Code_Aster
®
Version
5.0
Titrate:
Elements of absorbing border
Date:
02/04/01
Author (S):
G. DEVESA, V. TO MOW
Key:
R4.02.05-A
Page:
1/20
Manual of Reference
R4.02 booklet: Accoustics
HT-62/01/003/A
Organization (S):
EDF/RNE/AMV
Manual of Reference
R4.02 booklet: Accoustics
Document: R4.02.05
Elements of absorbing border
Summary
This document describes the establishment in Code_Aster of the elements of absorbing border. These elements of
type paraxial, which one describes the theory here, are assigned to borders of elastic ranges or fluid for
to deal with problems 2D or 3D of interaction ground-structure or ground-fluid-structure. They make it possible to satisfy
condition of Sommerfeld checking the assumption of anechoicity: the elimination of the elastic plane waves or
diffracted acoustics and not physiques coming from the infinite one.
Code_Aster
®
Version
5.0
Titrate:
Elements of absorbing border
Date:
02/04/01
Author (S):
G. DEVESA, V. TO MOW
Key:
R4.02.05-A
Page:
2/20
Manual of Reference
R4.02 booklet: Accoustics
HT-62/01/003/A
1 Introduction
1.1
Problems of a semi-infinite medium for the ISS
Standard problems of seismic answer and interaction ground-structure or ground-fluid-structure
bring to consider infinite or supposed fields such. For example, in the case of stoppings
subjected to the seism, there is often business with reserves of big size which enable us to make
the assumption of anechoicity: the waves which leave towards the bottom reserve “do not return” not. This has
for goal to reduce the size of the structure to be netted and to allow to pass from complex calculations
with average data processing current. One proposes on [Figure 1.1-a] below diagram which
described the type of situations considered.
S
'
S
F
B
Surface absorbing
rubber band
Surface absorbing
fluid
Fields modelized with the finite elements:
F
fluid field (for example retained stopping)
B
structure field (for example vault of stopping)
S
non-linear ground field
'
S
linear ground field
Appear 1.1-a: Field for the interaction ground-fluid-structure
In all the document, one considers that the border of the mesh finite elements of the ground is in
a field with the elastic behavior.
The elliptic system theory ensures simply the existence and the unicity of the solution of
acoustic or elastoplastic problems in the limited fields, under the assumption of conditions
with the limits ensuring the closing of the problem. It goes from there differently for the infinite fields. One must
to have recourse to a condition particular, known as of Sommerfeld, formulated in the infinite directions of
problem. This condition ensures in particular, in the case of the diffraction of a wave planes (elastic
or accoustics) by a structure, the elimination of the diffracted waves not physiques coming from the infinite one
that the conventional conditions on the edges of the field remotely finished are not enough to ensure.
Code_Aster
®
Version
5.0
Titrate:
Elements of absorbing border
Date:
02/04/01
Author (S):
G. DEVESA, V. TO MOW
Key:
R4.02.05-A
Page:
3/20
Manual of Reference
R4.02 booklet: Accoustics
HT-62/01/003/A
1.2
State of the art of the numerical approaches
The privileged method to treat infinite fields is that of the finite elements of border (or
integral equations). The fundamental solution used checks the condition automatically of
Sommerfeld. Only, the use of this method is conditioned by knowledge of this
fundamental solution, which is impossible in the case of a ground with complex geometry, for example,
or when the ground or the structure is nonlinear. It is thus necessary then to have recourse to the finite elements.
Consequently, of the conditions particular to the border of the mesh finite elements are necessary for
to prohibit the reflection of the outgoing diffracted waves and to reproduce the condition thus artificially of
Sommerfeld.
Several methods make it possible to identify boundary conditions answering our requirements.
Some lead to an exact resolution of the problem: they are called “consistent borders”.
They are founded on a precise taking into account of the wave propagation in the infinite field.
For example, if this field can be supposed elastic and with a simple stratigraphy far from
structure, one can consider a coupling finite elements - integral equations. One of the problems of
this solution is that it is not local in space: it is necessary to make a separating assessment on all the border
the finished field of the infinite field, which obligatorily leads us to a problem of
under-structuring. This not-locality in space is characteristic of the consistent borders.
To lead in the local terms of border in space, one can use the theory of the infinite elements
[bib1]. They are elements of infinite size whose basic functions reproduce as well as possible
elastic or acoustic wave propagation ad infinitum. These functions must be close to
solution because the conventional mathematical theorems do not ensure any more convergence of the result of
calculation towards the solution with such elements. In fact, one can find an analogy between the search for
satisfactory basic functions and that of a fundamental solution for the integral equations.
geometrical stresses are rather close but especially, this search presents a disadvantage
of size: it depends on the frequency. Consequently, such borders, local or not in space,
can be used only in the field of Fourier, which prohibits a certain category of
problems, with non-linearities of behavior or great displacements for example.
