2D - Element Quality Criteria

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In this article, we will look at the 2D element quality criteria, what they mean and their effects in a simulation. When it comes to 2D, elements these are some important criteria which have to be taken care of. 

Why do we need to take a lot of care when it comes to element quality? It is because the criteria make us understand how much the elements have deviated from the ideal element. Since they represent the geometry during a simulation, poor elements will misrepresent the geometry and will lead to inappropriate results.

 

Types of Quality Criteria in FEA

The different types of quality criteria in  FEA:

Aspect Ratio:

The aspect ratio is the ratio of maximum length to minimum length.

In simple terms, when you see a quad element with one side smaller than the remaining three sides, it indicates that your element has failed for aspect ratio. Since the shape of a perfect quad element has all the sides equal, the ideal value of aspect ratio is one.

Let us do a simple calculation to verify this, 

 

Since this is a perfect square, both the maximum length and the minimum length are going to be the same. The Aspect Ratio will be 1 if we solve the above formula.

For this quad element, we have a maximum length of 2 and a minimum length of 0.5. Now the aspect ratio becomes 4.

Due to geometrical constraints, not every time, you can achieve a perfect quad and this is exactly why you will be given a tolerance value. This tolerance value depends upon the accuracy required. Having a large aspect ratio will reduce the accuracy of your result.

Fix: 

The fix for this is, you need to make your element as close to the ideal quad. In most cases, you can remesh the elements or you can rectify this by moving the node of the smallest side so that the length between those two nodes is increased.

 

Skewness:

Skewness is the quality of the elements in terms of angles. For a Quad element,

In the case of an ideal quad.

                   

Whereas, when you have a distorted quad,

For this quad when we calculate the skewness will be 45*.

Now let's see how this skewness is calculated in a Tria element. Here the angle between the meridian line and the mid-node line is calculated.

The angle between the meridian and the mid-node line is 90*, when calculated for skew it becomes 0*

Let us also check for the distorted tria, 

The minimum angle here is 35*, if we calculate the skewness it becomes 55*. The ideal value of Skew is 0* but considering the geometrical constraints and accuracy needed, we can have a tolerance to this criteria.  

Fix: 

To get rid of this skewness in the element, you can remesh the elements or you can manually move the nodes to increase the angle.

 

Warpage:

Whenever you have a quad element in two different planes then you will have your element failing for Warpage. Ideally, your element should be in a single plane, when they are in two different planes if you split the quad along the diagonal and measure the angle between the normals it will give your element a warp angle. It is the angle of plane deviation in your element which is also known as warpage.

In the above image, the element is bent in an upward direction. The element may also bend in the downward direction and cause Warpage error in the element. 

When you have your element failing for warpage then the results of the simulation are going to be inaccurate. Features like fillets are more prone to have a warped element as the geometry itself will be in two different planes. Considering all these geometrical constraints and accuracy we can have a tolerance for these criteria.

Fix: 

You can move the nodes in the normal direction so that they stay in the same plane where the other nodes are, else you can split the element and have Trias. The Tria element will not have Warpage as they form in one plane.

 

Jacobian:

Every element will have its own coordinate system which is known as the Natural Coordinate system. From this coordinate system, it is difficult to derive a direct solution thus these elements' natural system has to be converted to a global system.

So in order to map the elements from their natural system to the global system, these elements have to be in ideal shape. 

As we know, not all the time we can have an ideal-shaped element in the geometry so tolerance is established. This deviation of the element from the ideal shape is measured through the Jacobian. The ideal value of a Jacobian is 1.

Fix: 

To fix the element failing for Jacobian we can remesh the failed elements or manually move the nodes of the element to reach close to the ideal element.

 

Chordal Deviation:

This chordal deviation is the measure of how well the elements captured the curves. When you have a small fillet and if your element size is bigger than the size of the fillet then you can expect this chordal deviation error. You can mesh with a smaller element to capture the fillet or you can increase the tolerance value of chordal deviation, considering the accuracy required.

Here the element size is more than the size of the fillet and the element captured the fillet as a straight surface. This element deviation is referred to as the Chordal Deviation.

Fix: 

To overcome this issue this same filler is meshed with a smaller element size to capture the fillet. Sometimes when you try to capture the fillet with small elements it might fail for minimum element size for your criteria, in those cases, you need to capture the fillet with bigger elements compromising on the curvature of the fillet.