Journal of Computational Acoustics,
f
c IMACS
A DISSIPATIONFREE TIMEDOMAIN DISCONTINUOUS GALERKIN METHOD
APPLIED TO THREEDIMENSIONAL LINEARIZED EULER EQUATIONS
AROUND A STEADYSTATE NONUNIFORM INVISCID FLOW
MARC BERNACKI
Projectteam Caiman,CERMICS and INRIA,
BP93,F06902 Sophia Antipolis Cedex,Marc.Bernacki@sophia.inria.fr
SERGE PIPERNO
Projectteam Caiman,CERMICS and INRIA,
BP93,F06902 Sophia Antipolis Cedex,Serge.Piperno@sophia.inria.fr
Received (to be inserted
Revised by Publisher)
We present in this paper a timedomain Discontinuous Galerkin dissipationfree method for the
transient solution of the threedimensional linearized Euler equations around a steadystate so
lution.In the general context of a nonuniform supporting ow,we prove,using the wellknown
symmetrization of Euler equations,that some aeroacoustic energy satises a balance equation with
source term at the continuous level,and that our numerical framework satises an equivalent bal
ance equation at the discrete level and is genuinely dissipationfree.In the case of P
1
Lagrange basis
functions and tetrahedral unstructured meshes,a parallel implementation of the method has been
developed,based on message passing and mesh partitioning.Threedimensional numerical results
conrm the theoretical properties of the method.They include testcases where KelvinHelmholtz
instabilities appear.
Keywords:aeroacoustics;acoustic energy;linearized Euler equations;nonuniformsteadystate ow;
Discontinuous Galerkin method;time domain;energyconservation.
1.Introduction
Aeroacoustics is a domain where numerical simulation meets great expansion.The min
imization of acoustic pollutions by aircrafts at landing and take o,or more generally by
aerospace and terrestrial vehicles,is now an industrial concern,related to more and more
severe norms.Dierent approaches coexist under the Computational Aeroacoustics activ
ity.The most widely used methods belong to classical Computational Fluid Dynamics and
consist in solving partial dierential equations for the uid,without distinction between the
supporting (possibly steadystate) ow and acoustic perturbations
1
.The equations model
ing the uid can be Euler or NavierStokes equations,possibly including extended models
like turbulence,LES techniques,etc
2
.One particular diculty of these approaches is the
dierence in magnitude between the ow and acoustic perturbations,then requiring very
Bernacki,Piperno
accurate { and CPUconsuming { numerical methods.
An alternative has developed recently with approaches consisting in separating the de
termination of the supporting steadystate ow and in modeling the generation of noise
(for example by providing equivalent acoustic sources),from the propagation of acoustic
perturbations
3;4;5
.For this problem,linearized Euler equations around the supporting ow
are to be solved and provide a good description of the propagation of aeroacoustic pertur
bations in a smoothly varying heterogeneous and anisotropic medium.This is not exactly
the case of more simple models based on Lighthill analogy
6
or of the thirdorder equation
of Lilley
7
(a clear description can be found in a more recent reference
8
).The noise source
modeled or derived from the steadystate ow are then dealt with as acoustic source terms
in the linearized Euler equations.
For direct approaches as well as for wave propagation approaches,the construction of
accurate absorbing boundary conditions required for bounding the computational domain
remains a concern.Many solutions have been proposed
10;11;13;14
but the construction of
an general,ecient,parameterfree,easytoimplement absorbing boundary condition in
timedomain aeroacoustics remains an active research domain
15
.
The work presented here is devoted to the numerical solution of linearized Euler equations
around steadystate discretized ows,obtained using a given Euler solver.The supporting
ow considered is always smooth and subsonic,it can be uniform or fully nonuniform.
Since we intend to consider complex geometries in three space dimensions,we consider
unstructured tetrahedral space discretizations.In this context,we propose a time do
main Discontinuous Galerkin dissipationfree method based on P
1
Lagrange elements on
tetrahedra.The method is derived from similar methods developed for threedimensional
timedomain Maxwell equations
16
.We use an elementcentered formulation with centered
numerical uxes and an explicit leapfrog time scheme.This kind of method provides a
dissipationfree approximation of propagation equations and allows for the accurate estima
tion of aeroacoustic energy variation,which is not possible with numerical methods (nite
volumes,discontinuous Galerkin,spectral elements) based on upwind numerical uxes.The
fact that centered numerical uxes in discontinuous Galerkin timedomain methods can lead
to discretization methods inducing no numerical dissipation is quite well known.This was
for example rst numerically shown on cartesian grids
12
,then later for DG methods on
arbitrary unstructured meshes
9;18
.
More precisely,the main results of this paper concern both the linearized Euler equations
at the continuous level,and the numerical DG method we propose.They can be summed
up the following way:
1.for a uniformsupporting ow,at the continuous level (i.e.before space discretization),
some quadratic energy veries a balance equation without source term.This means
energy is conserved (up to boundaries);
2.in this\uniform supporting ow"case,we are able to prove that our DG method
(with leapfrog timescheme and centered uxes) introduces no dissipation even on
A dissipationfree DGTD method for the threedimensional linearized Euler equations
unstructured simplicial meshes (some discrete energy is exacly conserved,or simply
nonincreasing when absorbing boundary conditions are used);therefore we claimthat
we\control energy variations in the uniform case";
3.accordingly,for a nonuniform supporting ow,at the continuous level (i.e.before
space discretization),we use the wellknown symmetrization of nonlinear Euler equa
tions
17
to derive an aeroacoustic energy which veries some balance equation with
source term.Because of this unsigned source term,aeroacoustic waves can be damped
or excited by the supporting ow.It is responsible for example for KelvinHelmholtz
instabilities.These instabilities are due to the model (linearized Euler equations),not
to the numerical method;
4.in the\non uniformsupporting ow"case,we are able to prove that,using an adapted
version of the same DG method on unstructured simplicial meshes,some\discrete"
energy balance equation with source termis also veried.We claim our method is still
nondissipative.The good point is that we are able to reproduce these instabilities.
The bad point is that we cannot damp themarticially (like methods based on upwind
uxes,which damp instabilities,in an uncontrolled way though);
5.we show nally that,by introducing an articial source term,we are able to\control
these instabilities".
The paper is organized as follows.In Section 2,we present linearized Euler equations
around a steadystate uniformor nonuniformsupporting ow,for which the symmetrization
of Euler equations is introduced to derive some aeroacoustic balance equation.In Section 3,
we present the main elements of the Discontinuous Galerkin timedomain method used,
with the main theoretical results.We give in Section 4 numerical results in two and three
space dimensions in dierent congurations,obtained with MPIbased Fortran parallel im
plementations of the method.We conclude in Section 5 with possible extensions of this
work.
