5.1. Finite Element Methods for Two-Dimensional Problems
107
TABLE 5.1. Basis functions in the local coordinate system.
Linear element
tPl = ~
tP2 ="
tP3=1-~-"
Quadratic element
tPl = 2~2 - ~
tP2 = 2,,2 - "
tP3 = (I - ~ - ,,)(1 - 2~ - 2,,)
tP4 = 4~"
tPs = 4(1 - ~ - "),,
tP6 = 4(1 - ~ - ,,)~
FIGURE 5.6. The third-order triangle.
Cubic element
tPl = tw~ - 1)(3~ - 2)
tP2 = t"(3,, - 1)(3" - 2)
tP3 = t(1 - ~ - ,,)(2 - 3~ - 3,,)(1 - 3~ - 3,,)
tP4 = ~~,,(3~ - I)
tPs = ~~"(3,, - I)
tP6 = ~,,(I - ~ - ")(3,, - I)
tP7 = ~,,(I - ~ - ,,)(2 - 3~ - 3,,)
tPs = ~W - ~ - ,,)(2 - 3~ - 3,,)
tP9 = ~W - ~ - ,,)(3~ - I)
tPlO = 27~,,(I - ~ - ,,)
3
6
~------~------~5~----~2·
Using Eq. (5.1.42), the Jacobian of the transformation can be calculated
and all the coefficients of Eqs. (5.1.28) to (5.1.30) can be integrated in the local
coordinate system. The integrands are polynomial expressions with e and ,.,
lower than the third order.
The cubic finite element may be derived in a manner similar to the quadratic element. In each triangle we define 10 nodes, and use a complete cubic
polynomial to approximately express the value of the unknown function in
the element, see Figure 5.6. Using the transformation in Eq. (5.1.42), element
(a) becomes a standard triangular element (a') in the e,., plane within the
local coordinate system. The relevant nodes are shown in Figure 5.7. The
third-order Galerkin basis functions in the local coordinate system are also
listed in Table 5.1.
With the expressions of basis functions, the values of coefficients Aij' B ij
and F; in element (a) can be obtained by numerical integration. The integrands
are all fourth-order polynomials of e and ,., in the local coordinate system.
The Gauss quadratic formula is often employed for this purpose.
Besides using 10 nodal values, we have other methods to determine the 10
coefficients of a cubic polynomial. For instance, take three corner points and
a center point of the triangie as nodes and assume that function values, as
weH as the partial derivatives with respect to x and y at the three vertices, are
107
TABLE 5.1. Basis functions in the local coordinate system.
Linear element
tPl = ~
tP2 ="
tP3=1-~-"
Quadratic element
tPl = 2~2 - ~
tP2 = 2,,2 - "
tP3 = (I - ~ - ,,)(1 - 2~ - 2,,)
tP4 = 4~"
tPs = 4(1 - ~ - "),,
tP6 = 4(1 - ~ - ,,)~
FIGURE 5.6. The third-order triangle.
Cubic element
tPl = tw~ - 1)(3~ - 2)
tP2 = t"(3,, - 1)(3" - 2)
tP3 = t(1 - ~ - ,,)(2 - 3~ - 3,,)(1 - 3~ - 3,,)
tP4 = ~~,,(3~ - I)
tPs = ~~"(3,, - I)
tP6 = ~,,(I - ~ - ")(3,, - I)
tP7 = ~,,(I - ~ - ,,)(2 - 3~ - 3,,)
tPs = ~W - ~ - ,,)(2 - 3~ - 3,,)
tP9 = ~W - ~ - ,,)(3~ - I)
tPlO = 27~,,(I - ~ - ,,)
3
6
~------~------~5~----~2·
Using Eq. (5.1.42), the Jacobian of the transformation can be calculated
and all the coefficients of Eqs. (5.1.28) to (5.1.30) can be integrated in the local
coordinate system. The integrands are polynomial expressions with e and ,.,
lower than the third order.
The cubic finite element may be derived in a manner similar to the quadratic element. In each triangle we define 10 nodes, and use a complete cubic
polynomial to approximately express the value of the unknown function in
the element, see Figure 5.6. Using the transformation in Eq. (5.1.42), element
(a) becomes a standard triangular element (a') in the e,., plane within the
local coordinate system. The relevant nodes are shown in Figure 5.7. The
third-order Galerkin basis functions in the local coordinate system are also
listed in Table 5.1.
With the expressions of basis functions, the values of coefficients Aij' B ij
and F; in element (a) can be obtained by numerical integration. The integrands
are all fourth-order polynomials of e and ,., in the local coordinate system.
The Gauss quadratic formula is often employed for this purpose.
Besides using 10 nodal values, we have other methods to determine the 10
coefficients of a cubic polynomial. For instance, take three corner points and
a center point of the triangie as nodes and assume that function values, as
weH as the partial derivatives with respect to x and y at the three vertices, are