Kantorovich Method Research Articles

Galerkin Kantorovich Method for Solving the Terzaghi’s One-Dimensional Soil Consolidation Equation

This paper presents the Galerkin-Kantorovich variational method for solving the Terzaghi’s one-dimensional consolidation equation for two-way drainage conditions. The solution was considered as an infinite series of known coordinate (shape) functions and unknown function of time which we sought such that the resulting functional is minimized. The shape functions satisfied the hydraulic boundary conditions at the boundary of the consolidating soil. Galerkin-Kantorovich variational integral equation was thus formulated for the initial boundary value problem using residual minimization principles. The solution resulted in a system of first order ordinary differential equations in which was solved for Orthogonalization principles were used to obtain the integration constants in terms of initial pore water pressure, thus yielding the general solution. Solutions for constant initial excess pore water pressure were obtained and found to be the closed-form solution. The solutions were presented in terms of global (average) degrees of consolidation and tabulated. The results obtained were exact and identical with results previously found using separation of variables techniques.

Dynamic buckling analysis of doubly curved orthotropic shallow shells via the Kantorovich and Rosenbrock methods

Dynamic buckling analysis of doubly curved orthotropic shallow shells via the Kantorovich and Rosenbrock methods

New algorithm on linearization-discretization solving systems of nonlinear integro-differential Fredholm equations

This article deals with a new strategy for solving a certain type of nonlinear integro-differential Fredholm equations with a weakly singular kernel. We build our new algorithm starting with the linearization phase using Newton's iterative process, then with the discretization phase we apply the Kantorovich's projection method. The discretized linear scheme will be approximated by the product integration method in the weak singular terms, and the other regular integrals will be approximated by the Nyström method. The process of convergence of our new algorithm is carried out under certain predefined and necessary conditions. Finally, we give practical examples where, the results show the efficiency of our new algorithm for solving systems of weakly singular nonlinear integro-differential equations.

The Nonlinear Bending of Sector Nanoplate via Higher-Order Shear Deformation Theory and Nonlocal Strain Gradient Theory

In this context, the nonlinear bending investigation of a sector nanoplate on the elastic foundation is carried out with the aid of the nonlocal strain gradient theory. The governing relations of the graphene plate are derived based on the higher-order shear deformation theory (HSDT) and considering von Karman nonlinear strains. Contrary to the first shear deformation theory (FSDT), HSDT offers an acceptable distribution for shear stress along the thickness and removes the defects of FSDT by presenting acceptable precision without a shear correction parameter. Since the governing equations are two-dimensional and partial differential, the extended Kantorovich method (EKM) and differential quadrature (DQM) have been used to solve the equations. Furthermore, the numeric outcomes were compared with a reference, which shows good harmony between them. Eventually, the effects of small-scale parameters, load, boundary conditions, geometric dimensions, and elastic foundations are studied on maximum nondimensional deflection. It can be concluded that small-scale parameters influence the deflection of the sector nanoplate significantly.

Buckling and free vibration of grid-stiffened composite conical panels using Extended Kantorovich Method

This paper develops very accurate closed-form solutions for buckling and free vibration of grid-stiffened composite conical panels using a semi-analytical method. On the base of first-order shear deformation theory (FSDT), the governing formulations are derived while an appropriate unit cell is used to derive the material properties of the lattice from their bulk properties. Multiple systems of ordinary differential equations (ODEs) are obtained by applying the Extended Kantorovich Method (EKM) to solve the governing system of partial differential equations (PDEs). The resulting ODEs are then solved using semi-analytical closed-form solutions to study buckling and free vibration of grid-stiffened panels. It is interesting to note that due to the general formulation of the conical panel, it is very easy to obtain other useful geometries including cylindrical panels, and even rectangular plates with general unit cell configurations. Regarding stability, fast convergence, and very low computing cost, the effectiveness of the proposed technique is investigated. The accuracy of the critical buckling loads and natural frequencies for various cases is studied which shows very good agreement in comparison with results obtained from the commercial finite element code ANSYS.

