# Optimal Discontinuous Galerkin Method Applications for Computational Aeroacoustics

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22 Φεβ 2014 (πριν από 7 χρόνια και 6 μήνες)

294 εμφανίσεις

Optimal
hp
Discontinuous Galerkin Method
Applications for Computational Aeroacoustics
Christophe Peyret and Philippe Delorme
ONERA/DSNA - 29 ave de la Division Leclerc
F92322 Châtillon cedex France
19
Dec 05
peyret@onera.fr
delorme@onera.fr
Introduction
Why Develop Methods based on DGM
to Compute Euler’s linearized equations ?
FEM faces
difﬁculties to
solve Linearized Euler’s equations
FDM faces difﬁculties
with complex geometries and boudary conditions
DGM
and
Disavantages
:
High Flexiblility
Complex Geometries
Variational Formulation
Adapted to Parallel Computation
Harder to program (OOP required)
CPU and RAM Expensive !
3
Galerk
i
n
Disc
ontinuous
Me
tho
d
F
or
Co
mputa
t
i
onal
A
er
o
ac
o
ustics
sym
metric,
it
is
d
iagonal
iz
ab
le
and
c
an
b
e
spli
tte
d
in
to
a
p
ositiv
e
(se
t
of
p
os
i
tiv
e
e
igen
v
alues)
and
a
n
e
gati
v
e
part
(se
t
of
n
e
gati
v
e
e
i
ge
n
v
alu
e
s)
:
A
i
n
i
=
[
A
i
n
i
]
+
+
[
A
i
.n
i
]
!
.
In
th
is
me
th
o
d
th
e
w
eak
for
m
ul
ation
to

nd
:
!
!
h
!
W
k
(
!
h
)
|
"
"
h
!
W
k
(
!
h
)
;
L
(
!
h
,
"
h
)
=
0
"
,
where:
L
(
!
h
,
"
h
)
=
#
!
"
h
.
"
t
!
h
+
#
!
"
h
.
A
i
"
i
!
h
+
#
!
"
h
.
B
!
h
+
\$
!
"
h
|
!
!
"
h
.
[
A
i
n
i
]
!
(
!
o
h
#
!
i
h
)
+
\$
!
"
h
"
!
!
"
h
.
(
M
!
h
#
g
)
#
#
!
"
h
.
g
.
(3.4)
Ex
press
ion
eq
(3.4)
c
an
b
e
spl
itted
as
:

th
e
ﬁr
s
t
term
is
the
b
lo
c-
d
iagon
al
l
o
c
al
mass
m
at
rix,

th
e
s
ec
on
d
term
i
s
the
b
lo
c-diagon
al
lo
cal
sti
!
ness
m
atr
ix
(v
anishes
f
or
fvm
),

th
e
thi
rd
term
i
s
a
matri
x
pro
duct
(also
b
lo
c-
d
iagonal
),

th
e
fou
rth
term
is
the
lo
cal
e
d
ge
s
ti
!
n
e
ss
matrix
conn
e
ctin
g
elem
en
ts,

th
e
ﬁf
th
term
in
tro
d
uce
s
t
he
b
oun
dar
y
cond
ition
s
usin
g
op
erator
M
,

th
e
s
i
xth
term
i
n
tro
d
uce
s
the
acoustic
sour
c
es
in
th
e
f
orm
u
lation
.
Settin
g
"
h
=
!
h
in
eq.
(3.4)
giv
es
aft
e
r
in
te
gr
ation
th
e
e
r
ror
es
timate:
|
!
#
!
h
|
L
2
([0
,T
]
,L
2
(
!
)
3
\$
C
(
T
,
"
)
h
k
+1
/
2
,
in
t
he
case
of
u
nstable
ﬂo
w,
constan
t
C
can
gro
w
to
i
nﬁ
nit
y
as
T
in
c
r
e
ase
s.
Numerical
exp
eri
m
en
ts
s
h
o
w
the
e
stimate
is
pr
obabl
y
b
e
tter
(
h
k
+1
).
T
o
obtai
n
th
is
e
stimate
s
ign
of
[
A
i
n
i
]
!
is
ess
en
tial.
3.3.
Bou
ndary
Condi
tions
Consider
an
elem
en
t
lo
calize
d
on
b
or
der
"
"
of
domain
"
s
imil
ar
to
t
he
on
e
exhib
ited
in
Fig.
2.
F
or
suc
h
an
ele
men
t
"
"
j
%
"
"
=
a
1
&
=
'
and
b
oun
dar
y
cond
ition
s
a
r
e
imp
ose
d
on
a
1
.
In
lo
cal
repres
en
tati
on,
the
v
ar
iation
al
form
ul
a
t
ion
#
!
j
t
"
I
(
"
t
!
+
A
i
"
i
!
+
B
!
)
+
\$
a
1
t
"
[
d
2
,d
3
]
M
!
[
d
2
,d
3
]
#
\$
a
1
t
"
[
d
2
,d
3
]
I
g
+
\$
a
2
t
"
[
d
3
,d
1
]
[
A
i
n
i
]
!
(
!
[
d
6
,d
7
]
#
!
[
d
3
,d
1
]
)
+
\$
a
3
t
"
[
d
1
,d
2
]
[
A
i
n
i
]
!
(
!
[
d
8
,d
9
]
#
!
[
d
1
,d
2
]
)
.
Symetric Friedrich System
Physical Modeling and Mathematical formulation
P
h
i
l
i
p
p
e
De
l
o
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m
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a
n
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b
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t
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w
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or

