In article method of build algorithm of modeling and rendering process of crystals growth is considered. This algorithm based on geometric methods and taking into account changes of crystals geometry.
Problem definition:
Sometimes standard optical devices can not be used for observation at experiment and we can not register some physical parameters. That is why modeling and rendering of physical processes often are only possible way to do it. Searching of detached parameters, its influences on the result of experiment in real condition can be long-term process. Relevance of solution of this problem is in reduction of financial costs and working time resources during conducting of experiment.
Analysis of achievements and publications:
In spite of the fact that the decision of this problem will allow to shorten the stage of preliminary researches before the practical raising of experiment, it now is not found. The review of literature shows that in the last decades this problem decided over thermodynamics theory without using of computing tools. Works using modern software-based and hardware-based computational power appeared in scientific literature. In this developments special emphasis lays on the modeling of the simplest physical interactions. But works like these are not very popular, and the problem decided in its are formulated mainly for the energy parameters of the system, instead of geometrical parameters of the system. [2,3]. That is why, in our apinion, we can use metods of geometrical modeling. It will allow to put in basis of model approach, taking into account changes of crystals geometry.
Statement of problem:
Properties of flexibility, bending strength, durability of products depend on a form, mutual position, size of crystals and their amount. Hereupon there is a task of development of algorithm of process crystallization in melt modeling and rendering, searching of main mechanisms of crystal interaction and deformation, building of boundary between them. Method of geometrical modeling is used for this. It permits to decline from linkage to certain lattice of growing crystal. Approach taking account crystal geometric changing is assumed as a basis of model. Solution of this task in three-dimensional space makes it possible to get model of surface of crystal or fractured surface. Solution of this task in two-dimensional space makes it possible to get picture of section by random plane.
Thus, comparing of program built image and photo from section surface may be used for checking model.
Three-dimensional model of process of crystallization.
It is suggested to set some parameters to description crystals. Physical features are taken into account by using in the model indexes of speed, growth of crystal, changes of representative volume. Geometrical features are set the form of nucleus of a crystal for different components. It will allow analyzing the process of crystallization with different compositions of incoming in mixture components, and similarly will allow checking up the built model’s adequacy by comparison of program built image and photo of section.
Nucleus of crystals in hard alloy has a form of polyhedrons [5] and characterized the geometrical parameters:
- cube: length of side, coordinates of center of object, critical size (conditional size that designating the minimum size of particle which can growth);
- prism: length of side of triangle in foundation, height of prism, center of object (the center of mass), critical size;
- plate: length, width, height of plate, critical size (used rarely);
- needle: length, size (used rarely).
Critical size for ever crystal is figure out depending on the area of his surface.
All crystals have an ideal form only in case when in the process of growth they do not adjoin. If there is interaction crystals are deformed, because on the border of contiguity growth is halted.
We will consider the spontaneous nucleation of crystallization’s centers.
Let:
n – number of nucleus of a crystals which arising in time unit in unit of metal’s volume.
c – linear rate of growth of every crystal
 - size of supercooling
 - energy of interface between hard and liquid phases ( - total surface of nucleus of a crystals,  - specific surface energy)
 (1)
(1) – total changing of free energy
 (2)
(2) - critical size
 (3)
In (3)  and  mean solidification heat and temperature of crystallization.
Nucleation rate is a rate with which in the casual points of space there are crystals which subsequent growth is possible for.
 (4)
In (4) Ns - number of atoms of liquid, Vl - frequency of vibrations of atoms, k – Boltzmann constant, Q – energy of activation of diffusion process in liquid phase,  =1/16 [2].
Let’s examine the boundaries’ forming process in the crystal growth. Let’s say that crystal boundary is an aggregation of crystal face fragments, which belong to two or more objects. Every crystal face grows in the direction of the surface normal. The modelling of boundaries’ forming process is constructed in the rectangular coordinate system.
Objects’ description is available with help of interfacial presentation. This type of presentation uses description of terminating surfaces for solid-core object specifying. Crystal surface is specified as a range of the facets. Every facet needs to be compactly described. For our situation it’s convenient to determine the facet as a polygon with “n” edges and “n” vertexes. One of the methods of acceleration processing of structure which keep information about an object there is arrangement of edges’ vertexes (for example, enumeration of vertexes clockwise).
Generating initial position for objects is possible in a random way. The proper numerical limitations of speed and critical size of nucleus of a crystal are used for this purpose. Practical realization is possible by the commands of generation of random numbers with an arbitrary turn about axis [1].
Growth of object will be realized by scaling. Scaling is realized by single-stepping. That is why we mast check up position of all objects at every step. Size of step is multiple of an atom size. Verification must be done in pairs of edges and faces. If objects have mutual points after scaling, we mast do supplementary research. That is determining vertex which belong to two faces, remove it from structure of face and include it in structure of a boundary.
If objects does not have mutual points after step of scaling an additional analysis is not required.
In a number of cases enough to consider interaction of objects on a plane. Correctness of such model is checked up by comparing of the programmatic built image to the photo of section of crystal (slice).
Two-dimensional model of process: We will consider interaction of objects in a plane. You can see general case of boundary build for two arbitrarily located crystals on figure 1. In this case we should analyzed interaction of polygons. Polygons can be both convex and non-convex, however without self-intersection.
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