Code_Aster ®
Version
3
Titrate:
Transitory response by traditional dynamic under-structuring
Date:
17/10/95
Author (S):
C. VARE
Key:
R4.06.04-A
Page:
1/16
Organization (S): EDF/EP/AMV
Handbook of Référence
R4.06 booklet: Under-structuring
Document: R4.06.04
Transitory response by under-structuring
traditional dynamics
Summary:
This document presents the theoretical bases of the two methods of calculation of answer transitory by
dynamic under-structuring implemented in Code_Aster.
The first method consists in carrying out a transitory calculation by under-structuring for which equations of
problem are projected on the bases associated with each substructure. The second method consists with
to determine the clean modes of the complete structure by under-structuring and to project on this basis them
equations of the transitory problem.
In both cases, only the case of an excitation per force imposed on the substructures is currently
available.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-61/95/072/A

Code_Aster ®
Version
3
Titrate:
Transitory response by traditional dynamic under-structuring
Date:
17/10/95
Author (S):
C. VARE
Key:
R4.06.04-A
Page:
2/16
Contents
1 Introduction ............................................................................................................................................ 3
2 transitory Calculation by projection on the basis of substructure ...................................................... 5
2.1 Dynamic equations checked by the substructures separately ............................................ 5
2.2 Assembly of the substructures .................................................................................................... 6
2.3 Dynamic equations checked by the total structure ................................................................ 6
2.4 Double dualisation of the boundary conditions ................................................................................. 7
2.5 Processing of the matrix of damping ...................................................................................... 8
2.6 Processing of the initial conditions .................................................................................................. 8
3 transitory Calculation on a modal basis calculated by under-structuring ............................................... 9
3.1 Calculation of the clean modes of the structure supplements by under-structuring ................................... 9
3.2 Dynamic equation checked by the total structure ..................................................................... 9
4 comparative Study of the two developed methods ......................................................................... 11
5 Implementation in Code_Aster .................................................................................................... 12
5.1 Study of the substructures separately ........................................................................................ 12
5.2 Assembly of the model generalized ............................................................................................... 12
5.3 Calculation of the modal base of the complete structure and projection .................................................. 13
5.4 Resolution and restitution about physical base ................................................................................... 13
6 ........................................................................................................................................... Conclusion 14
7 Bibliography ........................................................................................................................................ 15
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Code_Aster ®
Version
3
Titrate:
Transitory response by traditional dynamic under-structuring
Date:
17/10/95
Author (S):
C. VARE
Key:
R4.06.04-A
Page:
3/16
1 Introduction
The components of nuclear thermal power station are often of important size, complex geometry and
sometimes composed of an assembly of several elements. To model the dynamics of these
structures, the tools for vibratory analysis traditional then are badly adapted and it is necessary to have
resort to methods of reduction the such techniques of modal synthesis which were
developed in Code_Aster.
The methods of modal synthesis associate techniques resulting from the under-structuring and
modal recombination [bib4]. Thus, the field of study is cut out in several substructures and it
vibratory behavior of the complete structure is given according to the characteristics
vibratory of each one of them. In addition, each substructure, is represented by a base
of particular projection, made up of clean modes and static deformations of interface, on
which are projected the equations of the problem ([R4.06.02], [R4.06.03] and [bib4]).
From the purely data-processing point of view, these methods have two important advantages. Of one
leaves, they allow to limit the size memory necessary to the storage of the sizes used at the time
calculation and in addition, the calculating times are generally very reduced. From the point of view of
the organization of a draft study, the techniques of under-structuring are particularly
interesting because they make it possible to validate, stage by stage, the models of the substructures.
difficulties related to the modeling of a complex structure can thus be approached separately, it
who makes easier from there the resolutions.
Several computational tools by dynamic under-structuring are currently available in
Code_Aster. They make it possible to carry out modal calculations [R4.06.02] and calculations of answer
harmonic [R4.06.03]. Work completed to establish the harmonic calculation of answer by
dynamic under-structuring in Code_Aster, resulted in defining the processing of the vector of
forces external and of the matrix of viscous damping.
The object of this reference material is to present the theoretical bases of the two methods
of transitory calculation of response per dynamic under-structuring available in Code_Aster.
first consists in carrying out a transitory calculation by under-structuring for which equations of
problem are projected on the bases associated with each substructure. The difficulty lies in
double dualisation of the boundary conditions which leads to a matrix of singular mass. For
to use the diagram of integration explicit (which requires the inversion of the matrix of mass), it is necessary
thus to modify the processing of the interfaces in the operator of calculation of the transitory answer.
second method consists in determining the clean modes of the structure supplements by
under-structuring and to project on this basis the equations of the transitory problem. The stage of
restitution on the basis of physical final generalized result must thus take account of this double
projection.
The operator of transitory calculation of answer which receives the under-structuring is the operator
DYNA_TRAN_MODAL [U4.54.03]. Being based on methods of modal recombination, it was
conceived to solve transitory problems in generalized co-ordinates and it is very effective
for the problems of big size of which it makes it possible to reduce the number by degrees of freedom. Of other
leaves, it supports the taking into account of localized non-linearities (with the nodes) which one wishes
to generalize with the case of the under-structuring.
In this report/ratio, we present the two methods of calculation transitory per under-structuring
available in Code_Aster, like their implementation.
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Code_Aster ®
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3
Titrate:
Transitory response by traditional dynamic under-structuring
Date:
17/10/95
Author (S):
C. VARE
Key:
R4.06.04-A
Page:
4/16
General notations:
NR
:
A number of substructures
S
M
:
Stamp of mass resulting from modeling finite elements
K
:
Stamp rigidity resulting from modeling finite elements
C
:
Stamp damping exit of modeling finite elements
Q
:
Vector of the degrees of freedom resulting from modeling finite elements
F
:
Vector of the forces external with the system
ext.
F
:
Vector of the bonding strengths applied to the system
L

