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The basic principle of finite element lattice division

lattice division is an important part of establishing finite element model. It requires many problems to be considered and requires a large amount of work. The form of the divided lattice will have a direct impact on the calculation accuracy and scale. In order to establish a correct and reasonable finite element model, some basic principles that should be considered when dividing grids are introduced here

1 number of cells

the number of cells will affect the accuracy of the calculation results and the size of the calculation scale. Generally speaking, with the increase of the number of grids, the calculation accuracy will be improved, but at the same time, the calculation scale will also increase. Therefore, when determining the number of grids, two factors should be considered comprehensively

curve 1 in Figure 1 represents the general curve that the displacement in the structure converges with the number of grids, and curve 2 represents the change of calculation time with the number of grids. It can be seen that increasing the number of cells when there are fewer cells can significantly improve the calculation accuracy, but the calculation time will not be greatly increased. When the number of cells increases to a certain extent, and then continue to increase the number of cells, the accuracy improves little, but the calculation time increases significantly. Therefore, attention should be paid to increasing the economy of lattice. In practical application, the combination pad can be replaced at regular intervals to compare the calculation results of the two grid divisions. If the two calculation results differ greatly, you can continue to increase the grid, on the contrary, stop the calculation

Fig. 1 Variation of displacement accuracy and calculation time with the number of grids

the type of analysis data should be considered when determining the number of grids. In static analysis, if only the deformation of the structure is calculated, the number of grids can be reduced. If it is necessary to calculate the stress, a relatively large number of grids should be taken under the same requirements of professional software. Similarly, in the response calculation, the number of grids used to calculate the stress response should be more than the displacement response. When calculating the natural dynamic characteristics of the structure, if only a few low-order modes are calculated, fewer grids can be selected. If the calculated modal order is higher, more grids should be selected. In thermal analysis, the temperature gradient inside the structure is small, and a large number of internal units are not required. At this time, fewer grids can be divided

2 grid density

grid density refers to the use of grids of different sizes in different parts of the structure, which is to adapt to the distribution characteristics of the calculated data. In order to better reflect the law of data change, it is necessary to use dense grids in the parts with large gradient of calculation data (such as stress concentration). In order to reduce the scale of the model, the relatively sparse lattice should be divided in the part where the change gradient of the calculated data is small. In this way, the whole structure shows different lattice division forms of density

Figure 2 is a quarter model of a square plate with a circular hole in the center, and its lattice reflects different division principles of density. There is stress concentration near the small round hole, and a relatively dense lattice is used. The stress gradient around the plate is small and the lattice is thin. The difference in cell density in Figure B is larger, which is 48 less than that in figure a, but the difference in the calculated maximum stress at the hole edge is 1%, while the calculation time is reduced by 36%. It can be seen that the grid division with different density can not only maintain considerable calculation accuracy, but also reduce the number of grids. Therefore, the number of lattices should be increased to the key parts of the structure, and it is unnecessary and uneconomical to increase lattices in the secondary parts

Figure 2 countries with holes try to force the quarter model of square plates to reduce production capacity through coal production reduction

the division of grids with different densities is mainly used for stress analysis (including static stress and dynamic stress), while the calculation of inherent characteristics that stand out among many degradable materials tends to adopt a more uniform steel grid form. This is because the natural frequency and mode shape mainly depend on the structure mass distribution and stiffness distribution, and there is no similar phenomenon of stress concentration. The use of uniform lattice can make the elements of the structure stiffness matrix and mass matrix not too different, and can reduce the numerical calculation error. Similarly, the uniform lattice is also used in the calculation of structural temperature field

3 element order

many elements have linear, quadratic and cubic forms, of which quadratic and cubic forms are called high-order elements. The selection of high-order elements can improve the calculation accuracy, because the curve or surface boundary of high-order elements can better approach the curve and surface boundary of the structure, and the high-order interpolation function can approach the complex field function with higher accuracy. Therefore, high-order elements can be selected when the structure shape is irregular and the stress distribution or deformation is very complex. However, the number of nodes of high-order elements is large, and the scale of the model composed of high-order elements is much larger when the number of lattices is the same, so the calculation accuracy and time should be considered when using

Figure 3 shows the convergence of the top displacement with the number of grids when a cantilever beam is discretized by linear and quadratic triangular elements respectively. It can be seen that when the number of lattices is small, the calculation accuracy of the two elements is very different, and it is inappropriate to use low-order elements at this time. When the number of lattices is large, the accuracy difference between the two elements is not great, and it is not economical to use high-order elements at this time. For example, when discretizing details, due to the limitation of detail size, it is required that the lattice division near the details is very dense, so it is more appropriate to use linear elements

