# Numerical analysis and Simulation

#### At the present, the computer modeling of physical processes in engineering practice is often based on the finite element method (FEM) and similar approaches involving a grid approximation of the domain under consideration. These methods are implemented in the form of commercial software packages (ANSYS, Abacus, Comsol Multiphysics, etc.). They are considered as a universal modeling tool applicable to the objects of arbitrary shape and with arbitrary heterogeneity of physical properties. The IMMI team has experience in using and studying some finite element packages, and has now started to create its own procedures for FEM. In some aspects, the grid methods are poorly applicable to the propagation and diffraction of traveling waves problems. The complex spatial structure of wave fields requires many points for their satisfactory approximation. As a result, the using of FEM with extended waveguide structures, which are of primary interest in the study of traveling waves, leads to a fast increase in the required number of elements, which makes it too costly. In addition, FEM doesn’t give a clear physical representation of the wave structure of the solution.

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#### As an alternative, there are classical analytical and semi-analytical methods for solving wave problems, which give a physically visual description of the wave structure of the solution at almost negligible computational cost. For the development of semi-analytical models, IMMI uses application packages for analytical processing (Wolfram Mathematica, Maple, Matlab, Fortran, C++, Python). The gap between grid approximations and modal analysis closes the integral approach based on the representation of wave fields as path integrals containing the Greens matrix of the layered structure under consideration. Discretization of such boundary integral representations is equivalent to approximation of the solution within the boundary element method. This approach, called the layered elements method (LEM), as well as the integral approach, is actively developed by IMMI team. However, the field of its applicability in terms of geometry and elastic properties is still significantly narrower than that of FEM. In addition, the using of LEM in the case of obstacles with non-horizontal or non-vertical boundaries (inclined cracks, elliptical inclusion, etc.) leads to complex integral representations, their conclusion requires rigorous analytical calculations and high qualification of the researcher carrying out its computer implementation. To date, effective mathematical models based on the integral approach have already been developed and constructed. In addition, procedures have already been implemented that construct Green’s matrices and calculate wave fields in layered composites without defects for surface loads specified in areas of classical shape (stamp and perfect adhesion to the surface).