2.4 Extrapolation of Integration Point Values to the Nodes

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Solution of the governing finite element equations (2.27) yields the displacements at all

nodes. These can be used to calculate the strains (apply Equations (2.14) and (2.4)) and

the stresses (through Equation (2.2)) throughout each element. Because of the numerical

integration, the strains and stresses are more accurate at the integration points than anywhere

else. Therefore, the field variables are usually evaluated at the integration points and, if

LINEAR MECHANICAL APPLICATIONS 83

13

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

4

1

4

1

4

1

4

− 1

12

− 1

12

− 1

12 − 1

12

Figure 2.13 Relative nodal forces due to constant pressure

needed, extrapolated to the nodes. This extrapolation is done on an element basis, that is,

one obtains for a given node as many values as the number of elements it belongs to. These

values are usually discontinuous at the element borders. This is taken care of by calculating

the mean value over all elements the node belongs to.

Extrapolation toward the nodes is sometimes replaced by interpolation within patches

of elements. This is closely related to the very important topic of error estimation. For

further information the reader is referred to (Zienkiewicz and Zhu 1992a), (Zienkiewicz

and Zhu 1992b), (Gabald´on and Goicolea 2002) and (Prudhomme et al. 2003).

Now, extrapolation schemes will be presented for the three-dimensional elements introduced

in the previous sections.

2.4.1 The 8-node hexahedral element

The extrapolation of the field variables in the integration points toward the nodes for

the fully integrated linear hexahedral element is usually trilinear, that is, the shape functions

that are used for the displacements are also used for the stresses, strains and any

other dependent fields. Assume that the field variables are known in the nodes. Then,

the integration point values are obtained by (the stress σxx stands for any field variable)

σxxj =

8

_

i=1

ϕi(ξj, ηj, ζj) σxxi (2.118)

(i are the nodes, j are the eight integration points) or in matrix form

_σxx_integration points

= _A_ _σxx_nodes . (2.119)

84 LINEAR MECHANICAL APPLICATIONS

Consequently, the nodal values are found by inverting Equation (2.119)

_σxx_nodes

= _A_

−1 _σxx_integration points . (2.120)

_A_ is an 8 × 8 matrix and can be evaluated explicitly since both the shape functions

(Section 2.2.1) and the location of the integration points (Section 2.3.1) are known. Consequently,

the inverse matrix can also be coded explicitly into the finite element program.

For the node and integration point numbering of Figure 2.8 the matrix _A_

−1 satisfies

_A_

−1 =



+2.549 −0.683 −0.683 +0.183 −0.683 +0.183 +0.183 −0.049

−0.683 +2.549 +0.183 −0.683 +0.183 −0.683 −0.049 +0.183

+0.183 −0.683 −0.683 +2.549 −0.049 +0.183 +0.183 −0.683

−0.683 +0.183 +2.549 −0.683 +0.183 −0.049 −0.683 +0.183

−0.683 +0.183 +0.183 −0.049 +2.549 −0.683 −0.683 +0.183

0.183 −0.683 −0.049 +0.183 −0.683 +2.549 +0.183 −0.683

−0.049 +0.183 +0.183 −0.683 +0.183 −0.683 −0.683 +2.549

+0.183 −0.049 −0.683 +0.183 −0.683 +0.183 +2.549 −0.683



. (2.121)

Notice that Equation (2.120) defines the nodal values as a linear combination of the

integration point values. For the reduced integration 8-node element, there is only 1 integration

point, yielding one field value per element. This value is copied to the nodes and

corresponds to a constant function extrapolation.