One thus arrives at having to find borders absorbing powerful who are local in
space and in time to treat with the finished parts of the transitory problems posed on fields
infinite.
We will present in the continuation the theory of the paraxial elements which carry out absorption
sought with an effectiveness inversely proportional to their simplicity of implementation as well as
description of the stresses of implementation in Code_Aster. The developments are presented
to deal with 3D problems. Those for the cases 2D were carried out and their theory results
simply of modeling 3D.
Code_Aster
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Version
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Titrate:
Elements of absorbing border
Date:
02/04/01
Author (S):
G. DEVESA, V. TO MOW
Key:
R4.02.05-A
Page:
4/20
Manual of Reference
R4.02 booklet: Accoustics
HT-62/01/003/A
2
Theory of the paraxial elements
One presents in this part the principle of the paraxial approximation in the case of the elastodynamic one
linear. Two theoretical approaches make it possible to determine the spirit and the practical application elements
paraxial rubber bands: one owes the first in Cohen
and
Jennings
[bib2] and the second with MODARESSI
[bib3]. The application of the theory of the paraxial elements to the fluid case will be made in the part
following.
Subsequently, as presented on [Figure 1.1-a], one supposes that the border of the mesh of
ground is located in a field at the elastic behavior.
The approach of Modaressi established in Code_Aster at the same time makes it possible to build borders
absorbing and to introduce the incidental seismic field.
2.1
Spectral impedance of the border
To obtain the paraxial equation, we should initially determine the shape of the field of displacement
diffracted in the vicinity of the border. For that, one leaves the equations of the elastodynamic 3D:
2
2
11 2
2
2
12 2
2
3
22 2
2
32
0
U.E.
U
E
U
E
U
T
C
X
C
X X
C
X
-
-
-
=
'
'
With:
U
=
U
U
'
3
E
11
2
2
2
1
0
0
=
C
C
C
P
S
(
)
E
12
2
2
2
1
0 1
1 0
=
-
C C
C
P
S
E
22
2
2
2
1
0
0
=
C
C
C
S
P
The constant
C
, homogeneous at a speed, is introduced to return certain quantities
adimensional. The equations and their solutions are of course independent of this
constant.
One calls
x'
and
u'
directions and components of displacement in the tangent plan and
X
3
and
U
3
according to
E
3
, normal direction at the border.
One proceeds to two transforms of Fourier, one compared to time, the other compared to
variables of space in the plan at the border. One limits oneself to the case of a plane border and without corner:
The equations are written then:
(
)
C
C
I X
C
X
P
S
S
2
2
3
3
2
2
2
32
2
0
-
- +
+
- +
+
=
. !
!
!
!
U
U
U
U
(
)
C
C
I
X
X
C
X
P
S
S
2
2
3
2 3
32
2
2
2
32
3
2 3
0
-
-
+
+
- +
+
=
. !
!
!
!
U
U
U
U
where
!U
and
!U
3
the transforms of Fourier indicate and
the vector of wave associated with
x'
.
Code_Aster
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Version
5.0
Titrate:
Elements of absorbing border
Date:
02/04/01
Author (S):
G. DEVESA, V. TO MOW
Key:
R4.02.05-A
Page:
5/20
Manual of Reference
R4.02 booklet: Accoustics
HT-62/01/003/A
It is about a differential connection in
X
3
that one can solve by diagonalisant it. One deduces some:
(
)
(
)
=
-
!
.
exp
U
E
With
I
X
S
3
3
(
)
(
)
[
]
=
-
+
-
! .
exp
exp
U
With
I
X
With
I X
P
P
S
S
3
3
(
)
(
)
= -
-
-
-
!
exp
exp
U
3
3
3
With
I
X
With
I X
P P
P
S S
S
With:
P
P
C
=
-
2
2
2
and
S
S
C
=
-
2
2
2
To determine the constants
WITH A
With
S
P
,
and
, one supposes known
()
! ',
U
0
on the border of the field
finite elements. One expresses them according to
()
()
! ',
!
!
',
!
'
u'
U
U
U
0
0
0
3
30
=
=
and
.
One now will evaluate the vector forced on a breakage of normal
E
3
in
X
3
0
=
, which us
the impedance of the border will give. One subjects to
(
)
T X X
',
3
the same transform of Fourier in
space that for the equations of elastodynamic, so that:
(
)
(
)
! ,
! .
!
!
!
T
U
U.E.
U
U
=
+
+
+
+
µ
µ
X
I
X
X
I
3
3
3
3
3
3
2
One wishes to free oneself in
X
3
0
=
terms containing of derived in
X
3
. The system obtained
previously allows it to us according to
!
!
'
U
U
0
30
and
:
! .
!
=
U
U
0
3
2 30
X
I
(
)
!
.
!
.
= -
U
E
U
E
0
3
3
0
3
X
I
S
(
)
!
! .
!
U
U
U
30
3
0 2
30
X
I
P S
P
S
= -
+
+
One thus obtains the spectral impedance of the border:
!
has
B
C
T
E
E
0
0 3
0
0
3
=
+
+
where
has, B
C
0
0
0
and
are related to
and of
who depend linearly on
!
!
'
U
U
0
30
and
One can then write:
(
)
(
)
!
,
!
,
T
U
0
0
=
With
where
With
appoint the operator total spectral impedance. One returns to physical space by two
transforms of Fourier opposite.
Code_Aster
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Version
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Titrate:
Elements of absorbing border
Date:
02/04/01
Author (S):
G. DEVESA, V. TO MOW
Key:
R4.02.05-A
Page:
6/20
Manual of Reference
R4.02 booklet: Accoustics
HT-62/01/003/A
2.2
Paraxial approximation of the impedance
The spectral impedance calculated previously is local neither spaces some nor in time since it makes
to intervene
(
)
!
',
U
0
, transform of Fourier of
()
U
0
X T
',
for all
x'
and all
T
.
The idea is then to develop
P
and
S
according to the powers of
. This approximation will be good
either high frequency, or for
small.
Let us examine the dependence in
X
3
, for example of
!U
3
: one will have, for
(
)
U
3
3
X X T
',
terms of
form:
(
)
[
]
exp I
X
T
X
P
+
-
3
With the development of
P
:
P
P
P
C
C
=
-
+
1
2
…
One shows that for
small, there will be waves being propagated according to directions' close to
normal
E
3
at the border, because the exponential one is written:
exp I
T
X
C
I O
P
-
+
3
Consequently, with an asymptotic development of
P
and
S
, while multiplying by a power
suitable of
to remove this quantity with the denominator, one obtains:
(
)
(
)
With
,
!
With
,
!
0
0
1
0
=
T
U
where
With
0
and
With
1
are polynomial functions in
and
.
Maybe, after the two transforms of Fourier opposite:
With
,
With
,
0
0
1
0
=
X
T
X
T
T
U
One thus obtains the final form of the approximate transitory local impedance according to the last term
in
retained. One can find the calculation detailed of
With
I
in [bib5].
For example, for command 0:
T
E
U
0
3 3
=
+
C
U
T
C
T
P
S
This corresponds to viscous shock absorbers distributed along the border of the elements field
finished.
Code_Aster
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Version
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Titrate:
Elements of absorbing border
Date:
02/04/01
Author (S):
G. DEVESA, V. TO MOW
Key:
R4.02.05-A
Page:
7/20
Manual of Reference
R4.02 booklet: Accoustics
HT-62/01/003/A
With command 1:
(
)
(
)
T
U.E.
U
U.E.
U
U.E.
U
0
2 32 3
2
2
2 3 3
2
2
2 3 3
2
2
2
2
2
2
2
2
T
C
T
C
T
C
C
C
X T
C
C
X T
C C
C
X
C C
C
X
P
S
S
S
P
P
S
P
S
P
S
P
S
=
+
+
-
+
-
+
-
+
-
.
One sees appearing the derivative compared to the time of the vector forced. In the digital processing,
it will be necessary to have recourse to an integration of this term on the elements of the border.
To conclude, it will be retained that the paraxial approximation led to a transitory local impedance
utilizing that derivative in time and in the tangent plan at the border. In way
symbolic system, one writes:
T
U
T
U
U U
0
0
0
1
2
2
=
=
With
With
,
,
T
T
T
T
with command 0
with command 1
2.3
Taking into account of the incidental seismic field
It is reminded the meeting that the behavior of the ground is supposed to be elastic at least in the vicinity of the border. With
the infinite one, the total field
U
must be equal to the incidental field
U
I
(one of the consequences of the condition of
radiation of Sommerfeld). The diffracted field is thus introduced
U
R
such as:
U U
U
=
+
I
R
lim
X
R
=
U
0
At the border of the mesh finite elements, one writes the condition of absorption for the diffracted field:
()
()
T U
U
T U
U
U U
0
0
0
1
2
2
R
R
R
R
R
R
T
T
T
T
=
=
With
With
,
,
with command 0
with 'command 1
One deduces from it the total vector forced on the border from the mesh finite elements:
()
() () ()
T U
T U
T U
T U
U
U
0
0
0
0
0
0
=
+
=
+
-
I
R
I
I
T
T
With
With
with command 0
Code_Aster
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Version
5.0
Titrate:
Elements of absorbing border
Date:
02/04/01
Author (S):
G. DEVESA, V. TO MOW
Key:
R4.02.05-A
Page:
8/20
Manual of Reference
R4.02 booklet: Accoustics
HT-62/01/003/A
One thus obtains the variational formulation of the problem in the vicinity of the border for command 0:
() ()
()
2
2
0
0
U
U
T U
U
T v
U
v
T v
T
v
I
I
+
-
=
-
:
With
With
For any field
v
kinematically acceptable
For command 1, one preserves the conventional formulation:
() ()
()
2
2
0
U
U
v
U v
T v
T
+
-
=
:
where
()
T U
the law of following evolution follows:
()
()
T U
T U
U
U U
U
U U
T
T
T
T
T
T
I
I
I
I
=
+
-
With
,
,
With
,
,
1
2
2
1
2
2
The stress due to the incidental field appears explicitly in the case of command 0, but it is
contained in the law of evolution of
()
T U
for command 1.
