Computing 2D Periodic Centroidal Voronoi Tessellation

International audienceIn this paper, we propose an efficient algorithm to compute the centroidal Voronoi tessellation in 2D periodic space. We first present a simple algorithm for constructing the periodic Voronoi diagram (PVD) from a Euclidean Voronoi diagram. The presented PVD algorithm considers only a small set of periodic copies of the input sites, which is more efficient than previous approaches requiring full copies of the sites (9 in 2D and 27 in 3D). The presented PVD algorithm is applied in a fast Newton-based framework for computing the centroidal Voronoi tessellation (CVT). We observe that full-hexagonal patterns can be obtained via periodic CVT optimization attributed to the convergence of the Newton-based CVT computation

Using Centroidal Voronoi Tessellations to Scale Up the Multi-dimensional Archive of Phenotypic Elites Algorithm

The recently introduced Multi-dimensional Archive of Phenotypic Elites (MAP-Elites) is an evolutionary algorithm capable of producing a large archive of diverse, high-performing solutions in a single run. It works by discretizing a continuous feature space into unique regions according to the desired discretization per dimension. While simple, this algorithm has a main drawback: it cannot scale to high-dimensional feature spaces since the number of regions increase exponentially with the number of dimensions. In this paper, we address this limitation by introducing a simple extension of MAP-Elites that has a constant, pre-defined number of regions irrespective of the dimensionality of the feature space. Our main insight is that methods from computational geometry could partition a high-dimensional space into well-spread geometric regions. In particular, our algorithm uses a centroidal Voronoi tessellation (CVT) to divide the feature space into a desired number of regions; it then places every generated individual in its closest region, replacing a less fit one if the region is already occupied. We demonstrate the effectiveness of the new "CVT-MAP-Elites" algorithm in high-dimensional feature spaces through comparisons against MAP-Elites in maze navigation and hexapod locomotion tasks

Spatial distribution of pores in porous materials [Räumliche Verteilung von Poren in porösen Materialien]

Porous materials have been developed rapidly in recent years and have many particular properties due to their special geometries, such as lightweight, large specific surface area, good energy absorption and high heat-transfer capacity. A commonly used parameter to characterize porous structures is the number of pores per inch (PPI). In this work, PPI is computed in three-dimensional (3D) space rather than in two dimensions, for instance from a two-dimensional image. The studied structures are computer-generated or reconstructed from computed tomography (CT) scan images of real structures. The images segmented by the marker-based watershed algorithm (a part of the pore network model) are also used in this work. All the computations are conducted using the solver Parallel Algorithm for Crystal Evolution in 3D (Pace3D) for counting the number of pores accurately. The algorithm for counting the number of pores is proposed and modified according to the distribution of pores in different cases. This work is divided into three main stages: verifying the counting algorithm with an aligned structure, modifying it for complex structures, and applying it for real structures. The two modified algorithms are also validated by a periodic aligned structure, then two modified algorithms are used in complex structures to reduce the influence of periodic boundaries. The first modified algorithm is a subtraction of computed values and the second modified algorithm is an adjustment of the length of the measuring lines. The relationships between PPI and other important parameters, such as porosity, the total number of pores (Voronoi points) and strut radius, are observed. The effects of the stretching direction and degree of pores on PPI are also studied. The PPI of structures with different stretching factors in one or two directions is shown. In addition, the advantages and disadvantages of the two modified algorithms are also identified. By reconstructing the real structure and computing the PPI of the 3D models, the PPI of the real foams is found to be smaller than the value given by the manufacturer. This difference is particularly evident in the structure with a large PPI. By observing the PPI in different directions, it is possible to find the stretching of pores

Evaluating the performance of microstructure generation algorithms for 2-D foam-like representative volume elements