2.Linearization of Euler equations
We consider in this paper equations for the propagation of acoustic waves through a
steady smooth inviscid ow.Therefore,we linearize the threedimensional Euler equations
around a given steady ow and only take into account rstorder perturbation terms.For
a perfect inviscid gas,Euler equations read:
@
t
0
B
B
B
B
@
u
v
w
e
1
C
C
C
C
A
+@
x
0
B
B
B
B
@
u
u
2
+p
uv
uw
(e +p)u
1
C
C
C
C
A
+@
y
0
B
B
B
B
@
v
uv
v
2
+p
vw
(e +p)v
1
C
C
C
C
A
+@
z
0
B
B
B
B
@
w
uw
vw
w
2
+p
(e +p)w
1
C
C
C
C
A
= 0;(2.1)
Bernacki,Piperno
where ,~v =
t
(u;v;w),e and p denote respectively the density,the velocity,the volumic
total energy and the pressure,given by the perfect gas law p = ( 1)(e
1
2
k~vk
2
),where
is a xed constant ( > 1).
2.1.Linearization around a uniform ow
Arst step towards aeroacoustic wave propagation consists in linearizing Euler equations
around any uniform ow dened by some constant physical elds (
0
;~v
0
;p
0
),therefore being
a steadystate solution of Euler equations.Innitely small perturbations (;~v;p) of this
ow verify the following linear hyperbolic system of conservation laws:
@
t
~
W+A
x
@
x
~
W+A
y
@
y
~
W+A
z
@
z
~
W=
~
0;(2.2)
where
~
W=
t
(p c
2
0
;
t
0
c
0
~v;p),c
0
corresponds to the uniform sound speed given by
c
2
0
= p
0
=
0
.The variable
~
Whas been chosen such that the constant matrices A
x
,A
y
,and
A
z
are symmetric.They are given by:
A
x
=
0
B
B
B
B
@
u
0
0 0 0 0
0 u
0
0 0 c
0
0 0 u
0
0 0
0 0 0 u
0
0
0 c
0
0 0 u
0
1
C
C
C
C
A
;A
y
=
0
B
B
B
B
@
v
0
0 0 0 0
0 v
0
0 0 0
0 0 v
0
0 c
0
0 0 0 v
0
0
0 0 c
0
0 v
0
1
C
C
C
C
A
;A
z
=
0
B
B
B
B
@
w
0
0 0 0 0
0 w
0
0 0 0
0 0 w
0
0 0
0 0 0 w
0
c
0
0 0 0 c
0
w
0
1
C
C
C
C
A
In the particular case of the linearization around a uniform ow,one can dene a volumic
aeroacoustic energy E =
1
2
k
~
Wk
2
(a very simple expression in function of
~
W),which veries
the balance equation @
t
E +div
~
F = 0,where the energy ux
~
F is given by
F
s
=
1
2
t
~
WA
s
~
W;s 2 fx;y;zg:
Thus,away from boundaries,the aeroacoustic energy E is exactly conserved.
2.2.Linearization around a nonuniform ow
A more interesting framework consists in linearizing Euler equations around a non
uniform steadystate solution.In that case,the steady ow is dened by smoothly varying
physical quantities (
0
;~v
0
;p
0
).Linearizing straightforwardly Euler equations (2.1) yields:
@
t
~
W+@
x
A
0
x
~
W
+@
y
A
0
y
~
W
+@
z
A
0
z
~
W
= 0;(2.3)
where
~
Wnow denotes the perturbations of conservative variables (i.e.
~
W
T
= (;
0
~v +
~v
0
;p=( 1) +
0
~v
0
:~v +k~v
0
k
2
=2)) and the spacevarying matrices A
0
x
,A
0
y
,and A
0
z
are given in function of ~ = 1,
0
= c
2
0
=~ +k~v
0
k
2
=2,
0
= ( 2)kV
0
k
2
=2 c
2
0
=~ ,and
the canonical basis (~e
x
;~e
y
;~e
z
) of R
3
by
A
0
s
=
0
@
0
t
~e
s
0
~
2
k~v
0
k
2
~e
s
(~v
0
:~e
s
)~v
0
(~v
0
:~e
s
)I
3
~ ~e
s
t
~v
0
+~v
0
t
~e
s
~ ~e
s
0
(~v
0
:~e
s
)
0
t
~e
s
~ (~v
0
:~e
s
)
t
~v
0
(~v
0
:~e
s
)
1
A
;s 2 fx;y;zg:(2.4)
A dissipationfree DGTD method for the threedimensional linearized Euler equations
In this equation,the matrices A
0
x
,A
0
y
,and A
0
z
are not symmetric anymore,and it is
very dicult to deduce any aeroacoustic energy balance equation.Therefore,we con
sider other acoustic variables,derived from the quite classical symmetrization of Euler
equations.Assuming the ow is smooth enough,the change of variables (;~v;e)!
(
e~
p
+ +1 ln
p
;
~
p
~v;
~
p
) transforms Euler equations (2.1) into a symmetric sys
tem of conservation laws (i.e.Jacobians of uxes are symmetric matrices).Accordingly,
the linearization of these symmetrized Euler equations leads to more complex aeroacoustic
equations for perturbations of the new variables,which can be written as
A
0
0
@
t
~
V+@
x
~
A
0
x
~
V
+@
y
~
A
0
y
~
V
+@
z
~
A
0
z
~
V
= 0;(2.5)
where
~
V is given in function of variables
~
Wby
~
V = A
0
0
1
~
Wand
A
0
0
=
0
~
0
B
@
1
t
~v
0
0
c
2
0
=
~v
0
c
2
0
I
3
+~v
0
t
~v
0
0
~v
0
0
c
2
0
=
0
t
~v
0
2
0
c
4
0
= ( ~ )
1
C
A
;
~
A
0
s
= A
0
s
A
0
0
;s 2 fx;y;zg:(2.6)
A
0
0
clearly is symmetric and it can be proved that it is denite positive (and then not
singular).Eq.2.5 can also be obtained simply by replacing
~
Wby A
0
0
~
V in Eq.2.3 (and by
noting that @
t
A
0
0
= 0).Finally,the reader can also check that the symmetric matrices
~
A
s
(s 2 fx;y;zg) are given by
~
A
0
s
= (~v
0
:~e
s
) A
0
0
+
p
0
~
0
@
0
t
~e
s
(~v
0
:~e
s
)
~e
s
~e
s
t
~v
0
+~v
0
t
~e
s
(~v
0
:~e
s
)~v
0
+
0
~e
s
(~v
0
:~e
s
) (~v
0
:~e
s
)
t
~v
0
+
0
t
~e
s
2
0
(~v
0
:~e
s
)
1
A
:
Then,the volumic aeroacoustic energy E dened by E =
1
2
t
~
WA
0
0
1
~
W
1
2
t
~
VA
0
0
~
V veries
the following balance equation with source term:
@
t
E +div
~
F = S;with
(
F
s
=
t
~
V
~
A
0
s
~
V;s 2 fx;y;zg:
S =
1
2
t
~
V
h
@
x
(
~
A
0
x
) +@
y
(
~
A
0
y
) +@
z
(
~
A
0
z
)
i
~
V:
(2.7)
Thus the aeroacoustic energy is not conserved and the variations in the steady ow con
sidered can damp or amplify aeroacoustic waves,unless the source term vanishes (which is
the case for a uniform ow for example).In the sequel,we shall mainly discretize the con
servative form (2.3),but we shall need the equivalent symmetric form (2.5) for discussions
concerning energy conservation and stability.
