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. Fig. 1. Fragment of the convergence test of displacements of the middle of the free edge of the cylindrical shell at different sizes of the FE mesh

#### Closed circular cylindrical shell under the action of a self-equilibrated system of two concentrated loads (in russian)

 Mesh 4'4 8'8 16'16 32'32 64'64 Benchmark Displacements at point B for the full shell wB, ' -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. Fig. 2. Fragment of convergence test of vertical displacements in the center of the slab at different parameters of the FE mesh (square hinged and pinned bending slab loaded with a concentrated load in the center).

#### Verification of convergence of bending slab FE for rectangular region (in russian)

 Regular mesh Irregular mesh 4'4 8'8 16'16 32'32 64'64 Benchmark w '106, ' 12.271 11.813 11.664 10.679 11.295 11.512 wbenchmark '106, ' 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

## Triangulation methods in LIRA-SAPR

Three triangulation methods are used in LIRA-SAPR software

• a method based on the application of triangular finite elements (triangular triangulation). For 3D models, the analogs of these elements are tetrahedrons and triangular prisms;
• a method based on the maximum possible inclusion of rectangular and quadrangular FEs in the finite element mesh (quadrangular triangulation);
• a method based on the organization of regular inclusions in places of stress or force concentration (adaptive quadrilateral triangulation).

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 LIRA-SAPR software:

1. The outer contour of the physical area to be partitioned;
2. Information about the required finite element step with possible densification regions;
3. An array of internal contours (holes or voids in the triangulated region);
4. An array of additional points (coordinates of nodes that must be present in the resulting finite element mesh);
5. An array of additional segments (segments with which the edges of the finite elements must not intersect). Fig. 3. Triangulation settings and the result of generation of different meshes: triangular, adaptive quadrilateral - left and quadrilateral - right

Creating a consistent finite element mesh in LIRA-SAPR can be done in three ways (or a combination of them):

• contours triangulation by one of the automatic methods (simple contour, contour with openings, contour editor, fragment in SAPFIR);
• by sequentially entering firstly the nodes of the elements of the design model, and then by displaying the elements themselves (element input);
• using regular fragments to create the geometry of the design model. Fig. 4. Creation and triangulation of an arbitrary model fragment using SAPFIR system.
The main parameter of plate mesh generation is the step, which determines the maximum length of an edge of a triangle or quadrilateral of the mesh. The selection of the triangulation step is an important and difficult problem for the user. On the one hand, reducing the size of the finite element leads to a decrease in the discretization error, on the other hand, to an increase in rounding errors and errors associated with the deterioration of matrix conditioning.

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=lg10·, 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 15-r 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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• 111 #### Дмитрий Городецкий

Кандидат технических наук - специальность "САПР".
Руководитель проекта "МКЭ-процессор ЛИРА-САПР". Руководитель проекта МОНОМАХ-САПР.

More publications of the author #### Марина Ромашкина

Кандидат технических наук - специальность "Строительные конструкции, здания и сооружения".
Сопровождение программного комплекса ЛИРА-САПР.

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