This investigation evaluates various numerical algorithms; each designed to generate periodic 2-D Representative Volume Elements (RVEs) containing foam-like microstructures suitable for direct import into commercial finite element software for mechanical evaluation. The operation of each algorithm is discussed and the resulting RVEs are examined from both a mechanical and a morphological perspective. A basic Voronoi-based algorithm is found to be simple to implement but the method is shown to produce inherently anisotropic microstructures. Increasing the degree of irregularity of the microstructure reduces the anisotropy but at the cost of creating unrealistic microstructures, containing highly angular cells. A method of modifying such unrealistic microstructures using a centroidal tessellation relaxation algorithm is demonstrated, ultimately producing RVEs with relatively realistic mono-disperse microstructures. An alternative algorithm is also investigated the advantage of this algorithm is its ability to generate poly-disperse microstructures, with a controllable degree of poly-dispersity and an almost fully isotropic mechanical response

MODELING CHAIN PACKING IN COMPLEX PHASES OF SELF-ASSEMBLED BLOCK COPOLYMERS

Block copolymer (BCP) melts undergo microphase seperation and form ordered soft matter crystals with varying domain shapes and symmetries. We study the con- nection between diblock copolymer molecular designs and thermodynamic selection of ordered crystals by modeling features of variable sub-domain geometry filled with individual blocks within non-canonical sphere-like and network phases that together with layered, cylindrical and canonical spherical phases forms “natural forms” of self- assembled amphiphilic soft matter at large. First, we present a model to revise our understanding of optimal Frank-Kasper sphere-like morphologies by advancing the- ory to account for varying domain volumes. We then develop generic approaches to quantify local changes to domain thickness or packing frustration using medial sets and show its application to morphologies with arbitrary domain topologies and sym- metries in both theoretical models and experimental data. We further use medial sets as a proxy for terminal boundaries of blocks within different domains and revise thermodynamic models of BCP assembly in the strong segregation limit. Finally, we use this revised model to study effect of elastic stiffness asymmetry on relaxing packing frustration experienced by BCPs in tubular and matrix domains leading to equilibrium double gyroid network morphology in diblock copolymers

Three-dimensional random Voronoi tessellations: From cubic crystal lattices to Poisson point processes

We perturb the SC, BCC, and FCC crystal structures with a spatial Gaussian noise whose adimensional strength is controlled by the parameter a, and analyze the topological and metrical properties of the resulting Voronoi Tessellations (VT). The topological properties of the VT of the SC and FCC crystals are unstable with respect to the introduction of noise, because the corresponding polyhedra are geometrically degenerate, whereas the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. For weak noise, the mean area of the perturbed BCC and FCC crystals VT increases quadratically with a. In the case of perturbed SCC crystals, there is an optimal amount of noise that minimizes the mean area of the cells. Already for a moderate noise (a>0.5), the properties of the three perturbed VT are indistinguishable, and for intense noise (a>2), results converge to the Poisson-VT limit. Notably, 2-parameter gamma distributions are an excellent model for the empirical of of all considered properties. The VT of the perturbed BCC and FCC structures are local maxima for the isoperimetric quotient, which measures the degre of sphericity of the cells, among space filling VT. In the BCC case, this suggests a weaker form of the recentluy disproved Kelvin conjecture. Due to the fluctuations of the shape of the cells, anomalous scalings with exponents >3/2 is observed between the area and the volumes of the cells, and, except for the FCC case, also for a->0. In the Poisson-VT limit, the exponent is about 1.67. As the number of faces is positively correlated with the sphericity of the cells, the anomalous scaling is heavily reduced when we perform powerlaw fits separately on cells with a specific number of faces

Universal hidden order in amorphous cellular geometries

Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. Here we investigate the Quantizer problem, defined as the optimisation of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that tile space are preferred. We employ Lloyd’s centroidal Voronoi diagram algorithm to solve this problem and find that it converges to disordered states associated with deep local minima. These states are universal in the sense that their structure factors are characterised by a complete independence of a wide class of initial conditions they evolved from. They moreover exhibit an anomalous suppression of long-wavelength density fluctuations and quickly become effectively hyperuniform. Our findings warrant the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties