Marina L. Mozgaleva

Method of Extended Domain and General Principles of Mesh Approximation for Boundary Problems of Structural Analysis

The distinctive paper is devoted to general principles of mesh approximation for boundary problems of structural analysis with the use of so-called method of extended domain, proposed by Prof. Alexander B. Zolotov. In particular we studied questions dealing with approximation of domain, approximation of functions and approximations of operators. Mesh functions are introduced as well as corresponding mesh operations and various types of implementation. The results are rather efficient algorithms in respect to number of operations, computing time and required memory.

Correct Wavelet-based Multilevel Discrete-Continual Methods for Local Solution of Boundary Problems of Structural Analysis

The distinctive paper is devoted to correct wavelet-based multilevel discrete-continual methods for local solution of boundary problems of structural analysis. Initial discrete-continual operational formulation of the considering problem and corresponding operational formulation with the use of wavelet basis are presented. Due to special algorithms of averaging within multigrid approach, reduction of the problem is provided. Resultant multipoint boundary problem of structural mechanics for system of ordinary differential equations with piecewise-constant coefficients is given.

Correct Wavelet-Based Multilevel Numerical Method of Local Solution of Boundary Problems of Structural Analysis

The distinctive paper is devoted to correct wavelet-based multilevel numerical method of local solution of boundary problems of structural analysis. Operational and variational formulations of the problem (particularly with the use of wavelet basis) are presented. Computer-oriented algorithms of fast direct and inverse discrete Haar transforms are described. Due to special algorithms of averaging within multigrid approach, reduction of the problem is provided.

Wavelet-Based Multilevel Discrete-Continual Finite Element Method for Local Deep Beam Analysis

High-accuracy solution of the problem of deep beam analysis is normally required in some pre-known domains (regions with the risk of significant stresses that could potentially lead to the destruction of structure, regions which are subject to specific operational requirements). The distinctive paper is devoted to correct wavelet-based multilevel discrete-continual finite element method for local analysis of deep beams with regular (in particular, constant or piecewise constant) physical and geometrical parameters (properties) in one direction. Initial discrete-continual operational formulation of the considering problem and corresponding operational formulation with the use of wavelet basis are presented. Due to special algorithms of averaging within multigrid approach, reduction of the problem is provided. Resultant multipoint boundary problem of structural mechanics for system of ordinary differential equations with piecewise-constant coefficients is given.

Local High-Accuracy Plate Analysis Using Wavelet-Based Multilevel Discrete-Continual Finite Element Method

The distinctive paper is devoted to application of wavelet-based discrete-continual finite element method (WDCFEM), to analysis of plates with piecewise constant physical and geometrical parameters in so-called “basic” direction. Initial continual and discrete-continual formulations of the problem are presented. Due to special algorithms of averaging using wavelet basis within multigrid approach, reduction of the problem is provided. Resultant multipoint boundary problem for system of ordinary differential equations is given.

Application of discrete-continual finite element method for global and local analysis of multilevel systems

This paper is devoted to discrete-continual finite element method (DCFEM) of analysis of structures with regular (constant or piecewise constant) physical and geometrical parameters in some coordinate direction (so-called “basic” direction). Effective qualitative multilevel analysis of local and global stress-strain state of such structures is normally required in various technical problems. It is commonly known that defects and failures are mostly local in nature. However total load-bearing capacity of the structure, associated with the condition of limit equilibrium, is determined by its global behavior. Therefore multilevel approach within DCFEM is peculiarly relevant and apparently preferable in all aspects for qualitative and quantitative analysis of calculation data. Wavelet analysis (wavelet-based DCFEM) provides effective and popular tool for such researches.

Modified Wavelet-Based Multilevel Discrete-Continual Finite Element Method for Local Structural Analysis - Part 1: Continual and Discrete-Continual Formulations of the Problems

The distinctive paper is devoted to wavelet-based discrete-continual finite element method (WDCFEM) of structural analysis. Two-dimensional and three-dimensional problems of analysis of structures with piecewise constant physical and geometrical parameters along so-called “basic” direction are under consideration. High-accuracy solution of the corresponding problems at all points of the model is not required normally, it is necessary to find only the most accurate solution in some pre-known local domains. Wavelet analysis is a powerful and effective tool for corresponding researches. Initial continual and discrete-continual formulations of multipoint boundary problems of two-dimensional and three-dimensional structural analysis are presented.

Modified Wavelet-Based Multilevel Discrete-Continual Finite Element Method for Local Structural Analysis - Part 2: Reduced Formulations of the Problems in Haar Basis

The distinctive paper is devoted to wavelet-based discrete-continual finite element method (WDCFEM) of structural analysis. Discrete-continual formulations of multipoint boundary problems of two-dimensional and three-dimensional structural analysis are transformed to corresponding localized formulations by using the discrete Haar wavelet basis and finally, with the use of averaging and reduction algorithms, the localized and reduced governing equations are obtained. Special algorithms of localization with respect to each degree of freedom are presented.

About Verification of Wavelet-Based Discrete-Continual Finite Element Method for Three-Dimensional Problems of Structural Analysis Part 2: Structures with Piecewise Constant Physical and Geometrical Parameters along Basic Direction

This paper is devoted to verification of so-called wavelet-based discrete-continual finite element method (wDCFEM), proposed by authors, for three-dimensional problems of local structural analysis. Formulation of the problem for three-dimensional structure with piecewise constant physical and geometrical parameters along so-called its basic direction, solutions obtained by wDCFEM and discrete-continual finite element method (DCFEM) and their comparison are presented. It was confirmed that wDCFEM is rather effective in the most critical, vital, potentially dangerous areas of structure in terms of fracture (areas of the so-called edge effects), where some components of solution are rapidly changing functions and their rate of change in many cases can’t be adequately taken into account by the standard finite element method.

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.