3.A discontinuous Galerkin timedomain (DGTD) method
Discontinuous Galerkin methods have been widely used with success for the numerical
simulation of acoustic or electromagnetic wave propagation in the time domain
16;18
.The
very same type of methods can be used for the problems considered here,i.e.the propagation
Bernacki,Piperno
of aeroacoustic waves through a nonuniform ow
19;20;21;22
.In this section,we present the
DGTD method we use for the model equations (2.3).We recall the numerical properties
of the space discretization.Then we introduce the leapfrog time scheme and give some
details on properties related to energy conservation and stability.
In the whole paper,we assume we dispose of a partition of a polyhedral domain
(whose boundary @
is the union of physical boundaries of objects @
phys
and of the far
eld articial boundary @
1
).
is partitioned into a nite number of polyhedra (each
one having a nite number of faces).For each polyhedron T
i
,called"control volume"or
"cell",V
i
denotes its volume.We call face between two control volumes their intersection,
whenever it is a polyhedral surface.The union of all faces F is partitioned into internal faces
F
int
= F=@
,physical faces F
phys
= F
T
@
phys
and absorbing faces F
abs
= F
T
@
1
.For
each internal face a
ik
= T
i
T
T
k
,we denote by S
ik
the measure of a
ik
and by ~n
ik
the unitary
normal,oriented from T
i
towards T
k
.The same denitions are extended to boundary faces,
the index k corresponding to a ctitious cell outside the domain.Finally,we denote by V
i
the set of indices of the control volumes neighboring a given control volume T
i
(having a
face in common).We also dene the perimeter P
i
of T
i
by P
i
=
P
k2V
i
S
ik
.We recall the
following geometrical property for all control volumes:
P
k2V
i
S
ik
~n
ik
= 0.
Following the general principle of discontinuous Galerkin nite element methods,the
unknown eld inside each control volume is seeked for as a linear combination of local
basis vector elds ~'
ij
;1 j d
i
(generating the local space P
i
) and the approximate
eld is allowed to be fully discontinuous across element boundaries.Thus,a numerical ux
function has to be dened to approximate uxes at control volumes interfaces,where the
approximate solution is discontinuous.
This context is quite general.Actual implementations of the method have been consid
ered only on tetrahedral meshes,where control volumes are the tetrahedra themselves.We
shall only consider constant (P
0
) or linear (P
1
) approximations inside tetrahedra.
3.1.Time and space discretizations
We only consider here the most general case of aeroacoustic wave propagation in a non
uniform steady ow.Also,in order to limit the amount of computations,we restrict our
study to piecewise constant matrices A
0
s
(s 2 fx;y;zg) given in Eq.(2.4).For each control
volume T
i
,for s 2 fx;y;zg,we denote by A
i
s
an approximate for the average value of A
0
s
over T
i
.Dotmultiplying Eq.(2.3) by any given vector eld ~',integrating over T
i
and
integrating by parts yields
Z
T
i
~'
@
~
W
@t
=
Z
T
i
0
@
X
s2fx;y;zg
t
@
s
~'A
0
s
1
A ~
W
Z
@T
i
~'
0
@
X
s2fx;y;zg
n
s
A
0
s
~
W
1
A
:(3.8)
Inside volume integrals over T
i
,we replace the eld
~
W by the approximate eld
~
W
i
and
the matrices A
0
s
by their respective average values A
i
s
.For boundary integrals over @T
i
,
~
W
A dissipationfree DGTD method for the threedimensional linearized Euler equations
is discontinuous,and we dene totally centered numerical uxes,i.e.:
(
8i;8k 2 V
i
;
h
(n
ik
x
A
0
x
+n
ik
y
A
0
y
+n
ik
z
A
0
z
)
~
W
i
ja
ik
'
1
2
P
i
ik
~
W
i
+P
k
ik
~
W
k
;
with P
i
ik
= n
ik
x
A
i
x
+n
ik
y
A
i
y
+n
ik
z
A
i
z
;P
k
ik
= n
ik
x
A
k
x
+n
ik
y
A
k
y
+n
ik
z
A
k
z
:
(3.9)
Concerning the time discretization,we use a threelevel leapfrog scheme.The unknowns
~
W
i
are approximated at integer timestations t
n
= nt.Assuming we dispose of
~
W
n1
i
and
~
W
n
i
,the unknowns
~
W
n+1
i
are seeked for in P
i
such that,8~'2 P
i
,
Z
T
i
~'
~
W
n+1
i
~
W
n1
i
2t
=
Z
T
i
X
s2fx;y;zg
t
@
s
~'A
i
s
~
W
n
i
X
k2V
i
Z
a
ik
~'
P
i
ik
~
W
n
i
+P
k
ik
~
W
n
k
2
:(3.10)
Again,the time scheme above is almost explicit.Each time step only requires the inversion
of local symmetric positive denite matrices of size (d
i
d
i
).In the particular case where
P
i
is a complete linear (P
1
) representation,these 20 20 matrices are indeed made of 5
4 4 diagonal blocks (which are equal to the classical P
1
mass matrix).
3.2.Boundary conditions
Boundary conditions are dealt with in a weak sense.For the physical boundary,we
consider only a slip condition,which is set on both the steady ow and the acoustic per
turbations.This means that we assume that for any slip boundary face a
ik
belonging to
the control volume T
i
,the steady solution of Euler equations veries a slip condition at
the discrete level,i.e.~n
ik
~v
0
i
= 0.For the acoustic perturbations,we use a mirror c
titious state
~
W
k
in the computation of the boundary ux given in Eq.(3.10).We take
k
=
i
,p
k
= p
i
,and ~v
k
= ~v
i
2(~n
ik
~v
i
)~n
ik
(which implies (~v
k
~v
i
) ~n
ik
= 0 and
~v
k
:~n
ik
= ~v
i
:~n
ik
).
