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Boundary Element Techniques in Engineering by S. Walker, C. A. Brebbia

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70 HIGHER-ORDER ELEMENTS
and u
t
are the potentials at the nodes
1
to
4.
The same function can be
used
to
describe
the
geometry
of
the element,
i.e.
4
X = </>i*i +02*2+ </>3*3 + </>4*4= Σ
<t>i
X
i
(
3
'
34
)
i
= 1
and similarly
for y
and
z.
Note that once these functions are defined
we
can obtain the value
of
| G |
and
find
the derivatives of a function such
as u
with respect to x,
y
or z
using formula (3.22).
HIGHER-ORDER QUADRILATERAL ELEMENTS
We
can
improve
our
approximation
by
taking
a
second-order
quadrilateral surface element (Figure 3.12), such that,
8
8 8
*
= Σ
Φι
χ
ί
y=Σ
4nyi
z
= Σ ΦΛ
(
3
·
35
)
i
= 1 i = 1 i = 1
with,
Φι=έ(1-ίι)(1-«2)(-ίι-ί2-1)
02
= i(l
+ ί
1 )(1
-
ί
2
)(ί
1
-
«2
-
1)
03=iU+fl)(l+i2)(«l+{2-D
04
=
i(i-i
1
)(i
+
«2)(-ii
+
i2-i)
(136)
03
=
iU-tf)U-i2)
06
=
id-{?)(!+ίι)
07=*(l-tf)U+£2)
0e
=
i(i-i2)(i-fi)
The same type
of
function could
be
used
for u or q.
Higher-order quadrilateral elements
can be
used
as
boundary
elements,
such
as
the
cubic
type
(cubic
variation of the function on the
element side), shown
in
Figure 3.13(a). For this element one has two
additional nodes
on
each side,
but one can
also work with only
corner nodes
and
include
the
derivatives
as
nodal quantities (Fig-
ure 3.13(b)).
Figure 3.12 Eight-noded elements
(a)
15 6 2
Twelve-noded elements ( u are the unknowns)
1 i
(b)
1
Four-noded element with u and its derivatives
(-^>-4T
as
unknowns)
Figure 3.13 Higher-order elements
72 HIGHER-ORDER ELEMENTS
TRIANGULAR ELEMENTS
The simplest triangular element
is one as
shown
in
Figure 3.14. There
is a linear variation of the function over the element and one can work
with oblique coordinates
ξι
9
ξ
2
,
which vary from
0 to
1 over
the two
sides shown
in the
figure.
Figure
3.14
Simplest triangular element
The position vector
f of any
point
on the
triangle
is,
f=z
xi + yj + zk
=
*3*
+
y*f+
*3*+
Ί
ζ 1^1
+
l
2 ^2*2
(
3
·
37
)
where
HIGHER-ORDER ELEMENTS 73
Hence the position vector t
can
be defined as,
?=[x
3
+ (*i-*3Ki + (*2-*3K2!F+
+1>3 +
(>Ί
- y
3
Ki + to - Λ)δ2ΪΓ+
+ [z
3
+
(Zj
-
Z
3
K!
+
(Z
2
- Z
3
2
]T
= xf+yf+zk (3.39)
Thus we have that,
X = f 1*1 +
£
2
*2
+
(1
- £l - ^2)^3
y = tiyi
+
t
2
yi
+ (i-£i -W^
ζ = ^
1
ζ
1
2
ζ
2
+ (1-ξ
1
2
3
(3.40)
We can now define £
3
=
1
^
£
2
, where ξ
3
is a new but not
independent coordinate, such that,
£l+£
2
+ £3=l
Hence,
'
x
)
y
=
l
z
J
*1
x
2 *3
y\
yi y*
_ Z\ z
2
z
3
_
>
a
3
(3.41)
As ξ
3
is not independent we can invert the first two equations to
obtain,
ξι =
{2A
°
l
+ί>
ι
χ + α
ι>')
^ =
{2A
°
2 + b2X + a2y)
(3.42)
and obtain ξ
3
as
1
ξ
ι
ζ
2
, i.e.
1
ξ
3
=
—(2Α%
+ + α
3
γ)
(3.43)
The recurrence relationship for equations (3.42) and (3.43) is
<*i
= **-*,, bi = yj-y
k
, 2A? = Xjy
k
-x
k
yj (3.44)
74 HIGHER-ORDER ELEMENTS
i
=
1,2,3;
j =
2,3,1;
k
=
3,1,2. The area
A
is
Λ
=
ι(Μ
2
-Μι) (3.45)
and represents the area
of
the projection
of
the element
on
the
x,
y
plane.
The transformation
can now be
carried
out as
previously.
The
reason for using ξ
3
as a dependent variable is that it renders
a
simple
expression
for
the interpolation functions but care should be taken
not
to
confuse
it
with an independent variable.
Linear expressions for the coordinates or as u functions are now,
3
η
= η
ι
ξ
ι
+ η
2
ξ
2
+
η
ζ
ξ
ζ
=
Σ
u
i<t>i
(3.46)
X
=
X
1
£l+X
2
£
2
+
*3£3
= Σ
X
i<t>i
i=
1
Note that the interpolation functions are
φ
(
=
ξ^
The homogeneous triangular coordinates have
one
more advan-
tage.
This
is
that the integrals for polynomial
terms can be
carried out
using
a
simple rule,
i.e.
Jf
ξ
'
ΑΆΛΑ
-ν^ϊϋ
(3
·
47)
This formula greatly simplifies
the
algebra when numerical
in-
tegration
is not
used.
HIGHER-ORDER TRIANGULAR ELEMENTS
The next triangular model expresses
the
function
as a
complete
second degree polynomial, which has a mid-side node (Figure 3.15).
The model satisfies interelement compatibility and gives,
u
=
ΣΦΜ (3-48)
where
^
=
^(2^-1), tf>
4
=
4^
2
Φι
=
ξιΜι-ΙΙ
Φ5
=
2
ξ
3
(3.49)
Φ
3
=
ξ
3
(2ζ
3
-1),
φ
6
=
3
ξ
1

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