ABOUT WAVELET-BASED MULTIGRID NUMERICAL METHOD OF STRUCTURAL ANALYSIS WITH THE USE OF DISCRETE HAAR BASIS

he distinctive paper is devoted to so-called multigrid (particularly two-grid) method of structural analysis based on discrete Haar basis (one-dimensional, two-dimensional and three-dimensional problems are under consideration). Approximations of the mesh functions in discrete Haar bases of zero and first levels are described (the mesh function is represented as the sum in which one term is its approximation of the first level, and the second term is so-called complement (up to the initial state) on the grid of the first level). Special projectors are constructed for the spaces of vector functions of the original grid to the space of their approximation on the first-level grid and its complement (the refinement component) to the initial state. Basic scheme of the two-grid method is presented. This method allows solution of boundary problems of structural mechanics with the use of matrix operators of significantly smaller dimension. It should be noted that discrete analogue of the initial operator equation is a system of linear algebraic equations which is constructed with the use of finite element method or finite difference method. Block Gauss method can be used for direct solution.

Uncertainty product for Vilenkin groups

We study a localization of functions defined on Vilenkin groups. To measure the localization, we introduce two uncertainty products [Formula: see text] and [Formula: see text] that are similar to the Heisenberg uncertainty product. [Formula: see text] and [Formula: see text] differ from each other by the metric used for the Vilenkin group [Formula: see text]. We discuss analogs of a quantitative uncertainty principle. Representations for [Formula: see text] and [Formula: see text] in terms of Walsh and Haar basis are given.

WO-GRID METHOD OF STRUCTURAL ANALYSIS BASED ON DISCRETE HAAR BASIS. PART 1: ONE-DIMENSIONAL PROBLEMS

The distinctive paper is devoted to the two-grid method of structural analysis based on discrete Haar basis (in particular, the simplest one-dimensional problems are under consideration). A brief review of publications of recent years of Russian and foreign specialists devoted to the current trends in the use of wavelet analysis in construction mechanics is given. Approximations of the mesh functions in discrete Haar bases of zero and first levels are described (the mesh function is represented as the sum in which one term is its approximation of the first level, and the second term is so-called complement (up to the initial state) on the grid of the first level). Projectors are constructed for the spaces of vector functions of the original grid to the space of their approximation on the first-level grid and its complement (the detailing component) to the initial state. Basic scheme of the two-grid method is presented. This method allows solution of boundary problems of structural mechanics with the use of matrix operators of significantly smaller dimension. It should be noted that discrete analogue of the initial operator equation (defined on a given interval) is a system of linear algebraic equations (SLAE) constructed within finite difference method (FDM) or the finite element method (FEM). Next, the transition to the resolving SLAE is done. Block Gauss method is used for its direct solution (forward-backward algorithm is realized). We consider a numerical solution of the boundary problem of bending of the Bernoulli beam lying on an elastic foundation (within Winkler model) as a practically important one-dimensional sample. There is good consistency of the results obtained by the proposed method and by standard finite difference method.

TWO-STAGE GRID METHOD OF SOLUTION OF BOUNDARY PROBLEMS OF STRUCTURAL MECHANICS WITH THE USE OF DISCRETE HAAR BASIS

The distinctive paper is devoted to development of two-stage numerical method. At the first stage, the discrete problem is solved on a coarse grid, where the number of nodes in each direction is the same and is a pow-er of 2. Then the number of nodes in each direction is doubled and the resulting solution on a coarse grid using a discrete Haar basis is defined at the nodes of the fine grid as the initial approximation. At the second stage, we ob-tain a solution in the nodes of the fine grid using the most appropriate iterative method,. Test examples of the solu-tion of one-dimensional, two-dimensional and three-dimensional boundary problems are under consideration