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Mechanical

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28 Jun 2022

Skill-Lync

When the elements in a model deform but the strain energy is not computed for that deformation, it results in an Hourglass deformation effect or Hourglass effect. Usually, this can be clearly seen in the post-processing stage where the elements will have a zig-zag formation.

In order to understand why it is happening in an element, we should first have a clear understanding of how the results are calculated in an element.

In an Explicit scheme, when a force **F** is applied to node **N2,** first the acceleration of the node will be calculated by the solver, then the velocity, and finally the displacement. From the displacement, the solver will calculate the strain considering the material properties. While from the strain, the stress will be computed which will be applied as a force on the node next node **N5. **This goes on until the force is completely utilized by the model.

So, when the Stress and Strain are computed on the nodes, how are they calculated at the element? The answer is through the integration points. Now, whenever you are defining the property to a model or a component there is a parameter called Element Formulation (in Radioss as ISHELL and in LS-Dyna as ELFORM) this name changes for different solvers.

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Now let us understand what and how this integration point is helping in calculating the results in an element. These integration points are the points that are placed on the element to calculate the results from the nodes. Here the results in the elements are integrated using Gaussian Quadrature from the node through the integration points.

There are two types of integrated elements, reduced and fully integrated elements.

In the above image, the blue point refers to the integration points. Let's take a deeper look at how this integration point is helping in computing the results in the element.

When the force **F** is applied at node **2, **the nodal displacement and strains are calculated by the solver. The results from the displaced node **2 **and the other nodes **5,4,1 **will be integrated at the integration point in the element using Gaussian Quadrature. This is exactly how the stresses and strains are updated on the reduced element from the nodal results.

In the case of a fully integrated element, the results from nodes **1,2,4, **and **5 **will be calculated at the integration points respectively.

Since there are four integration points, the result computed at the element is going to be very accurate. On the other hand, the solver has to solve the result for four points because of which the computational cost is going to be very high.

In the reduced integration element, the solver will calculate the nodal result only at one point, and thus it is computationally inexpensive. In this case, there are two disadvantages, the first being the inaccuracy and the second, being the Hourglass Deformation.

The reason for the hourglass deformation to take place only in the reduced element is because of the less integration point.

Below are a few fixes for the effects of hourglass deformation:

- Stiffness can be added to the reduced integration element at the material level where this hourglass is occurring. In most of the solvers, this is done automatically provided you have activated the card.
- A damping force can be added at the deformation, depending upon the deformation, the force will be computed. This can be done for the problem with high velocity eg, blast simulations. This can also be done by activating the respective card from the solver.
- By using the fully integrated elements, this Hourglass effect can be avoided.
- Mesh refinement will also help in avoiding this Hourglass effect. Coarse mesh along with fewer integration points is more prone to Hourglass deformation.

There is another integration point in an element that does not contribute to the Hourglass deformation but to its accuracy. This point will be along with the thickness of the element.

When you look at the element from the side this is how the integration point will be distributed along with the thickness ( Blue dots ). These points will help in computing the change in strain and thickness when the element is subjected to bending or when they are plastically deformed. Again, when you use more points, it is more work for the solver and it is going to be computationally expensive.

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Anup KumarH S

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Skill-Lync

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