Computational Fluid Dynamics
Lecture 12
Finite Elements in 2D
Straightforward extension of 1D ideas
Simplest FE is rectangle (also consider triangles)
1.
Bilinear interpolation function:
C
C
1
2
3
4
x
C y
C xy
+
+
+
Coefficients determined by function values at four vertices.
Reduce to linear interpolation along
xy
axis.
The product of Chapeau functions in
x
and
y
called “Pagoda” functions.
Unity at central node.
And vanishing to zero at 8 neighboring nodes.
Consider 2D advection equation:
0
U
V
t
x
y
ψ
ψ
ψ
∂
∂
∂
+
+
=
∂
∂
∂
with Pagoda functions.
Define averaging operators
and
x
y
A
A
(
)
(
)
,
1,
,
1,
,
,
1
,
,
1
,
2
,
2
,
1
4
6
1
4
6
The advection equation becomes
0
x
i j
i
j
i j
i
j
y
i j
i j
i j
i j
i j
x
y
y
x
i j
x
y
i j
A a
a
a
a
A a
a
a
a
da
A A
UA
a
VA
a
dt
δ
δ
+
−
+
−
=
+
+
=
+
+
+
+
=
1
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where
1,
1,
2
,
,
1
,
1
2
,
2
2
i
j
i
j
x
i j
i j
i j
y
i j
a
a
a
x
a
a
a
y
δ
δ
+
−
+
−
−
=
∆
−
=
∆
The product of averaging operators
couples the time derivatives at 9 different nodes and
generates a band matrix with a wide band width.
x
y
A A
An efficient solution can be done by:
1.
Evaluate the spatial derivatives at every node point first by evaluating the tridiagonal system:
2
,
,
2
,
,
1
1
1,
,
1,
for
and
.
These are the independent "compact scheme" equations, i.e.
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 Spring '07
 Slinn
 Trigraph, ∂x, Computational fluid dynamics, ∂u ∂v

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