Statistic modeling and Monte Carlo methods
Various methods of specification of 3D geometry (3DG) are known, which are determined by a class of the problems being solved. In the problems, for the solution of which Monte Carlo technique (MCT) is used, the analytical set-theoretical method of specification of models geometry is applied. Usually the statement of the problem of 3DG specification is presented in the general case, that is the class of functions (surfaces) is determined, from which the limited volumes of continuum (Volumes) are constructed with the help of Boolean operations, which describe location and form of continua. The analytical way of specification has the obvious advantage – the relative briefness of description of complex geometries. However, some costs exist, it is clear that it is difficult to visualize the analytically described forms, the preprocessing is required. One algorithm (Contour) is proposed for finding and visualizing the contours and inside of arbitrary sections of 3D objects specified analytically, on the basis of functions, which are available in any MCT code. The algorithm consists of several constructive solutions. The algorithm idea: The additional regular grid is constructed in the section plane. Then the two auxiliary scalar quantities are calculated on this grid by a special method, the first quantity is formed in the result of the section plane tracing, for example, along horizontal guiding lines of the grid, and the second one — along vertical guiding lines. In the result we get two discrete scalar fields in the grid nodes. Then we use any correct algorithm of isolines tracking on the regular grid for auxiliary quantities equal to zero. The given algorithm should be modified in terms of considering the values of both quantities, by analyzing the intersection of isoline of horizontal or vertical cells edges. In the result the oriented piecewise-linear contour of the sections of the surfaces limiting volume is restored. Figure 1a presents the grid and the section contour subject to one quantity, under horizontal tracing, Figure 1b presents the quantities under vertical tracing and Figure 1c presents the integral result taking into account both quantities. Restriction: Algorithm CONTOUR operates correctly, if each cell edge has got no more than one point of intersection with the volume surface. Then we present the algorithm modification for the restriction elimination. If the volume surface intersects any cell edge more than once, then the notion of a specific cell (SC) is introduced. The classification of SC will be presented. The modification implies the specialized processing of SC of different types by generating of additional adaptive grid into SC by its binary or square division. Figure 2 presents the example of SC and its place in the grid and in the section, and the volume section place as well. Result: On the basis of generation of the special additive grid the algorithm CONTOUR and its modification is proposed for generation of the oriented contour of 3D flat section specified analytically. The algorithm ideas are generalized at a 3D case. The report will include many illustrations and a video reel about the algorithm operation.
Note. Abstracts are published in author's edition
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