u
i
d
d
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n
am
i
c
s
f
or
i
n
s
t
an
c
e
)
:
t
h
e
s
c
h
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m
e
i
s
d
i
s
s
i
p
at
i
f
an
d
t
h
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n
u
m
b
e
r
of
u
n
k
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s
(
d
e
gr
e
e
s
of
f
r
e
e
d
om
)
i
s
l
ar
ge
r
t
h
an
t
h
e
on
e
of
t
h
e
f
d
m
or
c
l
as
s
i
c
al
f
e
m
.
Not
e
t
h
at
t
h
e
d
i
s
c
r
e
p
an
c
y
d
e
c
r
e
as
e
s
w
h
e
n
t
h
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or
d
e
r
i
n
c
r
e
as
e
s
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s
o
w
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h
op
e
t
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i
s
m
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t
h
o
d
d
o
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s
w
or
k
e
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c
i
e
n
c
y
at
h
i
gh
or
d
e
r
s
.
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t
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ar
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on
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h
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w
a
y
t
o
q
u
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f
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t
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u
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or
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,
t
h
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g
d
m
p
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s
e
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t
s
s
om
e
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x
ot
i
c

w
a
y
s
t
o
e
x
p
l
or
e
l
i
k
e
t
h
e
s
p
at
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t
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m
p
or
al
as
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p
e
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t
s
,
t
h
e
ac
c
u
r
ac
y
r
e

n
e
m
e
n
t
,
t
h
e
u
s
e
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n
e
w
b
as
i
s
f
u
n
c
t
i
on
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d
t
h
e
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o-
c
on
f
or
m
u
n
s
t
r
u
c
-
t
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r
e
d
m
e
s
h
e
s
.
2.
P
h
y
s
i
c
a
l
m
o
d
e
l
T
h
e
u
s
u
al
l
i
n
e
ar
p
ar
t
i
al
d
i
"
e
r
e
n
t
i
al
e
q
u
at
i
on
s
s
y
s
t
e
m
u
s
e
d
f
or
ae
r
oac
ou
s
t
i
c
s
c
om
e
s
f
r
om
t
h
e
l
i
n
e
ar
i
z
at
i
on
of
E
u
l
e
r

s
e
q
u
at
i
on
s
an
d
r
e
p
l
ac
i
n
g
t
h
e
e
n
e
r
gy
e
q
u
at
i
on
w
i
t
h
t
h
e
e
n
t
r
op
y
e
q
u
at
i
on
.
I
t
i
s
v
al
i
d
f
or
s
u
b
s
on
i
c

o
w
s
.
M
or
o
v
e
r
,
as
s
u
m
i
n
g
e
n
t
r
op
y
p
e
r
u
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i
t
of
m
as
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f
or
m
i
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s
p
ac
e
at
i
n
i
t
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t
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,
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t
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e
m
ai
n
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n
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f
or
m
i
s
s
p
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an
y
t
i
m
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.
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h
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s
h
y
p
ot
h
e
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s
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s
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ot
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s
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m
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d
b
u
t
i
s
u
s
u
al
l
y
d
on
e
.
S
o
i
n
3D
(
r
e
s
p
.
2D
)
w
e
h
a
v
e
a
s
y
s
t
e
m
of
f
ou
r
e
q
u
at
i
on
s
(
r
e
s
p
.
t
h
r
e
e
)
:
on
e
f
or
t
h
e
m
as
s
an
d
3
(
r
e
s
p
.
2)
f
or
t
h
e
m
om
e
n
t
.
As
m
os
t
of
p
h
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s
i
c
al
p
ar
t
i
al
d
i
"
e
r
e
n
t
i
al
e
q
u
at
i
on
s
,
i
t
i
s
s
y
m
e
t
r
i
z
a
b
l
e
or
i
t
i
s
a
”F
r
i
e
d
r
i
c
h

s
s
y
s
t
e
m
”.
T
h
e
w
e
l
l
-
k
n
o
w
n
(
at
l
e
as
t
b
y
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h
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y
p
e
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ol
i
c
i
an
s
)
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h
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or
e
m
of
G
o
d
on
o
v
-
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o
c
k
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s
t
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s
)
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s
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s
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,
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ai
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m
:
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t
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=
0
w
h
e
r
e
:
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1
=
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0
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0
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a
0
0
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0
'
(
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2
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t
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o
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e
n
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e
d
i
n
t
h
i
s
p
ap
e
r
.
M
or
o
v
e
r
,
w
e
ar
e
ab
l
e
t
o
w
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i
t
e
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h
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p
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De
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o
r
m
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a
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C
h
r
i
s
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o
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e
P
e
y
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e
t
i
s
e
as
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l
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e
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d
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h
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c
al
c
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l
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s
c
an
b
e
r
e
al
i
z
e
d
on
p
c
-
c
l
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s
t
e
r
.
O
f
c
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r
s
e
t
h
e
m
e
t
h
o
d
p
r
e
s
e
n
t
s
s
om
e
d
i
s
v
e
n
t
age
s
w
i
t
c
h
ar
e
v
e
r
y
i
m
p
or
t
an
t
at
t
h
e
l
o
w
l
e
v
e
l
s
of
ac
c
u
r
ac
y
(
f
or