:
Stamp vectors of the base of the substructures

:
Vector of the generalized degrees of freedom
B
:
Stamp extraction of the degrees of freedom of interface
L
:
Stamp connection
Id
:
Stamp identity

:
Multipliers of Lagrange
Note:
The exhibitor K characterizes the sizes relating to the substructure S K and the sizes
generalized are surmounted by a bar: for example M K is the matrix of generalized mass
substructure S K.
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Code_Aster ®
Version
3
Titrate:
Transitory response by traditional dynamic under-structuring
Date:
17/10/95
Author (S):
C. VARE
Key:
R4.06.04-A
Page:
5/16
2
Transitory calculation by projection on the basis of under
structures
2.1
Dynamic equations checked by the substructures separately
That is to say a structure S made up of NS noted substructures S K. We suppose that each
substructure is modelled in finite elements. The vibratory behavior of the substructures results
forces external which are applied to him and of the bonding strengths which on them the others exert
substructures. Thus, for S K, we have:
M K qk + Ck qk + K K qk = F K + F K
!
!
ext.
L
éq 2.1-1
where:
Mk
is the matrix of mass resulting from modeling finite elements of S K,
Ck
is the matrix of damping resulting from modeling finite elements of S K,
K K
is the matrix of rigidity resulting from modeling finite elements of S K,
F kext
is the vector of the external forces applied to S K,
F kL
is the vector of the bonding strengths applied to S K,
qk qk and qk
!
!
are the vectors displacement, speed and acceleration resulting from modeling
finite elements.
The field of unknown displacement, resulting from modeling finite elements, is required on a space
adapted, of reduced size (transformation of Ritz) according to the formula:
qk
K K
=
éq 2.1-2
where:
K
is the vector of the generalized co-ordinates of S K,
K
is the matrix containing the modal vectors associated the dynamic clean modes and
with the static deformations of interface of S K.
The transformation of Ritz [éq 2.1-2], applied to the transitory dynamic equation of the substructure
[éq 2.1-1], allows to write:
M K K
+ Ck K
+ K K K
= F K + F K
!
!
ext.
L
éq 2.1-3
where:
Mk
K T Mk K
=
is the matrix of mass generalized of Sk,
Ck
K TCk K
=

is the matrix of generalized damping of S K,
K K
K T Kk K
=

is the matrix of generalized rigidity of S K,
F K
K T
= F K
is the vector of the generalized external forces applied to S K,
ext.
ext.
F K
K T
= F K
is the vector of the generalized bonding strengths applied to S K,
L
L
K K and K
!
!
are the vectors generalized displacement, speed and acceleration.
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Code_Aster ®
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Titrate:
Transitory response by traditional dynamic under-structuring
Date:
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Author (S):
C. VARE
Key:
R4.06.04-A
Page:
6/16
The problem defined by the equation [éq 2.1-3] is symmetrical. In addition, its dimension is given
by the number of modes taken into account (dynamic modes and static deformations). One is
thus brought to solve a traditional transitory problem but of reduced size.
2.2
Assembly of the substructures
After having studied each substructure separately, one proposes to establish the equations which
govern their assembly. Let us consider two substructures S K and SSL connected between them to the level
interface S K
SSL
. It is admitted that their respective grids are compatible [R4.06.02]. Thus, with
level of the interface, the nodes coincide and the meshs in opposite are identical. Consequently, the law
of action-reaction and the continuity of displacements to the interfaces, which represent the assembly of S K
and SSL, are written:
F K
= - F L
K
L
L
L
Q K
L = Q K
L
S K S L
S K SSL


