Vol 26, No 1 (2022)

An initial-boundary value problem is studied for a multidimensional loaded parabolic equation of general form with boundary conditions of the third kind. For a numerical solution, a locally one-dimensional difference scheme by A.A. Samarskii with order of approximation $O{h}^2+\tau$ is constructed. Using the method of energy inequalities, we obtain a priori estimates in the differential and difference interpretations, which imply uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem in the $L_2$ norm at a rate equal to the order of approximation of the difference scheme. An algorithm for the computational solution is constructed and numerical calculations of test cases are carried out, illustrating the theoretical calculations obtained in this work.

The paper deals with covariant constancy problem for tensors and pseudotensors of an arbitrary rank and weight in an Euclidean space. Requisite preliminaries from pseudotensor algebra and analysis are given. The covariant constancy of pseudotensors are separately considered. Important for multidimensional geometry examples of covariant constant tensors and pseudotensors are demonstrated. In particular, integer powers of the fundamental orienting pseudoscalar satisfied the condition of covariant constancy are introduced and discussed. The paper shows that the distortion and inversed distortion tensors are not actually covariant constant, contrary to the statements of those covariant constancy which can be found in literature on continuum mechanics.

A polar-symmetric elastic diffusion problem is considered for an orthotropic multicomponent homogeneous cylinder under uniformly distributed radial unsteady volumetric perturbations. Coupled elastic diffusion equations in a cylindrical coordinate system is used as a mathematical model. The model takes into account a relaxation of diffusion effects implying finite propagation speed of diffusion perturbations.

The solution of the problem is obtained in the integral convolution form of Green’s functions with functions specifying volumetric perturbations. The integral Laplace transform in time and the expansion into the Fourier series by the special Bessel functions are used to find the Green’s functions. The theory of residues and tables of operational calculus are used for inverse Laplace transform.

A calculus example based on a three-component material, in which two components are independent, is considered. The study of the mechanical and diffusion fields interaction in a solid orthotropic cylinder is carried out.

A search for optimal damping properties of structures using methods of numerical modelling is as a rule associated with a large number of computations. Alongside this an application of mechanical problem of natural vibrations of structures for this purpose allows estimating damping properties of structures regardless external force and kinematic impacts. This fact leads to sufficient decrease in computational costs. The results of the solution to the problem of natural vibrations of piecewise-homogeneous viscoelastic bodies are complex natural vibration frequencies, the real part of which is a frequency of vibrations and imaginary part is damping index (rate of vibration damping). A mechanical behavior of a viscoelastic material is described by the linear theory of Boltzman–Volterra. Within the frameworks of this theory mechanical properties of a viscoelastic material can be represented as complex dynamic moduli (shear modulus and bulk modulus). As a rule, these properties depend on frequency of external excitation. In current paper an algorithm which allows obtaining solution to the problem on natural vibrations, in case when components of complex dynamic moduli are frequency-dependent, is represented. The algorithm is based on using capabilities of the ANSYS software package and also the Mueller's method which allows solving partial problem of complex eigenvalues. An efficiency and productivity of the algorithm is demonstrated on the example of a two-layered cantilever plate. One layer of the plate is made of an elastic material and the second one is made of a viscoelastic material. Reliability of the obtained results is proved by comparison natural vibration frequencies obtained as a result of solution to the problem of natural vibrations and resonant frequencies at frequency response plots of the displacements obtained as a result of solution to the problem of forced steady-state vibrations using the ANSYS software package.

The paper considers the previously proposed method of numerical integration using the matrix calculus in the study of boundary value problems for nonhomogeneous linear ordinary differential equations of the second order with variable coefficients. According to the indicated method, when compiling a system of difference equations, an arbitrary degree of the Taylor polynomial in expanding the unknown solution of the problem into a Taylor series can be chosen while neglecting the approximation of the derivatives by finite differences.

Some aspects of the convergence of an unstable second-order difference boundary value problem are investigated. The concept of a pseudo-residual on a certain vector is introduced for an ordinary differential equation. On the basis of the exact solution of the difference boundary value problem, an approximate solution has been built, where the norm of pseudo-residuals is different from the trivial value.

It has been established theoretically that the estimate of the pseudo-residual norm decreases with an increase in the used degree of the Taylor polynomial and with a decrease in the mesh discretization step. The definitions of conditional stability and conditional convergence are given; a theoretical connection between them is established. The perturbed solution has been built on the basis of the found vector of pseudo-residuals, the estimate of the norm of its deviation from the exact solution of the difference boundary value problem has been calculated, which allows one to identify the presence of conditional stability. A theoretical relationship between convergence and conditional convergence is established.

The results of numerical experiments are presented.

In the framework of the Navier–Stokes equations, the flow of a viscous incompressible fluid between immovable parallel permeable walls is considered, on which only the longitudinal velocity component is equal to zero. Solutions are sought in which the velocity component transverse to the plane of the plates is constant. Both stationary and non-stationary solutions are obtained, among which there is a non-trivial solution with a constant pressure and a longitudinal velocity exponentially decaying with time. These solutions show the influence on the profile of the horizontal velocity component of the removal of the boundary layer into the depth of the flow from one plate with simultaneous suction of the boundary layer on the other plate. It is established that for stationary flows the removal of the boundary layer into the depth of the flow from one plate and, with simultaneous suction of the boundary layer on the other plate, leads to an increase in the drag compared to the classical Poiseuille flow. In the case of impermeable walls, an exact non-stationary solution is obtained, the velocity profile of which at fixed times differs from the profile in the classical Poiseuille flow and, in the limit (as time tends to infinity), corresponds to rest.