Nonlinear finite elements/Effect of mesh distortion

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An Example: Effect of Mesh Distortion[edit]

Consider the three-noded quadratic displacement element shown in Figure 1.

Figure 1. Reference and Current Configurations of a 3-noded element.

The shape functions for the parent element are

In matrix form,

The trial functions (in terms of the parent coordinates) are

The mapping from the Eulerian coordinates to the parent element coordinates is

In matrix form,

Therefore, the derivative with respect to is

In matrix form,

Now, the matrix is given by

Therefore,

The rate of deformation is then given by

The stress can be calculated using the relation

The internal forces are given by

Plugging in the expression for , and changing the limits of integration, we get

If node is midway between node and node ,

Then we have,

The rate of deformation becomes

which is a linear function of .

However, if node moves away from the middle during deformation, then is no longer constant and can become zero or negative. Under such situations the rate of deformation is either infinite or the element inverts upon itself since the isoparametric map is no longer one-to-one.

Let us consider the case where is zero. In that case, the Jacobian becomes

Similarly, when is negative, is negative. This implies that the conservation of mass is violated.

To find the location of when this happens, we set the relation for to zero. Then we get,

If at , then

This means that as node gets closer and closer to node , the rate of deformation become infinite at node and then negative.

If at , then

This means that as node gets closer and closer to node , the rate of deformation becomes infinite at node and then negative.

These effects of mesh distortion can be severe in multiple dimensions. That is the reason that linear elements are preferred in large deformation simulations.