Адаптивне гібридне чисельне моделювання хвильових процесів у багатошарових структурах на основі методів TMM та FEM

The object of research in this work is wave processes in multilayer thin films and methods of their numerical modeling using adaptive hybrid models. The research covers multilayer media with gradient distribution of physical parameters, including inhomogeneities. The problem addressed in this study is the enhancement of the accuracy and efficiency of numerical modeling of wave processes in complex multilayered structures while reducing computational costs. Traditional methods, such as the transfer matrix method or the finite element method, have limitations related to computational complexity, numerical stability, and the ability to account for intricate geometric features. The essence of the obtained results lies in the development and software implementation of an adaptive hybrid model that combines the transfer matrix method for wave propagation calculations in homogeneous regions and the finite element method for modeling complex geometries. The proposed approach optimizes computational resources by dynamically adjusting the grid resolution according to local variations in the refractive index. The use of adaptive discretization reduced the number of computational points by 40 % without compromising calculation accuracy. The relative error of the results obtained using the proposed model does not exceed 1 %, demonstrating its high precision. The achieved results can be attributed to the implementation of efficient adaptive algorithms that automatically adjust the grid resolution based on the gradient of physical parameters, as well as the application of consistent boundary conditions between computational domains using different methods. This ensures a smooth transition between different modeling zones and minimizes numerical errors at domain boundaries. The practical applications of these findings include optical technologies for the design and optimization of photonic devices, sensors, anti-reflective coatings, and nanostructured materials. The model can be utilized for the analysis of complex multilayered systems in nanotechnology, biomedical research, and the design of micro-optical elements. It is particularly useful in scenarios where it is necessary to account for structural inhomogeneities, complex geometries, and boundary conditions while maintaining minimal computational costs.