Towards Uniform Online Spherical Tessellations

The problem of uniformly placing N points onto a sphere finds applications in many areas. For example, points on the sphere correspond to unit quaternions as well as to the group of rotations SO(3) and the online version of generating uniform rotations (known as “incremental generation”) plays a crucial role in a large number of engineering applications ranging from robotics and aeronautics to computer graphics. An online version of this problem was recently studied with respect to the gap ratio as a measure of uniformity. The first online algorithm of Chen et al. was upper-bounded by 5.99 and later improved to 3.69, which is achieved by considering a circumscribed dodecahedron followed by a recursive decomposition of each face. In this paper we provide a more efficient tessellation technique based on the regular icosahedron, which improves the upper-bound for the online version of this problem, decreasing it to approximately 2.84. Moreover, we show that the lower bound for the gap ratio of placing at least three points is 1.618 and for at least four points is no less than 1.726

Towards Uniform Online Spherical Tessellations

The problem of uniformly placing N points onto a sphere finds applications in many areas. An online version of this problem was recently studied with respect to the gap ratio as a measure of uniformity. The proposed online algorithm of Chen et al. was upper-bounded by 5.99 and then improved to 3.69, which is achieved by considering a circumscribed dodecahedron followed by a recursive decomposition of each face. We analyse a simple tessellation technique based on the regular icosahedron, which decreases the upper-bound for the online version of this problem to around 2.84. Moreover, we show that the lower bound for the gap ratio of placing up to three points is 1+5√2≈1.618 . The uniform distribution of points on a sphere also corresponds to uniform distribution of unit quaternions which represent rotations in 3D space and has numerous applications in many areas

Frustration Propagation in Tubular Foldable Mechanisms

Shell mechanisms are patterned surface-like structures with compliant deformation modes that allow them to change shape drastically. Examples include many origami and kirigami tessellations as well as other periodic truss mechanisms. The deployment paths of a shell mechanism are greatly constrained by the inextensibility of the constitutive material locally, and by the compatibility requirements of surface geometry globally. With notable exceptions (e.g., Miura-ori), the deployment of a shell mechanism often couples in-plane stretching and out-of-plane bending. Here, we investigate the repercussions of this kinematic coupling in the presence of geometric confinement, specifically in tubular states. We demonstrate that the confinement in the hoop direction leads to a frustration that propagates axially as if by buckling. We fully characterize this phenomenon in terms of amplitude, wavelength, and mode shape, in the asymptotic regime where the size of the unit cell of the mechanism~$r$ is small compared to the typical radius of curvature~$\rho$. In particular, we conclude that the amplitude and wavelength of the frustration are of order $\sqrt{r/\rho}$ and that the mode shape is an elastica solution. Derivations are carried out for a particular pyramidal truss mechanism. Findings are supported by numerical solutions of the exact kinematics.Comment: 7 figures, added figures and references, corrected typo

Pore Network Modeling of Compressed Fuel Cell Components with OpenPNM

Pore network modeling is used to model water invasion and multiphase transport through compressed PEFC gas diffusion layers. Networks are created using a Delaunay tessellation of randomly placed base-points setting the pore locations and its compliment, the Voronoi diagram, is used to define the location of fibers and resultant pore and throat geometry. The model is validated in comparison to experimental capillary pressure curves obtained on compressed and uncompressed materials. Primary drainage is simulated with an invasion percolation algorithm that sequentially invades pores and throats separately with excellent agreement to experimental data, but required a slight modification to account for the higher aspect ratio of compressed pores. Compression is simulated by scaling the through-plane coordinates in a uniform manner representing a GDL wholly beneath the current-collector land. The relative permeability and diffusivity show some dependence on uniform compression. In-plane porosity variations introduced by land-channel compression are also investigated which have a marked effect on the limiting current. Saturation at breakthrough does not appear to be dependent on compression. However, a more important parameter, namely the peak saturation, is shown to influence the fuel cell performance and is dependent on the percolation inlet conditions

Glassy swirls of active dumbbells

The dynamics of a dense binary mixture of soft dumbbells, each subject to an active propulsion force and thermal fluctuations, shows a sudden arrest, first to a translational then to a rotational glass, as one reduces temperature $T$ or the self-propulsion force $f$. Is the temperature-induced glass different from the activity-induced glass? To address this question, we monitor the dynamics along an iso-relaxation-time contour in the $(T-f)$ plane. We find dramatic differences both in the fragility and in the nature of dynamical heterogeneity which characterise the onset of glass formation - the activity-induced glass exhibits large swirls or vortices, whose scale is set by activity, and appears to diverge as one approaches the glass transition. This large collective swirling movement should have implications for collective cell migration in epithelial layers.Comment: 13 pages, 11 figure

Packing of Softly Repulsive Particles in a Spherical Box —A Generalised Thomson Problem

We study the (near or close to) ground state distribution of N softly repelling particles trapped in the interior of a spherical box. The charges mutually interact via an inverse power law potential of the form $1/r^\gamma$. We study three regimes in which the charges form an single spherical shell at the edge of the box ($\gamma=1$), a series of concentric shells of increasing density ($\gamma=2$) and $\gamma=12$ for which the charges form shells with a more uniform charge distribution. We conduct numerical simulations for clusters containing up to 5000 charges and compare charge density across the system with continuum limit results. The agreement between numerical (discrete) results and the continuum limit is found to improve with increasing N

The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries

We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body