scholarly journals Effective Conductivity of Densely Packed Disks and Energy of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2161
Author(s):  
Wojciech Nawalaniec ◽  
Katarzyna Necka ◽  
Vladimir Mityushev
Keyword(s):  
Effective Conductivity ◽  
Periodic Structures ◽  
Voronoi Tessellation ◽  
Two Dimensional ◽  
Random Composites ◽  
Geometrical Information

The theory of structural approximations is extended to two-dimensional double periodic structures and applied to determination of the effective conductivity of densely packed disks. Statistical simulations of non-overlapping disks with the different degrees of clusterization are considered. The obtained results shows that the distribution of inclusions in a composite, as an amount of geometrical information, remains in the discrete corresponding Voronoi tessellation, hence, precisely determines the effective conductivity for random composites.

scholarly journals Classifying and analysis of random composites using structural sums feature vector

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180698 ◽  
Author(s):  
Wojciech Nawalaniec
Keyword(s):  
Materials Science ◽  
Feature Vector ◽  
Effective Conductivity ◽  
Feature Space ◽  
Two Dimensional ◽  
Learning Tools ◽  
Classification Problems ◽  
Random Composites ◽  
Random Structures ◽  
Computational Materials

The main goal of this paper is to present the application of structural sums, mathematical objects originating from the computational materials science, in construction of a feature space vector of two-dimensional random composites simulated by distributions of non-overlapping discs on the plane. Construction of the feature vector enables the immediate application of machine learning tools and data analysis techniques to random structures. In order to present the accuracy and the potential of structural sums as geometry descriptors, we apply them to classification problems comprising composites with circular inclusions as well as composites with shapes formed by discs. As an application, we perform the analysis of different models of composites in order to formulate the irregularity measure of random structures. We also visualize the relationship between the effective conductivity of two-dimensional composites and the geometry of inclusions.

Optimal and Manufacturable Two-dimensional, Kagomé-like Cellular Solids

2002 ◽  
Vol 17 (1) ◽  
pp. 137-144 ◽  
Author(s):  
S. Hyun ◽  
S. Torquato
Keyword(s):  
Heat Dissipation ◽  
Volume Fraction ◽  
Effective Properties ◽  
Effective Conductivity ◽  
Periodic Structures ◽  
Optimization Technique ◽  
Solid Volume Fraction ◽  
Cellular Solids ◽  
Two Dimensional ◽  
Wide Range

We used the topology optimization technique to obtain two-dimensional, isotropic cellular solids with optimal effective elastic moduli and effective conductivity. The overall aim was to obtain the best (simplest) manufacturable structures for these effective properties, i.e., single-length-scale structures. Three different but simple periodic structures arose due to the imposed geometric mirror symmetries: lattices with triangular-like cells, hexagonal-like cells, or Kagomé-like cells. As a general rule, the structures with the Kagomé-like cells provided the best performance over a wide range of densities, i.e., for 0 ≰ ф <0.6, where ф is the solid volume fraction (density). At high densities (ф > 0.6), Kagome-like structures were no longer possible, and lattices with hexagonal-like or triangular-like cells provide virtually the same optimal performance. The Kagomé-like structures were found to be a new class of cellular solids with many useful features, including desirable transport and elastic properties, heat-dissipation characteristics, improved mechanical strength, and ease of fabrication.

scholarly journals Optimal Random Packing of Spheres and Extremal Effective Conductivity

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1063
Author(s):  
Vladimir Mityushev ◽  
Zhanat Zhunussova
Keyword(s):  
Effective Conductivity ◽  
Dimensional Space ◽  
Elementary Function ◽  
Voronoi Tessellation ◽  
Random Packing ◽  
Periodicity Cell ◽  
Algebraic Equations ◽  
Optimal Packing ◽  
Linear Algebraic Equations ◽  
Initial Location

A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing and physical-conductivity problems are stated in a periodic toroidal d-dimensional space with an arbitrarily fixed number n of nonoverlapping spheres per periodicity cell. Energy E depends on Voronoi tessellation (Delaunay graph) associated with the centers of spheres ak (k=1,2,…,n). All Delaunay graphs are divided into classes of isomorphic periodic graphs. For any fixed n, the number of such classes is finite. Energy E is estimated in the framework of structural approximations and reduced to the study of an elementary function of n variables. The minimum of E over locations of spheres is attained at the optimal packing within a fixed class of graphs. The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations. Such an approach is useful for random optimal packing where an initial location of balls is randomly chosen; hence, a class of graphs is fixed and can dynamically change following prescribed packing rules. A finite algorithm for any fixed n is constructed to determine the optimal random packing of spheres in Rd.

scholarly journals Geometrical four-point functions in the two-dimensional critical Q-state Potts model: the interchiral conformal bootstrap

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yifei He ◽  
Jesper Lykke Jacobsen ◽  
Hubert Saleur
Keyword(s):  
Potts Model ◽  
Correlation Functions ◽  
Liouville Theory ◽  
Analytical Determination ◽  
Conformal Weight ◽  
Two Dimensional ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Point Correlation

Abstract Based on the spectrum identified in our earlier work [1], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the Q-state Potts model. Crucial in our approach is the existence of “interchiral conformal blocks”, which arise from the degeneracy of fields with conformal weight hr,1, with r ∈ ℕ*, and are related to the underlying presence of the “interchiral algebra” introduced in [2]. We also find evidence for the existence of “renormalized” recursions, replacing those that follow from the degeneracy of the field $$ {\Phi}_{12}^D $$ Φ 12 D in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.

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