2.2 The Shape Functions

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Equation (2.8) shows that within each element the displacement field is assumed to be a

continuous function of the displacements in the nodes. The question that arises is where

68 LINEAR MECHANICAL APPLICATIONS

X1

X2

X3

ξ

ζ η

1

1

2

2

3

3

4

4

5 5

6

6

8 7 8 7

Figure 2.1 8-node brick element

those nodes are located and what the shape functions look like. There is no unique answer

to this question. Rather, there are several schemes of which some have grown very popular

through time. As mentioned in the previous section, we will concentrate on formulations in

which the same shape functions are used for the displacements and the geometry (isoparametric

formulation).

2.2.1 The 8-node brick element

The most popular element form is the brick shape, with local coordinates satisfying −1 ≤

ξ, η, ζ ≤ 1. Figure 2.1 shows the 8-node brick element in local and global coordinates. At

each vertex of the brick there is a node. If the shape functions are such that

ϕi(ξj, ηj, ζj ) = δij (2.31)

then the Equations (2.11) and (2.12) are identically satisfied at the nodes. For the shape

functions, one often takes polynomials because of their mathematically simple form. The

lowest-order polynomial with eight unknown coefficients has the form

ϕi(ξ, η, ζ) = ai + biξ + ciη + diζ + eiξη + fiξζ + giηζ + hiξηζ. (2.32)

Equation (2.31) together with Equation (2.32) leads to eight equations in eight unknowns

for each shape function. These sets of equations uniquely determine the coefficients. One

obtains

ϕ1 = (1 − ξ)(1 − η)(1 − ζ)/8 (2.33)

ϕ2 = (1 + ξ)(1 − η)(1 − ζ)/8 (2.34)

LINEAR MECHANICAL APPLICATIONS 69

ϕ3 = (1 + ξ)(1 + η)(1 − ζ)/8 (2.35)

ϕ4 = (1 − ξ)(1 + η)(1 − ζ)/8 (2.36)

ϕ5 = (1 − ξ)(1 − η)(1 + ζ)/8 (2.37)

ϕ6 = (1 + ξ)(1 − η)(1 + ζ)/8 (2.38)

ϕ7 = (1 + ξ)(1 + η)(1 + ζ)/8 (2.39)

ϕ8 = (1 − ξ)(1 + η)(1 + ζ)/8. (2.40)

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). They look very attractive due to the simple shape functions and their intuitively

attractive form for meshing purposes; however, the element exhibits marked problematic

behavior such as shear locking, volumetric locking and hourglassing. This will be discussed

in Section 2.5.

2.2.2 The 20-node brick element

This element has the same shape in local coordinates as the 8-node brick, but contains 20

nodes instead of 8. They are located at the vertices and in the middle of the edges (see

Figure 2.2).

Using the following basis polynomials,

1

ξ η ζ

ηζ ξζ ξη

ξ 2 η2 ζ 2

ηζ 2 η2ζ ξζ 2 ξ 2ζ ξη 2 ξ 2η

ξηζ

ξ 2ηζ ξη2ζ ξηζ 2

(2.41)

one obtains for the shape functions, using Equation (2.31)

ϕ1 = −(1 − ξ)(1 − η)(1 − ζ)(2 + ξ + η + ζ)/8 (2.42)

ϕ2 = −(1 + ξ)(1 − η)(1 − ζ)(2 − ξ + η + ζ)/8 (2.43)

ϕ3 = −(1 + ξ)(1 + η)(1 − ζ)(2 − ξ − η + ζ)/8 (2.44)

ϕ4 = −(1 − ξ)(1 + η)(1 − ζ)(2 + ξ − η + ζ)/8 (2.45)

ϕ5 = −(1 − ξ)(1 − η)(1 + ζ)(2 + ξ + η − ζ)/8 (2.46)

ϕ6 = −(1 + ξ)(1 − η)(1 + ζ)(2 − ξ + η − ζ)/8 (2.47)

ϕ7 = −(1 + ξ)(1 + η)(1 + ζ)(2 − ξ − η − ζ)/8 (2.48)

70 LINEAR MECHANICAL APPLICATIONS

X1

X2

X3

ξ

ζ η

1

1

2

2

3

3

4

4

5 5

6

6

8 7 8 7

9

9

10

10

11 11

12

12

13 13

14 14

15 15

16 16

17

17 18

18

19

20 19 20

Figure 2.2 20-node brick element

ϕ8 = −(1 − ξ)(1 + η)(1 + ζ)(2 + ξ − η − ζ)/8 (2.49)

