On the problematic finite element behaviour

On the problematic finite element behaviour

1.   Introduction

Some of the finite elements exhibit anomalies under certain conditions. This invariably results from the finite element approximation. These are twofold: firstly the real displacement is approximated by shape functions and the continuous integration is replaced by summation at discrete points (i.e. numerical integration). This write up discusses in detail the most important anomalies that can occur in volume elements which include: shear locking, volumetric locking and hour-glassing.

 Shear locking is explained with reference to 8 noded brick elements in section 3 considering an example of a beam subjected to pure bending. To develop a good understanding of shear locking, a discussion on the shape functions of 8 noded bricks is presented in section 2.

 Section 4 provides an insight into the shape functions of a 4 noded quadrilateral element and its Jacobian (i.e. the determinant of the deformation gradient) which forms the basis of understanding volumetric locking discussed in section 5.

Section 6 is on hour-glassing.

2.  The 8 noded brick element

Shape functions:

It may be recalled that the shape functions of an 8 noded brick can be constituted considering the following polynomial:

In general;

Now to determine the constants (say) for the shape function N1, we need 8 equations. These are:


N1=1 at node 1 and 0 at all other nodes of the 8 noded brick shown in the Figure 1.

Similarly, to determine the constants  for the shape function N2, we need 8 equations. These are;

 N2=1 at node 2 and 0 at all other nodes;

Figure 1: 8 noded brick element with corner nodes

Doing this, we obtain the shape functions for 8 noded brick as below;

 

 

Variations of shape functions and hence the displacements along the edges of 8 noded bricks

The 8-node brick element is also called a linear brick element since the interpolation

functions along any edge are linear (keep two of the three local coordinates constant with

value ±1).

 

3.   Shear locking in an 8 noded brick element

 

Shear locking occurs in linear elements i.e. the 8 noded brick elements with full integration as shown in the figure 2. It results in deformations that are too stiff i.e. displacements are too small.

Figure 2: 8 noded brick element 2x2x2* scheme (the triangles- not at the corners denote the integration points)

3.1   Example: Shear locking

Shear locking can best be explained through an example of a beam subjected to pure bending as shown in the Figure 3.

 

Figure 3: Cantilever beam subjected to pure bending

In the beam shown in the Figure 3 above, shear force is 0 and the shearing strain is 0 everywhere. The shear force being zero indicates that the beam deflects respecting the Euler Bernoulli hypothesis i.e. plane sections remain plane and normal after bending.

 

However, the linear brick element cannot model the curvature appropriately and will approximate the deformed shape by a piecewise-linear curve (recall that the edges of an 8-node brick element are straight). Whereas in the real deformation cross sections remain perpendicular

to the beam axis (Figure 4) this is not necessarily the case in the finite element approximation (Figure 5).

Figure 4: Real deformation of a cantilever beam subjected to pure bending

Figure 5: Finite element approximation of cantilever beam subjected to pure bending using 2x2x2 integration scheme

Figures 6 and 7 show just one element from 5. If full integration is used (2 × 2 × 2 integration points) as shown in Figure 6, the shear strain at the integration points is not zero and a considerable amount of energy is absorbed by the fake shearing phenomenon, not leaving enough energy for bending: the displacements are too small.

 

The problem can be alleviated by using reduced integration (1 × 1 × 1 integration point) as shown in Figure 8: the shear strain at the integration point is zero and the correct displacements result.

Figure 6: The shear locking problem in an 8 noded brick with 2x2x2 integration scheme (fake shearing phenomenon)

Figure 7: The shear locking problem in an 8 noded brick alleviated using 1x1x1 integration scheme

 

 4. The 4 noded quadrilateral element: Shape functions and Jacobian

In order to understand volumetric locking discussed in the next section, let us look at the 4 noded quadrilateral element: its shape functions and Jacobian.

4.1  Shape functions

Figure 8: The actual finite element and the master element

The mapping from the (x,y) coordinate system to the (ξ, η) coordinate system is given in terms of shape functions which are written in terms of this master element

 The shape functions for a 4 noded quadrilateral element can be expressed as tensor product of 1-D first order (i.e. linear) Lagrangian shape functions along the edges parallel to ξ and η axis:

That is;

Thus,

The geometry can be interpolated using these shape functions (iso-parametric mapping) as indicated in the equation

Similarly, u can be expressed as (since the formulation is iso-parametric)

4.2  Jacobian: Determinant of the deformation gradient

The Jacobian i.e. the determinant of the deformation gradient is expressed as;

Since we only have a two dimensional element, the Jacobian becomes;

 

5. Volumetric locking

The problem of volumetric locking occurs for incompressible or nearly incompressible

behaviour.

In order to understand the concept of volumetric locking, let us consider a corner element of a certain domain with boundary conditions as shown in the Figure 9.

Figure 9: Locking behaviour in corner elements

Element 1 is fixed alongside 1–2 and 1–4. It is a standard two-dimensional quadrilateral element. The material is assumed to be incompressible. Accordingly, J = 1 everywhere.

Since u1 = u2 =u4 = v1 = v2 = v4 = 0, the displacements in the element amount to:

From section 4.1, we have;

 

For simplicity, let us assume that the global and local coordinates coincide. Therefore;

If we take 2x2x2 integration scheme, the above equation has to be specified at + or – 0.57.

 That is;

Which can only be specified when u3 = v3 =0. This results in a zero deformation field for the complete element: the element locks.

 6. Hourglassing

  As it was mentioned in section 3, the problem concerning shear locking can be alleviated by using reduced integration (1 × 1 × 1 integration point) as shown in Figure 8: the shear strain at the integration point is zero and the correct displacements result. But, this holds trye for the problem regarding pure bending as discussed the section 3.

 However, if one is dealing with a problem in which there exist considerable shear deformations and if the in-plane deformation of a linear quadrilateral element is estimated using only one integration point at the centre of the element, then there is no stiffness predicted to resist the shear mode shown below, as it causes no strain/curvature at the centre. IT may be recalled as mentioned in section 3 that this is because the shear strain at the integration point  which for 1x1x1 integration scheme is zero because the sections remain normal as shown in Figure 10.

Figure 10: In plane deformation of a linear quadratic element with one integration point.

Thus, there is no stiffness predicted to resist the shear, as it causes no strain at the centre. This results in a regular mesh, is a zig-zag pattern of deformation, dominating the solution as shown in the Figure 11 below;

Figure 11: Zig zag pattern of deformation due to hour glassing in quadratic elements with 1 integration point.