The influence of geometrical shape on the buckling of thin-walled axisymmetric shells

The most economical for long span structures are thin-walled covering consisting of shell elements. These include shells of cylindrical, spherical, conical and other shapes. Buckling analysis of thinwalled structures is important in the design of buildings and structures. In this case, the geometric shape of the shell used has a great effect on the critical load. A comparative analysis of the geometrically nonlinear deformation, buckling, and postbuckling behavior of rotation panels of the same v olume under static thermomechanical loading is carried out. Spherical and conical shells having the same thickness, rise and weight are considered. Preheated shells are loaded with uniform external pressure. The calculation method is based on geometrically nonlinear relations of the three-dimensional theory of thermoelasticity without the use of simplifying hypotheses of shell theory and the use of a moment finite element scheme. A universal 3D isoparametric finite element is used. The element allows you to model shells of stepwise variable thickness, with breaks in the middle surface and with other geometric features in thickness. The problem of nonlinear deformation, buckling, and postbuckling behavior of inhomogeneous shells is solved by a combined algorithm that employs the parameter continuation method, a modified Newton–Kantorovich method, and a procedure for automatic correction of algorithm parameters. The method has been justified numerically in the authors' articles. Research has revealed the behavioral features of the compared shallow panels.

Mathematical modeling of functionally graded porous geometrically nonlinear micro/nano cylindrical panels

Relevance. The study investigates the problem of stress-strain state and stability of porous functional-gradient size-dependent cylindrical panels. The composition and properties of alloys can differ and significantly affect the performance characteristics of products. Therefore, the research of material properties is relevant and contributes to the creation of new types of products demanded by the oil and gas industry. Aim. Development of a new model and creation of accurate methods for analyzing the stress-strain state of porous functional-gradient size-dependent micro/nano cylindrical panels taking into account geometrical nonlinearity. Methods. The method of variational iterations – the extended Kantorovich method is used to analyze the stress-strain state of cylindrical panels. The validity of the results is ensured by comparing the solutions obtained by the method of variational iterations in the first and second approximations with the solutions obtained by the authors, by the Bubnov–Galerkin method in higher approximations, by the finite difference method of the second order of accuracy, for which their convergence is investigated depending on a number of partitions of the integration area in the finite difference method and the number of series terms in the expansion of the basic functions in the Bubnov–Galerkin method. The results obtained by these methods are compared with the solutions obtained by other authors. It should be noted that the solutions obtained by the method of variational iterations for bending of functionally graded cylindrical panels under the action of transverse uniformly distributed load can be considered accurate. Results and conclusions. The authors have constructed the model of porous functional-gradient size-dependent cylindrical panels. Its use will allow studying the properties of alloys for producing drill pipes. The influence of material porosity type, porosity index, functional-gradient index, boundary conditions, size-dependent parameter, curvature parameters on the stress-strain state of cylindrical panels was analyzed using the developed method of variational iterations.

Machine learning-based modelling, feature importance and Shapley additive explanations analysis of variable-stiffness composite beam structures

In the current work, the displacement and inner stress state of orthotropic, composite, variable-stiffness beam structures is investigated, combining the extended Kantorovich method (EKM) with machine learning (ML) modeling and explainable artificial intelligence (AI) techniques. The complete displacement and inner stress fields of variable stiffness (VS) composite beams with a through-thickness variation of their material properties are computed for different orthotropy, material gradation and slenderness attributes. Both deep learning and tree-based machine learning architectures are considered, identifying high accuracy and low computational cost modeling architectures. Thereupon, the significance of underlying fundamental design features is assessed, classifying their importance for the first time. In particular, material orthotropy, stiffness gradation and slenderness are analyzed for different boundary conditions, elucidating their interdependence and relative significance. It is shown that reduced end kinematic constraints overall favor the importance of stiffness gradation. Differences in the importance of each design feature for the transverse displacement and inner normal and shear stress fields are identified and quantified using Shapley additive explanations (SHAP) analysis. It is found that material orthotropy is more important than stiffness gradation for the transverse displacement rather than for the normal or shear stress fields developed, for which their relative importance is inverted. What is more, evidence is provided that increased gradation and orthotropy values have opposite signed contributions to the stress fields developed at critical positions of the VS composite, while they uniquely yield equal-signed contributions to the transverse displacement field.

