Multipoint Boundary Problem Research Articles

A method to improve accuracy of definition and promptness of forecasting of spacecraft motion parameters

The subject of the presented paper is the methods used to define spacecraft motion parameters. The goal of the paper is to develop approaches to improve accuracy of definition and promptness of forecasting of motion parameters. The tasks are 1) to determine the aspects influencing the result accuracy at defining initial conditions of the spacecraft motion, 2) to reveal the drawbacks of traditional methods and 3) to suggest some possible ways that can be used to improve accuracy and promptness at defining the spacecraft motion parameters. During the research it is revealed that the result accuracy at defining the initial conditions of the spacecraft motion is influenced by three aspects – a random component caused by presence of random errors in trajectory measurements, a dynamic component due to dynamic errors of the used model of the spacecraft motion and the error defined by the size of convergence region of the minimization method that is used to solve the multi-point boundary problem. To receive the least inaccuracy at defining the spacecraft motion parameters it is necessary to process some optimal quantity of measuring information which should ensure sufficient compensation of random errors and at the same time prevent the dynamic error from influencing considerably. At it it should be taken into account that the amount of the measured information should not be less than it is necessary to ensure convergence of the algorithm used to define those motion parameters. The main results are as follows. For solving the task of increasing accuracy of spacecraft motion parameters definition the paper suggests a new method based on minimization with definition of initial approximation region. The suggested method implements a non-local approach to minimization of the goal function, which ensures better as compared to traditional methods convergence. To increase the promptness of spacecraft motion forecasting it is suggested to use the mathematical apparatus of differential transformations. It allows decreasing the computational expenditures 3 to 4 times as compared to the regular ballistic and navigational algorithms while the given accuracy is preserved. On the basis of the performed research the following conclusions can be made. To increase the accuracy of the spacecraft motion parameters definition it is necessary to optimize the required number of measurement orbit passes, for that it is appropriate to use the Nelder-Mead search method. At implementing the advanced coordinate methods of the spacecraft control, to increase the promptness of the spacecraft motion forecasting, it is necessary to considerably decrease the computational expenditures for the motion parameters definition. To achieve that it is reasonable to use the mathematical apparatus of the differential transformations.

БІФУРКАЦІЙНІ СТАНИ НАДДОВГИХ ОБЕРТОВИХ СТРИЖНІВ З ЦЕНТРУВАЧАМИ

The article deals with a super long elastic tubular vertical rod with centralizers embeded in its lower part and an internal fluid flow. The reasons of appearance of bifurcation protrusion of the rod are stated. Among these reasons, an internal longitudinal force, torque, inertia forces rotating rods, inertia forces of internal fluid flow are identified and described as main ones. The purpose of the study is to detect the beginning of bifurcation states of rods with centralizers. Research methods are based on the construction of singularity perturbed multy-point boundary-value problems determining the stress-strain state and stability of the rod. Applying decomposing approach, the beginning of bifurcation states of rods with centralizers are detected. KEY WORDS: SUPER LONG ROD, ROTATIONAL MOTION, INTERNAL FLUID FLOW, CENTRALIZERS, LONGITUDINAL FORCE, TORQUE, MULTI-POINT BOUNDARY PROBLEM, BIFURCATION STATES, METHOD OF DECOMPOSITION.

B-spline wavelet discrete-continual finite element method for the local solution to the two-dimensional problem of the theory of elasticity

Introduction. The distinctive article presents a local semi-analytical solution to the problem of the two-dimensional theory of elasticity. The corresponding structures, featuring the regularity (constancy) of physical and geometric parameters (the modulus of elasticity of the material of the structure, the Poisson’s ratio of the material of the structure, dimensions of the cross section of the structure) along one direction (dimension) are under consideration. This direction is conventionally called the basic direction.
 Materials and methods. The B-spline wavelet discrete-continual finite element method (DCFEM) is used. The initial ope­rational formulation of the problem was constructed using the theory of distribution and the so-called method of extended domain, proposed by Prof. Alexander B. Zolotov.
 Results. Some topical issues of construction of normalized basis functions of a B-spline are considered, the approximation technique for corresponding vector functions and operators within DCFEM is described. Along the basic direction, the problem remains continual and an exact analytical solution can be obtained, while along the non-basic direction the finite element approximation is used in combination with a wavelet analysis apparatus. As a result, we can obtain a discrete-continual formulation of the problem. Thus, we have a multi-point (in particular, a two-point) boundary problem for the first-order system of ordinary differential equations with constant coefficients. A special correct analytical method for the solution of such problems was developed, described and verified in numerous papers written by the authors. In particular, we consider the simplest sample analysis of a deep beam, fixed along the side faces and subjected to the load concentrated in the centre of the structure.
 Conclusions. The solution to the verification problem obtained using the proposed version of the wavelet-based DCFEM was in good agreement with the solution obtained using a conventional finite element method (corresponding solutions were constructed with localization and without localization; these solutions coincide almost completely, while the advantages of the numerical-analytical approach are quite obvious). It is shown that the use of B-splines of various degrees within the wavelet-based DCFEM leads to a significant reduction in the number of unknowns.