u
i
d
d
y
n
am
i
c
s
f
or
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n
s
t
an
c
e
)
:
t
h
e
s
c
h
e
m
e
i
s
d
i
s
s
i
p
at
i
f
an
d
t
h
e
n
u
m
b
e
r
of
u
n
k
n
o
w
s
(
d
e
gr
e
e
s
of
f
r
e
e
d
om
)
i
s
l
ar
ge
r
t
h
an
t
h
e
on
e
of
t
h
e
f
d
m
or
c
l
as
s
i
c
al
f
e
m
.
Not
e
t
h
at
t
h
e
d
i
s
c
r
e
p
an
c
y
d
e
c
r
e
as
e
s
w
h
e
n
t
h
e
or
d
e
r
i
n
c
r
e
as
e
s
an
d
s
o
w
e
h
op
e
t
h
i
s
m
e
t
h
o
d
d
o
e
s
w
or
k
e
!
c
i
e
n
c
y
at
h
i
gh
or
d
e
r
s
.
S
t
u
d
i
e
s
ar
e
on
t
h
e
w
a
y
t
o
q
u
an
t
i
f
y
t
h
i
s
as
s
e
r
t
i
on
.
F
u
r
h
e
r
m
or
e
,
t
h
e
g
d
m
p
r
e
s
e
n
t
s
s
om
e
”e
x
ot
i
c

w
a
y
s
t
o
e
x
p
l
or
e
l
i
k
e
t
h
e
s
p
at
i
o-
t
e
m
p
or
al
as
-
p
e
c
t
s
,
t
h
e
ac
c
u
r
ac
y
r
e

n
e
m
e
n
t
,
t
h
e
u
s
e
of
n
e
w
b
as
i
s
f
u
n
c
t
i
on
s
an
d
t
h
e
n
o-
c
on
f
or
m
u
n
s
t
r
u
c
-
t
u
r
e
d
m
e
s
h
e
s
.
2.
P
h
y
s
i
c
a
l
m
o
d
e
l
T
h
e
u
s
u
al
l
i
n
e
ar
p
ar
t
i
al
d
i
"
e
r
e
n
t
i
al
e
q
u
at
i
on
s
s
y
s
t
e
m
u
s
e
d
f
or
ae
r
oac
ou
s
t
i
c
s
c
om
e
s
f
r
om
t
h
e
l
i
n
e
ar
i
z
at
i
on
of
E
u
l
e
r

s
e
q
u
at
i
on
s
an
d
r
e
p
l
ac
i
n
g
t
h
e
e
n
e
r
gy
e
q
u
at
i
on
w
i
t
h
t
h
e
e
n
t
r
op
y
e
q
u
at
i
on
.
I
t
i
s
v
al
i
d
f
or
s
u
b
s
on
i
c

o
w
s
.
M
or
o
v
e
r
,
as
s
u
m
i
n
g
e
n
t
r
op
y
p
e
r
u
n
i
t
of
m
as
s
u
n
i
f
or
m
i
n
s
p
ac
e
at
i
n
i
t
i
al
t
i
m
e
,
i
t
w
i
l
l
r
e
m
ai
n
u
n
i
f
or
m
i
s
s
p
ac
e
at
an
y
t
i
m
e
.
T
h
i
s
h
y
p
ot
h
e
s
i
s
i
s
n
ot
e
s
s
e
n
t
i
al
f
or
t
h
e
m
e
t
h
o
d
b
u
t
i
s
u
s
u
al
l
y
d
on
e
.
S
o
i
n
3D
(
r
e
s
p
.
2D
)
w
e
h
a
v
e
a
s
y
s
t
e
m
of
f
ou
r
e
q
u
at
i
on
s
(
r
e
s
p
.
t
h
r
e
e
)
:
on
e
f
or
t
h
e
m
as
s
an
d
3
(
r
e
s
p
.
2)
f
or
t
h
e
m
om
e
n
t
.
As
m
os
t
of
p
h
y
s
i
c
al
p
ar
t
i
al
d
i
"
e
r
e
n
t
i
al
e
q
u
at
i
on
s
,
i
t
i
s
s
y
m
e
t
r
i
z
a
b
l
e
or
i
t
i
s
a
”F
r
i
e
d
r
i
c
h

