Protektor D. Non-stationary thermal processes in anisotropic solids.

The dissertation deals with the study of thermal processes in anisotropic solids by meshless method for solving three-dimensional non-stationary heat conduction problems. This dissertation presents a developed meshless method for solving three-dimensional non-stationary heat conduction problems in anisotropic solids. The family of the atomic radial basis functions of three independent variables, generated by Helmholtz-type operator was constructed. The functions extend the subclass of functions that are used as basis for the implementation of the meshless method for solving three-dimensional non-stationary heat conduction problems in anisotropic solids. The special properties of the function (infinite differentiability and compact support) had improved the accuracy and computational efficiency of the developed meshless method. The results of numerical solutions of non-stationary heat conduction problems in anisotropic solids of various geometric shapes in the presence of internal heat sources or sinks and various initial and boundary conditions obtained using the developed meshless method are demonstrated. Approximate solutions for the three-dimensional non-stationary heat conduction problem in the solid in the form of a cube in the presence of an internal heat source for the isotropic and anisotropic cases are obtained. The application of atomic functions as basis in the simulation of the non-stationary heat conduction in an anisotropic solid in the form of a plate in the presence of a moving point heat source is presented. Analytical solutions of these non-stationary heat conduction problems are obtained. The accuracy of approximate solutions of boundary-value problems is estimated by the values of the average relative error, the average absolute error, and the maximum error. A comparative analysis of the efficiency of approximation of non-stationary heat conduction problems by anisotropic radial basis functions and the atomic radial basis functions is performed. The software «AnisotropicHeatTransfer3D» for simulation of three-dimensional non-stationary heat conduction problems in anisotropic solids by the meshless method was developed, which is confirmed by the certificate of state registration of copyright for the work. The software «AnisotropicHeatTransfer3D» has been tested by numerical solution of the considered three-dimensional non-stationary heat conduction problems in anisotropic solids. The scientific novelty of the results of the dissertation research is as follows: for the first time the meshless method for solving three-dimensional non-stationary heat conduction problems in anisotropic solids based on a combination of the dual reciprocity method using anisotropic radial basis function and the method of fundamental solutions was developed; for the first time the distribution of non-stationary temperature fields in anisotropic solids of different geometric shapes in the presence of internal heat sources or sinks by meshless method was simulated; for the first time the family of atomic radial basis functions of three independent variables, generated by a Helmholtz-type differential operator, which includes the symmetric positive definite tensor of the second rank, that determines the anisotropy of the material was constructed; for the first time an algorithm for constructing the family of atomic radial basis functions of three independent variables, generated by a Helmholtz-type differential operator, which includes the symmetric positive definite tensor of the second rank, that determines the anisotropy of the material was programmatically implemented; for the first time the family of atomic radial basis functions of three independent variables was used as basis in the implementation of the meshless method for the numerical solution of heat conduction problems in anisotropic solids; for the first time the distribution of non-stationary temperature field in the anisotropic solid in the form of a plate in the presence of a moving point heat source by meshless method based on the use of atomic radial basis functions was simulated; for the first time the model of heat transfer in LiNbO3 single-crystal in the form of cylinder in interaction with continuous-wave laser radiation with the wavelength of 1064 nm at a time interval of 2 h 30 min by the meshless method was studied and the time required to achieve the steady-state heating mode of the LiNbO3 single-crystal was determined; for the first time the software «AnisotropicHeatTransfer3D» for simulation of non-stationary heat conduction in anisotropic solids as a GUI application, based on the use of the meshless method and the atomic radial basis function was developed, which is confirmed by the certificate of state registration of copyright for the work.