ϕ9 = (1 − ξ)(1 + ξ)(1 − η)(1 − ζ)/4 (2.50)

ϕ10 = (1 + ξ)(1 − η)(1 + η)(1 − ζ)/4 (2.51)

ϕ11 = (1 − ξ)(1 + ξ)(1 + η)(1 − ζ)/4 (2.52)

ϕ12 = (1 − ξ)(1 − η)(1 + η)(1 − ζ)/4 (2.53)

ϕ13 = (1 − ξ)(1 + ξ)(1 − η)(1 + ζ)/4 (2.54)

ϕ14 = (1 + ξ)(1 − η)(1 + η)(1 + ζ)/4 (2.55)

ϕ15 = (1 − ξ)(1 + ξ)(1 + η)(1 + ζ)/4 (2.56)

ϕ16 = (1 − ξ)(1 − η)(1 + η)(1 + ζ)/4 (2.57)

ϕ17 = (1 − ξ)(1 − η)(1 − ζ)(1 + ζ)/4 (2.58)

ϕ18 = (1 + ξ)(1 − η)(1 − ζ)(1 + ζ)/4 (2.59)

ϕ19 = (1 + ξ)(1 + η)(1 − ζ)(1 + ζ)/4 (2.60)

ϕ20 = (1 − ξ)(1 + η)(1 − ζ)(1 + ζ)/4. (2.61)

The 20-node brick elements are also called quadratic elements because the interpolation

along each edge is a quadratic function. Because of this, they can simulate curved boundaries

by a piecewise-quadratic approximation. Quadratic brick elements are usually well

behaved and in the author’s opinion they should be preferred to linear brick elements.

LINEAR MECHANICAL APPLICATIONS 71

A major disadvantage is that despite intensive research no satisfactory automatic meshers

are yet available (Tautges 2001). Therefore, the efficient generation of a good-quality

hexahedral mesh heavily relies on the expertise of the user.

If one of the faces of a 20-node brick element is collapsed, the element can simulate

singular strain and stress fields (Dhondt 1993). This is used in special applications such as

linear elastic fracture mechanics (Dhondt 2002).

2.2.3 The 4-node tetrahedral element

The 4-node tetrahedral element is characterized by linear interpolation functions within a

tetrahedron (see Figure 2.3). The local coordinates are such that

0 ≤ ξ, η, ζ ≤ 1 (2.62)

ξ + η + ζ ≤ 1. (2.63)

Nodes 2, 3 and 4 are characterized by ξ = 1, η = 1 and ζ = 1 respectively. In the

local coordinate system, the coordinates of a point P can be obtained by constructing

the tetrahedra T1, T2 and T3 extending from P to the faces 1 − 3 − 4 (opposite node 2),

1 − 2 − 4 (opposite node 3) and 1 − 2 − 3 (opposite node 4) respectively. Denoting the

volumes of T1, T2 and T3 by V1, V2 and V3 respectively, and the total volume by V one

can write

ξ = V1/V (2.64)

η = V2/V (2.65)

ζ = V3/V. (2.66)

X1

X2

X3

ξ

η

ζ

1

1

2

2

3

3

4

4

P

Figure 2.3 4-node tetrahedral element

72 LINEAR MECHANICAL APPLICATIONS

The shape functions take the form

ϕ1 = 1 − ξ − η − ζ (2.67)

ϕ2 = ξ (2.68)

ϕ3 = η (2.69)

ϕ4 = ζ. (2.70)

The shape functions are exceedingly simple. However, for stress calculations the element

is extremely stiff and lots of elements are needed to obtain acceptable results. It should

generally be avoided.