s
s
y
s
t
e
m
”.
T
h
e
w
e
l
l
-
k
n
o
w
n
(
at
l
e
as
t
b
y
t
h
e
h
y
p
e
r
b
ol
i
c
i
an
s
)
t
h
e
or
e
m
of
G
o
d
on
o
v
-
M
o
c
k
as
s
e
r
t
s
t
h
at
t
h
e
s
y
s
t
e
m
b
e
c
om
e
s
s
y
m
e
t
r
i
c
(
af
t
e
r
a
c
h
an
ge
of
t
h
e
u
n
k
n
o
w
s
)
i
f
a
m
at
h
e
m
at
i
c
al
e
n
t
r
op
y
e
x
i
s
t
s
.
T
h
i
s
i
s
t
h
e
c
as
e
of
t
h
e
p
h
y
s
i
c
al
p
r
ob
l
e
m
s
(
e
l
e
c
t
r
o-
m
agn
e
t
i
s
m
,
F
l
u
i
d
m
e
c
h
an
i
c
s
,
Ae
r
o-
ac
ou
s
t
i
c
s
,
q
u
an
t
i
c
m
e
c
h
an
i
c
s
.
.
.
)
.
F
or
ou
r
p
r
ob
l
e
m
i
t
m
e
an
s
,
i
f
on
e
u
s
e
s
t
h
e
n
e
w
v
ar
i
ab
l
e
s
:
!
=
!
"
u
1
v
1
a
0
!
1
/
!
0
#
\$
,
on
e
ob
t
ai
n
s
t
h
e
n
e
w
s
y
s
t
e
m
:
"
t
!
+
A
i
"
i
!
+
B
!
=
0
w
h
e
r
e
:
A
1
=
%
&
u
0
0
a
0
0
u
0
0
a
0
0
u
0
'
(
A
2
=
%
&
v
0
0
0
0
v
0
a
0
0
a
0
v
0
'
(
an
d
:
B
=
%
)
)
)
)
&
"
x
u
0
"
y
u
0
!
"
x
a
0
"
x
v
0
"
y
v
0
!
"
y
a
0
a
0
!
0
"
x
!
0
a
0
!
0
"
y
!
0
(
#
!
1)
(
"
x
u
0
+
"
y
v
0
)
'
*
*
*
*
(
t
h
e
m
at
r
i
x
-
v
al
u
e
d
s
y
m
b
ol
A
i
"
i
i
s
s
y
m
e
t
r
i
c
s
o
t
h
e
r
e
al
m
at
r
i
x
A
i
n
i
i
s
d
i
agon
al
i
z
ab
l
e
.
T
h
i
s
f
e
at
u
r
e
i
s
i
m
p
or
t
an
t
f
or
t
h
e
m
e
t
h
o
d
p
r
e
s
e
n
t
e
d
i
n
t
h
i
s
p
ap
e
r
.
M
or
o
v
e
r
,
w
e
ar
e
ab
l
e
t
o
w
r
i
t
e
Linearized Euler’s Equations
Galerk
i
n
Disc
ontinuous
Me
tho
d
F
or
Co
mputa
t
i
onal
A
er
o
ac
o
ustics
sym
metric,
it
is
d
iagonal
iz
ab
le
and
c
an
b
e
spli
tte
d
in
to
a
p
ositiv
e
(se
t
of
p
os
i
tiv
e
e
igen
v
alues)
and
a
n
e
gati
v
e
part
(se
t
of
n
e
gati
v
e
e
i
ge
n
v
alu
e
s)
:
A
i
n
i
=
[
A
i
n
i
]
+
+
[
A
i
.n
i
]
!
.
In
th
is
me
th
o
d
th
e
w
eak
for
m
ul
ation
to

nd
:
!
!
h
!
W
k
(
!
h
)
|
"
"
h
!
W
k
(
!
h
)
;
L
(
!
h
,
"
h
)
=
0
"
,
where:
L
(
!
h
,
"
h
)
=
#
!
"
h
.
"
t
!
h
+
#
!
"
h
.
A
i
"
i
!
h
+
#
!
"
h
.
B
!
h
+
\$
!
!
h
|
!
!
"
h
.
[
A
i
n
i
]
!
(
!
o
h
#
!
i
h
)
+
\$
!
"
h
"
!
!
"
h
.
(
M
!
h
#
g
)
#
#
!
"
h
.
g
.
(3.4)
Ex
press
ion
eq
(3.4)
c
an
b
e
spl
itted
as
:

th
e
ﬁr
s
t
term
is
the
b
lo
c-
d
iagon
al
l
o
c
al
mass
m
at
rix,

th
e
s
ec
on
d
term
i
s
the
b
lo
c-diagon
al
lo
cal
sti
!
ness
m
atr
ix
(v
anishes
f
or
fvm
),

th
e
thi
rd
term
i
s
a
matri
x
pro
duct
(also
b
lo
c-
d
iagonal
),

th
e
fou
rth
term
is
the
lo
cal
e
d
ge
s
ti
!
n
e
ss
matrix
conn
e
ctin
g
elem
en
ts,

th
e
ﬁf
th
term
in
tro
d
uce
s
t
he
b
oun
dar
y
cond
ition
s
usin
g
op
erator
M
,

