Geometry-based structural analysis and design via discrete stress functions

This PhD thesis proposes a direct and unified method for generating global static equilibrium for 2D and 3D reciprocal form and force diagrams based on reciprocal discrete stress functions. This research combines and reinterprets knowledge from Maxwell’s 19th century graphic statics, projective geometry and rigidity theory to provide an interactive design and analysis framework through which information about designed structural performance can be geometrically encoded in the form of the characteristics of the stress function. This method results in novel, intuitive design and analysis freedoms. In contrast to contemporary computational frameworks, this method is direct and analytical. In this way, there is no need for iteration, the designer operates by default within the equilibrium space and the mathematically elegant nature of this framework results in its wide applicability as well as in added educational value. Moreover, it provides the designers with the agility to start from any one of the four interlinked reciprocal objects (form diagram, force diagram, corresponding stress functions). This method has the potential to be applied in a wide range of case studies and fields. Specifically, it leads to the design, analysis and load-path optimisation of tension-and compression 2D and 3D trusses, tensegrities, the exoskeletons of towers, and in conjunction with force density, to tension-and-compression grid-shells, shells and vaults. Moreover, the abstract nature of this method leads to wide cross-disciplinary applicability, such as 2D and 3D discrete stress fields in structural concrete and to a geometrical interpretation of yield line theory

Quantization, Calibration and Planning for Euclidean Motions in Robotic Systems

The properties of Euclidean motions are fundamental in all areas of robotics research. Throughout the past several decades, investigations on some low-level tasks like parameterizing specific movements and generating effective motion plans have fostered high-level operations in an autonomous robotic system. In typical applications, before executing robot motions, a proper quantization of basic motion primitives could simplify online computations; a precise calibration of sensor readings could elevate the accuracy of the system controls. Of particular importance in the whole autonomous robotic task, a safe and efficient motion planning framework would make the whole system operate in a well-organized and effective way. All these modules encourage huge amounts of efforts in solving various fundamental problems, such as the uniformity of quantization in non-Euclidean manifolds, the calibration errors on unknown rigid transformations due to the lack of data correspondence and noise, the narrow passage and the curse of dimensionality bottlenecks in developing motion planning algorithms, etc. Therefore, the goal of this dissertation is to tackle these challenges in the topics of quantization, calibration and planning for Euclidean motions

Cohomology of Line Bundles: Applications

Massless modes of both heterotic and Type II string compactifications on compact manifolds are determined by vector bundle valued cohomology classes. Various applications of our recent algorithm for the computation of line bundle valued cohomology classes over toric varieties are presented. For the heterotic string, the prime examples are so-called monad constructions on Calabi-Yau manifolds. In the context of Type II orientifolds, one often needs to compute equivariant cohomology for line bundles, necessitating us to generalize our algorithm to this case. Moreover, we exemplify that the different terms in Batyrev's formula and its generalizations can be given a one-to-one cohomological interpretation. This paper is considered the third in the row of arXiv:1003.5217 and arXiv:1006.2392.Comment: 56 pages, 8 tables, cohomCalg incl. Koszul extension available at http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg

Symplectomorphism group relations and degenerations of Landau-Ginzburg models

In this paper, we describe explicit relations in the symplectomorphism groups of toric hypersurfaces. To define the elements involved, we construct a proper stack of toric hypersurfaces with compactifying boundary representing toric hypersurface degenerations. Our relations arise through the study of the one dimensional strata of this stack. The results are then examined from the perspective of homological mirror symmetry where we view sequences of relations as maximal degenerations of Landau-Ginzburg models. We then study the B-model mirror to these degenerations, which gives a new mirror symmetry approach to the minimal model program.Comment: 100 pages, 24 figure

Thermodynamic and Structural Phase Behavior of Colloidal and Nanoparticle Systems.

We design and implement a scalable hard particle Monte Carlo simulation toolkit (HPMC), and release it open source. Common thermodynamic ensembles can be run in two dimensional or three dimensional triclinic boxes. We developed an efficient scheme for hard particle pressure measurement based on volume perturbation. We demonstrate the effectiveness of low order virial coefficients in describing the compressibility factor of fluids of hard polyhedra. The second virial coefficient is obtained analytically from particle asphericity and can be used to define an effective sphere with similar low-density behavior. Higher-order virial coefficients --- efficiently calculated with Mayer Sampling Monte Carlo --- are used to define an exponential approximant that exhibits the best known semi-analytic characterization of hard polyhedron fluid state functions. We present a general method for the exact calculation of convex polyhedron diffraction form factors that is more easily applied to common shape data structures than the techniques typically presented in literature. A proof of concept user interface illustrates how a researcher might investigate the role of particle form factor in the diffraction patterns of different particles in known structures. We present a square-triangle dodecagonal quasicrystal (DQC) in a binary mixture of nanocrystals (NCs). We demonstrate how the decoration of the square and triangle tiles naturally gives rise to partial matching rules via symmetry breaking in layers perpendicular to the dodecagonal axis. We analyze the geometry of the experimental tiling and, following the ``cut and project'' theory, lift the square and triangle tiling pattern to four dimensional space to perform phason analysis historically applied only in simulation and atomic systems. Hard particle models are unsuccessful at explaining the stability of the binary nanoparticle super lattice. However, with a simple isotropic soft particle model, we are able to demonstrate seeded growth of the experimental structure in simulation. These simulations indicate that the most important stabilizing properties of the short range structure are the size ratio of the particles and an A--B particle attraction.PhDMaterials Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/120906/1/eirrgang_1.pd