2.2.4 The 10-node tetrahedral element

The 10-node tetrahedral element is characterized by quadratic interpolation functions within

the element. The extra degrees of freedom are taken care of by introducing nodes in the

middle of the element edges (Figure 2.4). The local coordinate system is the same as for

the 4-node tetrahedral element. The shape functions take the form

ϕ1 = [2(1 − ξ − η − ζ) − 1][1 − ξ − η − ζ ] (2.71)

ϕ2 = (2ξ − 1)ξ (2.72)

ϕ3 = (2η − 1)η (2.73)

ϕ4 = (2ζ − 1)ζ (2.74)

ϕ5 = 4(1 − ξ − η − ζ)ξ (2.75)

ϕ6 = 4ξη (2.76)

X1

X2

X3

ξ

η

ζ

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

Figure 2.4 10-node tetrahedral element

LINEAR MECHANICAL APPLICATIONS 73

ϕ7 = 4(1 − ξ − η − ζ)η (2.77)

ϕ8 = 4(1 − ξ − η − ζ)ζ (2.78)

ϕ9 = 4ξζ (2.79)

ϕ10 = 4ηζ. (2.80)

The 10-node tetrahedral element is a very flexible element due to its shape. Furthermore,

automatic reliable tetrahedral meshing routines have been developed that are able to cope

with nearly any structure (George and Borouchaki 1998), (Freitag and Knupp 2002). The

quality of the 10-node element is comparable to the 20-node brick element. Disadvantages

are the enormous amount of elements generated by automatic meshing routines and the

nontrivial quality check of the mesh. Indeed, owing to the irregular shape of the tetrahedra,

a visual check is nearly impossible and one has to rely on mathematical measures such as

the dihedral angle.

2.2.5 The 6-node wedge element

For this element type, the local coordinates are such that (Figure 2.5)

0 ≤ ξ, η ≤ 1, −1 ≤ ζ ≤ 1 (2.81)

ξ + η ≤ 1. (2.82)

For ζ = −1, one obtains the lower triangle 1 − 2 − 3. The values of ξ and η of a point A

are given by the surface ratio of triangle 1 − A − 3 and triangle 1 − A − 2 with respect to

triangle 1 − 2 − 3 respectively. The 6-node wedge element is linear, that is, the connection

of the nodes in Figure 2.5 is straight. Its shape functions take the form

ϕ1 = (1 − ξ − η)(1 − ζ)/2 (2.83)

ϕ2 = ξ(1 − ζ)/2 (2.84)

ϕ3 = η(1 − ζ)/2 (2.85)

ϕ4 = (1 − ξ − η)(1 + ζ)/2 (2.86)

ϕ5 = ξ(1 + ζ)/2 (2.87)

ϕ6 = η(1 + ζ)/2. (2.88)

2.2.6 The 15-node wedge element

The 15-node wedge element is the quadratic version of the 6-node wedge element (see

Figure 2.6). Equations (2.81) and (2.82) also apply here. The shape functions take the

form

ϕ1 = −(1 − ξ − η)(1 − ζ)(2ξ + 2η + ζ)/2 (2.89)

ϕ2 = ξ(1 − ζ)(2ξ − ζ − 2)/2 (2.90)

ϕ3 = η(1 − ζ)(2η − ζ − 2)/2 (2.91)

74 LINEAR MECHANICAL APPLICATIONS

ϕ4 = −(1 − ξ − η)(1 + ζ)(2ξ + 2η − ζ)/2 (2.92)

ϕ5 = ξ(1 + ζ)(2ξ + ζ − 2)/2 (2.93)

ϕ6 = η(1 + ζ)(2η + ζ − 2)/2 (2.94)

ϕ7 = 2ξ(1 − ξ − η)(1 − ζ) (2.95)

ϕ8 = 2ξη(1 − ζ) (2.96)

ϕ9 = 2η(1 − ξ − η)(1 − ζ) (2.97)

ϕ10 = 2ξ(1 − ξ − η)(1 + ζ) (2.98)

ϕ11 = 2ξη(1 + ζ) (2.99)

ϕ12 = 2η(1 − ξ − η)(1 + ζ) (2.100)

ϕ13 = (1 − ξ − η)(1 − ζ 2) (2.101)

ϕ14 = ξ(1 − ζ 2) (2.102)

ϕ15 = η(1 − ζ 2). (2.103)

Wedge elements are often used as fill-in elements by automatic hexahedral meshing codes.

Their quality is comparable to the 20-node brick and 10-node tetrahedral element.

X1

X2

X3

ξ

η

ζ

1

1

2

2

3

3

4

4 5

5

6

6

A

Figure 2.5 6-node wedge element

LINEAR MECHANICAL APPLICATIONS 75

X1

X2

X3

ξ

η

ζ

1

1

2

2

3

3

4

4 5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

Figure 2.6 15-node wedge element

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