th
e
s
i
xth
term
i
n
tro
d
uce
s
the
acoustic
sour
c
es
in
th
e
f
orm
u
lation
.
Settin
g
"
h
=
!
h
in
eq.
(3.4)
giv
es
aft
e
r
in
te
gr
ation
th
e
e
r
ror
es
timate:
|
!
#
!
h
|
L
2
([0
,T
]
,L
2
(
!
)
3
\$
C
(
T
,
"
)
h
k
+1
/
2
,
in
t
he
case
of
u
nstable
ﬂo
w,
constan
t
C
can
gro
w
to
i
nﬁ
nit
y
as
T
in
c
r
e
ase
s.
Numerical
exp
eri
m
en
ts
s
h
o
w
the
e
stimate
is
pr
obabl
y
b
e
tter
(
h
k
+1
).
T
o
obtai
n
th
is
e
stimate
s
ign
of
[
A
i
n
i
]
!
is
ess
en
tial.
3.3.
Bou
ndary
Condi
tions
Consider
an
elem
en
t
lo
calize
d
on
b
or
der
"
"
of
domain
"
s
imil
ar
to
t
he
on
e
exhib
ited
in
Fig.
2.
F
or
suc
h
an
ele
men
t
"
"
j
%
"
"
=
a
1
&
=
'
and
b
oun
dar
y
cond
ition
s
a
r
e
imp
ose
d
on
a
1
.
In
lo
cal
repres
en
tati
on,
the
v
ar
iation
al
form
ul
a
t
ion
#
!
j
t
"
I
(
"
t
!
+
A
i
"
i
!
+
B
!
)
+
\$
a
1
t
"
[
d
2
,d
3
]
M
!
[
d
2
,d
3
]
#
\$
a
1
t
"
[
d
2
,d
3
]
I
g
+
\$
a
2
t
"
[
d
3
,d
1
]
[
A
i
n
i
]
!
(
!
[
d
6
,d
7
]
#
!
[
d
3
,d
1
]
)
+
\$
a
3
t
"
[
d
1
,d
2
]
[
A
i
n
i
]
!
(
!
[
d
8
,d
9
]
#
!
[
d
1
,d
2
]
)
.
Variational formulation
is diagonalizable
Galerk
i
n
Disc
ontinuous
Me
tho
d
F
or
Co
mputa
t
i
onal
A
er
o
ac
o
ustics
where:
A
1
=
!
"
u
0
0
a
0
0
u
0
0
a
0
0
u
0
#
\$
A
2
=
!
"
v
0
0
0
0
v
0
a
0
0
a
0
v
0
#
\$
,
and
:
B
=
!
%
%
%
%
"
!
x
u
0
!
y
u
0
!
!
x
a
0
!
x
v
0
!
y
v
0
!
!
y
a
0
a
0
"
0
!
x
"
0
a
0
"
0
!
y
"
0
(
#
!
1)
(
!
x
u
0
+
!
y
v
0
)
#
&
&
&
&
\$
.
As
matrix
A
i
!
i
is
symm
etric,
the
r
e
al
m
atr
ix
A
i
n
i
is
d
iagonali
z
ab
le
.
Thi
s
r
e
mark
is
imp
or-
tan
t
f
or
the
metho
d
pres
en
ted
in
t
his
p
ap
e
r.
Mor
e
o
v
e
r
,
w
e
ar
e
abl
e
to
w
r
ite
t
he
mathematic
al
ener
gy
bal
ance
:
!
t
'
!
1
2
!
2
+
1
2
(
!
!
!
t
A
i
n
i
!
+
1
2
'
!
!
t
(
B
+
B
t
!
!
i
A
i
)
!
=
0
.
(2.1)
This
mathematic
al
ener
gy
bal
ance
h
as
a
p
h
ysical
s
ense.
Ind
e
ed,
it
i
s
th
e
sum
p
e
r
m
ass
uni
t
of
ki
netic
energy
(
u
2
1
+
v
2
1
)
/
2
and
acoustic
e
n
e
r
gy
a
2
0
"
2
1
/
"
2
0
.
In
(2.
1),
the
s
ec
on
d
term
i
s
ob
vi
ously
th
e
ﬂu
x
and
the
thi
rd
term
is
the
s
ou
rce
of
th
is
e
n
e
rgy
.
Matri
x
B
+
B
t
!
!
i
A
i
v
anishes
wh
e
n
ﬂo
w
is
u
nif
orm.
Otherwise
,
the
matrix
d
o
es
n
’t
ha
v
e
a
dete
r
m
in
e
d
s
i
gn.
It
me
an
s
e
n
e
r
gy
c
an
in
c
r
e
ase
in
deﬁn
itely
dep
end
ing
on
th
e
ﬂo
w
s
tab
ilit
y
.
3.
Galerkin
D
i
scon
tin
uous
Metho
d
Ou
r
aim
is
to
p
res
en
t
res
u
lts
dealing
with
ae
r
oac
ou
s
tics
(
not
mathem
at
ic
s).
O
nly
main
featu
re
s
of
th
e
m
etho
d
ar
e
prese
n
ted
in
the
p
ap
er
and
for
more
detail
s
w
e
in
v
ite
to
r
e
reference
s
10
,
11
.
F
or
a
k
-order
gd
m
,
te
st
and
in
terp
olation
f
un
c
ti
ons
ar
e
c
h
os
en
in
s
ame
fu
nction
al
s
p
ac
e
c
omp
os
ed
b
y
the
s
et
{
W
k
,
k
"
N
}
of
k
ord
e
r
p
ol
ynoms
i
nside
e
ac
h
ele
-
me
n
t
and
ma
y
b
e
disc
on
tin
u
ous
on
e
d
ge
s
b
orderi
ng
elem
en
ts
(p
ie
ce
wise
k
ord
e
r
p
olynomial
fu
nction
s
).
Th
e
fact
that
b
oth,
te
st
and
in
te
r
p
ol
ation
f
unction
s
are
discon
tin
u
ous
p
rev
e
n
t
fr
om
d
irec
tl
y
wri
ting
a
w
eak
for
m
ul
ation
for
al
l
domain.
Ind
e
ed,
deriv
ativ
es
of
d
is
con
ti
n-
uou
s
fu
nction
s
in
tro
du
c
e
Dir
ac
distri
bu
tions
on
elem
en
ts
b
ord
e
r.
A
fe
m
app
roac
h
l
ik
e
to
es
tab
lish
a
w
eak
for
m
u
lation
w
ou
ld
in
tr
o
du
c
e
se
n
s
ele
ss
pr
o
du
c
ts
b
e
t
w
e
en
distrib
uti
ons
and
di
s
con
tin
uous
fu
nction
s
on
e
d
ge
s.
Actuall
y
,
th
e
gd
m
is
a
patc
h
w
ork
of
w
eak
f
orm
u
lation
s
in
s
id
e
e
ac
h
elem
en
t
and
th
e
goal
is
to
c
on
nec
t
e
ac
h
elem
en
ts
to
its
neigh
b
ors
u
s
in
g
b
oun
dary
condi
tions.
Thi
s
is
realize
d
usin
g
b
ou
nd
ary
op
e
r
ator
M
th
at
giv
e
s
se
n
s
e
to
the
pr
o
du
c
ts
of
Dir
ac
di
s
tr
ibu
tion
and
disc
on
tin
u
ous
f
un
c
tion
s
.
3.1.
Mesh
G
ener
ati
on
Domain
!
is
me
shed
with
N
ele
me
n
ts
:
!
=
N
)
h
=1
\$
h
,
Galerk
i
n
Disc
ontinuous
Me
tho
d
F
or
Co
mputa
t
i
onal
A
er
o
ac
o
ustics
sym
metric,
it
is
d
iagonal
iz
ab
le
and
c
an
b
e
spli
tte
d
in
to
a
p
ositiv
e
(se
t
of
p
os
i
tiv
e
e
igen
v
alues)
and
a
n
e
gati
v
e
part
(se
t
of
n
e
gati
v
e
e
i
ge
n
v
alu
e
s)
:
A
i
n
i
=
[
A
i
n
i
]
+
+
[
A
i
.n
i
]
!
.
In
th
is
me
th
o
d
th
e
w
eak
for
m
ul
ation
to

