**Published on** March 9th, 2018 |
*by ozgun*

# Is Your Mesh Good Enough?

Finite element solutions are becoming vital for complex engineering systems that are not simple to analyze by using old traditional methods. Comparing to theoretical approaches, FE models are capable to deal with much more intricate multiphysics problems. However, two crucial steps, meshing and its quality, play an important role in your results.

### How do I know if my mesh is good enough?

Finite element preprocessors have come a long way over the years, to the point where users with minimal training can create meshes that appear “good”. But, how can you really know if the mesh is good enough for your analysis? Meshes that are “good enough” are ones that produce results with an acceptable level of accuracy, assuming that all other inputs to the model are accurate. Mesh density is a significant metric used to control accuracy (element type and shape also affect accuracy). Assuming no singularities are present, a high-density mesh will produce results with high accuracy. However, if a mesh is too dense, it will require a large amount of computer memory and long run times, especially for multiple-iteration runs that are typical of nonlinear and transient analyses.

There are several ways to check your mesh quality;

- Software mesh control features (you can check the global mesh control in Ansys here)
- Convergence Analysis
- Test data or to theoretical values

Unfortunately, test data and theoretical results are often not available. So, other means of evaluating mesh quality are needed.

### Mesh quality check Ansys

The quality of the mesh plays a significant role in the accuracy and stability of the numerical computation. Regardless of the type of mesh used in your domain, checking the quality of your mesh is essential. One important indicator of mesh quality that ANSYS Fluent allows you to check is a quantity referred to as the orthogonal quality. In order to determine the orthogonal quality of a given cell, the following quantities are calculated for each face;

- the normalized dot product of the area vector of a face () and a vector from the centroid of the cell to the centroid of that face ():

**(1–1)** - the normalized dot product of the area vector of a face () and a vector from the centroid of the cell to the centroid of the adjacent cell that shares that face ():

**(1–2)**

The minimum value that results from calculating *Equation 1–1* and *Equation 1–2* for all of the faces is then defined as the orthogonal quality for the cell. Therefore, the worst cells will have an orthogonal quality closer to 0 and the best cells will have an orthogonal quality closer to 1. *Figure 1.1: The Vectors Used to Compute Orthogonal Quality* illustrates the relevant vectors, and is an example where *Equation 1–2* produces the minimum value and therefore determines the orthogonal quality.

Another important indicator of the mesh quality is the aspect ratio. The aspect ratio is a measure of the stretching of a cell. It is computed as the ratio of the maximum value to the minimum value of any of the following distances: the normal distances between the cell centroid and face centroids (computed as a dot product of the distance vector and the face normal), and the distances between the cell centroid and nodes. For a unit cube (see *Figure 1.1: Calculating the Aspect Ratio for a Unit Cube*), the maximum distance is 0.866, and the minimum distance is 0.5, so the aspect ratio is 1.732. This type of definition can be applied to any type of mesh, including polyhedral.

**Figure 1.2: Calculating the Aspect Ratio for a Unit Cube**

### Convergence Analysis

The most fundamental and accurate method for evaluating mesh quality is to refine the mesh until a critical result, such as the maximum stress in a specific location converges (i.e. it doesn’t change significantly with each refinement). An example is shown in Figure 1, where a 2D bracket model is constrained at its top end and subjected to a shear load at the edge on the lower right. This generates a peak stress in the fillet, as shown. The curve shows that as the mesh density increases, the peak stress in the fillet increases. Ultimately, increasing the mesh density further produces only minor increases in peak stress. In this case, an increase from 1134 elements per unit area to 4483 elements per unit area yields only a 1.5% increase in stress.

The problem with this method is that it requires multiple remeshing and re-solving operations. While this method is fine for simple models, it can be very time-consuming for complex models. However, in Ansys you can easily perform this operation automatically by using Convergence tool option.

Basically, the convergence tool increases the mesh density and checks the results between each step. You can easily see how your results change depending on the element quantity.

Researcher

Solid Mechanics, Finite Element and Material Science