Discretisation and Interpolation - The Boundary Element Method with Programming - page 52

3.9 DIFFERENTIAL GEOMETRY

In the boundary element method it will be necessary to work out the direction normal to
a line or surface element.

[

V

v

3

Figure 3.27 Vectors normal and tangential to a one-dimensional element

The best way to determine these directions is by using vector algebra. Consider a
one-dimensional quadratic boundary element (Figure 3.27). A vector in the direction of
[ can be obtained by

w

V x

(3.45)

[

w

By the differentiation of equation (3.4) we get

3

w w N

¦

n e

V x x

(3.46)

[ n

w [ w [

n 1

A vector normal to the line element, V 3 , may then be computed by taking the cross-
product of V [ with a unit vector in the z-direction ( v z ):

V V u v

(3.47)

3 [ z

This vector product can be written as:

- ½

dx - dy ½

° °

° °

d [ d [

- ½

V 0

- ½

3 x

° °

® ¾ dy ° dx °

® ¾

V V u 0

(3.48)

® ¾ ® ¾

3 3 y

d [ d [

¯ ¿ ° ° ¯ ¿ ° °

1

V

° ° ° °

3 z

dz 0

d [

¯ ¿

¯ ¿