3
Anechoic fluid elements in transient
This part presents the main part of the general stresses of implementation of fluid elements
anechoic of border absorbents with the paraxial approximation of command 0 in Code_Aster.
For reasons of simplicity related to the handling of scalar sizes such as the pressure or
potential of displacement, in opposition to the vector quantities like displacement, one
be interested initially in the fluid elements.
3.1 Formulation
standard
One takes again here the reasoning of Modaressi by adapting it to an acoustic fluid field. In one
the first time, one is interested in the only data of the size pressure in this fluid. One will return
then on this modeling to adapt to the stresses of Code_Aster, by underlining them
adjustments to be made.
That is to say thus following configuration, by taking again conventions of the preceding part in the vicinity
border:
The definition of a local reference mark on the level of the element makes it possible to bring back for us systematically in
such a situation.
X
1
X
3
X
2
Border locally orthogonal with axis X
3
Mesh elements
finished
Fluid field
infinite
Code_Aster
®
Version
5.0
Titrate:
Elements of absorbing border
Date:
02/04/01
Author (S):
G. DEVESA, V. TO MOW
Key:
R4.02.05-A
Page:
9/20
Manual of Reference
R4.02 booklet: Accoustics
HT-62/01/003/A
3.1.1 Finite elements formulation
Pressure
p
check the equation of Helmholtz in all the field
modelized with the finite elements, it
who gives, for any virtual field of pressure
Q
:
- -
+
=
p Q C T pq
p
N Q
.
1
0
2
2
2
represent the border of the field
.
Size to be estimated on
thanks to the paraxial approximation is here
p
N
.
3.1.2 Approximation
paraxial
In the configuration suggested, the term
p
N
corresponds to
p
X
3
.
Let us consider a wave consequently planes harmonic being propagated in the fluid:
(
)
[
]
p
With
I K X
K X
K X
T
=
+
+
-
exp
1 1
2 2
3 3
While replacing in the equation of Helmholtz, one obtains:
(
)
K
C
C
K
K
3
2
2
12
22
1
=
-
+
One obtains the following development then, for high frequencies (
large) or in the vicinity of
the border (
K
1
and
K
2
small):
(
)
K
C
C
K
K
3
2
2
12
22
1 2
=
-
+
Maybe, while multiplying by
to make disappear this quantity with the denominator and after one
transform of Fourier reverses in space and time:
2
3
2
2
2
12
2
22
1
1
2
p
X
T
C
p
T
C
p
X
p
X
= -
+
+
As had presented it Modaressi, this equation utilizes the derivative compared to the time of
surface term. Within the framework of this part, one is interested only at the end of command 0, that is to say, after one
integration in time, which makes disappear the awkward derivative:
p
X
C
p
T
3
1
= -
or more generally:
p
N
C
p
T
= -
1
It is this relation of impedance which we will discretize on the border of the finite elements field.
Code_Aster
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Version
5.0
Titrate:
Elements of absorbing border
Date:
02/04/01
Author (S):
G. DEVESA, V. TO MOW
Key:
R4.02.05-A
Page:
10/20
Manual of Reference
R4.02 booklet: Accoustics
HT-62/01/003/A
Note:
Taking into account the disappearance of the term of command 1 in the development of the square root, the order
minimal of approximation for the paraxial fluids is in fact 1 and not 0. We will preserve
the name of elements of command 0 for coherence with the solid. However, one speaks about elements
fluids of command 2 at the time to consider elements of a strictly positive nature.
3.2
Impedance of the vibroacoustic elements in Code_Aster
Code_Aster has vibroacoustic elements. One recalls in this paragraph the choices of
formulation made at the time of their implementation. One is inspired to present existing it of
reference material of Code_Aster [bib6].
3.2.1 Limits of the formulation out of p
In the framework of the interaction fluid-structure in harmonic, the formulation in pressure only of
acoustic fluid led to nonsymmetrical matrices. Indeed, the total system is expressed, under
variational form, in the following way:
C
U
v
U v
statement N
ijkl
K L I J
S I I
I
I
S
S
.
.
,
,
-
-
=
2
0
for the structure
1
0
2
2
F
I
I
p Q K
p Q
U N Q
F
F
-
-
=
.
.
for the fluid
with
K
C
=
, a number of wave for the fluid,
v
and
Q
two virtual fields in the structure and in
fluid respectively.
