Nonlinear finite elements/Effect of mesh distortion

An Example: Effect of Mesh DistortionEdit

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.