About the B-spline wavelet discrete-continual finite element method of the local plate analysis

Introduction. This distinctive paper addresses the local semi-analytical solution to the problem of plate analysis. Isotropic plates featuring the regularity (constancy) of physical and geometric parameters (modulus of elasticity of the plate material, Poisson’s ratio of the plate material, dimensions of the cross section of the plate) along one direction (dimension) are under consideration. This direction is conventionally called the basic direction.
 Materials and methods. The B-spline wavelet discrete-continual finite element method (DCFEM) is used. The initial operational formulation of the problem was constructed using the theory of distribution and the so-called method of extended domain, proposed by Prof. Alexander B. Zolotov.
 Results. Some relevant issues of construction of normalized basis functions of the B-spline are considered; the technique of approximation of corresponding vector functions and operators within DCFEM is described. The problem remains continual if analyzed along the basic direction, and its exact analytical solution can be obtained, whereas the finite element approximation is used in combination with a wavelet analysis apparatus in respect of the non-basic direction. As a result, we can obtain a discrete-continual formulation of the problem. Thus, we have a multi-point (in particular, two-point) boundary problem for the first-order system of ordinary differential equations with constant coefficients. A special correct analytical method of solving such problems was developed, described and verified in the numerous papers of the co-authors. In particular, we consider the simplest sample analysis of a plate (rectangular in plan) fixed along the side faces exposed to the influence of the load concentrated in the center of the plate.
 Conclusions. The solution to the verification problem obtained using the proposed version of wavelet-based DCFEM was in good agreement with the solution obtained using the conventional finite element method (the corresponding solutions were constructed with and without localization; these solutions almost completely coincided, while the advantages of the numerical-analytical approach were quite obvious). It is shown that the use of B-splines of various degrees within wavelet-based DCFEM leads to a significant reduction in the number of unknowns.

Discrete-Continual Finite Element Method for Semianalytical Analysis of Plates on Two-Parameter Elastic Foundation. Part 2: Discrete-Continual Formulation of the Problem, Numerical Samples

This paper is devoted to application of discrete-continual finite element method (DCFEM) for semianalytical solution of problems of analysis of plates on two-parameter elastic foundation. Corresponding resultant multipoint boundary problem for system of ordinary differential equations of the first order with piecewise-constant coefficients is presented. Numerous samples of problems of structural mechanics have been considered. These samples allow demonstration of some advantages of application of the suggested approach to semianalytical analysis of plates on elastic foundation.

Discrete-Continual Finite Element Method for Semianalytical Analysis of Plates on Two-Parameter Elastic Foundation. Part 1: Continual Formulations of the Problem and Approximations

This paper is proposed discrete-continual finite element method (DCFEM) for semianalytical analysis of plates on two-parameter elastic foundation. First of all, the operational and discrete-continual formulations of the problem are described, then the approximation of unknown functions and its partial derivatives are obtained. Later the internal forces, local stiffness matrices and local load vectors are evaluated, then these local vectors and matrices are assembled to obtain resultant multipoint boundary problem and in the final step the problem is solved by the exact analytical method. This method allowed to obtain exact analytical solutions of boundary problems along the regular direction, and these solutions remain exact for arbitrary influences such as any force or moment, soil parameters, intermediate constraints and connections. Using this method, the problem remains continual in the one direction (basic), while the discrete (finite element) approximation is carried out with respect to the non-basis direction. The structural discontinuities in the analytical direction can be taken into account, by addition of new appropriate boundary conditions at the relevant section. The DCFEM increases the accuracy of the solution and significantly reduces the computational efforts, especially within analysis of extended plates such as strip foundations.

Solution of boundary problems of structural mechanics with the combined application use of Discrete-Continual Finite Element Method and Finite Element Method

In most structural problems the object is usually to find the distribution of stress in elastic body produced by an external loading system. The theory of elasticity is a methodology that creates a linear relation between the imposing force (stress) and resulting deformation (strain), for the majority of materials, which behave fully or partially elastically. This paper is devoted to combined semianalytical and numerical static analysis of three-dimensional structures. The stress-strain or constitutive behavior is given for isotropic materials. Solution of multipoint (particularly, two-point) boundary problem of three-dimensional elasticity theory based on combined application of finite element method (FEM) and discrete-continual finite element method (DCFEM) is under consideration. The given domain, occupied by structure, is embordered by extended one within method of extended domain. The application field of DCFEM comprises fragments of structure (subdomains) with regular (constant or piecewise constant) physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element mesh approximation for non-basic dimensions of extended domain while in the basic dimension problem remains continual (corresponding correct analytical solution is constructed). FEM is used for approximation of all other subdomains. Discrete (within FEM) and discrete-continual (within DCFEM) approximation models for subdomains and coupled multilevel approximation model for extended domain are constructed.

