Browsing by Subject "Lattices"

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    Helly numbers of subsets of ℝᵈ and sampling techniques in optimization
    (University of Minnesota. Institute for Mathematics and Its Applications, 2015-04) De Loera, Jesús A.; La Haye, R. N.; Oliveros, D.; Roldán-Pensado, E.
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    Topological Metamaterials: Beyond the Kagome Lattice
    (2024-02) Charara, Mohammad
    Metamaterials are architected solids composed of networks of individual building blocks, whose ensemble yields emergent properties that transcends those of the individual components. Within the broad family of metamaterials, Maxwell lattices are a type of lattice material composed of a network of interconnected bonds such that the number of degrees of freedom and constraints are equal in the bulk. Topological Maxwell lattices are a special subclass of this family that display the ability to localize floppy modes (zero-energy deformation modes) and stress on opposing edges of a finitedomain, a property referred to as topological polarization. This behavior, analogous to topological insulators in electronic and quantum systems, is protected by the topology of the lattice’s band structure (referred to as k-space topology) and formally captured by a topological invariant. When interpreted as structural systems, where the bonds are replaced by structural elements that can deform flexurally, and the ideal hinges are replaced by internal clamps of finite-thickness ligaments that can store bending energy, the polarization, and resulting asymmetry between the edges, is maintained, such that soft modes are still localized on one edge, but the zero mode mechanisms rise in the spectrum and morph into finite frequency phonons. While topological polarization has been well documented in the literature, most investigations are limited to configurations that can be regarded as variations of the kagome and square lattices. In this dissertation, we aim to expand the design space for topologically polarized systems through a series of strategies that increase the geometric complexity of the lattice unit cells. In the first presented strategy, we introduce complexity by extending the problem of polarization, native to 2D elastodynamics, to out-of-plane flexural mechanics, resulting in plate-like 2D-periodic lattices embedded in 3D space. We design bilayer systems which couple in-plane and flexural mechanics with the goal of exporting the topological polarization of kagome lattices to the flexural response. The second strategy introduces a framework for unit cell augmentation which increases the geometric complexity of a Maxwell lattice via a series of mirror operations on a kagome cell, resulting in macrocells with higher kinematic complexity, dubbed “superkagome” cells, and we study the circumstances under which these augmented configurations display polarization. For both strategies, we show that, in the limit of ideal lattice conditions, the obtained configurations enjoy topological protection. For their structural lattice counterparts, we document the signature of polarization via finite-element simulations and laser vibrometry experiments.