Computation of Compact Distributions of Discrete Elements

In our daily lives, many plane patterns can actually be regarded as a compact distribution of a number of elements with certain shapes, like the classic pattern mosaic. In order to synthesize this kind of pattern, the basic problem is, with given graphics elements with certain shapes, to distribute a large number of these elements within a plane region in a possibly random and compact way. It is not easy to achieve this because it not only involves complicated adjacency calculations, but also is closely related to the shape of the elements. This paper attempts to propose an approach that can effectively and quickly synthesize compact distributions of elements of a variety of shapes. The primary idea is that with the seed points and distribution region given as premise, the generation of the Centroidal Voronoi Tesselation (CVT) of this region by iterative relaxation and the CVT will partition the distribution area into small regions of Voronoi, with each region representing the space of an element, to achieve a compact distribution of all the elements. In the generation process of Voronoi diagram, we adopt various distance metrics to control the shape of the generated Voronoi regions, and finally achieve the compact element distributions of different shapes. Additionally, approaches are introduced to control the sizes and directions of the Voronoi regions to generate element distributions with size and direction variations during the Voronoi diagram generation process to enrich the effect of compact element distributions. Moreover, to increase the synthesis efficiency, the time-consuming Voronoi diagram generation process was converted into a graphical rendering process, thus increasing the speed of the synthesis process. This paper is an exploration of elements compact distribution and also carries application value in the fields like mosaic pattern synthesis.

Nonflatness and totality

We interpret finite types as domains over nonflat inductive base types in order to bring out the finitary core that seems to be inherent in the concept of totality. We prove a strong version of the Kreisel density theorem by providing a total compact element as a witness, a result that we cannot hope to have if we work with flat base types. To this end, we develop tools that deal adequately with possibly inconsistent finite sets of information. The classical density theorem is reestablished via a ‘finite density theorem,’ and corollaries are obtained, among them Berger's separation property.

A note on compact elements of the lattice of solubly saturated formations

It is proved that every [Formula: see text]-closed solubly saturated formation contained in a compact element of the lattice of all [Formula: see text]-closed saturated formations of finite groups is also contained in some compact element of the lattice of all [Formula: see text]-closed solubly saturated formations of finite groups.

THE MOVING FRICTION CONE APPROACH FOR THREE-DIMENSIONAL CONTACT SIMULATIONS

A finite element contact approach based on the Moving Friction Cone (MFC) formulation is presented herein. The formulation is based on the contact constraint described using a single gap vector. Such a simplification, in comparison with the standard approach where normal and tangential gap vectors are used, results in significantly simpler, shorter and faster element code. The associated penalty is formulated to include large deformations and displacements. Within this approach a triangular contact element is developed using a high abstract mathematical level of symbolic description. Using this technique, a consistent linearization is obtained which leads to quadratic rates of convergence. Furthermore, the new technique results in algorithmic robustness, fast evaluation time, as well as a compact element code.