Continuation Method For Solution Research Articles

Estimating the Domain of Absolute Stability of a Numerical Scheme Based on the Method of Solution Continuation with Respect to a Parameter for Solving Stiff Initial Value Problems

Estimating the Domain of Absolute Stability of a Numerical Scheme Based on the Method of Solution Continuation with Respect to a Parameter for Solving Stiff Initial Value Problems

Estimating the Domain of Absolute Stability of a Numerical Scheme Based on the Method of Solution Continuation with Respect to a Parameter for Solving Stiff Initial Value Problems

The modeling of physical and technological processes often involves solving stiff initial value problems. In most cases, their exact solution is difficult to find, while numerical schemes sometimes fail to produce a sufficiently accurate solution in acceptable computation time. Moreover, for some classes of problems, numerical solution schemes are unsuitable because of their insufficient stability. This paper deals with numerical methods based on solution continuation with respect to arguments of various types that make it possible to enhance the stability of explicit numerical schemes. Most frequently, the used best argument is hardly applicable to problems in which the integral curves grow at a superpower or nearly exponential rate. Previously, the authors proposed a modification of the best argument that alleviates these disadvantages. In the present paper, we estimate the domain of absolute stability of the explicit Euler scheme as applied to problems transformed to a modified best argument of special form and refine the proof of a similar estimate for initial value problems transformed to the best argument. A test initial value problem is used to verify the resulting theoretical estimates and to analyze the application of the modified best argument of solution continuation.

Passage through Limiting Singular Points by Applying the Method of Solution Continuation with Respect to a Parameter in Inelastic Deformation Problems

Numerical methods are developed for solving the Cauchy problem for systems of ordinary differential equations with a single limiting singular point lying on the right boundary of the considered range of the argument. Such initial value problems arise in a variety of areas, such as inelastic deformation of metal structures at various temperatures and stresses under creep conditions, the computation of strength characteristics and the estimation of residual strain in the design of nuclear reactors, the construction and aerospace industries, and mechanical engineering. Since such initial value problems are ill-conditioned, their numerical solution faces considerable difficulties. Conventional explicit methods can be used for problems of this class, but only outside a neighborhood of limiting singular points, where the error of the numerical solution grows sharply. As a result, the step size has to be reduced significantly, which increases the computation time and makes explicit methods computationally expensive. For the numerical solution, it is possible to use the method of solution continuation, whereby the original argument of the problem is replaced by a new one such that the transformed problem is better conditioned. However, the best argument does not give desirable numerical advantages in this case, since the form of the original problem is complicated considerably. As a more suitable approach, we propose a specialized argument for solution continuation, which is called a modified best argument. For tubular samples of X18H10T steel subjected to tension to fracture, the advantages of the method of solution continuation with respect to a modified best argument are demonstrated by comparing it with traditional explicit methods for the Cauchy problem and with the best parametrization. The reliability of the results is confirmed by comparing them with experimental measurements and data of other authors.

Numerical solving of initial value problems for ordinary differential equations with internal limiting singular points

In the present paper, we investigate the numerical methods for solving initial value problems for systems of nonlinear ordinary differential equations with several limiting singular points. Similar tasks have great practical importance in such areas as hydrodynamics, chemical kinetics, combustion theory, computational geometry, and nuclear reactors. However, the exact solution may be constructed to nonlinear initial value problems only in exceptional cases. Numerical methods also do not give the desired result. Explicit methods make it possible to obtain a solution only if there is one limiting singular point on the right boundary of the considered interval of the argument change. Implicit methods also have usage restrictions. They consist of the difficulty or impossibility of overcoming the internal limiting singular points. For solving the considered class of problems, the paper proposes to use the method of solution continuation, which allows passing any limiting singular points effectively. Features of the application of various numerical methods are shown by solving one test initial value problem. The reliability of the obtained numerical results is confirmed by comparison with the finding analytical solution of the considered test problem.

Simulation of Nonlinear Oscillations in a Micro-Generator of Clock Frequency

In this paper, we consider a mathematical model of a new-type micro- generator. The model is based on excitation by electrostatic forces of oscillations of a mobile electrode in a micro gap. The principle of operation of the micro-generator is similar to the well-known theory of clock with a striking escapement mechanism. The difference is that in the equation of motion, the form of the right-hand side takes into account the electrostatic nature of the pulsed action. The numerical analysis shows that with time the bounded oscillations in the phase plane tend to a stable limit cycle, and thus the emerging oscillations are stable towards external perturbations. Studying periodic oscillations in dependence on model parameters relies on the boundary value problem solution for an equation with a discontinuous right-hand side transformed so that to enable application of the method of solution continuation with respect to a parameter. In this way, the domain in the plane of model parameters in which stable limit cycles exist is defined.

