and Computational Methods

doi: 10.18698/2309-3684-2020-2-2645

The paper is devoted to the development of a method for calculating the nonlinear dielectric properties of composites with a periodic structure. Methods for predicting of the nonlinear dielectric properties of composites play an important role in the design of dielectric materials with specified properties, in particular for heterogeneous ferroelectrics, which are widely used to create various devices and electrical devices, for example, to create memory storage devices for computers. A quasi-static problem of the distribution of an electric charge in an inhomogeneous polarizable medium with a periodic structure and nonlinear dielectric properties is considered. To solve this nonlinear problem, the asymptotic homogenization method proposed by N.S. Bakhvalov, E. Sanchez-Palencia, B.E. Pobedria. As a result, local nonlinear problems of electrostatics on the periodicity cell are formulated, an algorithm for calculating effective nonlinear constitutive relations for dielectric properties, and an averaged problem for a composite with effective properties are proposed. For the case of a composite with a layered structure, the solution of local problems is obtained, and effective defining relations for the nonlinear dielectric properties of the composite are constructed. It is shown that a laminated composite is a transversely isotropic nonlinear dielectric material if it is isotropic materials. A numerical example of calculating the nonlinear properties of a 2-layer composite based on barium titanate and ferroelectric ceramic varicond VK4 is considered. A model is proposed that describes the nonlinear dependence of the dielectric constant of these materials on the vector of the electric field strength. It is shown that the nonlinear dependence of the dielectric constant tensor of the composite on the strength vector differs significantly for the direction of the field in the plane of the layers and in the transverse direction. It is shown that the developed technique can serve as a basis for designing nonlinear dielectric composite materials with anisotropic properties.

Димитриенко Ю.И., Губарева Е.А., Зубарев К.М. Моделирование нелинейных диэлектрических свойств композитов на основе метода асимптотической гомогенизации. Математическое моделирование и численные методы. 2020. № 2. с. 26–45

doi: 10.18698/2309-3684-2018-1-4154

The problem of the electrophysical parameter recovery of layered media, which is the inverse problem of mathematical physics, is solved on the basis of the electromagnetic field measurement results. Various optimization methods for its solution are formulated. The mathematical model is proposed for a horizontally layered medium with specified parameters consistent with real values. The algorithm is developed for solving the direct problem allowing finding an analytical solution for various environmental parameter values. For solving inverse problems the complete enumeration and Hook - Jeeves methods as well as the developed modified method of complete enumeration are used. According to the results of solving the direct problem, the characteristic features of the medium are found for various values of the electrophysical parameters. When solving the inverse problem using various optimization methods, the features of each algorithm are described.

Краснов И.К., Зубарев К.М., Иванова Т.Л. Численное решение задачи восстановления электрофизических параметров по результатам зондирования переменным током. Математическое моделирование и численные методы, 2018, № 1, с. 41-54