Puzzling the 120-cell

We introduce Quintessence: a family of burr puzzles based on the geometry and combinatorics of the 120-cell. We discuss the regular polytopes, their symmetries, the dodecahedron as an important special case, the three-sphere, and the quaternions. We then construct the 120-cell, giving an illustrated survey of its geometry and combinatorics. This done, we describe the pieces out of which Quintessence is made. The design of our puzzle pieces uses a drawing technique of Leonardo da Vinci; the paper ends with a catalogue of new puzzles.Comment: 25 pages, many figures. Exposition and figures improved throughout. This is the long version of the shorter published versio

On the combinatorics of quivers, mutations and cluster algebra exchange graphs

Over the last 20 years, cluster algebras have been widely studied, with numerous links to different areas of mathematics and physics. These algebras have a cluster structure given by successively mutating seeds, which can be thought of as living on some graph or tree. In this way one can use various combinatorial tools to discover more about these cluster structures and the cluster algebras themselves. This thesis considers some of the combinatorics at play here. Mutation-finite quivers have been classified, with links to triangulations of surfaces and semi-simple Lie algebras, while comparatively little is known about mutation-infinite quivers. We introduce a classification of the minimal types of these mutation-infinite quivers before studying their properties. We show that these minimal mutation-infinite quivers admit a maximal green sequence and that the cluster algebras which they generate are equal to their related upper cluster algebras. Automorphisms of skew-symmetric cluster algebras are known to be linked to automorphisms of their exchange graphs. In the final chapter we discuss how this idea can be extended to skew-symmetrizable cluster algebras by using the symmetrizing weights to add markings to the exchange graphs. This opens possible opportunities to study orbifold mapping class groups using combinatoric graph theory

Curvilinear Interface Methodology for Finite-Element Applications

Recent trends in design and manufacturing suggest a tendency toward multiple centers of specialty which results in a need for improved integration methodology for dissimilar finite element or CFD meshes. Since a typical finite element or CFD analysis requires about 50% of an engineers effort to be devoted to modeling and input, there is a need to advance the state-of-the-art in modeling, methodology. These two trends indicate a need to for the capability to combine independently-modeled configurations in an automated and robust way without the need for global remodeling. One approach to addressing this need is the development of interfacing methodology which will automatically integrate independently modeled subdomains. The present research included the following objectives: (i) to develop and implement computational methods for automatically remodeling non-coincident finite element models having a pre-defined interface, (ii) to formulate and implement a parametric representation of general space curves and surfaces with a well-defined orientation, and (iii) to demonstrate the computational methodology with representative two- and three-dimensional finite element models. Methodology for automatically remodeling non-coincident subdomains was developed and tested for two- and three-dimensional, independently modeled subdomains. Representative classes of applications have been solved which gave good agreement with reference solutions obtained with conventional methods. The two-dimensional classes of problems solved included flat and curved membranes multiple subdomains having large gaps between the subdomains and general space curves representing an interface for re-modeling the portions of subdomains adjacent to the interface. The three-dimensional classes of problems solved includes multiple three-dimensional subdomains having large three-dimensional gap between previously modeled subdomains. The interface was represented by general surfaces with a well-defined orientation and having curvature in possibly more than one direction. The results demonstrated the re-modeling methodology to be general, flexible in use, highly automated, and robust for a diverse class of problems. The research reported represents an important advancement in the area of automated re-modeling for computational mechanics applications

Multi-scale active shape description in medical imaging

Shape description in medical imaging has become an increasingly important research field in recent years. Fast and high-resolution image acquisition methods like Magnetic Resonance (MR) imaging produce very detailed cross-sectional images of the human body - shape description is then a post-processing operation which abstracts quantitative descriptions of anatomically relevant object shapes. This task is usually performed by clinicians and other experts by first segmenting the shapes of interest, and then making volumetric and other quantitative measurements. High demand on expert time and inter- and intra-observer variability impose a clinical need of automating this process. Furthermore, recent studies in clinical neurology on the correspondence between disease status and degree of shape deformations necessitate the use of more sophisticated, higher-level shape description techniques. In this work a new hierarchical tool for shape description has been developed, combining two recently developed and powerful techniques in image processing: differential invariants in scale-space, and active contour models. This tool enables quantitative and qualitative shape studies at multiple levels of image detail, exploring the extra image scale degree of freedom. Using scale-space continuity, the global object shape can be detected at a coarse level of image detail, and finer shape characteristics can be found at higher levels of detail or scales. New methods for active shape evolution and focusing have been developed for the extraction of shapes at a large set of scales using an active contour model whose energy function is regularized with respect to scale and geometric differential image invariants. The resulting set of shapes is formulated as a multiscale shape stack which is analysed and described for each scale level with a large set of shape descriptors to obtain and analyse shape changes across scales. This shape stack leads naturally to several questions in regard to variable sampling and appropriate levels of detail to investigate an image. The relationship between active contour sampling precision and scale-space is addressed. After a thorough review of modem shape description, multi-scale image processing and active contour model techniques, the novel framework for multi-scale active shape description is presented and tested on synthetic images and medical images. An interesting result is the recovery of the fractal dimension of a known fractal boundary using this framework. Medical applications addressed are grey-matter deformations occurring for patients with epilepsy, spinal cord atrophy for patients with Multiple Sclerosis, and cortical impairment for neonates. Extensions to non-linear scale-spaces, comparisons to binary curve and curvature evolution schemes as well as other hierarchical shape descriptors are discussed