МЕТОД НЕПРЕРЫВНОГО ПРОДОЛЖЕНИЯ РЕШЕНИЯ ПО НАИЛУЧШЕМУ ПАРАМЕТРУ ПРИ РАСЧЕТЕ ОБОЛОЧЕЧНЫХ КОНСТРУКЦИЙ

МЕТОД НЕПРЕРЫВНОГО ПРОДОЛЖЕНИЯ РЕШЕНИЯ ПО НАИЛУЧШЕМУ ПАРАМЕТРУ ПРИ РАСЧЕТЕ ОБОЛОЧЕЧНЫХ КОНСТРУКЦИЙ

The Parametrization of the Cauchy Problem for Nonlinear Differential Equations with Contrast Structures

Introduction. The paper provides an analysis of numerical methods for solving the Cauchy problem for nonlinear ordinary differential equations with contrast structures (interior layers). Similar equations simulate various applied problems of hydro- and aeromechanics, chemical kinetics, the theory of catalytic reactions, etc. An analytical solution to these problems is rarely obtained, and numerical procedure is related with significant difficulties associated with ill-conditionality in the neighborhoods of the boundary and interior layers. The aim of the paper is the scope analysis of traditional numerical methods for solving this class problems and approbation of alternative solution methods. Materials and methods. The traditional explicit Euler and fourth-order Runge-Kutta methods, as well as the implicit Euler method with constant and variable step sizes are used for the numerical solution of the Cauchy problem. The method of solution continuation with respect to the best argument is suggested as an alternative to use. The solution continuation method consists in replacing the original argument of the problem with a new one, measured along the integral curve of the problem. The transformation to the best argument allows obtaining the best conditioned Cauchy problem. Results. The computational difficulties arising when solving the equations with contrast structures by traditional explicit and implicit methods are shown on the example of the test problem solution. These difficulties are expressed in a significant decrease of the step size in the neighborhood of the boundary and interior layers. It leads to the increase of the computational time, as well as to the complication of the solving process for super stiff problems. The authenticity of the obtained results is confirmed by the comparison with the analytical solution and the works of other authors. Conclusions. The results of the computational experiment demonstrate the applicability of the traditional methods for solving the Cauchy problem for equations with contrast structures only at low stiffness. In other cases these methods are ineffective. It is shown that the method of solution continuation with respect to the best argument allows eliminating most of the disadvantages inherent to the original problem. It is reflected in decreasing the computational time and in increasing the solution accuracy. Keywords: contrast structures, method of solution continuation, the best argument, illconditionality, the Cauchy problem, ordinary differential equation For citation: Kuznetsov E. B., Leonov S. S., Tsapko E. D. The Parametrization of the Cauchy Problem for Nonlinear Differential Equations with Contrast Structures. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2018; 28(4):486–510. DOI: https://doi.org/10.15507/0236-2910.028.201804.486-510 Acknowledgements: This work was supported by the Russian Science Foundation, project no. 18-19-00474.

Strength of orthotropic cylindrical panels with account for geometric nonlinearity

This paper describes the influence of geometric nonlinearity in the analysis of the strength of orthotropic cylindrical panels. The values of maximum permissible loads in the linear and nonlinear versions of calculations of structures made of unidirectional carbon plastics are given, and the loads at which stability loss occurs are determined. The mathematical model accounts for transverse shifts and geometric nonlinearity and stated as the full potential strain energy functional. The calculations are carried out on the basis of the method of solution continuation with respect to the parameter. The strength is estimated by using the maximum stress criterion.

Strength and stability of geometrically nonlinear orthotropic shell structures

The article presents a methodology for the study of shell structure strength and stability. The basis of the study is a geometrically nonlinear mathematical model, which takes into account the transverse shifts and orthotropy of material. The model is presented in dimensionless parameters in the form of the total energy potential functional and can be used for different types of shells of revolution.The model is studied by using an algorithm based on the Ritz method and the method of solution continuation according to the best parameter (MSCBP), which allows for obtaining the values of the upper and lower critical loads and examining the supercritical behavior of designs.In accordance with an algorithm, the computer program has been developed and a comprehensive study of the strength and stability of shallow shells (which are square in plane), cylindrical, and conical panels has been explored. The load loss of strength and buckling load values have been obtained, and their relationship to one another has been demonstrated.