FEM for 3D Solids (Finite Element Method) Part 2

Hexahedron Element

Strain Matrix

Consider now a 3D domain, which is divided in a proper manner into a number of hexahedron elements with eight nodes and six surfaces, as shown in Figure 9.9. Each hexahedron element has nodes numbered 1, 2, 3, 4 and 5, 6, 7, 8 in a counter-clockwise manner, as shown in Figure 9.10.

As there are three DOFs at one node, there is a total of 24 DOFs in a hexahedron element. It is again useful to define a natural coordinate system (ξ,η,ζ) with its origin at the centre of the transformed cube, as this makes it easier to construct the shape functions and to evaluate the matrix integration.Like the quadrilateral element, shape functions are also used to interpolate the coordinates from the nodal coordinates:

Figure 9.9. Solid block divided into eight-nodal hexahedron elements.

Figure 9.10. An eight-nodal hexahedron element and the coordinate systems.

 

The shape functions are given in the local natural coordinate system as

or in a concise form,

where (^i, ηi, ζi) denotes the natural coordinates of node I.

From Eq. (9.36), it can be seen that the shape functions vary linearly in the ξ, η and ζ directions. Therefore, these shape functions are sometimes called tri-linear functions. The shape function Ni is a three-dimensional analogy of that given in Eq. (7.54). It is very easy to directly observe that the tri-linear elements possess the delta function property. In addition, since all these shape functions can be formed using the common set of eight basis functions of

which contain both constant and linear basis functions. Therefore, these shape functions can expect to possess both partitions of the unity property as well as the linear reproduction property.

In a hexahedron element, the displacement vector U is a function of the coordinates x, y and z, and as before, it is interpolated using the shape functions

where the nodal displacement vector, de is given by

in which

is the displacement at node i. The matrix of shape functions is given by

in which each sub-matrix, Ni, is given as

In this case, the strain matrix defined by Eq. (9.17) can be expressed as

whereby

As the shape functions are defined in terms of the natural coordinates, ξ, η and ζ, to obtain the derivatives with respect to x, y and z in the strain matrix, the chain rule of partial differentiation needs to be used:

which can be expressed in the matrix form

where J is the Jacobian matrix defined by

Recall that the coordinates, x, y and z are interpolated by the shape functions from the nodal coordinates. Hence, substitute the interpolation of the coordinates, Eq. (9.34), into Eq. (9.47), which gives

or

Equation (9.46) can be re-written as

which is then used to compute the strain matrix, B, in Eqs. (9.43) and (9.44), by replacing all the derivatives of the shape functions with respect to x, y and z to those with respect to ξ, η and ζ.

Element Matrices

Once the strain matrix, B, has been computed, the stiffness matrix, ke, for 3D solid elements can be obtained by substituting B into Eq. (3.71):

Note that the matrix of material constant, c, is given by Eq. (2.9). As the strain matrix, B, is a function of ξ, η and ζ, evaluating the integrations in Eq. (9.51) can be very difficult. Therefore, the integrals are performed using a numerical integration scheme. The Gauss integration scheme discussed in Section 7.3.4 is often used to carry out the integral. For three-dimensional integrations, the Gauss integration is sampled in three directions, as follows:

To obtain the mass (inertia) matrix for the hexahedron element, substitute the shape function matrix, Eq. (9.41), into Eq. (3.75):

The above integral is also usually carried out using Gauss integration. If the hexahedron is rectangular with dimensions of a x b x c, the determinate of the Jacobian matrix is simply given by

and the mass matrix can be explicitly obtained as

where

or

in which

As an example, m33 is calculated as follows:

The other components of the mass matrix for a rectangular hexahedron element are:

Note that the equalities in the above equation can be easily figured out by observing the relative geometric positions of the nodes in the cube element. For example, the relative geometric positions of nodes 1-2 is equivalent to the relative geometric positions of nodes 2-3, and the relative geometric positions of nodes 1-7 is equivalent to the relative geometric positions of nodes 2-8. If we write the portion of the mass matrix corresponding to only one translational direction, say the x direction, we have

The mass matrices corresponding to only the y and z directions are exactly the same as me.

The nodal force vector for a rectangular hexahedron element can be obtained using Eqs. (3.78), (3.79) and (3.81). Suppose the element is loaded by a distributed force fs on edge 3-4 of the element, as shown in Figure 9.10; the nodal force vector becomes

If the load is uniformly distributed, fsx, fsx and fsz are constants, and the above equation becomes

where l3-4 is the length of edge 3-4. Equation (9.63) implies that the distributed forces are equally divided and applied at the two nodes. This conclusion suggests also to evenly distribute surface forces applied on any face of the element, and to evenly distribute body forces applied on the entire body of the element.

Figure 9.11. A hexahedron broken up into five tetrahedrons.

Using Tetrahedrons to form Hexahedrons

An alternative method of formulating hexahedron elements is to make use of tetrahedron elements. This is built upon the fact that a hexahedron can be said to be made up of numerous tetrahedrons. Figure 9.11 shows how a hexahedron can be made up of five tetrahedrons. Of course, this is not the only way that a hexahedron can be made up of five tetrahedrons, and it can also be made up of six tetrahedrons, as shown in Figure 9.12. Similarly, there is more than one way of dividing a hexahedron into six tetrahedrons. In this way, the element matrices for a hexahedron can be formed by assembling all the matrices for the tetrahedron elements, each of which is developed in Section 9.2.2. The assembly is done in a similar way to the assembly between elements.