FINITE ELEMENT ANALYSIS OF THE DIFFUSION MODEL OF THE BIOCLOGGING OF THE GEOBARRIER

The distribution of an organic chemical and the filtration process in the soil which contains a thin geochemical barrier are considered. Microorganism colonies develop in the presence of organic chemicals in the soil which leads to the so-called phenomenon of bioclogging of the pore space. As a result, the conductivity characteristics of both the soil as a whole and the geochemical barrier change. Conjugation conditions as a component of the mathematical model of chemical filtration in the case of inhomogeneity of porous media and the presence of fine inclusions were modified for the case of bioclogging. The numerical solution of the corresponding nonlinear boundary value problem with modified conjugation conditions was found by the finite element method. The conditions of the existence of a generalized solution of the corresponding boundary value problem are indicated. The results on the theoretical accuracy of finite element solutions are presented. Differences in the value of pressure jumps at a thin geochemical barrier were analyzed for the case considered in the article and the classical case on a model example of filtration consolidation of the soil in the base of solid waste storage. The excess pressure in 600 days after the start of the process reaches 25 % of the initial value when taking into account the effect of bioclogging, while is only 6 % for the test case disregarding the specified effect.

Interfacing conditions in the pipeline mathematical model with a kink of profile

Сформулирована замкнутая краевая задача расчета напряженно-деформированного состояния трубопровода как оболочки Власова с линией излома поверхности. Выведены разрешающие уравнения оболочки в перемещениях в избранной криволинейной системе координат; в локальных координатах, связанных с линией излома, выведены кинематические условия сопряжения; на линии излома поверхности наложены и доказаны условия сопряжения для моментов и усилий в оболочке. Условия сопряжения выведены в перемещениях оболочки на линии излома, не являющейся координатной линией. Доказано наличие сингулярности в условиях сопряжения. Установлена согласованность результатов численного анализа с известными результатами. A closed-ended formulation of the boundary-value problem of calculating the pipeline stress-strain state as a Vlasov shell with a kink line of surface was given. The resolving equations of the shell in displacements in the chosen curvilinear coordinate system were derived; in the local coordinates associated with the kink line, the kinematic conjugation conditions on this line were derived; conjugation conditions for moments and forces in the shell on the surface kink line were stated and proved. All conjugation conditions were deduced in the displacements of the shell on the kink line, which is not a coordinate line. The presence of a singularity in the obtained conjugation conditions was proved. The consistency of the numerical analysis results with known results was established.

ПРО КРАЙОВІ ЗАДАЧІ ДЛЯ РІВНЯННЯ ПУАССОНА У БАГАТОЛИСТІЙ ОБЛАСТІ, СКЛАДЕНІЙ ІЗ РІЗНИХ КРУГОВИХ СЕГМЕНТІВ

The non-classical boundary problem of the mathematical physics for the two-dimensional Poisson equation is considered. As the area, in which the solution is sought, the area, made up of different circular segments, folded into a multi-sheet plate of a book structure, is taken. All sheets are different from each other, both in their physical properties and in geometric dimensions, and are interconnected by a chord common to all sheets. The problem statement is given and its exact solution is obtained.The solution to the problem is considered in bipolar coordinate systems, each of which is associated with one of the segments. In this case, all coordinate systems have a common parameter - the length of the rectilinear segment boundary. As a method for solving the problem, the classical method of separation of variables is used – the Fourier method. Although the Dirichlet problem is considered as a basic one, however, the proposed method can be applied in the case when conditions of other types are given on the arcs of separate circles: Neumann or the third main problem.The statement of the considered problem differs from the classical one in that the conjugation conditions of fields on the line of connection of segments are added to the traditional boundary conditions. These conditions represent the equality of the values of the functions and the equality to zero of the sum of linear combinations of their normal derivatives. The solution is constructed (selected) in such a way that the first of the field conjugation conditions is fulfilled automatically for any choice of unknown functions. The boundary conditions on the segments and the second conjugation condition make it possible to determine all the unknown functions of the problem. To apply the Fourier method, it is necessary that all boundary functions are equal to zero at the corner points of the segments. If this condition is violated, a modification of the method that allows one to obtain an exact solution in this case is proposed. As an application, such problems are considered: a) on the torsion of a composite rod, the cross-section of which is two different segments; b) the stationary heat conductivity problem for two glued half-segment with sources of heat inside the area. Exact analytical solutions to these new problems have been obtained.

Mixed problem for a one-dimensional wave equation with conjugation conditions and second-order derivatives in boundary conditions

In this paper, we consider the boundary problem for the half-strip on the plane for the case of two independent variables. This mixed problem is solved for a one-dimensional wave equation with Cauchy conditions on the basis of the half-strip and boundary conditions for lateral parts of the area border containing second-order derivatives. Moreover, the conjugation conditions are specified for the required function and its derivatives for the case when the homogeneous matching conditions are not satisfied. A classical solution to this problem is found in an analytical form by the characteristics method. This solution is approved to be unique if the relevant conditions are fulfilled.

Inhomogeneous boundary-value problem of radiation heat transfer for a multicomponent medium

An analysis of the solvability of an inhomogeneous boundary value problem for the equations of radiative heat transfer with the Fresnel conjugation conditions is presented. The nonlocal unique solvability of the boundary value problem is proved.

Calculation of thin-walled axisymmetrically loaded structures of the AIC taking into account FEM-based physical nonlinearity

The article describes an algorithm for calculating an axisymmetrically loaded shell structure with a branching meridian, taking into account elastic-plastic deformations when loading based on the deformation theory of plasticity without assuming that the material is incompressible during plastic deformations. The correct relations which determine the static conjugation conditions of several revolution shells in the joint assembly are used. A comparative analysis of finite element solutions is presented for various options plasticity matrix development at the loading stage.