For an absorbing boundary face a
ik
,upwinding is used to select outgoing waves only.
Before discretization in time,classical upwinding leads to a boundary ux F
ik
given by
F
ik
=
P
i
ik
+
~
W
i
,where for any diagonalizable matrix Q = S
1
DS with D diagonal,Q
+
=
(Q+jQj)=2 and terms of the diagonal matrix jDj are the moduli of the eigenvalues.This
general idea leads to P
k
ik
~
W
k
= jP
i
ik
j
~
W
i
.However,for this intuitive numerical ux,it is
very dicult to prove that the resulting timescheme is stable and that energy is actually
sent in the exterior domain.We then consider the numerical ux based on the following
ctitious state:P
k
ik
~
W
n
k
=
p
A
i
0
p
A
i
0
1
P
i
ik
p
A
i
0
p
A
i
0
1
~
W
n1
i
+
~
W
n+1
i
2
,where
p
A
i
0
is the positive
square root of the symmetric denite positive matrix A
i
0
.Indeed,this expression derives
from the intuitive upwind ux for the symmetrized equations (2.5).It leads to timescheme
which is locally implicit near absorbing boundaries (i.e.independent linear systems are to
be solved inside elements having at least one absorbing face,at each time step).It leads
to a globally secondorder timeaccurate scheme.A less accurate explicit version is also
available
22
.It takes the form P
k
ik
~
W
n
k
=
p
A
i
0
p
A
i
0
1
P
i
ik
p
A
i
0
p
A
i
0
1
~
W
n1
i
.
Bernacki,Piperno
3.3.Energy balance and stability
In order to investigate stability,we dene a discrete aeroacoustic energy F
n
by:
F
n
=
1
2
X
i
Z
T
i
t
~
W
n
i
A
i
0
1
~
W
n
i
+
t
~
W
n+1
i
A
i
0
1
~
W
n1
i
+
t
4
X
a
ik
2F
abs
Z
a
ik
t
A
i
0
1
~
W
n1
i
M
ik
A
i
0
1
~
W
n1
i
+
~
W
n+1
i
;
with M
ik
=
p
A
i
0
p
A
i
0
1
P
i
ik
p
A
i
0
p
A
i
0
p
A
i
0
p
A
i
0
1
~
P
i
ik
p
A
i
0
1
p
A
i
0
.One can show that the
matrices M
ik
are symmetric and positive.We give in Annex 1 the expression of the variation
through one time step of the aeroacoustic energy (i.e.F
n+1
F
n
):
F
n+1
F
n
=
t
2
X
a
ik
2F
int
Z
a
ik
t
~
V
n
i
(
~
P
k
ik
~
P
i
ik
)
~
V
n+1
k
+
t
~
V
n+1
i
(
~
P
k
ik
~
P
i
ik
)
~
V
n
k
t
4
X
a
ik
2F
abs
Z
a
ik
t
~
V
n1
i
+
~
V
n+1
i
M
ik
~
V
n1
i
+
~
V
n+1
i
:(3.11)
where we have used the auxiliary variables
~
V
n
i
A
i
0
1
~
W
n
i
;8i;8n.The rst term is a
discrete version of the source term appearing in Eq.(2.7).The second term is negative and
shows that our absorbing boundary conditions actually absorbs energy.This results also
shows that the slip boundary condition has no in uence on the global energy balance.
In order to prove stability,we show in Annex 2 that F
n
is a quadratic positive denite
formof numerical unknowns (
~
W
n1
i
;
~
W
n
i
) under some CFLlike sucient stability condition
on the timestep t:
8i;8k 2 V
i
;t (2
i
i
+
ik
ik
) <
2V
i
P
i
;(3.12)
where
i
and
ik
are dimensionless regularity coecients depending of basis functions and
element aspect ratio,
i
= ju
i
0
j +jv
i
0
j +jw
i
0
j +3c
i
0
,and
ik
= j~v
i
0
~n
ik
j +c
i
0
for a boundary face
and
2
ik
= sup
j~v
i
0
~n
ik
j +c
i
0
)
2
A
k
0
A
i
0
1
;
j~v
k
0
~n
ik
j +c
k
0
2
A
i
0
A
k
0
1
for an internal
face ( here denotes the spectral radius of a matrix).
In the case of a uniform ow,we have P
i
ik
= P
k
ik
.Thus the aeroacoustic energy is
nonincreasing (and exactly conserved if no absorbing boundary is present,which shows
the scheme is genuinely nondiusive) and the scheme is stable under a CFLtype stability
condition depending on the size of elements and sup
i
(k~v
i
k +c
0
i
).
4.Numerical results
We dispose of a threedimensional parallel implementation of the DGTD method pre
sented in the previous section.Any subsonic steady ow can be considered,even with strong
ow gradients.However,the ow,given as the output of a nonlinear Euler equations solver,
has to be postprocessed:average of the ow over tetrahedra must be computed and the
A dissipationfree DGTD method for the threedimensional linearized Euler equations
nonslip condition must be enforced on physical boundaries.We present in this section test
cases in two and three space dimensions,in order to validate the method on benchmark
problems,test the method on complex ows and congurations,and nally evaluate the
performance of the parallel Fortran 77 implementation,based on the MPICH implementa
tion of MPI.Parallel computations were performed on a 16 node cluster (2GHzPentium4
1GbRDRAM memory biprocessor each).In this section,tables give performance results
for 64 bit arithmetic computations:N
p
is the number of processes for the parallel execution,
REAL denotes the total (wall clock) simulation time and CPU denotes the corresponding
total CPU time taken as the maximum of the per process values.Finally,% CPU denotes
the ratio of the total CPU time to the total wall clock time.This ratio clearly allows an
evaluation of the CPU utilization and yields a metric for parallel eciency.
4.1.Acoustic perturbations in a threedimensional uniform ow
This quite classical testcase
23;24
is aimed at validating the DGTD method proposed by
mixing acoustic,entropic and vorticity perturbations,in a uniform horizontal ow (with
Mach number 0.5).In that case,the exact solution is well known.The geometry is a unit
cube with absorbing boundaries.Two unstructured tetrahedral meshes have been used,
whose characteristics are given in Table 1 below.Plane cuts in the numerical solution
Table 1.Tam's testcase.Characteristics of meshes
Mesh#vertices#tetrahedra#absorbing faces
M1 68,921 384,000 19,200
M2 531,441 3,072,000 76,800
are shown on Fig.1.The result agree with the exact solution.The absorbing boundary
condition is not perfect,but the L2norm of the error at time t = 50 is limited to 2% for
the coarse mesh M1 and around 1% for the ner mesh M2.Articial re ections at the
absorbing boundaries are limited to 1% (of the amplitude at the corresponding time) for
mesh M1 and around 0;5% for M2.The time evolution of the energy inside the domain is
given on Fig.2.As expected,since the steady ow is uniform,the total acoustic energy is
rst exactly conserved,and then decreasing,once perturbations have reached the absorbing
boundaries (at t = 15).Computation times are given in Table 2.Parallel eciency reaches
a satisfying level and total CPU time are really acceptable.