After discretization by finite elements, one obtains the following matric system:
K
C
H
M
C
Q
-
-
=
0
0
0
2
2
U
p
C
U
p
F
T
where:
K
and
M
are the matrices of rigidity and mass of the structure
H
and
Q
are the fluid matrices obtained respectively starting from the bilinear forms:
p Q
F
.
and
p Q
F
C
is the matrix of coupling obtained starting from the bilinear form:
p U N
I
I
.
The nonsymmetrical character of this system does not make it possible to use the algorithm of resolution
conventional of Code_Aster. This justifies the introduction of an additional variable into description
fluid.
Code_Aster
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Version
5.0
Titrate:
Elements of absorbing border
Date:
02/04/01
Author (S):
G. DEVESA, V. TO MOW
Key:
R4.02.05-A
Page:
11/20
Manual of Reference
R4.02 booklet: Accoustics
HT-62/01/003/A
3.2.2 Symmetrical formulation out of p and phi
The new introduced size is the potential of displacements
, such as
X
=
. According to [bib6],
one obtains the new variational form of the system coupled fluid-structure:
C
U
v
U v
v N
ijkl
K L I J
S I I
F
I
I
S
S
.
.
,
,
-
-
=
2
2
0
for the structure
(
)
1
1
0
2
2
2
F
F
F
I I
C
pq
C
Q
p
U N
F
F
F
-
+
- +
=
.
for the fluid
With:
p
F
=
2
in the fluid and
a field of potential of virtual displacement
This leads us to the symmetrical matric system:
K
M
M
M
M
M
M
H
0
0
0
0
0
0
0
0
0
0
0
2
2
2
2
F
F
F
fl
F
T
fl
T
F
C
U
p
C
C
U
p
-
=
where:
K
and
M
are the matrices of rigidity and mass of the structure
M
is the matrix of coupling obtained starting from the bilinear form
U N
I I
MR. M
F
fl
,
and
H
are the fluid matrices obtained starting from the bilinear forms:
pq
F
,
p
F
(or
Q
F
) and
.
F
3.2.3 Imposition of an impedance with the formulation out of p and phi
Generally, a relation of impedance at the border of the fluid is expressed as follows:
p
Z v N
=
.
where:
Z
is the imposed impedance
v N
.
is the outgoing normal speed of the fluid particles
One deduces some, according to the law of behavior of the fluid, which connects the pressure to the displacement of
fluid particles for an acoustic fluid
-
=
p
U
T
F
2
2
0
:
F
Z
p
T
p
N
=
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The discretization of such an equation leads to a nonsymmetrical term in a formulation in
p
and
. One prefers to formulate the condition compared to the potential of displacement, that is to say:
+
=
F
Z
T
0
One obtains then like expression for the term of edge associated with the relation with impedance:
F
F
T
N
T
Z
2
2
3
3
2
=
One then notes the appearance (somewhat artificial) of a term in derived third compared to
time. In harmonic, which is the privileged applicability of the vibroacoustic elements in
Code_Aster, that does not pose a problem. One treats a term in
3
without difficulty. For calculation
transient, rather than to introduce an approximation of a derivative third into the diagram of
Newmark implemented in the operators of direct integration in dynamics in Code_Aster
DYNA_LINE_TRAN
[U4.53.02] and
DYNA_NON_LINE
[U4.53.01], one prefers to operate a simple correction
of the second member, which returns in fact to consider the impedance explicitly. Conditions of
stability of the diagram of Newmark are not rigorously any more the same ones, but the experiment has us
shown that it is simple to arrive at convergence starting from the old conditions.
This choice of an explicit correction of the second member will be also justified at the time of
implementation of paraxial elements of command 1, which it makes easier definitely.
3.2.4 Formulation
detailed
One proposes here the precise formulation for an acoustic fluid modelized on a field
with one
anechoic condition on a part
has
border
field. Apart from that, one
break up the border into a free face and a part in contact with a rigid solid. The introduction
stresses external or the presence of an elastic structure is modelized easily by
current methods. The elements of volume and surface are formulated in
p
and
.
The equations in the fluid are:
F
C p
+
=
1
0
2
in volume
éq 3.2.4-1
p
T
F
=
2
2
in volume
éq 3.2.4-2
p
=
0
on the free face
éq 3.2.4-3
N
=
0
on the rigid wall
éq 3.2.4-4
p
N
C
p
T
= -
1
on the part of the border with anechoic condition
éq 3.2.4-5
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One multiplies the equation [éq 3.2.4-1] by a field of virtual potential
and one integrates in
:
(
)
1
0
2
2
2
2
2
C p
T
T
N
F
F
F
+
+
=
.
according to the formula of Green
Maybe, with the boundary conditions on
and the equation [éq 3.2.4-2]:
(
)
1
0
2
2
2
C p
T
p
N
F
F
F
has
+
+
=
.