ABOUT SOLUTION OF MULTIPOINT BOUNDARY PROBLEMS OF THREE-DIMENSIONAL STRUCTURAL ANALYSIS WITH THE USE OF COMBINED APPLICATION OF FINITE ELEMENT METHOD AND DISCRETE-CONTINUAL FINITE ELEMENT METHOD. PART 1: FORMULATION AND BASIC PRINCIPLES OF APPROXIMATION

The distinctive paper is devoted to formulation and basic principles of approximation of multipoint boundary problem of static analysis of three-dimensional structure with the use of combined application of finite element method and discrete-continual finite element method. Basic notation system, design model, general formulation of the problem (based on three-dimensional theory of elasticity), basic principles of domain approxima­tion, rule of numbering of subdomains, rule of numbering of finite elements, rule of numbering of discrete- continual finite elements are considered. Construction of discrete (finite element) and discrete-continual approx­imation models for subdomains is under consideration as well

Combined Semianalytical and Numerical Static Plate Analysis. Part 2: Resultant Multipoint Boundary Problem

The distinctive paper is devoted to solution of multipoint boundary problem of plate analysis (Kirchhoff model) based on combined application of finite element method (FEM) and discrete-continual finite element method (DCFEM). As is known the Kirchhoff-Love theory of plates is a two-dimensional mathematical model that is normally used to determine the stresses and deformations in thin plates subjected to forces and moments. The given domain, occupied by considering structure, is embordered by extended one. The field of application of DCFEM comprises fragments of structure (subdomains) with regular (constant or piecewise constant) physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element mesh approximation for non-basic dimension of extended domain while in the basic dimension problem remains continual. FEM is used for approximation of all other subdomains (it is convenient to solve plate bending problems in terms of displacements). Coupled multilevel approximation model for extended domain and resultant multipoint boundary problem are constructed. Brief information about software systems and verification samples are presented as well.

Combined Semianalytical and Numerical Static Plate Analysis. Part 1: Formulation of the Problem and Approximation Models

The distinctive paper is devoted to solution of multipoint (particularly, two-point) boundary problem of plate analysis (Kirchhoff model) based on combined application of finite element method (FEM) and discrete-continual finite element method (DCFEM). As is known the Kirchhoff-Love theory of plates is a two-dimensional mathematical model that is normally used to determine the stresses and deformations in thin plates subjected to forces and moments. The given domain, occupied by considering structure, is embordered by extended one. The field of application of DCFEM comprises fragments of structure (subdomains) with regular (constant or piecewise constant) physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element mesh approximation for non-basic dimension of extended domain while in the basic dimension problem remains continual. FEM is used for approximation of all other subdomains (it is convenient to solve plate bending problems in terms of displacements). Discrete (within FEM) and discrete-continual (within DCFEM) approximation models for subdomains are under consideration.

ABOUT SOLUTION OF MULTIPOINT BOUNDARY PROBLEMS OF TWO-DIMENSIONAL STRUCTURAL ANALYSIS WITH THE USE OF COMBINED APPLICATION OF FINITE ELEMENT METHOD AND DISCRETE-CONTINUAL FINITE ELEMENT METHOD PART 2: SPECIAL ASPECTS OF FINITE ELEMENT APPROXIMATION

As is well known, the formulation of a multipoint boundary problem involves three main components: a description of the domain occupied by the structure and the corresponding subdomains; description of the conditions inside the domain and inside the corresponding subdomains, the description of the conditions on the boundary of the domain, conditions on the boundaries between subdomains. This paper is a continuation of another work published earlier, in which the formulation and general principles of the approximation of the multipoint boundary problem of a static analysis of deep beam on the basis of the joint application of the finite element method and the discrete-continual finite element method were considered. It should be noted that the approximation within the fragments of a domain that have regular physical-geometric parameters along one of the directions is expedient to be carried out on the basis of the discrete-continual finite element method (DCFEM), and for the approximation of all other fragments it is necessary to use the standard finite element method (FEM). In the present publication, the formulas for the computing of displacements partial derivatives of displacements, strains and stresses within the finite element model (both within the finite element and the corresponding nodal values (with the use of averaging)) are presented. Boundary conditions between subdomains (respectively, discrete models and discrete-continual models and typical conditions such as “hinged support”, “free edge”, “perfect contact” (twelve basic (basic) variants are available)) are under consideration as well. Governing formulas for computing of elements of the corresponding matrices of coefficients and vectors of the right-hand sides are given for each variant. All formulas are fully adapted for algorithmic implementation.

