Pavel A. Akimov

Correct Discrete-Continual Finite Element Method of Structural Analysis Based on Precise Analytical Solutions of Resulting Multipoint Boundary Problems for Systems of Ordinary Differential Equations

The distinctive paper is devoted to correct discrete-continual finite element method (DCFEM) of structural analysis based on precise analytical solutions of resulting multipoint boundary problems for systems of ordinary differential equations with piecewise-constant coefficients. Corresponding semianalytical (discrete-continual) formulations are contemporary mathematical models which currently becoming available for computer realization. Major peculiarities of DCFEM include uni-versality, computer-oriented algorithm involving theory of distributions, computational stability, optimal conditionality of resulting systems and partial Jordan decompositions of matrices of coeffi-cients, eliminating necessity of calculation of root vectors.

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 Method of Analytical Solution of Multipoint Boundary Problems of Structural Analysis for Systems of Ordinary Differential Equations with Piecewise Constant Coefficients

This paper is devoted to correct method of analytical solution of multipoint boundary problems of structural analysis for systems of ordinary differential equations with piecewise constant coefficients. Its major peculiarities include universality, computer-oriented algorithm involving theory of distributions, computational stability, optimal conditionality of resultant systems and partial Jordan decomposition of matrix of coefficients, eliminating necessity of calculation of root vectors.

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.

Weight and Time Recursions in Dynamic State Estimation Problem With Mixed-Norm Cost Function

The mixed-norm cost functions arise in many applied optimization problems. As an important example, we consider the state estimation problem for a linear dynamic system under a nonclassical assumption that some entries of state vector admit jumps in their trajectories. The estimation problem is solved by means of mixed l1/l2-norm approximation. This approach combines the advantages of the well-known quadratic smoothing and the robustness of the least absolute deviations method. For the implementation of the mixed-norm approximation, a dynamic iterative estimation algorithm is proposed. This algorithm is based on weight and time recursions and demonstrates the high efficiency. It well identifies the rare jumps in the state vector and has some advantages over more customary methods in the typical case of a large amount of measurements. Nonoptimality levels for current iterations of the algorithm are constructed. Computation of these levels allows to check the accuracy of iterations.

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.

State Estimation via l1-norm Approximation: Application to Inertial Navigation

Abstract The problem of the detection of jumps in the biases of the sensors of a strapdown inertial navigation system at bench tests is considered. This problem is set as a state estimation problem for a dynamic system under the presence of outliers in object disturbances. The estimation method is based on l 1 -norm approximation and is robust to abrupt changes in signals.

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.