Nonlinear deformations of size-dependent porous functionally graded plates in a temperature field

In this paper, the Variational Iteration Method (VIM) or Extended Kantorovich Method (EKM) is formulated for the first time for flexible porous functionally graded (PFGM) plates with different boundary conditions and geometric nonlinearity according to the theory of Theodore von Karman. Its accuracy and efficiency are demonstrated. The plate is subjected to a uniform transversal load and temperature field. The displacement field of the plate is approximated based on the classical plate theory (CTP) or Kirchhoff's plate theory. The governing equations are derived using Hamilton's principle. Modified Coupled Stress Theory (MCST) is used to account for size-dependent effects. The material properties vary with thickness and are temperature dependent. Four porosity distribution patterns are considered in this study. Several examples are solved to demonstrate the proposed algorithm. The results obtained are compared with solutions obtained by the Bubnov-Galerkin Method (BGM) in higher approximations, the Finite Difference Method (FDM) of second order accuracy, as well as with results obtained by the Finite Element Method (FEM) of other authors. The results include an analysis of the effect of size dependent parameters, porosity type pattern, porosity index, functionally graded index, temperature field and different types of boundary conditions on the stress–strain state and bending deflection of plates.

Galerkin-Kantorovich variational method for solving saint venant torsion problems of rectangular bars

The unrestrained torsional analysis of bars is an important theme in elasticity theory, first solved by Saint-Venant using semi-inverse methods. It has been considered and solved by several others using analytical methods and numerical procedures due to the importance in the design of machine parts under torsional moments. In this paper, the Saint Venant torsion problem is solved for rectangular prismatic bars using Galerkin-Kantorovich variational method (GKVM). The work presents a detailed theoretical framework of the problem, deriving using first principles considerations the stress compatibility equation in terms of the Prandtl stress function ϕ(x,y). The derived domain equation which is required to be satisfied over the rectangular cross-sectional domain is a partial differential equation of the Poisson type. GKVM is adopted as the solution method for finding the solution to the domain equation. The unknown Prandtl stress function ϕ(x,y) is assumed, following Kantorovich method to be a product of an unknown function for fx sought to minimize the Galerkin-Kantorovich variational functional (integral) (GKVF) and a known function (y2-b2) which satisfies the boundary conditions at all boundary points in the y-direction, that is, at y=±b. The resulting GKVF is a simplified functional whose integral is a second order inhomogeneous ordinary differential equation (ODE) in fx. The integrand is solved to find fx leading in a full determination of the Prandtl stress function. The expression for stresses, torsional moments and torsional parameters are then found and they satisfy the boundary conditions and the domain equation. The results for the torsional moments and torsional parameters are identical to previous results obtained using double finite sine transform method (DFSTM), and analytical methods. The merit of GKVM is that it has led to the exact solution of the unrestrained torsion problems.