Fig. 3 convergence of elements with different orders

increasing the number of lattices and the order of elements can improve the calculation accuracy. Therefore, when the accuracy is certain, the appropriate number of lattices should be selected when using high-order element discrete structure. Too many lattices can not significantly improve the calculation accuracy, but will greatly increase the calculation time. In order to take into account the calculation accuracy and calculation amount, the same structure can use elements of different orders, that is, the important parts with high accuracy requirements use high-order elements, and the secondary parts with low accuracy requirements use low-order elements. Different order elements are connected by special transition elements or by multi-point constraint equations

4 lattice mass

lattice mass refers to the rationality of lattice geometry. The quality will affect the calculation accuracy. The lattice with poor quality will even stop the calculation. Intuitively, the quality of the lattice is better when the edges or internal angles of the lattice are not much different, the lattice surface is not too distorted, and the edge nodes are near the equal points of the boundary. Lattice quality can be measured by slenderness ratio, taper ratio, internal angle, warpage, stretching value, position deviation of edge nodes and other indicators

when dividing grids, it is generally required that the quality of grids can meet some index requirements. In the key parts of the structure, we should ensure the division of high-quality lattices. Even individual lattices with poor quality will cause great local errors. In the secondary part of the structure, the quality of the lattice can be appropriately reduced. When there is a lattice with poor quality (called deformed lattice) in the model, the calculation process cannot be carried out. Figure 4 shows three common deformed lattices, in which the nodes of element a are cross numbered, the internal angle of element B is greater than 180 °, the two pairs of nodes of element C coincide, and the lattice area is zero

Figure 4 several common deformed lattices

5 lattice interfaces and dividing points

some special interfaces and special points in the structure should be divided into lattice boundaries or nodes in order to define material properties, physical properties, load and displacement constraints. That is, the lattice form should meet the characteristics of boundary conditions, rather than the boundary conditions should be adapted to the lattice. Common special interfaces and special points include material interface, sudden change surface of geometric dimension, dividing line (point) of distributed load, action point of concentrated load and action point of displacement constraint. Figure 5 shows the structure and lattice division form with the above interfaces

Figure 5 special interface and special lattice division

6 displacement coordination

displacement coordination refers to that the forces and moments on the element can be transmitted to adjacent elements through nodes. In order to ensure displacement coordination, the node of an element must also be the node of adjacent elements, not the inner point or boundary point. The common nodes of adjacent elements have the same degree of freedom properties. Otherwise, multi-point constraint equations or constraint elements must be used for constraint processing between elements. Figure 6 shows the lattice division of two kinds of displacement incongruity. Node 1 in figure a belongs to only one element, and material cracks or overlaps will occur after deformation. The degree of freedom properties of the nodes of the plane element and the beam element in Figure B are different, and the moment of the beam element cannot be transferred to the plane element

Figure 6 lattice division with incompatible displacement

7 lattice layout

when the structural shape is symmetrical, its lattice should also be divided into symmetrical lattice, so that the model shows the corresponding symmetrical characteristics (such as the symmetry of the lumped matrix). Asymmetric layout will cause certain errors. For example, in Figure 7, the cantilever beam section is symmetrical relative to the y-axis. Under the action of symmetrical load, the deflection values of two symmetrical nodes 1 and 2 at the free end should be equal. However, if the asymmetric lattice shown in sub figure B, the calculated y1=0.0346, y2=0.0350. If the lattice shown in Figure C is used instead, Y1 and Y2 are exactly the same

Fig. 7 Influence of grid layout on calculation results

8 node and element number

the number of nodes and elements affects the bandwidth and wavefront number of the total structural stiffness matrix, thus affecting the calculation time and storage capacity. Therefore, a reasonable number is conducive to improving the calculation speed. However, for complex models and automatic division, it is difficult to determine a reasonable number manually. At present, many finite element analysis software comes with optimizers, which can optimize the bandwidth and wavefront after lattice division, so as to reduce human labor intensity

References:

[1] Du Ping'an Structural finite element analysis modeling method Beijing China Machine Press, 1998

［2］eel. Applied Finite Element Modeling. New York: Eastman Kodak Company Rochester, 1989.

[3] Xu Shangxian Finite element method in mechanical structure Beijing: Higher Education Press, 1992 (end)

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