Kozak A. Numerical implementation of boundary integral equation method in problems of nonstationary vibrations of elastic elements of structures

New effective numerical techniques for research of nonstationary 2D dynamic fields of displacements and stresses in elastic media are developed in the dissertation on the basis of the theoretical apparatus of the potential method. After the computer implementation the techniques were applied for solving test, model and real technical problems. Nonstationary anti-plane shear vibrations of solids with holes were considered. Resolving relations are obtained and an algorithm for numerical solution of the problem using the time-dependent boundary integral equations method is constructed. On the basis of the developed algorithm the technique for the numerical analysis of dynamic stress-strain state parameters is developed and implemented in computer code. A new solution of the problem of nonstationary anti-plane shear vibrations of a continuum weakened by two cylindrical holes is obtained. The dependence of the numerical results on the distance between the holes is investigated. Studies of nonstationary vibrations of massive structural elements that are in a state of planar deformation have been carried out. Two techniques for numerical research of dynamic fields of displacements and stresses in such objects have been developed and implemented in computer codes. One technique uses as an algorithmic basis the relations of the time-dependent boundary integral equations method, and the other is based on the transition to the frequency domain by decomposing the load into a Fourier series. The same efficiency of both approaches is established. To implement the second method, a new numerical-analytical approach to the calculation of singular components of boundary integrals in problems of harmonic oscillations of elastic solids in the state of plane deformation is proposed and implemented. The approach is based on an approximate representation of the integral equations kernels components by an initial segment of the power series. The first term of this representation has the strongest singularity and coincides with the corresponding component of the static kernel, the integration of which is not difficult. The problem of mutual influence of two closely spaced cylindrical holes, the boundary of one of which is subjected to a pulse load, is solved. The applied problem of the dynamic stress-strain state of the elastic solid on which the pipeline rests is solved. The solid is in a state of plane deformation, and parts of its surface are exposed to semi-sinusoidal pulses. It is established that technological conditions of functioning of the pipeline are not broken as a result of the applied dynamic loading.