The Boundary Element Method - Mechanics of Structures: Variational and Computational Methods - page 552

N i v i ,

For the linear element, the shape function for ω takes the form of Eq. (9.30), u =

N i q i

and q = where

1
2 (

1
2 (

N i = [ N 1 N 2 ] with N 1 = 1 − ξ) ,N 2 = 1 + ξ)

v i

2 ] T ,

q i q 2 ] T

= ω ω =

[ [ q 1

1

This element is shown in Fig. 9.6a.
Follow the procedure from Eq. (9.32) to (9.34) to obtain

M M

1 H ij 1 G ij q i

i

c ω j = v =

i = i =

r k a k
r 2

where H ij N i dS =−

S i

ln 1
r

N i dS

G ij

=− .

S i

After the warping function is found, the torsional constant is calculated using Eq. (15) of
Example 9.2. The results of the computations are shown in the following table. The numbers
in the brackets indicate the percentage error.

Torsional Constant

Number of Constant Linear

Elements Element Element

12 0.1366 (2.8) 0.1540 (9.55)

20 0.1393 (0.9) 0.1456 (3.57)

28 0.14008 (0.35) 0.1432 (1.87)

40 0.1404 (0.126) 0.1418 (0.87)

Exact 0.140577

Solution

Although the accuracy of both elements is reasonable in this case, the constant element
gives better results than the linear element.

9.2.3 Indirect Formulation
The unknown variables in the boundary integral equation in the previous sections are the
variables that can be used directly for the computation of the desired physical quantities.
Such a formulation is referred to as being direct . In an alternative boundary element formu-
lation, the indirect formulation , the variables sought are not directly the physical variables.
Thus the unknown variables in the indirect formulation will not be u and q but some other
quantities which are used to express u and q . The physical variables are calculated after
these quantities are found.
The indirect formulation starts from the investigation of the solution of the equation

∇

2 u =

0 in the space outside V which is denoted by V . Multiply

∇

2 u

δ =

u ∗

by u of

Eq. (9.22), with x in V and ξ in V , and integrate over V to obtain

∂
∂

u ∗ ∂ u ∗ ∂ u ∗ ∂ dV

u
∂

u
∂

u
∂

+ ∂
∂

+ ∂
∂

dV =

u ∗ ( ∇

2 u )

x x y y z z

V V

∂ dV

u
∂

u ∗
∂

u
∂

u ∗
∂

u
∂

u ∗
∂ x ∂ x + ∂ y ∂ y + ∂ z ∂

− (9.41)

z

V