Poor element shapes can result in inaccurate results, but so far, there is no general standard to determine the quality of element shapes. One element shape may produce incorrect results in one analysis, but may be completely acceptable in another analysis. Therefore, the quality of element shapes and the accuracy of results depend entirely on the user’s judgment and analysis based on experience or relevant industry standards.
Therefore, when we use ANSYS for finite element analysis, a very important step is to check the mesh quality. Even if the mesh is successfully generated, if the quality is poor, the results may not be reliable. Therefore, good mesh quality is the primary condition for ensuring accurate analysis results. In this article, I will briefly introduce the methods for checking mesh quality in ANSYS and discuss my own methods for evaluating mesh quality.
This article first discusses eight common parameters for evaluating mesh quality, which I believe are familiar to students who frequently perform analysis. They are aspect ratio, parallel deviation, maximum corner angle, Jacobian ratio, element skewness, mesh quality index, distortion, and orthogonality quality index. I will briefly introduce the meaning of each parameter below.
1. Aspect Ratio
You may encounter this term frequently when dividing a mesh or performing calculations. For example:
“Brick element 91 has an aspect ratio of 1.E+20, which exceeds the error limit of 1000000.”
This means that the aspect ratio of the element exceeds the limit, and the mesh needs to be improved. By default, the software will issue a warning when the aspect ratio reaches 20, and an error will be raised when it exceeds 10^6.
The aspect ratio is calculated in two situations: for triangles and quadrilaterals. For solid elements, the midline of each face is selected to form a triangle, and then the calculation is performed one by one, as shown in the figure above.
The limitation of aspect ratio means that when dividing a mesh, the sides of each element should be as evenly sized as possible. If the sizes of the elements in different parts of a component vary greatly, a transition section should be set as much as possible. The following figures show the deformed shapes of triangles and quadrilaterals with aspect ratios of 1 and 20, respectively:
In other words, the further the element shape is from an equilateral triangle or square, the larger the aspect ratio, and the best value is 1.
2. Parallel Deviation
This parameter mainly applies to quadrilaterals and describes the angle between two opposite sides. The calculation principle is to calculate the unit vectors parallel to each side, and then sequentially solve the cosine value between the unit vectors of the opposite sides. Finally, the angle between the two angles is taken as the deviation angle.
The following figures show different element shapes under various angles:
The default settings are as follows:
1).When there are no intermediate nodes, a warning is issued at 70 degrees and an error is raised at 150 degrees.
2). When there are intermediate nodes, a warning is issued at 100 degrees and an error is raised at 170 degrees.
Therefore, the closer the shape of the element is to a rectangle, the smaller the deviation angle, and the better the element shape. The best value is 0.
3. Maximum Corner Angle
This parameter mainly checks the maximum internal angle of the element, which is easy to understand. The following figure shows the element shapes under different parameter values:
The default settings are as follows:
- For triangles, a warning will be issued at 165 degrees and an error will be issued at 179.9 degrees.
- For quadrilaterals without a central node, a warning will be issued at 155 degrees and an error will be issued at 179.9 degrees.
- For quadrilaterals with a central node, a warning will be issued at 165 degrees and an error will be issued at 179.9 degrees.
Therefore, for quadrilaterals, 90 degrees is the optimal angle, and for triangles, 60 degrees is the optimal angle.
4. Jacobian Ratio
Jacobian Ratio is an important coefficient in the unit shape check, which can be understood as a reliability indicator of finite element simulation and the actual situation. The larger the Jacobian Ratio, the more unreliable the unit simulation. The calculation of this coefficient is complex, and interested friends can refer to relevant finite element monographs.
By default, the Jacobian Ratio is calculated at the corner points of the unit, with a warning value of 30 and an error value of 1000. When the sample point is set to the integration point, the warning value is 10 and the error value is 40. This can be enabled using the command flow [shpp,lstet,on]. The Jacobian Ratio calculated by sampling at the integration point is smaller than that calculated by sampling at the corner point, which is suitable for linear analysis. In general, the corner point sampling calculation is used, which is the software default.
Below are the unit shapes of triangles, quadrilaterals, and quadrilaterals with intermediate nodes under different Jacobian Ratios.
5. Warping Factor
Warping Factor is also a commonly encountered problem in mesh partitioning. The excessive distortion of quadrilateral shell units, hexahedral units, the quadrilateral surface of the wedge body, and the surface of pyramid units need to be calculated and detected for the warping coefficient.
For shell elements, the limit value of the warping coefficient is related to the element type and solution setting. The default warning value for shell elements is 1.0, and the error value is 5.0. When large deformation is turned on and thin shells are set, the warning value becomes 0.1, and the error value is 1.0, which can be set to the software default value.
For solid elements, the warning value is 0.2, and the error value is 0.4. The best value for the warping coefficient is 0. The larger the value, the more severe the unit warping.
In addition to the above five coefficients, there are three more coefficients available for reference in Workbench.
6. Element Quality
Element Quality is calculated to evaluate the size of the mesh. When this coefficient is 1, it indicates the best mesh quality, and when it is 0, it indicates the worst. In practical analysis, the average value should be no less than 0.7.
Skewness is used to determine whether the mesh shape is close to the ideal state. The value of 0 indicates the closest to the ideal state and the best mesh quality, and the value of 1 indicates the worst mesh quality. For planar problems, this coefficient should not exceed 0.5, and for three-dimensional analysis problems, most meshes should be less than 0.5.
8. Orthogonal Quality
Orthogonal Quality is of particular interest to fluid mechanics enthusiasts. The best value for this coefficient is 1, and the worst value is 0.
In summary, the ideal values for each standard are:
- Aspect Ratio: 1
- Parallel Deviation: 0
- Maximum Corner Angle: for triangles: closer to 60 degrees is better, and for quadrilaterals: closer to 90 degrees is better.
- Jacobian Ratio: 1
- Warping Factor: 0
- Mesh Metric: 1
- Skewness: 0
- Orthogonal Quality: 1