2.4.2 The 20-node hexahedral element

For the fully integrated 20-node element, a similar scheme as for the fully integrated 8-node

element is proposed: the field variables are interpolated using the shape functions of the

element:

σxxj =

20

_

i=1

ϕi (ξj, ηj, ζj) σxxi, j= 1, . . . , 27 (2.122)

(i are the nodes, j are the integration points). This, however, leads to 27 equations in 20

unknowns (the nodal values σxxi) and cannot be inverted: the system is overdetermined. A

standard procedure to solve overdetermined systems is the least-squares method. Writing

Equation (2.122) as

bj =

20

_

i=1

ajixi, j= 1, . . . , 27 (2.123)

corresponds to minimizing

LINEAR MECHANICAL APPLICATIONS 85

I :=

27

_

j=1

_ 20

_

i=1

ajixi − bj

_

2

. (2.124)

The solution can be found by differentiation:

∂I

∂xk

= 2

27

_

j=1

__ 20

_

i=1

ajixi − bj

_

ajk

_

= 0, k= 1, . . . , 20 (2.125)

is equivalent to

20

_

i=1

27

_

j=1

ajiajk

xi

=

27

_

j=1

ajkbj, k= 1, . . . , 20 (2.126)

or

20

_

i=1

ckixi = dk, k= 1, . . . , 20 (2.127)

where

cki =

27

_

j=1

ajkaji, k= 1, . . . , 20 (2.128)

dk =

27

_

j=1

ajkbj, k= 1, . . . , 20. (2.129)

Equation (2.127) is a system of 20 equations in 20 unknowns. Let _b1_ be a unit vector

with a unit value in its first row. Then, the solution _x1_ contains the nodal values for a

unit value in the first integration point and zero in all other integration points. This can be

repeated for all other integration points. One finally obtains the 20 × 27 matrix _B_ in the

equation

_σxx_nodes

= _B_ _σxx_integration points . (2.130)

It takes the form

_B_ = __x1_ _x2_ . . . _x27__ . (2.131)

The numerical values can be found in the CalculiXcode (CalculiX 2003).

For the reduced integration element, there are only 8 integration point values. The same

scheme as for the fully integrated 8-node element is used to obtain the vertex nodal values.

The values of the middle nodes are obtained by taking the mean of the neighboring vertex

nodal values.

86 LINEAR MECHANICAL APPLICATIONS

2.4.3 The tetrahedral elements

For the linear tetrahedral element, there is only 1 integration point and its value is simply

copied to the nodes.

The quadratic tetrahedral element contains 4 vertex nodes and 4 integration points.

Consequently, exactly the same procedure can be used as for the reduced integrated 20-

node elements: one takes the displacement shape functions of the corresponding linear

element – the linear tetrahedron – and writes Equations (2.118) to (2.119). The inversion

of _A_ yields the vertex nodal values by Equation (2.120). Here, _A_

−1 takes the

form

_A_

−1 =



+1.92705 −0.30902 −0.30902 −0.30902

−0.30902 +1.92705 −0.30902 −0.30902

−0.30902 −0.30902 +1.92705 −0.30902

−0.30902 −0.30902 −0.30902 +1.92705



(2.132)

for the node and integration point numbering of Figure 2.10. The values in the middle

nodes are obtained by taking the mean of the neighboring vertex nodal values.

2.4.4 The wedge elements

For the linear wedge element, the values in the two integration points are linearly extrapolated

toward the nodes in the upper and lower triangle.

The quadratic wedge element has 15 nodes and 9 integration points. It is underdetermined.

However, there are only 6 vertex nodes. Consequently, one can apply a

least-squares scheme for the vertex nodes. It results in the following 6 × 9 _B_ matrix

(cf Equation (2.130)):

_B_ =



+1.6314 −0.3263 −0.3263 +0.5556 −0.1111 −0.1111 −0.5203 +0.1041 +0.1041

−0.3263 +1.6314 −0.3263 −0.1111 +0.5556 −0.1111 +0.1041 −0.5203 +0.1041

−0.3263 −0.3263 +1.6314 −0.1111 −0.1111 +0.5556 +0.1041 +0.1041 −0.5203

−0.5203 +0.1041 +0.1041 +0.5556 −0.1111 −0.1111 +1.6314 −0.3263 −0.3263

+0.1041 −0.5203 +0.1041 −0.1111 +0.5556 −0.1111 −0.3263 +1.6314 −0.3263

+0.1041 +0.1041 −0.5203 −0.1111 −0.1111 +0.5556 −0.3263 −0.3263 +1.6314



(2.133)

for the node and integration points numbering of Figure 2.11. The midnode values are

obtained by taking the mean of the neighboring vertex nodal values.

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