59 research outputs found
Experimental study of energy-minimizing point configurations on spheres
In this paper we report on massive computer experiments aimed at finding
spherical point configurations that minimize potential energy. We present
experimental evidence for two new universal optima (consisting of 40 points in
10 dimensions and 64 points in 14 dimensions), as well as evidence that there
are no others with at most 64 points. We also describe several other new
polytopes, and we present new geometrical descriptions of some of the known
universal optima.Comment: 41 pages, 12 figures, to appear in Experimental Mathematic
Elastostatics of star-polygon tile-based architectured planar lattices
We showed a panoptic view of architectured planar lattices based on
star-polygon tilings. Four star-polygon-based lattice sub-families were
investigated numerically and experimentally. Finite element-based
homogenization allowed computation of Poisson's ratio, elastic modulus, shear
modulus, and planar bulk modulus. A comprehensive understanding of the range of
properties and micromechanical deformation mechanisms was developed. By
adjusting the star angle from to the uniqueness limit ( to
), our results showed an over 250-fold range in elastic modulus,
over a 10-fold range in density, and a range of to for
Poisson's ratio. Additively manufactured lattices showed good agreement in
properties. The additive manufacturing procedure for each lattice is available
on www.fullcontrol.xyz/#/models/1d3528. Three of the four sub-families
exhibited in-plane elastic isotropy. One showed high stiffness with auxeticity
at low density with a primarily axial deformation mode as opposed to bending
deformation for the other three lattices. The range of achievable properties,
demonstrated with property maps, proves the extension of the conventional
material-property space. Lattice metamaterials with Triangle-Triangle, Kagome,
Hexagonal, Square, Truncated Archimedean, Triangular, and Truncated Hexagonal
topologies have been studied in the literature individually. We show that all
these structures belong to the presented overarching lattices
The Surface Evolver
The Surface Evolver is a computer program that minimizes the energy of a surface subject to constraints. The surface is represented as a simplicial complex. The energy can include surface tension, gravity, and other forms. Constraints can be geometrical constraints on vertex positions or constraints on integrated quantities such as body volumes. The minimization is done by evolving the surface down the energy gradient. This paper describes the mathematical model used and the operations available to interactively modify the surface
A formula for crossing probabilities of critical systems inside polygons
In this article, we use our results from Flores and Kleban (2015 Commun. Math. Phys. 333 389-434, 2015 Commun. Math. Phys. 333 435-81, 2015 Commun. Math. Phys. 333 597-667, 2015 Commun. Math. Phys. 333 669715) to generalize known formulas for crossing probabilities. Prior crossing results date back to Cardy's prediction of a formula for the probability that a percolation cluster in two dimensions connects the left and right sides of a rectangle at the percolation critical point in the continuum limit (Cardy 1992 J. Phys. A: Math. Gen. 25 L201-6). Here, we predict a new formula for crossing probabilities of a continuum limit loop-gas model on a planar lattice inside a 2N-sided polygon. In this model, boundary loops exit and then re-enter the polygon through its vertices, with exactly one loop passing once through each vertex, and these loops join the vertices pairwise in some specified connectivity through the polygon's exterior. The boundary loops also connect the vertices through the interior, which we regard as a crossing event. For particular values of the loop fugacity, this formula specializes to FK cluster (resp. spin cluster) crossing probabilities of a critical Q-state random cluster (resp. Potts) model on a lattice inside the polygon in the continuum limit. This includes critical percolation as the Q = 1 random cluster model. These latter crossing probabilities are conditioned on a particular side-alternating free/fixed (resp. fluctuating/fixed) boundary condition on the polygon's perimeter, related to how the boundary loops join the polygon's vertices pairwise through the polygon's exterior in the associated loop-gas model. For Q is an element of{2, 3, 4}, we compare our predictions of these random cluster (resp. Potts) model crossing probabilities in a rectangle (N = 2) and in a hexagon (N = 3) with high-precision computer simulation measurements. We find that the measurements agree with our predictions very well for Q is an element of{2, 3} and reasonably well if Q = 4.Peer reviewe
- ā¦