Computer-Aided Geometry Modeling

Techniques in computer-aided geometry modeling and their application are addressed. Mathematical modeling, solid geometry models, management of geometric data, development of geometry standards, and interactive and graphic procedures are discussed. The applications include aeronautical and aerospace structures design, fluid flow modeling, and gas turbine design

On the Boundary Conditions and Internal Mechanics of Parallel Wire Strands

This dissertation analyzes the internal mechanics of parallel wire strands as found in the main cables of suspension bridges. Parallel wire strands of reduced order (7-wire, 19-wire, and 61-wire strands made of steel and aluminum) are fabricated and subjected to various boundary conditions and external loads (tension, clamping, twist, etc.). Neutron diffraction is used as an elastic strain measurement tool for its ability to penetrate bulk materials and/or layers of a multi-body system without disturbing the sample. Firstly, this thesis aims to quantify the development length – the distance over which a broken wire within a strand regains near-full service strain – as a function of various boundary conditions and failure scenarios. The feasibility of using neutron diffractometers to measure in situ elastic strains on civil-engineering-scale samples under both tensile load and radial confinement is validated using strands fabricated from steel bridge wire. Results from various 7-wire strands indicate that friction and mechanical interference on the microscopic level play a significant role in the load partitioning. Furthermore, wires that have been broken – either pre-cracked or fractured live and in situ during tensile loading – are capable of regaining significant stresses from their neighbors over a distance of tens of centimeters. The contribution of both friction force and mechanical interference on elastic strain redevelopment in broken wires should be included in analytical models designed to simulate failure processes. The second part of this thesis aims to measure the internal mechanics of larger parallel wire strands in response to various confinement (clamping) forces. 19 and 61 aluminum wire strands are fabricated and the internal strains of all constituent wires mapped in three orthogonal directions under various clamping loads. The strain distributions for both 19-wire and 61-wire strands show a surprising degree of heterogeneity. An increase in clamping force homogenizes the distribution to a degree, but only at unfeasibly high clamping forces. The results suggest that microscale variations in wire diameter dominate the internal mechanics of parallel wire strands. The stochastic distribution of wire sizes due to manufacturing tolerances throughout a strand cross-section creates a randomly ordered network of over- and under-sized wires. This imperfectly packed lattice results in large wire-to-wire variations in clamping constraint. The up-scaling in strand size from 19 to 61 wires increases the resolution of the experiment but does not reduce the heterogeneity of the strain distribution. Ergo, the assumption of perfect hexagonal packing in parallel wire strands is weak, and mean field distributions do not accurately describe the internal mechanics of such structures

Modeling of ground excavation with the particle finite element method

The present work introduces a new application of the Particle Finite Element Method (PFEM) for the modeling of excavation problems. PFEM is presented as a very suitable tool for the treatment of excavation problem. The method gives solution for the analysis of all processes that derive from it. The method has a high versatility and a reasonable computational cost. The obtained results are really promising.Postprint (published version

Mixed finite elements with independent strain interpolation for isotropic and orthotropic damage