On Approximate Solutions of a Nonlinear Periodic Boundary-Value Problem with Switching in the Case of Parametric Resonance by the Newton–Kantorovich Method

On Approximate Solutions of a Nonlinear Periodic Boundary-Value Problem with Switching in the Case of Parametric Resonance by the Newton–Kantorovich Method

EQUILIBRIUM OF A NORMALLY LOADED ELASTIC SHALLOW CYLINDRICAL SHELL WITH DISLOCATIONS AND DISCLINATIONS

We consider the problem of the equilibrium of a normally loaded elastic shallow rectangular circular cylindrical shell, which contains sources of internal stress in the form of fields of continuously distributed edge dislocations and wedge disclinations or other sources, for example, thermal ones. Based on the modified system of Karman equations for the equilibrium of an elastic plate with dislocations and disclinations, a system of nonlinear equilibrium equations for the shell was constructed. The resulting system of equations differs from the Karman system of equilibrium equations for an elastic shallow cylindrical shell under the action of a normal load by the presence on the right side of the continuity equation of a nonzero function, called the incompatibility function, which is expressed through the densities of edge dislocations and wedge disclinations. The edges of the shell are freely clamped or hinged. In the absence of internal stress sources, this system transforms into a system of equilibrium equations for a shallow shell, and in the case of an infinitely large radius of curvature (and the presence of internal stress sources) into a modified system of Karman equilibrium equations for a flexible elastic plate with dislocations and disclinations. To solve the system of nonlinear shell equilibrium equations, the Newton – Kantorovich method in combination with the difference method is proposed. For the case of small values of the normal load and small values of the incompatibility function, linearization is carried out with respect to the deflection and stress function. As a result, a linear boundary value problem was obtained in the form of a system of differential equations for the deflection and the stress function, which is solved by the difference method. The solution of the linearized system is used as an initial approximation in the implementation of the Newton – Kantorovich method. Examples of numerical calculations of deflection and stress function for given values of normal load and incompatibility function are given, and corresponding graphs are constructed.

DYNAMIC CONTACT PROBLEMS FOR COMPOSITE MEDIA WITH ANISOTROPIC STRUCTURE

The paper for the first time develops a method for solving dynamic contact problems on the effect of a rigid die in the form of a strip of finite width on a layered anisotropic composite. By applying Mandelstam's principle of marginal absorption, the initial boundary value problem is reduced to a boundary value problem with absorption having a single solution. The symbol of the integral equation has no singularities on the real axis. The contact problem is reduced to solving a two-dimensional integral equation with a difference kernel. The application of the Fourier transform along the coordinate along the strip reduces the integral equation to a one-dimensional one containing a free real parameter of the Fourier transform. Integral equations with an exact solution are introduced, with symbols majoring above and below the symbol of the integral equation. Using the factorization method, the initial integral equation is reduced to two integral equations of the second type, the operator of which turns out to be compressive with a sufficiently large bandwidth. By applying the Newton–Kantorovich method to this integral equation, an exact solution of the integral equation is constructed in an operator form. The presence of the majorant symbol of the integral equation allows us to obtain an upper and lower estimate of the constructed exact solution of the integral equation containing the parameter of the Maldenstam limit absorption principle. After that, in the constructed solution, this parameter rushes to zero from above. The solution of the integral equation is obtained in an analytical form and allows us to identify all its singular features. The result is important when searching for harbingers of an increase in seismicity in mountainous areas. The method is applicable in all cases when it is possible to construct a Green function for a non-mixed boundary value problem in a layered anisotropic composite.

Stress-strain state of a porous flexible rectangular FGM size-dependent plate subjected to different types of transverse loading: Analysis and numerical solution using several alternative methods

In this study, a mathematical model of flexible porous functionally graded (PFG) Kirchhoff size-dependent plates on a rectangular plane is constructed. Hamilton's principle provides new size-dependent governing equations, taking into account geometrical nonlinearity according to the model of Theodor von Kármán, as well as boundary conditions and initial conditions for the displacement of the plates. Modified Coupled Stress Theory (MCST) is used to account for size-dependent effects. Numerical investigations analysing the stress-strain state of flexible Kirchhoff functionally graded size-dependent porous plates on a rectangular plane are based on the application of the Variational Iteration Method (VIM) or the Extended Kantorovich Method (EKM). The efficiency and high accuracy of the VIM method are demonstrated. The validity and reliability of the solutions obtained by VIM are discussed and compared with solutions obtained by other methods, such as the BGM in higher approximations, FDM of second-order accuracy and the FEM. Additionally, the solutions are compared with results obtained by other authors in the study of porous functionally graded macro, micro, and nanoplates. The solutions obtained are considered accurate. The results include an analysis of the influence of size-dependent parameters, type of material porosity, porosity index, functionally graded index, and different types of boundary conditions on the stress-strain state of plates under the action of different types of transverse loading.

