In-plane Density Gradation of Shoe Midsoles for Optimized Cushioning Performance

Midsoles are important components in footwear as they provide shock absorption and stability, thereby improving comfort and effectively preventing certain foot and ankle injuries. A rationally tailored midsole can potentially mitigate plantar pressure, improving performance and comfort levels. Despite the importance of midsole design, the potential of using in-plane density gradation in midsole has been rarely explored in earlier studies. The present work investigates the effectiveness of in-plane density gradation in shoe midsoles using a new class of polyurea foams as the material candidate. Their excellent cushioning properties justify the use of polyurea foams. Different polyurea foam densities, ranging from 95 to 350 kg/m3 are examined and tested to construct density-dependent correlative mathematical relations required for the optimization process. An optimization framework is then created to allocate foam densities at certain plantar zones based on the required cushioning performance constrained by the local pressures. The interior-point algorithm was used to solve the constrained optimization problem. The optimization algorithm introduces a novel approach, utilizing the maximum specific energy absorption as the objective function. The optimization process identifies specific foam densities at various plantar regions for maximum biomechanical energy dissipation without incurring additional weight penalties. Our results suggest midsole design can benefit from horizontal (in-plane) density gradation, leading to potential weight reduction and localized cushioning improvements. With local plantar peak pressure data analysis, the optimization results indicate low-density polyurea foams (140 kg/m3) for central and lateral phalanges, whereas stiffer foams (185-230 kg/m3) are identified as suitable candidates for metatarsal and arch regions in an in-plane density graded midsole design.Comment: 31 pages, 6 figures This work is submitted for consideration at the Journal of Sports Engineering and Technology: Part P and currently under peer review process. Data will be available upon request from the corresponding autho

Computing Fast and Scalable Table Cartograms for Large Tables

Given an m x n table T of positive weights and a rectangle R with an area equal to the sum of the weights, a table cartogram computes a partition of R into m x n convex quadrilateral faces such that each face has the same adjacencies as its corresponding cell in T, and has an area equal to the cell's weight. In this thesis, we explored different table cartogram algorithms for a large table with thousands of cells and investigated the potential applications of large table cartograms. We implemented Evans et al.'s table cartogram algorithm that guarantees zero area error and adapted a diffusion-based cartographic transformation approach, FastFlow, to produce large table cartograms. We introduced a constraint optimization-based table cartogram generation technique, TCarto, leveraging the concept of force-directed layout. We implemented TCarto with column-based and quadtree-based parallelization to compute table cartograms for table with thousands of cells. We presented several potential applications of large table cartograms to create the diagrammatic representations in various real-life scenarios, e.g., for analyzing spatial correlations between geospatial variables, understanding clusters and densities in scatterplots, and creating visual effects in images (i.e., expanding illumination, mosaic art effect). We presented an empirical comparison among these three table cartogram techniques with two different real-life datasets: a meteorological weather dataset and a US State-to-State migration flow dataset. FastFlow and TCarto both performed well on the weather data table. However, for US State-to-State migration flow data, where the table contained many local optima with high value differences among adjacent cells, FastFlow generated concave quadrilateral faces. We also investigated some potential relationships among different measurement metrics such as cartographic error (accuracy), the average aspect ratio (the readability of the visualization), computational speed, and the grid size of the table. Furthermore, we augmented our proposed TCarto with angle constraint to enhance the readability of the visualization, conceding some cartographic error, and also inspected the potential relationship of the restricted angles with the accuracy and the readability of the visualization. In the output of the angle constrained TCarto algorithm on US State-to-State migration dataset, it was difficult to identify the rows and columns for a cell upto 20 degree angle constraint, but appeared to be identifiable for more than 40 degree angle constraint

Framework for The Generation and Design of Naturally Functionally Graded Lattice Structures

Functionally Graded Lattice (FGL) Structures have shown improved performance over uniform lattice structures in different fields. Another form of functional grading can be seen in materials in nature, where the cellular structure can vary in both cell porosity and size. To distinguish between lattice structures that vary in porosity only and lattice structures that vary in both, we will refer to the latter in this research as Naturally Functionally Graded Lattice (NFGL) structures. Research into NFGL structures' performance against FGL structures in the literature is lacking. Furthermore, the current methods in the literature to generate these structures are severely limited and suffer from multiple drawbacks. This research aims to develop a framework, namely the NFGL Framework, to generate NFGL structures without the drawbacks that exist in current methods and to improve the performance of the generated structures using the NFGL Framework against existing FGL structures. The NFGL Framework uses a novel method to generate nodes for NFGL structures from using a developed simplified sphere packing algorithm to generate conformal NFGL structures in a deterministic and computationally efficient manner. Furthermore, the NFGL Framework can perform a similarity analysis using a modified Mean Structural Similarity (MSSIM) index to improve the performance of the generated NFGL structure. The generated structures using the NFGL Framework were tested against the existing methods and showed to overcome the drawbacks of these methods with improved performance and computational time. Furthermore, the generated NFGL structures were tested against FGL structures and the results showed a performance gain from the use of NFGL structures over FGL structures with a reduced computational cost.Ph.D