Tesi per compendi de publicacionsThe numerical modelling of fracture has been an active topic of research for over five decades. Most of the approaches employed rely on the use of the Finite Element Method, which has shown to be an effective and cost-efficient tool for solving many physical phenomena. However, the issue of the spurious dependency of the computed solution with the mesh orientation in cracking problems has raised a great concern since its early reports in the 1980s. This matter has proved to be a major challenge in computational solid mechanics; it affects numerous methods employed to solve the problem, in which the computed crack trajectories are spuriously dependent on the arrangement of the finite element (FE) mesh employed. When performing a structural analysis and, in particular, when computing localized failure, it is fundamental to use a reliable and mesh objective method to be able to trust the results produced by the FE code in terms of the fracture paths, bearing capacity, collapse mechanism and nonlinear responses. In this doctoral thesis, the mixed e/u strain/displacement finite element method is used together with multiple isotropic and orthotropic damage constitutive laws for the numerical modelling of quasi-brittle fracture with mesh objectivity. The independent interpolation of the strains increases the accuracy of the computed solution, guaranteeing the local convergence of the stress and strain fields. This feature is a crucial improvement over the standard FE formulation in solid mechanics where the strains are computed as local derivatives of the displacements and the local convergence of the resulting stresses and strains is not ensured. The enhanced precision provided by the mixed formulation in the area near the crack tip is decisive for obtaining unbiased numerical results with regard to the orientation of the FE mesh. The strain-driven format of the mixed formulation enables to readily consider different constitutive laws defined in a stress-strain structure in the numerical simulations. The thesis includes the study of the effect of the material model employed in the resulting crack trajectories as well as the analysis of the relative performance of several isotropic and orthotropic damage behaviors in mode I, mode II, mode III and mixed mode fracture problems. In this work specific isotropic and orthotropic damage laws are proposed for the numerical modelling of fracture under cyclic loading, which include tensile and compressive damage, stiffness recovery due to crack closure and reopening, as well as irreversible strains. Also, the capacity of the proposed model in reproducing the structural size effect is examined, which is an essential requirement for models aiming at computing quasi-brittle behavior. In this thesis, a comprehensive comparison of the mixed FE formulation with other techniques employed for computing fracture, specifically the Extended Finite Element Method (XFEM) and the Phase-field model, is made, revealing the cost-efficiency of the proposed Mixed Finite Element Method for modelling quasi-brittle cracking with mesh objectivity. This allows to perform the analysis of real-scale structures, in 2D and 3D, with enhanced accuracy, demonstrating the applicability of this method in the engineering practice. The validation of the model is performed with an extensive comparison of computed results with existing experimental tests and numerical benchmarks. The capacity of the mixed formulation in reproducing force-displacement curves, crack trajectories and collapse mechanisms with enhanced accuracy is demonstrated in detail.En esta tesis doctoral, el método de los elementos finitos mixtos e/u deformación/desplazamiento es utilizado junto con varias leyes constitutivas de daño isótropo y ortótropo para la modelización numérica de la fractura cuasi-frágil de forma objetiva con respecto a la orientación de la malla. La interpolación independiente de las deformaciones aumenta la precisión de la solución calculada, garantizando la convergencia local de los campos de tensiones y deformaciones. Esta característica representa una mejora crucial con respecto a la formulación estándar de elementos finitos de la mecánica de sólidos, donde las deformaciones se calculan como derivadas locales de los desplazamientos y la convergencia local de las tensiones y deformaciones resultantes no está garantizada. La mayor precisión aportada por la formulación mixta en la zona cercana a la punta de la fisura es decisiva para obtener resultados numéricos que no presenten una dependencia espuria con la orientación de la malla de elementos finitos. El formato expresado en función de la deformación de la formulación mixta permite considerar directamente diferentes leyes constitutivas que siguen una estructura tensión-deformación para su uso en las simulaciones numéricas. La tesis incluye el estudio del efecto que tiene la ley constitutiva utilizada en la trayectoria de las fisuras resultantes, así como el análisis del desempeño relativo de varias leyes de daño isótropas y ortótropas en problemas de fractura en modo I, modo II, modo III y modo mixto. En este trabajo se proponen leyes de daño isótropo y ortótropo específicas para la modelización numérica de la fractura bajo carga cíclica, que incluyen daño a tracción y a compresión, recuperación de la rigidez por el cierre y reapertura de fisuras, así como deformaciones irreversibles. Además, se comprueba la capacidad del modelo propuesto para reproducir el efecto tamaño, que es un requisito esencial para los modelos que tengan como objetivo calcular el comportamiento cuasi-frágil de los materiales. En la tesis se realiza una comparación exhaustiva de la formulación mixta de elementos finitos con otras técnicas que se utilizan para calcular el problema, específicamente el Método de los Elementos Finitos Extendidos (XFEM) y el modelo Phase-field, revelando la eficiencia computacional del Método de los Elementos Finitos Mixtos propuesto para modelizar la rotura cuasi-frágil de forma objetiva con respecto a la malla. Ello permite realizar el análisis de estructuras de tamaño real, en 2D y 3D, con mayor precisión, demostrando la aplicabilidad del método a problemas reales de ingeniería. La validación del modelo se realiza con una comparación de resultados calculados con ensayos de laboratorio existentes y con simulaciones de casos teóricos de referencia. Se demuestra la capacidad de la formulación mixta para reproducir curvas fuerza-desplazamiento, trayectorias de fisuras y mecanismos de colapso con precisión mejorada.Postprint (published version