4.2.Acoustic perturbations in a threedimensional subsonic Couette ow
This second testcase is more complex and is classically used to evaluate the merits of a
numerical method for nonuniform shear ows
25
.The steady ow considered is horizontal,
with ~v
0
=
t
(0:9z;0;0).At the beginning of the simulation,the same (acoustic,entropic,
and vorticity) perturbations are mixed.The domain considered is parallelepipedic with a
larger dimension in the direction of the supporting ow.Although the geometry recalls a
Bernacki,Piperno
Fig.1.Tam's testcase. contour lines at times t = 10:7,25,39:3,and 50.
0.5
1
1.5
2
2.5
3
3.5
0
5
10
15
20
25
30
35
40
45
50
55
F
time
Fig.2.Tam's testcase.Time evolution of acoustic energy.
A dissipationfree DGTD method for the threedimensional linearized Euler equations
Table 2.Tam's testcase.Parallel CPU times and eciency.
Mesh N
p
CPU REAL % CPU S(N
p
)
M1 1 33381 s 33447 s 100% 1
 4 8167 s 8412 s 97% 4
 8 4286 s 4465 s 96% 7.5
 16 2195 s 2381 s 92% 15.2
M2 8 34012 s 35651 s 95% 1
 16 17203 s 18760 s 92% 1.98
waveguide,the aim of this testcase is to validate the accurate propagation and convection
of the waves by the shear ow,and the behavior of the absorbing boundary condition,which
is set on all boundaries of the domain.Two unstructured tetrahedral meshes have been
generated.Their characteristics are given in Table 3.Plane cuts and volumic contours for
Table 3.Couette ow testcase.Characteristics of meshes.
Mesh#vertices#tetrahedra#absorbing faces
M1 73,629 405,600 23,504
M2 522,801 3,000,000 90,000
in the numerical solution are shown on Fig.3.The accuracy of the numerical results
is very similar to the one observed for a uniform ow:in the numerical results,articial
re ections fromabsorbing boundaries have a relative amplitude of 1:5% for the coarse mesh
M1 (respectively 1% for the ner mesh M2) compared to maximal amplitude at the same
time in the whole domain.In this testcase,the initial acoustic perturbation propagates
and gets quickly out of the domain,while the entropic perturbation is only convected by
the supporting ow (thus the bottom part of this perturbation travels slowly with the
supporting ow,whose velocity vanishes in the plane z = 0).
The time evolution of the global acoustic energy is given on Fig.4.As expected,the en
ergy is not conserved anymore because the nonuniform ow provides energy to the acoustic
perturbations.The energy decreases when perturbations reach the absorbing boundaries.
A part of the energy corresponding to the bottompart of the entropic perturbation remains
in the domain.Computation times are given in Table 4 and reveal similar parallel eciency
and execution times.
4.3.Twodimensional KelvinHelmholtz instabilities
It is wellknown that particular steadystate solutions of Euler equations can lead to
unstable solutions for the corresponding linearized Euler equations.These instabilities are
known as KelvinHelmholtz instabilities and are proved to appear for example for shear
ows.These instabilities are present in the linearized equations.In the original Euler
equations,some limiting nonlinear term must be playing the role of a limiter for the overall
Bernacki,Piperno
Fig.3.Couette ow testcase. contour lines (left:contours in plane cuts;right:volumic contours) at
times t = 0,0:5,and 5:75.
A dissipationfree DGTD method for the threedimensional linearized Euler equations
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0
1
2
3
4
5
Energy
Time (s)
Fig.4.Couette ow testcase.Time evolution of the acoustic energy.
Table 4.Couette ow testcase.Parallel CPU times and eciency.
Mesh N
p
CPU REAL % CPU S(N
p
)
M1 4 12830 s 13008 s 99% 1
 8 6428 s 6590 s 98% 2
 16 3230 s 3373 s 96% 3.9
M2 8 46359 s 47743 s 97% 1
 16 23412 24959 s 94% 1.98
Bernacki,Piperno
acoustic energy in the ow.When linearized Euler equations are solved using numerical
methods with numerical dissipation,these instabilities might be eliminated.It might be an
positive feature in some cases.However,there is no control on the numerical dissipation and
the accuracy of the solution obtained.In our case,the numerical method we have proposed
is genuinely nondissipative.Therefore,KelvinHelmholtz instabilities should appear easily,
and more easily as transverse gradients in the steady ow increase.This is the case for
example in two space dimensions with ~v
0
= 0:25 tanh [151:51(y 0:25) +0:5] ~e
x
and
0
= 1,
p
0
= 1= .In this ow,u
0
goes through an in ection point at y = 0:25 (and k~v
0
k = 0:5
on the in ection point).In this particular ow,the acoustic energy is amplied along the
in ection line and at the same time,perturbations are convected along this line,which
leads to an instability.The behavior is completely dierent from the one obtained with a
shear ow of the form ~v
0
= (y +0:25)~e
x
,where no instability occurs.We have introduced
a periodic in time,Gaussian in space source term to excite possible unstable modes and
numerical results obtained are compared on Fig.5.The instability appears quite quickly
for the shear ow with an in ection point.Dierent means are under current investigation
to control this kind of instability.
Fig.5.Contours of k~vk at t=0.257 for the linear (left) and in ection (right) shear ows with source term
4.4.Twodimensional instabilities past a NACA airfoil prole
This testcase is aimed at evaluating the occurrence of KevinHelmholtztype instabilities
for steadystate ows with sharp gradients.We have considered a testcase past a NACA
type airfoil prole (courtesy of ONERA
19
).The steadystate solution at M
1
= 0:5 obtained
by our nitevolume nonlinear Euler solver on the 33046vertex 65580triangle unstructured
mesh is shown on Fig.6.A Gaussian in space and periodic in time (T = 1ms) source term
has been set up for the aeroacoustic problem,where a slip boundary condition is used on
the prole,and our absorbing boundary conditions in the far eld boundary.For this test
case,we have observed an instability,i.e.the energy source term has a positive sign and
some aeroacoustic blow up occurs.A detail on contours of k~vk are shown on Fig.7.Some
numerical treatments are possible to overcome these diculties and get rid of instabilities.
