Prof. Pavel A. Akimov, Prof. Marina L. Mozgaleva

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Wavelet-Based Numerical and Semianalytical Methods of Local Structural Analysis in Engineering Prof. Pavel A. Akimov, Prof. Marina L. Mozgaleva Research & Development Center “StaDyO”, Moscow, Russia National Research Moscow State University of Civil Engineering, Moscow, Russia International Conference “Mathematical Modeling and Computational Physics, 2017” (MMCP2017) 1 1

Basic formulas of fast direct and inverse discrete Haar transforms and averaging

Discrete Haar functions for one-dimensional problems Basic formulas of fast direct and inverse discrete Haar transforms and averaging Discrete Haar functions for one-dimensional problems

Basic formulas of fast direct and inverse discrete Haar transforms and averaging

Basic formulas of fast direct and inverse discrete Haar transforms and averaging

Basic formulas of fast direct and inverse discrete Haar transforms and averaging

Basic formulas of fast direct and inverse discrete Haar transforms and averaging

Two-dimensional rectangular domain Basic formulas of fast direct and inverse discrete Haar transforms and averaging Two-dimensional rectangular domain

Basic formulas of fast direct and inverse discrete Haar transforms and averaging

Basic formulas of fast direct and inverse discrete Haar transforms and averaging

Basic formulas of fast direct and inverse discrete Haar transforms and averaging

Basic formulas of fast direct and inverse discrete Haar transforms and averaging

Basic formulas of fast direct and inverse discrete Haar transforms and averaging

Basic formulas of fast direct and inverse discrete Haar transforms and averaging

About Correct Multilevel Wavelet-Based Numerical Method of Local Structural Analysis

About Correct Multilevel Wavelet-Based Numerical Method of Local Structural Analysis

About Correct Multilevel Wavelet-Based Numerical Method of Local Structural Analysis

About Correct Multilevel Wavelet-Based Numerical Method of Local Structural Analysis

NUMERICAL AND SEMIANALYTICAL METHODS OF LOCAL STRUCTURAL ANALYSIS The most dangerous from the standpoint of strength is the stress- strain state of the structure in places of various kinds of concentrations (cracks, joints of structural elements, etc.), local changes within the reconstruction of the object (needling, demolition of supports that interfere with the functional purpose of the structure and etc.) or in areas of local strengthening (reinforcing beams, metal ties, etc.).

NUMERICAL METHOD OF LOCAL STRUCTURAL ANALYSIS Local numerical simulation is based on a modern, highly efficient apparatus of multi-scale wavelet analysis. Finite element method (FEM) Discrete formulations of boundary problems of structural analysis (within FEM) Reduced wavelet-based discrete formulations of boundary problems of structural analysis Solution of resultant reduced system of linear algebraic equations, return to the original basis Main result: a significant reduction in the dimension of the problem, high accuracy of computing of stress-strain state in the most dangerous areas of the structure y = 0.47 м

SEMIANALYTICAL METHOD OF LOCAL STRUCTURAL ANALYSIS Local semianalytical simulation is based on a modern, highly efficient apparatus of multi-scale wavelet analysis. Discrete-continual finite element method Discrete-continual formulations of boundary problems of structural analysis Reduced wavelet-based discrete-continual formulations of boundary problems of structural analysis Exact analytical solution of resultant reduced multipoint boundary problem, return to the original basis Comparison for the first discrete-continual finite element Main result: a significant reduction in the dimension of the problem, high accuracy of computing of stress-strain state in the most dangerous areas of the structure

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