An Energy-Minimization Finite-Element Approach for the Frank-Oseen Model of Nematic Liquid Crystals: Continuum and Discrete Analysis

This paper outlines an energy-minimization finite-element approach to the computational modeling of equilibrium configurations for nematic liquid crystals under free elastic effects. The method targets minimization of the system free energy based on the Frank-Oseen free-energy model. Solutions to the intermediate discretized free elastic linearizations are shown to exist generally and are unique under certain assumptions. This requires proving continuity, coercivity, and weak coercivity for the accompanying appropriate bilinear forms within a mixed finite-element framework. Error analysis demonstrates that the method constitutes a convergent scheme. Numerical experiments are performed for problems with a range of physical parameters as well as simple and patterned boundary conditions. The resulting algorithm accurately handles heterogeneous constant coefficients and effectively resolves configurations resulting from complicated boundary conditions relevant in ongoing research.Comment: 31 pages, 3 figures, 3 table

Collisions at infinity in hyperbolic manifolds

For a complete, finite volume real hyperbolic n-manifold M, we investigate the map between homology of the cusps of M and the homology of $M$. Our main result provides a proof of a result required in a recent paper of Frigerio, Lafont, and Sisto

Computing equilibrium states of cholesteric liquid crystals in elliptical channels with deflation algorithms

We study the problem of a cholesteric liquid crystal confined to an elliptical channel. The system is geometrically frustrated because the cholesteric prefers to adopt a uniform rate of twist deformation, but the elliptical domain precludes this. The frustration is resolved by deformation of the layers or introduction of defects, leading to a particularly rich family of equilibrium configurations. To identify the solution set, we adapt and apply a new family of algorithms, known as deflation methods, that iteratively modify the free energy extremisation problem by removing previously known solutions. A second algorithm, deflated continuation, is used to track solution branches as a function of the aspect ratio of the ellipse and preferred pitch of the cholesteric.Comment: 9 pages, 6 figure

Sustainable City Planning: A Data-Driven Approach for Mitigating Urban Heat

Urban areas are expected to triple by 2030 in order to accommodate 60% of the global population. Anthropogenic landscape modifications expand coverage of impervious surfaces inducing the urban heat island (UHI) effect, a critical twenty first century challenge associated with increased economic expenditure, energy consumption, and adverse health impacts. Yet, omission of UHI measures from global climate models and metropolitan planning methodologies precludes effective sustainable development governance. We present an approach that integrates Earth observation and climate data with three-dimensional urban models to determine optimal tree placement (per square meter) within proposed urban developments to enable more effective localized UHI mitigation. Such data-driven planning decisions will enhance the future sustainability of our cities to align with current global urban development agendas

Peripheral separability and cusps of arithmetic hyperbolic orbifolds

For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X-hyperbolic (n+1)-orbifold. A principal tool in the proof of this classification theorem is a subgroup separability result which may be of independent interest.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-32.abs.htm

Design ideation through improvised comedy processes

We argue that the processes of creating successful comedy are comparable to the processes of designing an innovative product. Our research explores how constructs of humour may be applied to the early phase of engineering design, when divergent thinking is assumed to be most valuable. During a series of exploratory workshops, the principles and processes of creating improvised comedy presented an opportunity to reinvigorate the design process, and overcome some of the common barriers to effective group brainstorming. This paper discusses the link between improvised comedy and design creativity, and the early development of a new improvisation-based approach to design ideation