One of them
4
consists in adding a source term which leads to energy dissipation.As we
are not specialists of physics,we do not discuss here the validity of such a model,but we
can observe that the instability disappears after treatment.Contours of k~vk are shown on
the right part of Fig.8 and can be compared to classical acoustic scattering (uniform still
ow).The comparison of contours near the trailing edge for acoustics and aeroacoustics
is shown on Fig.9,where the shear ow at the trailing edge has a visible in uence on the
A dissipationfree DGTD method for the threedimensional linearized Euler equations
Mach, min = 0.0231943, max = 0.889346
Fig.6.Mach number contours for the steadystate ow past the airfoil prole at M
1
= 0:5
Fig.7.Contours of k~vk at t = 1:57ms for the original aeroacoustic model.
Bernacki,Piperno
propagation of acoustics (and generation of vortices).
Fig.8.Contours of k~vk for the stabilized aeroacoustic model (left) and classical acoustics (right).
Fig.9.Tailing edge zoom { k~vk contours for stabilized aeroacoustics (left) and acoustics (right).
4.5.Threedimensional Aeroacoustic wave propagation past a sphere
We consider the case of a steady threedimensional ow past a sphere and the propa
gation of waves emitted by an acoustic source (Gaussian prole,periodic source term with
period T = 0:2s.The subsonic ow with M
1
= 0:5 is obtained by a threedimensional
MUSCLextended nitevolume solver of threedimensional Euler equations on tetrahedral
grids.The steadystate solution presents Mach numbers between 0:002 and 0:8 and con
tours for
0
are presented on Fig.10.Details on the computational tetrahedral mesh are
given on Table 5.The unit sphere is centered inside the 10mside cubic computational
domain.A slip boundary condition is set on the sphere,while the absorbing boundary
A dissipationfree DGTD method for the threedimensional linearized Euler equations
Figure 10:Aeroacoustic propagation past a sphere { steadystate density
0
condition is set on far eld boundaries.The aeroacoustic simulation was performed on 16
processors,with computational times reported on Table 5.We have shown on Fig.11 the
Table 5:Aeroacoustic propagation past a sphere { computational mesh and times
#vertices#elements#abs.faces#slip faces N
p
CPU REAL % CPU
324,471 1,870,288 10,092 46,080 16 50h4mn 51h18mn 97,6%
resulting propagated waves for this aeroacoustic case in a steady ow past a sphere (after
10 and 25 periods).The numerical results (scattering and re ection patterns) as well as
the computational times are encouraging.One can notice that the periodic regime is not
reached after 10 periods.These results lead us to consider in the near future more complex
testcases and more realistic congurations.
5.Conclusion
In this paper,we have presented a time domain Discontinuous Galerkin dissipation
free method for the transient solution of the threedimensional linearized Euler equations
around a steadystate solution,based on P
1
Lagrange elements on tetrahedra.In the more
general context of a nonuniform supporting ow,we have proved,using the wellknown
symmetrization of Euler equations,that the aeroacoustic energy satises a balance equa
tion with source term at the continuous level,and that our numerical method satises
an equivalent balance equation at the discrete level.This shows that the method is gen
uinely dissipationfree,which is conrmed by numerical results,including testcases where
KelvinHelmholtz instabilities appear.
Further works can concern many dierent aspects.On the modeling side,it is possible to
Bernacki,Piperno
Figure 11:Aeroacoustic propagation past a sphere { after 10T (left) and 25T (right)
design models for dealing with natural KevinHelmholtz instability,for example by adding
some source terms.This has to be done in cooperation with physicists.Anyway,the
numerical framework proposed here provides a valuable tool for investigating this kind of
instability in complex ows and geometries.On the numerical side,the overall accuracy
could be enhanced either by considering morethanlinear basis functions (P
k
Lagrange
elements with k > 1) or by dealing with a more accurate description of the supporting
ow (currently,it is only P
0
).Higherorder accuracy in absorbing boundary conditions and
timescheme should also be seeked for.
Annex 1
We shall prove the result given in Eq.(3.11).Since A
i
0
is symmetric,we have
F
n+1
F
n
=
1
2
X
i
Z
T
i
t
~
W
n
i
A
i
0
1
~
W
n+2
i
~
W
n
i
+
t
~
W
n+1
i
A
i
0
1
~
W
n+1
i
~
W
n1
i
+
t
4
X
a
ik
2F
abs
Z
a
ik
t
A
i
0
1
~
W
n
i
M
ik
A
i
0
1
~
W
n+2
i
+
~
W
n
i
t
A
i
0
1
~
W
n1
i
M
ik
A
i
0
1
~
W
n+1
i
+
~
W
n1
i
:
The variational formof the scheme (3.10) allows to consider any eld in P
i
,thus for example
A dissipationfree DGTD method for the threedimensional linearized Euler equations
the elds
~
W
n1
i
or
~
W
n
i
themselves.We obtain that F
n+1
= F
n
+t(T
1
1
2
T
2
1
2
T
3
) with
T
1
=
X
i
Z
T
i
X
s2fx;y;zg
t
@
s
~
V
n
i
~
A
i
s
~
V
n+1
i
+
t
@
s
~
V
n+1
i
~
A
i
s
~
V
n
i
T
2
=
X
a
ik
2F
int
Z
a
ik
t
~
V
n
i
~
P
i
ik
~
V
n+1
i
+
~
P
k
ik
~
V
n+1
k
+
t
~
V
n
k
~
P
k
ki
~
V
n+1
k
+
~
P
i
ki
~
V
n+1
i
+
X
a
ik
2F
int
Z
a
ik
t
~
V
n+1
i
~
P
i
ik
~
V
n
i
+
~
P
k
ik
~
V
n
k
+
t
~
V
n+1
k
~
P
k
ki
~
V
n
k
+
~
P
i
ki
~
V
n
i
T
3
=
X
a
ik
2F
slip
Z
a
ik
t
~
V
n
i
~
P
i
ik
~
V
n+1
i
+
~
P
i
ik
H
ik
~
V
n+1
i
+
t
~
V
n+1
i
~
P
i
ik
~
V
n
i
+
~
P
i
ik
H
ik
~
V
n
i
+
X
a
ik
2F
abs
Z
a
ik
"
t
~
V
n
i
~
P
i
ik
~
V
n+1
i
+M
ik
~
V
n
i
+
~
V
n+2
i
2
!