One can consequently apply the condition of impedance formulated in pressure:
F
F
p
N
C
p
T
has
has
= -
1
Moreover, to arrive to a symmetrical formulation of the terms of volume, one multiplies the equation
[éq 3.2.4-2] by a virtual field of pressure Q and one integrate in
:
pq
C
T
Q
C
F
F
F
2
2
2
2
0
-
=
By summoning the two variational equations, one obtains:
(
)
1
1
1
0
2
2
2
2
F
F
F
F
C
pq
T
C
Q
p
C
p
T
F
F
F
has
+
+
-
-
=
.
Matriciellement:
M
With
M
M
H
F
fl
fl
T
F
p
C
p
C
C
p
0
0
0
1 0
0 0
0
0
2
2
-
+
=
""
""""
where submatrices
MR. M
H
F
fl
,
and
the same bilinear forms discretize as previously.
The submatrix
With
discretize the term
F
p
T
has
. The matrix of damping obtained is not
symmetrical, as one had predicted higher. This is why one rejects this term with the second member.
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3.2.5 Direct temporal integration
In our case, because of nonthe symmetry of the matrix of impedance, one chooses to consider it
anechoic term explicitly as we evoked before. That amounts calculating it
at the moment T and to place it among the stresses at the time of the expression of dynamic balance at the moment
T
T
+
.
One solves:
M
M
M
H
With
F
T
T
T
T
fl
fl
T
F
T
T
T
T
T
T
p
C
C
p
C
p
0
0
0
0
1 0
0 0
2
2
+
=
+
+
+
+
""""
""
éq 3.2.5-1
Instead of:
M
With
M
M
H
F
T
T
T
T
T
T
T
T
fl
fl
T
F
T
T
T
T
p
C
p
C
C
p
0
0
0
1 0
0 0
0
0
2
2
-
+
=
+
+
+
+
+
+
""
""""
Thus, there is not a nonsymmetrical matrix to treat in the system giving
X
at the moment
T
T
+
.
Note:
In a nonlinear calculation, one reactualizes the second member with each internal iteration.
Calculation can thus prove more exact and more stable in this case.
3.3
Use in Code_Aster
The taking into account of anechoic fluid elements and the calculation of their impedance require one
specific modeling on the absorbing borders:
·
in 2D with modeling
“2d_FLUI_ABSO”
on the finite elements of type MEFASEn (n=2
or 3) on the absorbing edges with N nodes.
·
in 3D with modeling
“3d_FLUI_ABSO”
on the finite elements of MEFA_FACEn type
(n=3, 4, 6, 8 or 9) on the absorbing faces with N nodes.
In harmonic analysis with the operator
DYNA_LINE_HARM
[U4.53.11], one as a preliminary is calculated
mechanical impedance by the option
IMPE_MECA
of the operator
CALC_MATR_ELEM
[U4.61.01] and one
inform in
DYNA_LINE_HARM
(key word
MATR_IMPE_PHI
).
In transitory analysis, the taking into account of the correct force due under the terms of impedance is
automatic with modelings of elements absorbents in the operators
DYNA_LINE_TRAN
[U4.53.02] and
DYNA_NON_LINE
[U4.53.01].
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4
Elastic elements absorbents in Code_Aster
This part presents the main part of the general stresses of implementation of elastic elements of
border absorbents with the paraxial approximation of command 0 in Code_Aster. The relation is pointed out
of paraxial impedance of command 0 such as it was established by Modaressi for an elastic range
linear:
()
T U
U
U
=
+
C
T
C
T
p
S
//
U
U
U
T
U
becomes
and
becomes
3
//
/
4.1
Adaptation of the seismic loading to the paraxial elements
One presented in the first part the principle of taking into account of the incidental field thanks to
paraxial elements. It is advisable here to present the methods of modeling of the seismic loading
in Code_Aster to be able to adapt the data to the requirements of the paraxial elements.
The fundamental equation of dynamics associated with an unspecified model 2D or 3D discretized in
finite elements of continuous medium or structure and in the absence of external loading is written in
identify absolute:
MX
CX
KX
""
“
has
has
has
+
+
=
0
One breaks up the movement of the structures into a movement of drive
X
E
and a movement
relative
X
R
.
Appear 4.1-a: Decomposition of the movement of the structures
X
has
X
E
X
R
X
S
X
S
(
has
)
X
S
: imposed displacement of the supports
X
has
: absolute movement
X
E
: movement of drive
X
R
: relative movement
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Thus,
X
X
X
has
R
E
=
+
·
X
has
is the vector of displacements in the absolute reference mark,
·
X
R
is the vector of relative displacements, i.e. the vector of displacements of
structure compared to the deformation which it would have under the static action of displacements
imposed on the level of the supports
X
S
.
X
R
is thus null at the points of anchoring,
·
X
E
is the vector of displacements of drive of the structure produces statically by
the imposed displacement of the supports
X
S
:
X
X
E
S
=
,
·
is the matrix of the static modes. The static modes represent the answer of
structure with a unit displacement imposed on each degree of freedom of connection (others
being locked), in the absence of external forces. Thus,
K
=
0
, i.e.,
K X
E
=
0
.