nd
:
!
!
h
!
W
k
(
!
h
)
|
"
"
h
!
W
k
(
!
h
)
;
L
(
!
h
,
"
h
)
=
0
"
,
where:
L
(
!
h
,
"
h
)
=
#
!
"
h
.
"
t
!
h
+
#
!
"
h
.
A
i
"
i
!
h
+
#
!
"
h
.
B
!
h
+
\$
!
!
h
|
!
!
"
h
.
[
A
i
n
i
]
!
(
!
o
h
#
!
i
h
)
+
\$
!
"
h
"
!
!
"
h
.
(
M
!
h
#
g
)
#
#
!
"
h
.
g
.
(3.4)
Ex
press
ion
eq
(3.4)
c
an
b
e
spl
itted
as
:

th
e
ﬁr
s
t
term
is
the
b
lo
c-
d
iagon
al
l
o
c
al
mass
m
at
rix,

th
e
s
ec
on
d
term
i
s
the
b
lo
c-diagon
al
lo
cal
sti
!
ness
m
atr
ix
(v
anishes
f
or
fvm
),

th
e
thi
rd
term
i
s
a
matri
x
pro
duct
(also
b
lo
c-
d
iagonal
),

th
e
fou
rth
term
is
the
lo
cal
e
d
ge
s
ti
!
n
e
ss
matrix
conn
e
ctin
g
elem
en
ts,

th
e
ﬁf
th
term
in
tro
d
uce
s
t
he
b
oun
dar
y
cond
ition
s
usin
g
op
erator
M
,

