Algebra of shape similarity and transversal subdivisions
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1MSTU STANKIN, 127055, Moscow, Vadkovsky per. 1, 8 (499) 972-95-00, firstname.lastname@example.org
The paper investigates the possibility to use local self-similarity while subdivide computational grids used in solving boundary value problems . It is shown that self-similariry can be implimented in "cubic" and "hexagonal" subdivision algorithms in the presence of mathematical means for calculating the similarity relations between the cell of the partition, its neighborhood and the nucleus.
A method of packing cells with $2^n$ (power-of-two) vertices (edges, faces, etc) into a shape vector is proposed, and it is shown that: 1) for any pair of shape vectors, a similarity relation can be
defined "by example" using an attached basis; 2) a rational basis can be uniquely attached to any shape vector by means of permutation symmetries; 3) any rational basis can be transformed into an orthonormal one by changing its structure, in particular, by breaking symmetry.
An algorithm for such a transformation is presented. Rules for calculating complex similarity relations are formulated. It is shown that the main requirement is an explicit distinction between the vector as operand and the matrix as operator