 Discretization
 Initially, all objects of structural mechanics represent a continuous environment. As a rule, it is not possible to determine SSS parameters (displacements, stresses, forces) in all points of a continuous environment, since the number of these points is generally infinite. All numerical methods are based on the assignment of a finite number of points at which the desired SSS parameters are located. Replacement of an infinite number of points of a continuous environment by a finite number of points (nodes) is called discretization.
 Triangulation
 The continuous environment is replaced by a set of subareas (triangular, quadrangular, tetrahedron, parallelepiped, etc.) with assigned nodes at their vertices. Thus, the continuous environment is replaced by a discrete environment consisting of individual subareas. Such replacement is called TRIANGULATION (TRIANGULATION  fr om Latin triangulus triangle  determining the mutual location of points on a surface by means of constructing a network of triangles).
The construction of finite element meshes is an important stage of solving the problem of determining the SSS of structures. This stage is associated with satisfaction of a number of contradictory requirements.
On the one hand, a sufficiently dense mesh allows to achieve the required accuracy of the problem solution. On the other hand, an excessively dense mesh increases the time of the problem solution and can lead to poor conditioning of the matrix of canonical FEM equations, and, consequently, to large errors in the factorization of this matrix.

Mesh  
4'4  8'8  16'16  32'32  64'64  Benchmark  
Displacements at point B for the full shell w_{B}, '  3.357*10^{2}  3.561*10^{2}  3.644*10^{2}  3.657*10^{2}  3.663*10^{2}  3.70*10^{2} 
Error, %  9.243  3.757  1.514  1.162  1.000   
An important factor is the shape of the finite elements. Thus, quadrilateral finite elements are more accurate than triangular finite elements. Equilateral finite elements are more preferable to elements with strongly unequal sides. The last ones worsen the conditionality of the matrix of canonical FEM equations, therefore, equilateral finite elements should be preferred for triangulation.
Regular mesh  Irregular mesh  
4'4  8'8  16'16  32'32  64'64  Benchmark  
w '10^{6}, '  12.271  11.813  11.664  10.679  11.295  11.512 
w_{benchmark} '10^{6}, '  11.60  11.60  11.60  11.60  11.60  11.60 
Error, %  5.78  1.84  0.55  7.94  2.63  0.76 
Three triangulation methods are used in LIRAFEM software
For 3D models, the analog of quadrilateral elements is parallelepipeds and octagonal 3D FEs.
The following information is used as input data for realization of triangulation methods in LIRAFEM software:
Creating a consistent finite element mesh in LIRAFEM can be done in three ways (or a combination of them):
There is a rough estimate of the loss of accuracy during rounding: the number of decimal places that are lost as a result of Gaussian factorization of the matrix r=lg_{10}·, where · is the size of the matrix of canonical equations. 'Thus, if the initial values of the matrix are made with double precision (approximately 15 decimal places), the result will contain 15r correct signs.
As a purely engineering recommendation, it may be advisable to assign the triangulation step such that the most characteristic span contains at least 10 triangulation nodes.
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