th
e
s
i
xth
term
i
n
tro
d
uce
s
the
acoustic
sour
c
es
in
th
e
f
orm
u
lation
.
Settin
g
"
h
=
!
h
in
eq.
(3.4)
giv
es
aft
e
r
in
te
gr
ation
th
e
e
r
ror
es
timate:
|
!
#
!
h
|
L
2
([0
,T
]
,L
2
(
!
)
3
\$
C
(
T
,
"
)
h
k
+1
/
2
,
in
t
he
case
of
u
nstable
ﬂo
w,
constan
t
C
can
gro
w
to
i
nﬁ
nit
y
as
T
in
c
r
e
ase
s.
Numerical
exp
eri
m
en
ts
s
h
o
w
the
e
stimate
is
pr
obabl
y
b
e
tter
(
h
k
+1
).
T
o
obtai
n
th
is
e
stimate
s
ign
of
[
A
i
n
i
]
!
is
ess
en
tial.
3.3.
Bou
ndary
Condi
tions
Consider
an
elem
en
t
lo
calize
d
on
b
or
der
"
"
of
domain
"
s
imil
ar
to
t
he
on
e
exhib
ited
in
Fig.
2.
F
or
suc
h
an
ele
men
t
"
"
j
%
"
"
=
a
1
&
=
'
and
b
oun
dar
y
cond
ition
s
a
r
e
imp
ose
d
on
a
1
.
In
lo
cal
repres
en
tati
on,
the
v
ar
iation
al
form
ul
a
t
ion
#
!
j
t
"
I
(
"
t
!
+
A
i
"
i
!
+
B
!
)
+
\$
a
1
t
"
[
d
2
,d
3
]
M
!
[
d
2
,d
3
]
#
\$
a
1
t
"
[
d
2
,d
3
]
I
g
+
\$
a
2
t
"
[
d
3
,d
1
]
[
A
i
n
i
]
!
(
!
[
d
6
,d
7
]
#
!
[
d
3
,d
1
]
)
+
\$
a
3
t
"
[
d
1
,d
2
]
[
A
i
n
i
]
!
(
!
[
d
8
,d
9
]
#
!
[
d
1
,d
2
]
)
.
Matrix is symetric
Galerk
i
n
Disc
ontinuous
Me
tho
d
F
or
Co
mputa
t
i
onal
A
er
o
ac
o
ustics
where:
A
1
=
!
"
u
0
0
a
0
0
u
0
0
a
0
0
u
0
#
\$
A
2
=
!
"
v
0
0
0
0
v
0
a
0
0
a
0
v
0
#
\$
,
and
:
B
=
!
%
%
%
%
"
!
x
u
0
!
y
u
0
!
!
x
a
0
!
x
v
0
!
y
v
0
!
!
y
a
0
a
0
"
0
!
x
"
0
a
0
"
0
!
y
"
0
(
#
!
1)
(
!
x
u
0
+
!
y
v
0
)
#
&
&
&
&
\$
.
As
matrix
A
i
!
i
is
symm
etric,
the
r
e
al
m
atr
ix
A
i
n
i
is
d
iagonali
z
ab
le
.
Thi
s
r
e
mark
is
imp
or-
tan
t
f
or
the
metho
d
pres
en
ted
in
t
his
p
ap
e
r.
Mor
e
o
v
e
r
,
w
e
ar
e
abl
e
to
w
r
ite
t
he
mathematic
al
ener
gy
bal
ance
:
!
t
'
!
1
2
!
2
+
1
2
(
!
!
!
t
A
i
n
i
!
+
1
2
'
!
!
t
(
B
+
B
t
!
!
i
A
i
)
!
=
0
.
(2.1)
This
mathematic
al
ener
gy
bal
ance
h
as
a
p
h
ysical
s
ense.
Ind
e
ed,
it
i
s
th
e
sum
p
e
r
m
ass
uni
t
of
ki
netic
energy
(
u
2
1
+
v
2
1
)
/
2
and
acoustic
e
n
e
r
gy
a
2
0
"
2
1
/
"
2
0
.
In
(2.
1),
the
s
ec
on
d
term
i
s
ob
vi
ously
th
e
ﬂu
x
and
the
thi
rd
term
is
the
s
ou
rce
of
th
is
e
n
e
rgy
.
Matri
x
B
+
B
t
!
!
i
A
i
v
anishes
wh
e
n
ﬂo
w
is
u
nif
orm.
Otherwise
,
the
matrix
d
o
es
n
’t
ha
v
e
a
dete
r
m
in
e
d
s
i
gn.
It
me
an
s
e
n
e
r
gy
c
an
in
c
r
e
ase
in
deﬁn
itely
dep
end
ing
on
th
e
ﬂo
w
s
tab
ilit
y
.
3.
Galerkin
D
i
scon
tin
uous
Metho
d
Ou
r
aim
is
to
p
res
en
t
res
u
lts
dealing
with
ae
r
oac
ou
s
tics
(
not
mathem
at
ic
s).
O
nly
main
featu
re
s
of
th
e
m
etho
d
ar
e
prese
n
ted
in
the
p
ap
er
and
for
more
detail
s
w
e
in
v
ite
to
r
e
reference
s
10
,
11
.
F
or
a
k
-order
gd
m
,
te
st
and
in
terp
olation
f
un
c
ti
ons
ar
e
c
h
os
en
in
s
ame
fu
nction
al
s
p
ac
e
c
omp
os
ed
b
y
the
s
et
{
W
k
,
k
"
N
}
of
k
ord
e
r
p
ol
ynoms
i
nside
e
ac
h
ele
-
me
n
t
and
ma
y
b
e
disc
on
tin
u
ous
on
e
d
ge
s
b
orderi
ng
elem
en
ts
(p
ie
ce
wise
k
ord
e
r
p
olynomial
fu
nction
s
).
Th
e
fact
that
b
oth,
te
st
and
in
te
r
p
ol
ation
f
unction
s
are
discon
tin
u
ous
p
rev
e
n
t
fr
om
d
irec
tl
y
wri
ting
a
w
eak
for
m
ul
ation
for
al
l
domain.
Ind
e
ed,
deriv
ativ
es
of
d
is
con
ti
n-
uou
s
fu
nction
s
in
tro
du
c
e
Dir
ac
distri
bu
tions
on
elem
en
ts
b
ord
e
r.
A
fe
m
app
roac
h
l
ik
e
to
es
tab
lish
a
w
eak
for
m
u
lation
w
ou
ld
in
tr
o
du
c
e
se
n
s
ele
ss
pr
o
du
c
ts
b
e
t
w
e
en
distrib
uti
ons
and
di
s
con
tin
uous
fu
nction
s
on
e
d
ge
s.
Actuall
y
,
th
e
gd
m
is
a
patc
h
w
ork
of
w
eak
f
orm
u
lation
s
in
s
id
e
e
ac
h
elem
en
t
and
th
e
goal
is
to
c
on
nec
t
e
ac
h
elem
en
ts
to
its
neigh
b
ors
u
s
in
g
b
oun
dary
condi
tions.
Thi
s
is
realize
d
usin
g
b
ou
nd
ary
op
e
r
ator
M
th
at
giv
e
s
se
n
s
e
to
the
pr
o
du
c
ts
of
Dir
ac
di
s
tr
ibu
tion
and
disc
on
tin
u
ous
f
un
c
tion
s
.
3.1.
Mesh
G
ener
ati
on
Domain
!
is
me
shed
with
N
ele
me
n
ts
:
!
=
N
)
h
=1
\$
h
,
Fully Upwind
Scheme
4
Boundary Conditions
Rigid Wall
Non-reﬂecting
Lilley’s Solution (reference)
Lined Wall
Lined Wall
5
Perfectly Matched Layers
p
1
-1
-0.5
0
0.5
1
Pa
Part One
Optimal
hp
DGM principles
“Optimal” Functional Basis

Easy to Build (and Program)

