SERENDIPITY RECTANGULAR ELEMENTS

Serendipity elements are obtained as follows. First the number of nodes defining a 1D polynomial of a given degree along each side is chosen.

Then, the minimum number of nodes within the element is added so that a complete and symmetrical 2D polynomial of the same degree as the 1D polynomial chosen along the sides is obtained. Figure 5.8 shows some of the more popular Serendipity elements and the polynomial terms contained in the shape functions. The simplest element of the Serendipity family, i.e.

the 4-noded rectangle, coincides with the same element of the Lagrange family. Also note that the quadratic and cubic elements of 8 and 12 nodes,

Fig. 5.8 Some Serendipity elements and terms contained in their shape functions respectively, have not interior nodes, whereas the 17 node element requires a central node to guarantee the complete quartic approximation, as is explained next.

The derivation of the shape functions for Serendipity elements is not as straightforward as for Lagrange elements. In fact, some ingenuity is needed and this explains the name Serendipity, after the ingenuous discoveries of the Prince of Serendip quoted in the romances of Horace Walpole in the eighteenth century [EIZ,ZTZ].

5.4.1 Eigth-noded quadratic Serendipity rectangle

The shape functions for the side nodes are readily obtained as the product of a second degree polynomial in ξ (or η) and another one in η (or ξ). It can be checked that this product contains the required complete quadratic terms (Figure 5.9). For these nodes we obtain

N_i(ξ, η) = 1

2(1 + ξξ_i)(1 − η²) ; i = 4, 8 N_i(ξ, η) = 1

2(1 + ηη_i)(1 − ξ²) ; i = 2, 6

(5.18)

Unfortunately this strategy can not be applied for the corner nodes, since in this case the product of two quadratic polynomials will yield a zero value at the center and thus the criterion of Eq.(5.9) would be violated.

Consequently, a different procedure is followed as detailed below.

Step 1 . The shape function for the corner node is initially assumed to be bi-linear, i.e. for node 1 (Figure 5.9) we have

N₁^L= 1

4(1 − ξ)(1 − η) (5.19)

This shape function takes the value one at the corner node and zero at all the other nodes, except for the two nodes 2 and 8 adjacent to node 1 where it takes the value 1/2.

Step 2 . The shape function is made zero at node 2 by subtracting from N₁^L one half of the quadratic shape function of node 2:

N₁(ξ, η) = N₁^L−1

2N₂ (5.20)

Step 3 . Function N₁ still takes the value 1/2 at node 8. The final step is to substract from N₁ one half of the quadratic shape function of node 8

N₁(ξ, η) = N₁^L−1

2 N₂− 1

2N₈ (5.21)

Fig. 5.9 8-noded quadratic Serendipity rectangle. Derivation of the shape functions for a mid-side node and a corner node

The resulting shape function N₁satisfies the conditions (5.8) and (5.9) and contains the desired (quadratic) polynomial terms. Therefore, it is the shape function of node 1 we are looking for.

Following the same procedure for the rest of the corner nodes yields the following general expression

N_i(ξ, η) = 1

4(1 + ξξ_i)(1 + ηη_i)(ξξ_i+ ηη_i− 1) ; i = 1, 3, 5, 7 (5.22) Figure 5.9 shows that the shape functions for the 8-noded Serendipi-ty element contain a complete quadratic polynomial and two terms ξ²η and ξη² of the cubic polynomial. Therefore, this element has the same approximation as the 9-noded Lagrange element and it has one node less.

This makes the 8-noded quadrilateral in principle more competitive for practical purposes (see Section 5.9.2 for further details).

5.4.2 Twelve-noded cubic Serendipity rectangle

This element has four nodes along each side and a total of twelve side nodes which define the twelve terms polynomial approximation shown in Figure 5.8. The shape functions are derived following the same procedure explained for the 8-noded element. Thus, the shape functions for the side nodes are obtained by the simple product of two Lagrange cubic and linear polynomials. For the corner nodes the starting point is again the bilinear approximation. This initial shape function is forced to take a zero value at the two side nodes adjacent to the corner node by subtracting the shape functions of those nodes weighted by the factors 2/3 and 1/3. Figure 5.10 shows the expression of the shape functions which can be derived by the reader as an exercise.

It is simple to check that the element satisfies conditions (5.8) and (5.9). Figure 5.8 shows that the shape functions contain a complete cubic approximation plus two terms (ξ³η, ξη³) of the quartic polynomial. This element compares very favourably with the 16-noded Lagrange element, since both have a cubic approximation but the Serendipity element has fewer nodes (12 nodes versus 16 nodes for the cubic Lagrange rectangle).

5.4.3 Seventeen-noded quartic Serendipity rectangle

The quartic Serendipity rectangle has five nodes along each side and a total of seventeen nodes (sixteen side nodes plus a central node, Figure 5.10). The central node is necessary to introduce the “bubble” function (1 − ξ²)(1 − η²) as shown in Figure 5.5. This function contributes the term ξ²η² to complete a quartic approximation.

The derivation of the shape functions follows a procedure similar to that for the 8 and 12 node Serendipity elements. The shape functions

Fig. 5.10 Shape functions for the cubic (12 nodes) and quartic (17 nodes) Seren-dipity rectangles

for the side nodes are obtained by the product of a quartic and a linear polynomial. An exception are nodes 3, 7, 11 and 15 for which the function 1/2 (1 − ξ²)(1 − η²) is subtracted from that product so that the resulting shape function takes a zero value at the central node. The starting point for the corner nodes is the bilinear function to which a proportion of the shape functions of the side nodes is subtracted so that the final shape function takes a zero value at these nodes. Finally, the shape function for the central node is the bubble function. Figure 5.10 shows the shape functions for this element.

Figure 5.8 shows that the shape functions contain a complete quartic approximation plus two additional terms (ξ⁴η and ξη⁴) from the quintic polynomial. The corresponding quartic Lagrange element has 25 nodes (Figure 5.3) and hence the 17-noded Serendipity rectangle is more eco-nomical for practical purposes.

5.5 SHAPE FUNCTIONS FOR C

CONTINUOUS TRIANGULAR