S S
S S
where:
F kL
is the vector of the bonding strengths applied to the substructure S K, the level of
S K S L

the interface S K
SSL
.
qk
K
L
resulting from modeling
S K SSL

is the vector of the degrees of freedom of the interface S
S
finite elements of the substructure S K.
Let us introduce the matrices of extraction of the degrees of freedom of the interface S K
SSL
:
qk K
L = B K K
L qk
S S
S S
ql K
L = Bl K
L ql
S S
S S
By using the transformation of Ritz and the formulation applied above to the two substructures,
one obtains:
Bk
K K
K
L
= Bl
L L

S
S
S K SSL


That is to say:
Lk
K
K
L
= L
L

S
S
S K SSL


where: Lk
K
L
of Sk.
S K SSL
is the matrix of connection associated with the interface S
S
2.3
Dynamic equations checked by the total structure
The matric writing of the dynamic equation checked by the total structure, is simply written to leave
dynamic equations checked by each substructure:
M1
1
! C1
1
! K1

1 1
F
1
F

ext.
L


.


… ……










K
K
K K K
Mk
! +
Ck
! +
K K
= F
+ F






ext. L










… ……









MR. NR
NR

NR
NR
NR
S
S



!
C NR

!
K NR
S
S
S

NS F S
S
ext.
fL
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Code_Aster ®
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Titrate:
Transitory response by traditional dynamic under-structuring
Date:
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Author (S):
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Key:
R4.06.04-A
Page:
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Which, it is necessary to add the equations of connection:
K, L
K
K
L
= L
L
K
L
K
L
L K
L
F L
= - F
S S
S S
L
S K SSL
S K S L
This system can be written in the condensed form:
M! + C! + K = fext + fL
éq 2.3-1
L = 0
éq 2.3-2
K, L
K
L
F L
= - fL
éq 2.3-3
S K S L
S K SSL
2.4
Double dualisation of the boundary conditions
The condensed problem, given above, arises in the form of a transitory system to which is
associated a linear equation of constraint (in force and displacement). In Code_Aster, this type
of problem is traditional and it is solved by double dualisation of the boundary conditions [R3.03.01],
i.e. by the introduction of auxiliary variables still called multipliers of Lagrange for
dualiser boundary conditions. After introduction of the multipliers of Lagrange, the system
matric puts itself in the form:
0

0
0!
0

0 0! - Id L
Id 0
1
1
1


T
T


0 M 0! + 0 C 0! + L
K
L
= F






ext.
éq 2.4-1
0

0
0

!

0

0 0

!
Id L



- Id
0

2
2
2

where: 1 and 2 is the multipliers of Lagrange.
It is noted, the introduction of the multipliers of Lagrange makes singular the matrix of mass. As of
at the time, the use of the diagram of integration explicit developed in operator DYNA_TRAN_MODAL
[U4.54.03] of Code_Aster is impossible because they require the inversion of the matrix of mass.
To make the matrix nonsingular, it is enough to dualiser with the same multipliers of Lagrange,
the condition on the derivative second of the equations of connection.
Thus the condition of continuity of displacements [éq 2.3-2] is modified by the equivalent system
[éq 2.4-2]:
L (+
! ) = 0
T L = 0
éq 2.4-2
L °=

0 and L! °=

0
where
° and!° is initial generalized displacement and speed.
The matric system which results from this is form [bib7]:
- Id L
Id! 0 0

0!
1

1
- Id L
Id 0

1
T
T

T
T


L
M
L! + 0 C
0! + L
K
L = fext éq 2.4-3





Id L - Id!2 0 0 0!2 Id L - Id2 0
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Transitory response by traditional dynamic under-structuring
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It is noted that the matrix of mass to the same form as the matrix of stiffness. It is thus
invertible. This system is thus perfectly equivalent to the equation [éq 2.4-1] (it checks at any moment
the conditions of connection) and it can be treated, in this form, by operator DYNA_TRAN_MODAL.
2.5
Processing of the matrix of damping
It is noted that the condition of continuity of displacements, formulated in the equation [éq 2.4-2],
translated by an equation of the second command not deadened. At the time of the resolution by step of time of one
transitory problem, any numerical error is likely car-to discuss, thus decreasing the stability of
the algorithm. To optimize the damping of the numerical error, it is enough to dualiser the condition on
derived first from the equations of connection with the same multipliers of Lagrange multiplied by 2
(so as to make this damping critical):
L (+ 2! +!) = 0
T L = 0
éq 2.5-1
L °=

