For the calculation of flat shells by the numerical analytical method of boundary elements

The application of the numerical-analytical boundary elements method (NA BEM) to the calculation of shallow shells is considered. The method is based on the analytical construction of the fundamental system of solutions and the Green’s function for the differential equation of the problem under consideration. The theory of calculation of a shallow shell proposed by V. Z. Vlasov, which for the problem under consideration leads to an eighth-order partial differential equation. The problem of bending a shallow shell is two-dimensional, and in the numerical-analytical boundary elements method, the plate and shell are considered in the form of generalized one-dimensional modules, therefore, the Fourier separation method and the Kantorovich-Vlasov variational method were applied to this equation, which made it possible to obtain ordinary differential equations of the eighth order. It is noted that until recently, the main problem in the subsequent implementation of the algorithm of the numerical-analytical boundary element method was due to the fact that all analytical expressions of the method (fundamental functions, Green’s functions, vectors of external loads) are very cumbersome, and intermediate transformations are associated with determinants of the eighth order. It is proposed to use the direct integration method at the first stage, when, along with the original differential equation, an equivalent system of equations for the unknown shell state vector is considered. In this case, the calculations of some analytic expressions associated with determinants of higher orders can be avoided by using the Jacobi formula. As a result, the calculation of the determinant at an arbitrary point is reduced to its calculation at a zero value of the argument, which leads to a significant simplification of all intermediate transformations and analytical expressions of the numerical-analytical boundary elements method.

Optimization of the Geometric Parameters of the Thermal Insulation of the Heating System by Using Multi-Pipe Circular Thermal Insulation

The publication presents a design solution for circular multi-pipe thermal insulation and an example of an existing heating installation consisting of six individual heating pipes in the building of the Bialystok University of Technology. In the paper, the arrangement of six heating system pipes in circular thermal insulation was designed in such a way that one heating pipe is centrally located in the circular thermal insulation, the other five heating pipes are located at the vertices of a regular pentagon inside the circular thermal insulation. Heat loss calculations were made using the Boundary Elements Method (BEM) with the actual boundary conditions in the room where the existing heating installation is located. Additionally, the ecological effect was determined in the form of reduction of pollutants emitted into the atmosphere resulting from heat losses for the developed multi-pipe thermal insulation. The calculation results showed a significant reduction in heat losses as a result of the use of multi-pipe thermal insulation in relation to the existing single heating installation. The use of multi-pipe insulation undoubtedly follows the trend of energy-saving heat transport and is an alternative to the commonly used single pipes.

The solution of the plane problem of the theory of elasticity by the boundary elements method

An approach to solving the biharmonic equation of the plane problem of the theory of elasticity by the numerical-analytical method of boundary elements is developed. The reduction of the two-dimensional problem to the one-dimensional one was carried out by the KantorovichVlasov method. Systems of fundamental orthonormal functions and the Green function are constructed without any restrictions on the nature of the boundary conditions. A numerical example of solving a plane problem by the boundary element method for a rectangular plate is considered. The results are compared with the data of finite element analysis in the ANSYS program and those obtained by A.V. Aleksandrov.

The solution of the problem of free circulation of circular arcs by numerical analytical boundary elements method

The work is devoted to determining the natural frequencies and vibration modes of a circular arch using the numerical-analytical boundary elements method. A differential equation of the natural vibrations of the arch is obtained, a complete system of its fundamental solutions is determined, a transcendental frequency equation is constructed, the arch is calculated based on the obtained analytical dependencies, and the results are compared with the results of finite element analysis. For 10 possible combinations of the roots of the equation of the characteristic equation, 360 fundamental functions are calculated. An example of calculating a circular arch for free vibrations by the method of the authors is considered. The search for the roots of the transcendental frequency equation can be carried out by the method of successive approximations, using any programming environment. Here used MATLAB. As a result of the calculation, the first five frequencies of the natural vibrations of the arch were obtained. Their comparison with those calculated in the ANSYS program shows that the spectrum of natural frequencies calculated by the boundary element method is slightly lower (except for the first frequency) than the spectrum calculated by the finite element method, which indicates greater reliability of the results of the boundary element method.

The Solution of the Shells Theory Problems by the Numerical-Analytical Boundary Elements Method

The application of the numerical-analytical boundary elements method (NA BEM) to the calculation of shells is considered. The main problem here is due to the fact that most of the problems of statics, dynamics and stability of shells are reduced to solving an eighth-order differential equation. As a result, all analytical expressions of the NA BEM (fundamental functions, Green functions, external load vectors) turn out to be very cumbersome, and intermediate transformations are associated with eighth-order determinants. It is proposed along with the original differential equation to consider an equivalent system of equations for the unknown state vector of the shell. In this case, calculations of some analytical expressions related to high-order determinants can be avoided by using the Jacobi formula. As a result, the calculation of the determinant at an arbitrary point reduces to its calculation at the point , which leads to a significant simplification of all analytical expressions of the numerical-analytical boundary elements method. On the basis of the proposed approach, a solution is obtained of the problem of bending a long cylindrical shell under the action of an arbitrary load, the stress-strain state of which is described by an eighth-order differential equation. The results can be applied to other types of shells.