KANTOROVICH VARIATIONAL METHOD FOR THE ELASTIC BUCKLING ANALYSIS OF KIRCHHOFF PLATES WITH TWO OPPOSITE SIMPLY SUPPORTED EDGES

We present the Kantorovich variational method for the elastic buckling solutions of Kirchhoff plates with two opposite simply supported edges. Uniform compressive loading is applied on the two simply supported edges, x = 0 and x = a. Three cases of edge support considered along the y = 0 and y = b edges are (i) both are clamped (ii) both are simply supported, and (iii) edge y = 0 is simply supported while edge y = b is free. The Kantorovich method which is based on seeking to minimize the total potential energy functional P with respect to the unknown deflection w(x, y) assumes the deflection in variable separable form as an infinite series of the product of an unknown function fm(y) and a known sinusoidal function that satisfies the Dirichlet boundary conditions along the simply supported edges. Euler-Lagrange differential equation is used to obtain the fourth order ordinary differential equation which fm(y) must satisfy to minimize P. The general solution for fm(y) is obtained as a combination of hyperbolic and trigonometric functions with four unknown integration constants. The conditions for nontrivial solutions are applied to find the characteristic stability equations in each considered case. Exact solutions are obtained for the elastic buckling equations which are identical with previously obtained solutions in the literature. The elastic buckling equations are solved by iteration methods and by closed form solution methods for the Kirchhoff plate to obtain the elastic buckling loads expressed in terms of the elastic buckling load coefficients. It was found that the method yields exact elastic buckling load and exact elastic buckling load coefficient for each boundary condition considered in this study.

Two-dimensional analytical solutions for multi-segmented piezoelectric panels: An EKM approach

A 2-D closed-form analytical solution for the multi-segmented piezoelectric panel has been developed using the extended Kantorovich method (EKM). In this work, a study on the bending behavior of axially segmented piezoelectric panel with different materials have been carried out. This work consider two equally segmented piezoelectric panels made of piezoelectric fiber-reinforced composite (PFRC-2) and lead zirconate titanate (PZT-5A). Furthermore, a particular case of four equal segmented piezoelectric panel is also considered, which imitates an axially graded piezoelectric actuator. The panels are subjected to various boundary conditions and an uniform pressure loading at the top surface. Along the X-axis, interfaces are assumed to be perfectly bonded to satisfy the continuity of displacement and stress at each segment interface. 2-D finite element (FE) solutions using ABAQUS are also obtained to validate displacement and stresses. A good agreement between the analytical and 2-D FE results is observed for all the cases. The current development can be used as a benchmark solution for axially stacked plates or other complex problems.

Comparative analysis of dynamic stability of cylindrical and conical shells under periodic axial compression