+
t
~
V
n+1
i
~
P
i
ik
~
V
n
i
+M
ik
~
V
n1
i
+
~
V
n+1
i
2
!#
1
2
X
a
ik
2F
abs
Z
a
ik
t
~
V
n
i
M
ik
~
V
n
i
+
~
V
n+2
i
t
~
V
n1
i
M
ik
~
V
n1
i
+
~
V
n+1
i
:
Since
~
P is symmetric,and since
~
P
i
ik
=
~
P
i
ki
and
~
P
k
ik
=
~
P
k
ki
,we obtain that
T
2
= 2
X
a
ik
2F
int
Z
a
ik
t
~
V
n
i
~
P
i
ik
~
V
n+1
i
+
t
~
V
n
k
~
P
k
ki
~
V
n+1
k
+
X
a
ik
2F
int
Z
a
ik
t
~
V
n
i
~
P
k
ik
~
P
i
ik
~
V
n+1
k
+
t
~
V
n
k
~
P
k
ik
~
P
i
ik
~
V
n+1
i
;and
T
2
+T
3
= 2
X
i
X
k2V
i
Z
a
ik
t
~
V
n
i
~
P
i
ik
~
V
n+1
i
+
X
a
ik
2F
slip
Z
a
ik
t
~
V
n
i
~
P
i
ik
H
ik
~
V
n+1
i
+
t
~
V
n+1
i
~
P
i
ik
H
ik
~
V
n
i
+
X
a
ik
2F
int
Z
a
ik
t
~
V
n
i
~
P
k
ik
~
P
i
ik
~
V
n+1
k
+
t
~
V
n
k
~
P
k
ik
~
P
i
ik
~
V
n+1
i
+
1
2
X
a
ik
2F
abs
Z
a
ik
t
~
V
n1
i
+
~
V
n+1
i
M
ik
~
V
n1
i
+
~
V
n+1
i
:
In the expressions above,the matrix H
ik
denotes the matrix used for slip boundary con
ditions,in the sense that the ctitious state
~
W
n
k
is given by
~
W
n
k
= H
ik
~
W
n
i
(H
ik
does not
change the density and energy perturbations,and operates an orthogonal symmetry on the
velocity perturbation).One can also show that H
2
ik
= I and that H
ik
commutes with A
i
0
.
Bernacki,Piperno
We nally get that F
n+1
= F
n
+t(T
4
+T
5
+T
6
) with
T
4
=
X
i
2
4
Z
T
i
X
s2fx;y;zg
t
@
s
~
V
n
i
~
A
i
s
~
V
n+1
i
+
t
@
s
~
V
n+1
i
~
A
i
s
~
V
n
i
X
k2V
i
Z
a
ik
t
~
V
n
i
~
P
i
ik
~
V
n+1
i
3
5
T
5
=
1
2
X
a
ik
2F
ref
Z
a
ik
t
~
V
n
i
~
P
i
ik
H
ik
+
t
~
P
i
ik
H
ik
~
V
n+1
i
T
6
=
1
4
X
a
ik
2F
abs
Z
a
ik
t
~
V
n1
i
+
~
V
n+1
i
M
ik
~
V
n1
i
+
~
V
n+1
i
1
2
X
a
ik
2F
int
Z
a
ik
t
~
V
n
i
~
P
k
ik
~
P
i
ik
~
V
n+1
k
+
t
~
V
n
k
~
P
k
ik
~
P
i
ik
~
V
n+1
i
Since the matrices
~
A
s
are symmetric and constant over control volumes,we get by integrat
ing by parts that T
4
= 0.Also,since the matrix
~
P
i
ik
H
ik
is skewsymmetric,we get T
5
= 0.
Using the expression of M
ik
and using the
~
Wvariables,we get the result announced.
Annex 2
We shall prove that F
n
is a positive denite quadratic form of unknowns
~
W
n1
i
and
~
W
n
i
under the condition given in Eq.(3.12).Let us introduce the following denitions.
Denition 1 8
~
X 2 P
i
,we denote by k
~
Xk
T
i
= (
R
T
i
k
~
Xk
2
)
1=2
the L
2
norm of
~
X over T
i
.
For any interface a
ik
2 F,we also dene k
~
Xk
a
ik
= (
R
a
ik
k
~
Xk
2
)
1=2
.Finally,the matrix A
i
0
being positive denite,we dene a second norm by kj
~
Xkj
T
i
= k
p
A
i
0
1
~
Xk
T
i
.
Denition 2 We assume some regularity for the basis functions ~'
ij
;1 j d
i
.More
precisely,we assume that,for each control volume T
i
,there exist dimensionless positive
constants
i
and
ik
(for each k 2 V
i
) such that
8
~
X2 P
i
;
(
8s 2 fx;y;zg;k@
s
~
Xk
T
i
i
P
i
V
i
k
~
Xk
T
i
;
8a
ik
2 F;k
~
Xk
2
a
ik
ik
k~n
ik
k
V
i
k
~
Xk
2
T
i
:
(5.13)
Denition 3 We choose here some notations.We dene
~
Z
n
i
=
p
A
i
0
1
~
W
n
i
.We introduce
i
=
P
s
A
i
s
(and we recall
A
i
s
= j~e
s
:~v
i
0
j + c
i
0
).For boundary faces a
ik
,we recall
that some neighboring ctitious control volume T
k
is imagined,and we take by convention:
k
~
W
n
k
k
T
k
= k
~
W
n
i
k
T
i
,kj
~
W
n
k
kj
T
k
= kj
~
W
n
i
kj
T
i
,
ki
=
ik
,V
k
= V
i
.Finally,we dene
ik
for
any face a
ik
by:
8
>
<
>
:
a
ik
2 F
int
;
2
ik
= sup
j~v
i
0
~n
ik
j +c
i
0
2
A
k
0
A
i
0
1
;
j~v
k
0
~n
ik
j +c
k
0
2
A
i
0
A
k
0
1
a
ik
2 F
slip
;
ik
=
P
i
ik
= j~v
i
0
~n
ik
j +c
i
0
a
ik
2 F
abs
;
ik
= 0;
A dissipationfree DGTD method for the threedimensional linearized Euler equations
In order to prove that F
n
is a positive denite quadratic form of unknowns
~
W
n1
i
and
~
W
n
i
under the condition (3.12),we rst show the following result:
F
n
i
1
2
kj
~
W
n
i
kj
2
T
i
+kj
~
W
n1
i
kj
2
T
i
i
P
i
i
t
V
i
kj
~
W
n
i
kj
T
i
kj
~
W
n1
i
kj
T
i
t
4
X
k2V
i
ik
ik
k~n
ik
k
V
i
kj
~
W
n1
i
kj
2
T
i
+
ki
ki
k~n
ki
k
V
k
kj
~
W
n
k
kj
2
T
k
:(5.14)
We can show easily that the above lemma gives the result.Using the above denitions and
assumptions,we have F
n
i
P
k2V
i
k~n
ik
k(X
ik
ki
ki
t
4V
k
kj
~
W
n
k
kj
2
T
k
) with
X
ik
=
1
2P
i
kj
~
W
n
i
kj
2
T
i
+
1
2P
i
ik
ik
t
4V
i
kj
~
W
n1
i
kj
2
T
i
i
i
t
V
i
kj
~
W
n
i
kj
T
i
kj
~
W
n1
i
kj
T
i
1
2P
i
i
i
t
2V
i
kj
~
W
n
i
kj
2
T
i
+
1
2P
i
ik
ik
t
4V
i
i
i
t
2V
i
kj
~
W
n1
i
kj
2
T
i
Finally,F
n
X
a
ik
2F
int
k~n
ik
kY
ik
+
X
a
ik
2F
slip
S
F
abs
k~n
ik
kZ
ik
with
Y
ik
=
1
2P
i
ik
ik
t
4V
i
i
i
t
2V
i
kj
~
W
n
i
kj
2
T
i
+kj
~
W
n1
i
kj
2
T
i
+
1
2P
k
ki
ki
t
4V
k
k
k
t
2V
k
kj
~
W
n
k
kj
2
T
k
+kj
~
W
n1
k
kj
2
T
k
;
Z
ik
=
1
2P
i
ik
ik
t
4V
i
i
i
t
2V
i
kj
~
W
n
i
kj
2
T
i
+kj
~
W
n1
i
kj
2
T
i
:
Therefore,the energy F
n
is a positive denite quadratic form of unknowns under the su
cient CFLtype stability condition proposed in Eq.(3.12).