In the case of the mono-support (all the supports undergo the same imposed movement),
is
a rigid mode of body.
Assumption in Code_Aster:
It is supposed that the damping dissipated by the structure is of viscous type i.e.
force damping is proportional to the relative speed of the structure. Thus,
CX
“
E
=
0
.
The fundamental equation of dynamics in the relative reference mark is written then:
MX
CX
KX
MR. X
""
“
""
R
R
R
S
+
+
= -
The operator
CALC_CHAR_SEISME
[U4.63.01] the term calculates
-
M
, or more exactly
-
M D
,
where
D
is an unit vector such as
()
X
D
S
T
=
.f
with
F
a scalar function of time.
One distinguishes two types of seismic loadings introduced into Code_Aster thanks to the operator
CALC_CHAR_SEISME
:
1) The loading of the type
MONO_APPUI
, for which
is the matrix identity (the modes
statics are modes of rigid body),
2) The loading of the type
MULTI_APPUI
, for which
is unspecified.
According to the method of taking into account of the incidental field with the paraxial elements presented
in the first part, it is necessary for us to know on the border displacement and the stresses due to
incidental field. For the loading of the type
MULTI_APPUI
, only displacement is directly
accessible at any moment. It thus seems difficult to allow the use of such a load pattern
with paraxial elements in the ground. Moreover, if such a loading modelizes displacements
imposed supports, it does not require a modeling of the ground since all the influence is taken in
count to these displacements.
The case
MONO_APPUI
can be perceived differently. It represents an overall acceleration
applied to the model. Consequently, the propagation of wave in the ground can have a role to play in
behavior of the structure, since the movements of the interface ground-structure are not
imposed. Moreover, the paraxial elements are usable with this type of loading because it does not create
no stresses at the border of the mesh (a rigid mode of body does not create deformations).
Consequently, one has all the data necessary to calculation of the impedance absorbing on
border.
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HT-62/01/003/A
Notice 1:
In the case of a seismic stress
MONO_APPUI
, dynamic calculation is done in
relative reference mark. If one amounts on the term discretizing on the paraxial elements (see first
part), one notices that
U
I
corresponds exactly to the displacement of drive
X
E
presented higher. Thus,
U U
-
I
corresponds to the relative displacement calculated during calculation.
Consequently, the relation to be taken into account on the paraxial elements in such
configuration is simply:
()
T U
U
=
With
0
T
Notice 2:
In the case of a calculation of interaction ground-fluid-structure with infinite fluid, pressure to be taken
in account for the calculation of the anechoic impedance in the fluid is well the pressure
absolute, if there is not an incidental field in the fluid (what is often the case).
correction which one could exempt to make for the ground must then be made for
fluid paraxial elements.
4.2
Implementation of the elements in transient and harmonic
4.2.1 Implementation in transient
The mode of implementation of the elastic paraxial elements in transient is very close to that
presented for the fluid elements. The difference comes primarily from the need for breaking up
displacement in a component according to the normal with the element, corresponding to a wave P, and one
component in the plan of the element, corresponding to a wave S. One is then capable
to discretize the relation of impedance introduced into the first part:
()
T U
U
U
=
+
C
T
C
T
p
S
3
/
One does not reconsider the diagram of temporal integration which one already described in the part
the preceding one, knowing that one always considers the relation of impedance explicitly by one
correction of the second member.
4.2.2 Implementation in harmonic
The fluid acoustic elements of Code_Aster propose already the possibility of taking into account
an impedance imposed on the border of the mesh in harmonic. That corresponds to the processing of one
term in
3
in the equations, like city higher. It is trying to introduce the possibility
to impose an impedance absorbing for an elastic problem in harmonic.
For a harmonic calculation of response of an infinite structure, the taking into account of the impedance
absorbing as a correction of the second member is obviously not applicable. However,
relation of impedance to command 0 expresses the surface terms according to the speed of the nodes
element. One can thus build a pseudo-matrix of viscous damping translating
presence of the infinite field.
Decomposition of the relation of impedance according to components' normal or tangential of
displacement on the constrained element us to build the matrix of impedance in a local reference mark on
the element. One defines this local reference mark in the elementary routine as well as the matrix of passage which
the return to the total base allows.
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Note:
In the case of elastic paraxial elements of command 0, to create a matrix of damping
could have allowed us to solve the problem in transient without deteriorating the stability of
diagram of Newmark, contrary to the taking into account clarifies that we retained.
However, we showed the problems that that created for the fluid elements and
we wished to keep the homogeneity of the modes of implementation. Moreover, processing
elements of a nature 1 making the channel explicit obligatory, the whole seems coherent. That
also allows to take into account at the same time an infinite field with elements
paraxial and a damping of modal the damping type for the structure.