ABOUT SOLUTION OF MULTIPOINT BOUNDARY PROBLEMS OF TWO-DIMENSIONAL STRUCTURAL ANALYSIS WITH THE USE OF COMBINED APPLICATION OF FINITE ELEMENT METHOD AND DISCRETE-CONTINUAL FINITE ELEMENT METHOD PART 1: FORMULATION AND BASIC PRINCIPLES OF APPROXIMATION

The distinctive paper is devoted to formulation and basic principles of approximation of multipoint boundary problem of static analysis of deep beam with the use of combined application of finite element method and discrete-continual finite element method. Design model, general formulation of the problem, basic principles of domain approximation, rule of numbering of subdomains, rule of numbering of finite elements, rule of numbering of discrete-continual finite elements are considered. Construction of discrete (finite element) and discretecontinual approximation models for subdomains is under consideration as well.

About solution of multipoint boundary problem of static analysis of deep beam with the use of combined application of finite element method and discrete-continual finite element method. part 1: formulation of the problem and general principles of approximation

This paper is devoted to formulation and general principles of approximation of multipoint boundary problem of static analysis of deep beam with the use of combined application of finite element method (FEM) discrete-continual finite element method (DCFEM). The field of application of DCFEM comprises structures with regular physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element approximation for non-basic dimension while in the basic dimension problem remains continual. DCFEM is based on analytical solutions of resulting multipoint boundary problems for systems of ordinary differential equations with piecewise-constant coefficients.

About solution of multipoint boundary problem of static analysis of deep beam with the use of combined application of finite element method and discrete-continual finite element method. part 2: boundary conditions

The formulation of considering multipoint boundary problem includes three main components: a description of the domain occupied by the structure and the corresponding subdomains; description of the conditions inside the domain and inside subdomains; description of boundary conditions (for boundaries of domain and boundaries between subdomains). These boundary conditions (interface conditions) in under consideration in the distinctive paper.

Solution of Multipoint Boundary Problem of Two-Dimensional Theory of Elasticity Based on Combined Application of Finite Element Method and Discrete-Continual Finite Element Method

The distinctive paper is devoted to solution of multipoint boundary problem of two-dimensional theory of elasticity (static analysis of two-dimensional structure) based on combined application of finite element method (FEM) and discrete-continual finite element method (DCFEM). The given domain, occupied by considering structure, is embordered by extended one. The field of application of DCFEM comprises fragments of structure (subdomains) with regular (constant or piecewise constant) physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element mesh approximation for non-basic dimension of extended domain while in the basic dimension problem remains continual. Analytical solution for such typical subdomain is apparently preferable in all aspects for qualitative analysis of calculation data. It allows investigator to consider boundary effects when some components of solution are rapidly varying functions. Due to the abrupt decrease inside of mesh elements in many cases their rate of change can’t be adequately considered by conventional numerical methods while analytics enables study. FEM is used for approximation of all other subdomains. Discrete (within FEM) and discrete-continual (within DCFEM) approximation models for subdomains and coupled multilevel approximation model for extended domain are under consideration. Generally, discrete-continual formulations are contemporary mathematical models which currently becoming available for computer realization. Brief information about software systems and verification samples are presented as well.

Iterative positive solutions for singular nonlinear fractional differential equation with integral boundary conditions

In this article, we study the existence of iterative positive solutions for a class of singular nonlinear fractional differential equations with Riemann-Stieltjes integral boundary conditions, where the nonlinear term may be singular both for time and space variables. By using the properties of the Green function and the fixed point theorem of mixed monotone operators in cones we obtain some results on the existence and uniqueness of positive solutions. We also construct successively some sequences for approximating the unique solution. Our results include the multipoint boundary problems and integral boundary problems as special cases, and we also extend and improve many known results including singular and non-singular cases.

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.

Semianalytical analysis of shear walls with the use of discrete-continual finite element method. Part 1: Mathematical foundations

The distinctive paper is devoted to the two-dimensional semi-analytical solution of boundary problems of analysis of shear walls with the use of discrete-continual finite element method (DCFEM). This approach allows obtaining the exact analytical solution in one direction (so-called “basic” direction), also decrease the size of the problem to one-dimensional common finite element analysis. The resulting multipoint boundary problem for the first-order system of ordinary differential equations with piecewise constant coefficients is solved analytically. The proposed method is rather efficient for evaluation of the boundary effect (such as the stress field near the concentrated force). DCFEM also has a completely computer-oriented algorithm, computational stability, optimal conditionality of resultant system and it is applicable for the various loads at an arbitrary point or a region of the wall.

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.

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.