SMA-HPC '2002 MIT
2-D Integration
(from 3-D problems)
Reminder
Calculating Matrix Elements
Panel j
i
c
x
Collocation
point
(
,
,
i
cij
panel j
Gx xA dS
′′
=
∫
(
,
,
i
c entroii dj
Area G x xA ≈ ?
One point
quadrature
Approximation
x
y
z
(
int
4
,
1
*
4
i
oij c
j
p
Ar
A
ea
Gx x
=
≈
∑
Four point
quadrature
Approximation
)
)
j
c
)
,
j
SMA-HPC '2002 MIT
Symmetrically
Normalized 2-D
Problem
Quadrature Scheme
General n-point formula
,
i
xy
n points, n weights
2n parameters
-1
1
1-1
y
x
( (
1
11
11
,,
n
i i
i
wf xfx yydx
=
??
∫∫
∑
.
i
) )
i
SMA-HPC '2002 MIT
Symmetrically
Normalized 2-D
Problem
Quadrature Scheme
2-D Gaussian Quadrature
( (
11
1
1
1
, ,
n
il i i
i
l
wy pxx yp x
=
??
=
∑
∫∫
l-th order 2-D poly definition
Exactness for l-th order polys
(
(
,
,
i
l j
ij l
p xy x yα
+≤
=
∑
( ( )
1
Number of terms =
2
l +
) )
d
)
)
j
i
)
2l +
SMA-HPC '2002 MIT
Symmetrically
Normalized 2-D
Problem
Quadrature Scheme
Product Method
1) Get a 1-d formula
()
(
1
1
1
1
m
d
i
i
wfxfxdx
=
?
∫
∑
.
2) Form a product grid
1
1
d
x
1d
m
x
1d
m
x
1
1
d
x
1
,
d
i m m
x y xx=
)
1 d
i
1
,
d
i
SMA-HPC '2002 MIT
Symmetrically
Normalized 2-D
Problem
Quadrature Scheme
Product Method Continued
3) Determine the weights
(
(
11
1
,
1
1
1
1
, ,
m
d
ij i j
i
wfx xf xydx
=
??
∫
∑∑
∫
.
1
,
d
ij i j
w w=
Note that n = mxm
)
)
1
m
d
i =
1 d
w
SMA-HPC '2002 MIT
Symmetrically
Normalized 2-D
Problem
Quadrature Scheme
Product Method Theorem
Theorem: The product method is exact
for all 2-d polys up to order 2m
Proof:
(
(
11 11
,
11 1 1
,
i
m j
ij m
px y dxdy x y dxdyα
+≤
?? ? ?
=
∑
∫∫ ∫∫
Using the l-th order poly def
)
)
j
i
SMA-HPC '2002 MIT
Symmetrically
Normalized 2-D
Problem
Quadrature Scheme
Product Theorem Cont.
( ( )
11 1 1
,
11 1 1
i i j
ij ij
ij m i j m
x y dxdy x dx y dyα
+≤ + ≤
?? ? ?
?
=
?
?
∑
∫∫ ∫ ∫
Using the properties of integration
(
1
,
1
i
ij
ij m
xdx y dyα
+≤
?
? ? ?
=
? ? ?
? ? ?
∑
∫
)
,
j
α
?
?
?
∑
)
1
1
j
?
?
?
?
∫
SMA-HPC '2002 MIT
Symmetrically
Normalized 2-D
Problem
Quadrature Scheme
Product Theorem Cont.
Since the 1-d quadrature is exact for polys
of order less than 2m
(
1
,
1
i
ij
ij m
xdx y dyα
+≤
?
? ? ?
=
? ? ?
? ? ?
∑
∫
(
( (
1 1 1
,
1
m
i
d d
ij k k l l
ij m k l
w w α
+≤ = =
? ? ?
? ? ?
? ? ?
∑ ∑
)
1
1
j
?
?
?
?
∫
)
) )
1
1
m
j
d d
x x
?
?
?
∑
SMA-HPC '2002 MIT
Symmetrically
Normalized 2-D
Problem
Quadrature Scheme
Product Theorem Cont.
Rearranging the sums
(
( (
1 1
,
1
m
i
d d
ij k k l l
ij m k l
w w α
+≤ = =
? ? ?
=
? ? ?
? ? ?
∑ ∑
(
(
(
1 1 1
,
1
m
i
d d d
k i j k l
l i j m
ww x xα
= + ≤
=
∑∑ ∑
(
1 1 1
1
,
m
d d d
k m k l
l
ww p x x
=
∑∑
Which proves the theorem
)
) )
1 1
1
m
j
d d
x x
?
?
?
∑
)
)
)
1
1
m
j
d
l
k =
)
1
1
m
d
l
k =
SMA-HPC '2002 MIT
3-D Laplace s
Equation
Basis Function Approach
Calculating Self-Term
Panel i
i
c
x
Collocation
point
,
1
i
ii
pa cnel i
x
A dS
′
′
?
=
∫
,
0
i
ii
c
Panel Area
A
x ?
≈
%(&('
One point
quadrature
Approximation
x
y
z
,
is an integrable singularity
1
i
ii
panel i c
x
A S
′
?
′
=
∫
x
i
c
x
x
d
SMA-HPC '2002 MIT
3-D Laplace s
Equation
Basis Function Approach
Calculating Self-Term
Tricks of the trade
Panel i
i
c
x
Collocation
point
,
1
i
ii
pa cnel i
x
A dS
′
′
?
=
∫
x
y
z
Disk
surrounding
collocation point
,
1
i
c
ii
disk rest of panel
A dS dS
x x
′
?
′
=
∫
Disk Integral has
singularity but has
analytic formula
Integrate in two
pieces
2
0
1
2
1
i
R
dk cis
dS rdrd R
r
x
π
θ
′
′
=
?
∫ ∫
x
of radius R
1
i
c
x x
′
?
′
+
∫
0
x
π =
∫
SMA-HPC '2002 MIT
3-D Laplace s
Equation
Basis Function Approach
Calculating Self-Term
Other Tricks of the trade
Panel i
i
c
x
Collocation
point
Integrand is singular
,
1
i
ii
panel i c
x
A dS
′
′
=
?
∫
%(&('
x
y
z
2) Curve panels can be handled with projection
1) If panel is a flat polygon, analytical formulas exist
x