## 1 Introduction
### Modeling in Continuum Mechanics  
Mathematical models describing a continuous medium usually constitute systems of integral or partial differential equations. This is due to the fact that in a continuous medium, variables are dependent on at least one spatial variable and time (if the object tested is variable in time). The mathematical model is supplemented with appropriate boundary and initial conditions. As a result, a boundary or boundary-initial problem is obtained. Only in simple problems is it possible to obtain a precise analytical solution to a given mathematical problem. Most often, the solution requires the use of approximate methods. In practice, approximate numerical methods are used most often.  
Numerical methods of an approximate solution of continuous boundary-initial problems include the finite difference method MRS [46], the method of boundary elements MEB [47], the finite element method FEM [48; 49; 50; 51; 52; 53], various meshless methods, such as the fuzzy particle method (smoothed particle hydrodynamics, SPH) [54], the free point method [55] and the material point method, also known as the material point method or the particle-in-cell-method [56; 57].  
The essence of the finite difference method is to replace derivatives with appropriate difference quotients determined on a discrete set of points. The finite difference method was initially applied to differential equations (local formulation), later the range of its applications was extended to problems formulated in variational forms (global formulation). The standard MRS formulation used regular (square, rectangular or cubic) node grids.  
In advanced formulations, an arbitrary mesh of nodes is used. Difference quotients can be built on arbitrarily generated points, owing to which this MRS formulation can be classified as a gridless method [58]. It is also used in the mechanics of solids, e.g., a well-known program for solving problems in geomechanics FLAC [59] is based on the finite difference method.  
In this paper, the finite element method (FEM) was used to discretize a continuous problem. The essence of FEM is a division of the continuous area under consideration into a finite number of sub-areas (finite elements) connected to each other at node points and an approximation of the solution in the area of elements using interpolation functions (shape functions) and values in nodes [60]. FEM equations are obtained by means of a formulation of an integral (global) problem using the principle of variation or the method of weighted residuals.  
The method of boundary elements [61] consists in reducing the system of differential equations with given boundary conditions to the system of integral equations defined on the boundary of the area considered. Contrary to the finite element method and the finite difference method, its application does not require any discretization of the interior of the area, but only its edge. This is the main advantage of this method.  
One of the disadvantages of using FEM is the need to generate an appropriate mesh of elements, which in the case of complex three-dimensional geometries is not easy for some types of elements. In an analysis of large deformations, the mesh of elements is subject to excessive distortions, which often makes it necessary to re-mesh the computational domain many times. Avoiding these problems is one of the reasons for the growing popularity of meshless methods. There are many meshless methods. The meshless finite difference method was mentioned above. Other meshless methods include the smoothed particle hydrodynamics method (SPH), the free point method, the finite point method [62] and the element-free Galerkin method [63]. The fuzzy particle method was initially developed for discrete problems in astrophysics [64]. Later, it was applied as a discretization method for continuous problems.