A comparative analysis of the dynamic stability of cylindrical and conical shells with the same geometric and mechanical characteristics under periodic uniformly distributed axial compression was presented. The study of the stability of steady periodic vibrations of thin elastic shells was based on the joint use of the method of curvilinear grids, the projection method and the parameter continuation method combined with the Newton–Kantorovich method. Geometrically nonlinear relations of the thin elastic shells theory are formulated on the basis of the vector approximation of the displacements function in the general curvilinear coordinate system in tensor form and satisfy the Kirchhoff-Love hypothesis. The discretization of the differential equations of the steady forced vibrations in the direction of the generating shells using the method of curvilinear grids was carried out. The components of the elements displacement vectors of the shells middle surface in the circular directionare approximated by trigonometric series. Reduction of the number of generalized coordinates of the discrete dynamic model of shells steady forced vibrations was performed by the Bubnov-Galerkin basis reduction method. A transition from vector ordinary differential equations to a nonlinear system of algebraic equations was made. The construction of a mathematical model of the dynamic stability of steady forced nonlinear vibrations of thin elastic shells was performed according to Floquet's theory using the projection method. The criterion for the loss of stability was the equality to zero of the determinant of the matrix of linearized equations of steady forced nonlinear vibrations of shells according to the Lyapunov theorem. A comparative analysis of frequencies and modes of natural vibrations of cylindrical and conical shells with the same geometric and mechanical characteristics and boundary conditions was performed. Nonlinear steady vibrations of the shells due to periodic axial compression were studied. The critical values of the dynamic load and the corresponding forms of loss of shell stability in the range of lower frequencies of their natural vibrations were obtained.

Stochastic trading of storage systems in short term electricity markets considering intraday demand response market

In recent years, the renewable energy sources are used in the power system for reducing environmental pollution. One of these renewable energy sources is wind farm that is used widely. However, they increase the uncertainty of the power system. To solve this problem, plenty researches have proposed several methods that have some disadvantages. This paper tries to decline the effects of uncertainty of wind turbines by using of the energy storage systems and optimal demand response programs. A stochastic planning of energy storage systems and wind generation are presented in the short term electricity markets considering the trading day-head and intraday market of demand response. In order to formulate an accurate model of various components of system, all needed constraints are considered. In this problem, wind turbines, energy storage systems, and demand response aggregator work coordinately that is called virtual power plant. For better comparison, two states consist of coordinated and uncoordinated working of all components are considered and assessed. The various scenarios for electricity market price, adjustment market price, imbalance energy price, and the generated power of wind turbine are considered. These scenarios are generated with hybrid neural network and coati optimization algorithm. Also, the iterative fast progressive Kantorovich method is used for scenario reduction. The results show the better response of the proposed method in optimal operation of virtual power plant.

Application of Variational Iterations Method for Studying Physically and Geometrically Nonlinear Kirchhoff Nanoplates: A Mathematical Justification

We have proposed a development of the variational iteration method (VIM), or extended Kantorovich method, by studying physically nonlinear (FN) or geometrically nonlinear (GN) Kirchhoff nanoplates as an example. The modified couple stress theory was used for modeling size-dependent factors of the Kirchhoff nanoplates. Nested one into the other iteration procedures of the Birger method of variable elasticity parameters, of the variational iteration method (VIM), and of the Newton–Raphson method for physically nonlinear (FN) Kirchhoff nanoplates were constructed. The solution of problems for geometrically nonlinear (GN) Kirchhoff nanoplates was carried out on the basis of the variational iteration method and the Newton–Raphson method. The validity of the results was ensured by the coincidence of the results obtained via several methods of reducing partial differential equations to ordinary differential equations and via the finite difference method. The computational effectiveness of the proposed iterative procedure was demonstrated in terms of both accuracy and performance. A comparison of the results obtained showed that the variational iteration method (VIM) is the most efficient and fastest of all the methods considered both for problems with physical nonlinearity and for geometrically nonlinear problems.

An Approximation Solution of Linear Differential Equation using Kantorovich Methods

In our work, we constructed a numerical approximations method to deal with approximations of a linear differential equation. We explained the general framework of the projection method which helps to clarify the basic ideas of the Kantorovich methods. We applied the iterative projection methods and presented a theorem to show the convergence of the constructed solutions to the exact solution. Also, most of the expressions encountered earlier can be used to define functions. Here are some illustrations. A great deal of information can be learned about a functioning relationship by studying its graph. A fundamental objective of section 4, is to acquaint with the graphs of some important functions and develop basic graphing procedures.