ceramics as either the spiral or matrix phase. Particulate composites with the same compositions were fabricated to compare to the materials in this study. Spiral additions were formed using powder loaded polymers followed by a single and/or multi-filament co-extrusion. For 25 vol% SiC spiral additions to TiB2, boron nitride was added to the SiC spiral to alter the bonding at the interface and reduce thermal residual stresses. All samples were hot-pressed to near full density at 1980 ⁰C. Hot pressed multi-filament co-extrusion of 2.4 mm / 1 mm resulted in the smallest, consistent spirals ~50 µm in diameter. For the SiC spirals in TiB2 study, the room temperature flexure strength was 193 ± 17 MPa, with the particulate composite being 488 ± 45 MPa. The fracture toughness for the spiral material was as high as 7.5 ± 0.6 MPa·m1/2 with the particulate composite being 5.3 ± 0.4 MPa·m1/2. Spiral length was studied with TiB2 spirals in a SiC matrix. The resulting average room temperature flexure strength was 313 ± 11 MPa and 417 ± 41 MPa for spiral and monolithic samples, respectively. Fracture toughness was increased from 4.2 ± 0.2 MPa·m1/2 for the monolithic to 6.2 ± 0.4 MPa·m1/2 with the addition of spirals. The higher fracture toughness is a result of crack deflection in and around the spiral inclusions. Wear testing resulted in a loss of 1.1 mm3 and 3.3 mm3 per 6000 revolutions for monolithic and uniaxial specimens, respectively. While more wear was observed, the strength of the uniaxial samples after wear increased 16% whereas monolithic strength decreased 18% --Abstract, page iv