Higher Order

Quadratures for Num Integrations

Ortogonal Basis
High Order Lagrangian Elements
n
01
n
02
n
03
n
04
n
05
n
06
n
07
n
08
n
09
n
10
n
11
n
12
n
13
n
14
n
15
n
16
n
17
n
18
n
19
n
20
Rigid or Lined W
all
Rigid or Lined W
all
Mode
Non
Reﬂ
ecting
3.00 m
0.50 m
Inﬁnite 2D Duct with constant cross section
a
0
=
3
4
0
m
.
s
!
1
p
0
=
1
0
1
3
2
5
P
a
!
0
=
1
.
2
3
k
g
.
m
!
3
Air
!
=
0
.
1
0
m
n
=
0
f
=
3
.
4
k
H
z
Approximated
Non Reﬂecting condition
works perfectly
p
(
x
,
y
)
=
s
in
(
2
!
f
t
!
2
!
"
x
)
Analytical Solution for Rigid Wall No Flow
x
y
30 periods for 3m length
9
P4
CPU (s)
deg
0
0
.
5
1
1
.
5
2
2
.
5
3
-
1
-
0
.
5
0
0
.
5
1
10.55
5430
P5
CPU (s)
deg
0
0
.
5
1
1
.
5
2
2
.
5
3
-
1
-
0
.
5
0
0
.
5
1
28.43
7602
P6
CPU (s)
deg
0
0
.
5
1
1
.
5
2
2
.
5
3
-
1
-
0
.
5
0
0
.
5
1
68.61
10136
30 intervals
5
intervals
h
!
!
( CPU are obtained on a Dual Apple G5 2 GHz )
10
P1
CPU (s)
deg
0
.
0
0
.
5
1
.
0
1
.
5
2
.
0
2
.
5
3
.
0
-
1
.
0
-
0
.
5
0
.
0
0
.
5
1
.
0
1047.85
110658
300 x 50 intervals
P1
CPU (s)
deg
0
.
5
1
.
0
1
.
5
2
.
0
2
.
5
3
.
0
-
1
.
0
-
0
.
5
0
.
0
0
.
5
1
.
0
4709.81
216684
420 x70 intervals
h
!
!
/
1
0
h
!
!
/
1
4
P1
CPU (s)
deg
0
.
5
1
.
0
1
.
5
2
.
0
2
.
5
3
.
0
-
1
.
0
-
0
.
5
0
.
0
0
.
5
1
.
0
2288.29
159660
360 x 60 intervals
h
!
!
/
1
2
11
Mesh Reﬁnement / Element Order
element
n
(
P
i
)
m
(
P
i
)
!
(
P
i
)
h
m
i
n
(
!
5
%
)
h
m
i
n
(
!
1
0
%
)
P0
1
0
λ
/
2
0
λ
/40
P1
3
1
1.0
λ
/14
λ
/12
P2
6
4
0.5
λ
/4
λ
/4
P3
10
9
0.37
λ
/3
λ
/2
P4
15
16
0.31
λ
/2
λ
/3
P5
21
25
0.28
λ
P6
28
36
0.26
λ
12
Remeshing Tool
Original mesh
np: 67 108   nt: 134 212
np: 9 796  nt: 19 588
13
Optimal
hp
DGM CAA
Meshes
Optimal Mesh: 3729 triangles
CFD Mesh: 17024 triangles
14
P
0
P
1
P
2
P
3
P
4
P
5
P
6
P
0
P
1
P
2
P
3
P
4
P
5
P
6
Optimal
hp
DGM CAA Orders
f
=
2
k
H
z
15
Optimal
hp
DGM CAA Results
iter
1456
dt
4,46
μ
s
t
6504
μ
s
CPU
580’’
RAM
300 MO
iter
1200
dt
5,42
μ
s
t
6507
μ
s
CPU
195’’
RAM
200 MO
f
=
2
k
H
z
16
Cut 1
Cut 2
Optimal
hp
DGM CAA Validations
!
2
=
n
!
i
=
1
"
¯
p
1
(
i
)
!
˜
p
1
(
i
)
#
2
n
!
i
=
1
"
¯
p
1
(
i
)
#
2
=
0
.
8
3
%
–1
0
1
2
3
4
5
–4
.
10
–5
–2
.
10
–5
0
2
.
10
–5
4
.
10
–5
x (m)
Cut 1
hp
p2
–0.50
–0.25
0.00
0.25
0.50
–4
.
10
–5
–2
.
10
–5
0
2
.
10
–5
4
.
10
–5
Cut 2
hp
p2
Optimal
hp
DGM CAA
f
=
3
k
H
z
18
Part Two
High Performance Computing
HPC = High Performance Computing
Simulations for 3D geometries
500 000 Tetraedra Mesh
4 GBytes
First Idea for 3D Simulation:
HPC or HCC
Formal Calculation
Vectorization
Massively Parallel Computation (MPI+OMP)
20
High Performance Computing
21
Massively Parallel Computation
open mpi 1.0.1
Message Passing Interface
ParMetis: Parallel Graph Partitioning
22
Falcon CAA HPC Computation
500 000 Tetras
64 domains
TetMesh (INRIA/Simulog)
23
Falcon CAA HPC Computation
24
Cluster - 4 nodes Apple G5 2 GHz - 48 hours CPU
HPC Optimal
hp
DGM
Mesh for Computation
72 Tetra with P5 approx
Mesh for Visualization
with solution
Part Three
Optimal
h
Gaussian Distribution of Sources
T
F
(
e
!
a
2
t
2
)
=
1
a
!
2
e
!
f
2
4
a
2
-1
!
1
0
4
-7
5
0
0
-5
0
0
0
-2
5
0
0
0
2
5
0
0
5
0
0
0
7
5
0
0
1
!
1
0
4
2
,
5
!
1
0
-4
5
!
1
0
-4
7
,
5
!
1
0
-4
0
,
0
0
1
-0
,
0
0
4
-0
,
0
0
3
-0
,
0
0
2
-0
,
0
0
1
0
0
,
0
0
1
0
,
0
0
2
0
,
0
0
3
0
,
0
0
4
0
,
2
5
0
,
5
0
,
7
5
1
a
=
1
0
3
Conclusion
DGM
DGM is able to solve most CAA problems (and many others)
DGM is expensive (especially for lower order elements)
hp
DGM
hp
DGM mixes element orders and results a much less expensive cost
With
hp
DGM, CFD and CAA computations are handled on same mesh
Introduction of the doppler effect when determining local orders
HPC DGM
Computation on clusters make big conﬁgurations possible
(
hp
+ HPC) DGM
Balance of the Processes to optimize cluster efﬁciency
31