POTENTIAL PROBLEMS

35

is given

by

u* —

—\UVLV

(c)

The source formulation gives

u= au*dr- bu*dQ

(d)

Equation (d) can be applied

at

points

x = 0

and

x =

1,

to

obtain,

"o

=

o"i(isin l) +

i x

sin

x dx = 0

u

i

—

σ

ο(ι^

η

l) + i

x s

^

n

(1 —^)dx

= 0

Jo

This system gives,

1 cosl

4

<7

0

=

^—r-1,

*!=-—--1

(

f

)

sin

1

sin 1

Note that these two values for the sources are the same as the q

0

and

q

x

of

Example

1.7.

2.4 MATRIX FORMULATION

We will now explain how the boundary element techniques described

in Sections 2.2 and 2.3 can be interpreted

in

matrix form. The direct

case given by relationship (2.23) will be considered in detail. This case

is more general than equations (2.35) and (2.36) which can also

be

rendered into matrix form.

Consider equation (2.23) specialised

for

the boundary points and

without

the

term

in p for

simplicity (the case with

p is

considered

later),

i.e.

K+

uq*

dr

=

qu*dr

(2.41)

Assume

for

simplicity that

the

body

is

two dimensional and

its

boundary

is

divided into

N

'segments'

or

'boundary' elements,

as

shown

in

Figure 2.4.

The

points where

the

unknown values

are

36 POTENTIAL PROBLEMS

Figure 2.4 Boundary elements: (a) constant;

(b) linear; (c) quadratic

(b)

considered are called 'nodes' and taken to be in the middle of each

segment for the so-called 'constant' elements (Figure 2.4(a)). These

are going to be the only elements considered in this section, but in the

next chapter we will also discuss the case of linear elements (i.e.

elements for which the nodes are at the intersection between elements

as shown in Figure 2.4(b)) and curved elements such as the quadratic

ones shown in Figure 2.4(c). For the latter case an extra mid-element

node is required and the elements are called quadratic.

For the constant element case the boundary is discretised into N

elements, of which N

i

are assumed to belong to Γ

ί

andN

2

tor

2

.The

values of u and q are assumed to be constant on each element and

equal to the values at the mid-node of the element. Note that on each

POTENTIAL PROBLEMS

37

element

one

of

the two

variables

(u or q)

is

known.

Equation (2.41)

can

be

discretised

for a

given point

T

before

applying

any

boundary conditions,

as

follows,

N

r N r

fa*

Σ uq*dr=

X

qu*dr (2.42)

j =

i Jrj j

=

l

Jr,.

Note that

for

this type

of

element

the

boundary

is

always 'smooth'

hence

the

multiplier

of

w,

is j.

Γ

}

is

the

boundary

of

element

'/.

Formula (2.42) applies for a particular node

T. The u

}

and

q

}

values

can

be

taken

out of the

integrals

as

they

are

assumed

to be

constant

over each element. This gives,

ΐ".

+

Σ

(

[

9*

dr

)»J =

Σ

(J

"*

dr

\J

(2·

43

)

The integrals

j

q*

άΓ

relate

the

V

node with

the

segment

'/

over

which

the

integral

is

carried

out. We

shall call these integrals

Η^. The

integrals

on the

right-hand side

are

of

the

type

J

u*

άΓ and

will

be

called

G

tJ

.

Hence

we can

write (2.43)

as,

±11,+

Σ

#0";=

Σ

Giili

(

2

·

44

)

j

= i j

=

i

The integrals

can

easily

be

calculated analytically

but for

higher-

order elements they become more difficult

to

evaluate

and

they

are

usually evaluated numerically.

We can rewrite equation (2.44)

for

each i node under consideration.

Let

us now

call

j Η^ when

i

Φ

j

[Hij

+

i

when

i =7

Hence equation (2.44)

can now be

written

as,

Σ H

u

uj= Σ Gijlj

(

2

·

4

*)

7=1

7=1

The whole

set

of equations can also

be

expressed

in

matrix form as,

H

U

=

G

Q

(2.47)

Note that

N

x

values

of

u

and

N

2

values of q are known

on Γ,

hence

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