Numerical Solution of the Two-Phase Stefan Problem in the Enthalpy Formulation with Smoothing the Coefficients

To simulate heat transfer processes with phase transitions, the classical enthalpy model of Stefan is used, accompanied by phase transformations of the medium with absorption and release of latent heat of a change in the state of aggregation. The paper introduces a solution to the two-phase Stefan problem for a one-dimensional quasilinear second-order parabolic equation with discontinuous coefficients. A method for smearing the Dirac delta function using the smoothing of discontinuous coefficients by smooth functions is proposed. The method is based on the use of the integral of errors and the Gaussian normal distribution with an automated selection of the value of the interval of their smoothing by the desired function (temperature). The discontinuous coefficients are replaced by bounded smooth temperature functions. For the numerical solution, the finite difference method and the finite element method with an automated selection of the smearing and smoothing parameters for the coefficients at each time layer are used. The results of numerical calculations are compared with the solution of Stefan’s two-phase self-similar problem --- with a mathematical model of the formation of the ice cover of the reservoir. Numerical simulation of the thawing effect of installing additional piles on the existing pile field is carried out. The temperature on the day surface of the base of the structure is set with account for the amplitude of air temperature fluctuations, taken from the data of the Yakutsk meteorological station. The study presents the results of numerical calculations for concrete piles installed in the summer in large-diameter drilled wells using cement-sand mortars with a temperature of 25 °С. The distributions of soil temperature are obtained for different points in time

Penalization for a PDE with a nonlinear Neumann boundary condition and measurable coefficients

We consider a system of semilinear partial differential equations (PDEs) with measurable coefficients and a nonlinear Neumann boundary condition. We then construct a sequence of penalized PDEs, which converges to our initial problem. Since the coefficients we consider may be discontinuous, we use the notion of solution in the [Formula: see text]-viscosity sense. The method we use is based on backward stochastic differential equations and their [Formula: see text]-tightness. This work is motivated by the fact that many PDEs in physics have discontinuous coefficients. As a consequence, it follows that if the uniqueness holds, then the solution can be constructed by a penalization.

Stability analysis for the Lienard equation with discontinuous coefficients

A nonlinear mechanical system, whose dynamics is described by a vector ordinary differential equation of the Lienard type, is considered. It is assumed that the coefficients of the equation can switch from one set of constant values to another, and the total number of these sets is, in general, infinite. Thus, piecewise constant functions with infinite number of break points on the entire time axis, are used to set the coefficients of the equation. A method for constructing a discontinuous Lyapunov function is proposed, which is applied to obtain sufficient conditions of the asymptotic stability of the zero equilibrium position of the equation studied. The results found are generalized to the case of a nonstationary Lienard equation with discontinuous coefficients of a more general form. As an auxiliary result of the work, some methods for analyzing the question of sign-definiteness and approaches to obtaining estimates for algebraic expressions, that represent the sum of power-type terms with non-stationary coefficients, are developed. The key feature of the study is the absence of assumptions about the boundedness of these non-stationary coefficients or their separateness from zero. Some examples are given to illustrate the established results.

SYNTHESIS OF HIGH-CLASS MECHANISMS OF VARIABLE STRUCTURE BASED ON DYNAMIC ANALYSIS

This paper presents the model of a high class variable structure mechanism. These mechanisms have not gained widespread acceptance in practice, despite obvious improvements in the ability to transmit motion and power. The article presents a synthesized high class mechanism. The block diagram of the formation of the mechanism is shown, the results of synthesis are presented. It is shown that with a relatively compact kinematic scheme, it is possible, with one drive, to create a system with several working links providing various technological operations, that is, on the basis of one mechanism, it becomes possible to create automatic mechanisms. In these studies, the development of a methodology for the dynamic design of the variable structure mechanism based on the analysis of dynamics and assessment of the influence of nonlinear factors on the quality of the machine as a whole was carried out. To study the motion of such variable structure mechanism, mathematical modeling of dynamics with discontinuous coefficients and nonlinear external forces is used.