End of the proof.We now give the proof for the preliminary result (5.14).We rst have
F
n
i
=
1
2
kj
~
W
n
i
kj
2
T
i
+kj
~
W
n1
i
kj
2
T
i
+
t
2
(X
n
i
+Y
n
i
) with
X
n
i
=
1
2
X
a
ik
2F
abs
\@T
i
Z
a
ik
t
A
i
0
1
~
W
n1
i
M
ik
A
i
0
1
~
W
n1
i
+
~
W
n+1
i
=
1
2
X
a
ik
2F
abs
\@T
i
Z
a
ik
t
~
V
n1
i
M
ik
~
V
n1
i
+
~
V
n+1
i
Y
n
i
=
1
t
Z
T
i
t
~
W
n1
i
A
i
0
1
~
W
n+1
i
~
W
n1
i
= 2
Z
T
i
X
s2fx;y;zg
t
@
s
~
V
n1
i
~
A
i
s
~
V
n
i
X
k2V
i
Z
a
ik
t
~
V
n1
i
~
P
i
ik
~
V
n
i
+
~
P
k
ik
~
V
n
k
=
Z
T
i
X
s2fx;y;zg
t
@
s
~
V
n1
i
~
A
i
s
~
V
n
i
t
~
V
n1
i
~
A
i
s
@
s
~
V
n
i
X
k2V
i
Z
a
ik
t
~
V
n1
i
~
P
k
ik
~
V
n
k
Bernacki,Piperno
Using H
ik
and eliminating terms corresponding to absorbing boundaries leads to
X
n
i
+Y
n
i
=
Z
T
i
X
s2fx;y;zg
t
@
s
~
V
n1
i
~
A
i
s
~
V
n
i
t
~
V
n1
i
~
A
i
s
@
s
~
V
n
i
X
a
ik
2F
int
Z
a
ik
t
~
V
n1
i
~
P
k
ik
~
V
n
k
X
a
ik
2F
slip
Z
a
ik
t
~
V
n1
i
~
P
i
ik
H
ik
~
V
n
i
:
Then we deduce that X
n
i
+Y
n
i
= T
1
+T
2
+T
3
with
T
1
=
Z
T
i
X
s2fx;y;zg
t
@
s
~
Z
n1
i
p
A
i
0
1
~
A
i
s
p
A
i
0
1
~
Z
n
i
t
~
Z
n1
i
p
A
i
0
1
~
A
i
s
p
A
i
0
1
@
s
~
Z
n
i
;
T
2
=
X
a
ik
2F
int
\@T
i
Z
a
ik
t
~
Z
n1
i
p
A
i
0
1
p
A
k
0
p
A
k
0
1
~
P
k
ik
p
A
k
0
1
~
Z
n
k
;
T
3
=
X
a
ik
2F
slip
\@T
i
Z
a
ik
t
~
Z
n1
i
p
A
i
0
1
~
P
i
ik
p
A
i
0
1
p
A
i
0
H
ik
p
A
i
0
1
~
Z
n
i
:
The matrix
p
A
i
0
1
~
A
i
s
p
A
i
0
1
is symmetric and
p
A
i
0
1
~
A
i
s
p
A
i
0
1
=
A
i
s
,then we have
jT
1
j
2
i
i
P
i
V
i
k
~
Z
n1
i
k
T
i
k
~
Z
n
i
k
T
i
=
2
i
i
P
i
V
i
kj
~
W
n1
i
kj
T
i
kj
~
W
n
i
kj
T
i
:
The matrix
p
A
k
0
1
~
P
k
ik
p
A
k
0
1
is also symmetric and
p
A
k
0
1
~
P
k
ik
p
A
k
0
1
=
P
k
ik
,then
jT
2
j
X
a
ik
2F
int
\@T
i
P
k
ik
r
p
A
k
0
A
i
0
1
p
A
k
0
k
~
Z
n1
i
k
a
ik
k
~
Z
n
i
k
a
ik
X
a
ik
2F
int
\@T
i
P
k
ik
r
A
k
0
A
i
0
1
kj
~
W
n1
i
kj
a
ik
kj
~
W
n
i
kj
a
ik
:
Since the matrices A
i
0
and H
ik
commute and H
2
ik
= Id,we have
jT
3
j
X
a
ik
2F
ref
\@T
i
P
i
ik
k
~
Z
n1
i
k
a
ik
k
~
Z
n
i
k
a
ik
=
X
a
ik
2F
ref
\@T
i
P
i
ik
kj
~
W
n1
i
kj
a
ik
kj
~
W
n
i
kj
a
ik
:
Using the above denitions and inequalities of the form ab
1
2
a
2
+b
2
,rewriting F
n
i
1
2
kj
~
W
n
i
kj
2
T
i
+kj
~
W
n1
i
kj
2
T
i
t
2
(jT
1
j +T
2
j +T
3
j) leads to the intermediate result (5.14).
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