4.3
Seismic load pattern per plane wave
In complement of the methods of taking into account of the seismic loading already available and in
reason of the inadequacy of the mode
MULTI_APPUI
with the paraxial elements, it seems interesting
to introduce a principle of loading per plane wave. That corresponds to the loadings
classically met during calculations of interaction ground-structure by the integral equations.
4.3.1 Characterization of a wave planes in transient
In harmonic, a wave planes elastic is characterized by its direction, its pulsation and its type
(wave P for the waves of compression, waves SV or HS for the waves of shearing). In
transient, the data of the pulsation, corresponding to a standing wave in time, must be
replaced by the data of a profile of displacement which one will take into account the propagation with
run from time in the direction of the wave.
More precisely, one will consider a plane wave in the form:
()
(
)
U X
K X
K
,
F
.
T
C T
p
=
-
for a wave P (with
K
unit)
()
(
)
U X
K X
K
,
F
.
T
C T
S
=
-
for a wave S (with always
K
unit)
F
then represent the profile of the wave given according to the direction
K
.
O
K
Function
F
“Main” face of wave
corresponding at the origin
profile
H
H is the distance from the origin to the main face of wave. To initialize calculation, the distance should be given
H
0
who separates the main face of wave from the origin at moment 0.
Code_Aster
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4.3.2 User data for the loading by plane wave
In accordance with the theory exposed in first part, it is necessary for us to calculate the stress at the border
mesh due to the incidental wave and the term of impedance corresponding to incidental displacement,
that is to say:
()
T U
I
and
With
T
I
0
U
To express the stresses, it is necessary for us to have the deformations due to the incidental wave, the law of
behavior of material allowing us to pass from the ones to the others.
On the elements of border, one can express the tensor of the deformations linearized in each
node by the conventional formula:
()
()
()
[
]
X T
X T
X T
T
,
,
,
=
+
1
2
U
U
Finally, to consider the stresses due to the incidental field, we thus should determine them
derived
()
U
I J
K
X
for
J
and
K
traversing the three directions of space. One obtains these quantities with
to leave the definition of the incidental plane wave:
()
(
)
U
K X
I J
K
K
m
J
X
K
C T K
=
-
F
.
with
m
= S or P
With regard to the term of impedance, it is necessary for us
U
I
T
. One also obtains it starting from the wave
plane:
(
)
U
K X
K
I
m
m
T
C
C T
=
-
F
.
always with m = S or P
It is seen whereas the important function for a loading by wave planes with elements
paraxial of command 0 is not the profile of the wave
F
, but its derivative
f'
. It is thus a this function which
the user must enter like data of calculation.
One can consequently recapitulate the parameters to enter for the definition of a loading per plane wave
in transient:
Type of the wave
: P, SV or HS
Direction of the wave
: K
X
, K
y
, K
Z
Outdistance main face of wave with
the origin at the initial moment
: H
0
Derived from the profile of the wave
: f' (X) for X
]
[
- +
,
Code_Aster
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4.4
Use in Code_Aster
The taking into account of elastic elements absorbents and the calculation of their impedance requires one
specific modeling on the absorbing borders:
·
in 2D with modeling
“D_PLAN_ABSO”
on the finite elements of type MEPASEn (n=2
or 3) on the absorbing edges with N nodes.
·
in 3D with modeling
“3d_ABSO”
on the finite elements of type MEAB_FACEn (n=3, 4,
6, 8 or 9) on the absorbing faces with N nodes.
In harmonic analysis with the operator
DYNA_LINE_HARM
[U4.53.11], one as a preliminary is calculated
mechanical cushioning by the option
AMOR_MECA
of the operator
CALC_MATR_ELEM
[U4.61.01] and one
informs in
DYNA_LINE_HARM
(key word
MATR_AMOR
).
In transitory analysis, the taking into account of the correct force due under the terms of impedance is
automatic with modelings of elements absorbents in the operators
DYNA_LINE_TRAN
[U4.53.02] and
DYNA_NON_LINE
[U4.53.01].
5 Bibliography
[1]
J. Mr. CREPEL, “three-dimensional Modeling of the interaction ground-structure by elements
stop and infinite.“, Thesis doctor-engineer, Central School of Paris (1983))
[2]
Mr. COHEN, PC JENNINGS, “Silent boundary methods for acoustic and elastic wave
equations.“, S.S.A. (1977.
[3]
H. MODARESSI, “numerical Modeling of the wave propagation in the mediums
porous rubber bands.“, Thesis doctor-engineer, Central School of Paris (1987.
[4]
D. CLOUTEAU, A.S. BONNET-BEN DHIA, “Propagation of waves in the solids.”, Course
central School of Paris
[5]
B. ENGQUIST, A. MAJDA, “Absorbing boundary conditions for the numerical simulation off
waves.“, Mathematics off Computation (1977)
[6]
Fe. WAECKEL, “vibroacoustic Elements.”, Reference document. Code_Aster
[R4.02.02]