0 and L! °=

0
The matric system which results from this is form [bib7]:
- Id L
Id!
1
-.
2 Id
.
2 L
.
2 Id!
1
- Id L
Id 0

1
T
T

T
T
T
T


L
M
L! + .2L
C
.
2 L! + L
K
L = fext





Id L - Id!2 .2Id .2L - .2Id!2 Id L - Id2 0
It is thus noted that the processing of the numerical error on the equations of connection results in modifying
the matrix of damping of the transitory problem. This modification is completely comparable with that
who is carried out on the matrix of mass.
On the other hand, we did not wish to generalize this treatment the case of the resolution of the problems
not deadened transients. That would have led us to create a matrix of temporary damping. One
could have feared to increase the calculating times, without real benefit. Moreover, it is completely
possible with the user to define a matrix of damping whose coefficients are null.
modification of this one being automatic, the transitory system not deadened will be actually solved,
while optimizing the processing of any numerical error intervening on the equations of connection.
2.6
Processing of the initial conditions
Let us consider a substructure S K characterized by its base of projection K made up of modes
normal and of static deformations. It is supposed that initially the substructure S K is subjected to one
field of displacement or speed (that does not modify of anything the demonstration) noted: qko.
transformation of Ritz enables us to write:
qk
K K
O = O
where: KB is the vector of displacements (or speeds) generalized (E) S of S K to be determined.
The vector of displacements (or speeds) generalized (E) S initial (ales) is given as follows:
qk
K
K
K T
=
=> qk
K T
K
K
O
O
O = (
) .o
éq 2.6-1
K
K T
K
K T
=> =
-
() 1 qk
O
O
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Code_Aster ®
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Titrate:
Transitory response by traditional dynamic under-structuring
Date:
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Author (S):
C. VARE
Key:
R4.06.04-A
Page:
9/16
3
Transitory calculation on a modal basis calculated by under
structuring
3.1 Calculation of the clean modes of the structure supplements by
under-structuring
The second developed method consists in solving the transitory problem on the basis of modal
complete structure calculated by under-structuring.
Each substructure S K is represented by a base of projection, composed of clean modes
dynamic and of static deformations, which we noted: K. The base of projection of
structure supplements which results from it is noted: .
The modal base of the complete structure is calculated by under-structuring. Each mode obtained is
thus linear combination of the vectors of the bases of projection of the substructures:
NS

K
K
p = =

éq 3.1-1
K =1
where:

is the matrix of the clean modes of the complete structure,
p

is the matrix of the generalized modal co-ordinates of the structure.
Projection of the matrices and the vectors constitutive of the transitory problem, on the basis of mode
clean of the complete structure calculated by modal synthesis allows to determine:
· the matrix of generalized mass:
M = TM
· the matrix of generalized rigidity:
K = T C
· possibly the matrix of generalized damping:
C = T C
· the vector of the generalized external forces:
F
T
= F
ext.
ext.
Because of orthogonality of the clean modes of the structure calculated by modal synthesis, report/ratio
with the matrices M and K, the matrices of generalized mass and rigidity obtained above are
diagonals:
M = TM = (
T
) M = TM
p
p
éq 3.1-2
K = TK = (
T
) K = TK
p
p
3.2
Dynamic equation checked by the total structure
The complete structure is subjected to the external forces which are applied to him. Thus, we can
to write:
Mq
! + Cq! + Kq = fext
éq 3.2-1
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Titrate:
Transitory response by traditional dynamic under-structuring
Date:
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Author (S):
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Key:
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Page:
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The field of unknown displacement, resulting from modeling finite elements, is replaced by its
projection on the basis of clean mode of the structure, according to the formula:
Q = p
éq 3.2-2
where: is the vector of the generalized co-ordinates of the structure.
The transformation of Ritz [3.2-2], applied to the transitory dynamic equation of the structure [3.2-1],
allows to write:
M! + C! + K = fext
éq 3.2-3
The stage of restitution on physical basis requires to take account of the double projection (on the basis
modal of the complete structure, then on the basis of projection of the substructures - cf éq 3.2-4).
Q = p =



éq 3.2-4
The problem defined by the equation [éq 3.2-3] is of completely traditional form. One is brought to solve
a symmetrical transitory problem whose dimension is determined by the number of calculated modes
by under-structuring and whose matrices of mass and rigidity are diagonal.
Let us note finally that the processing of the initial conditions is identical to the case of transitory calculation by
projection on the basis of substructure (cf § 2.6).
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Transitory response by traditional dynamic under-structuring
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Author (S):
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Key:
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4
Comparative study of the two developed methods
Theoretical bases, associated the two methodologies implemented in Code_Aster for
to carry out a transitory calculation of response by using the techniques of under-structuring, were
presented in the preceding chapters. We specify, here, their essential characteristics.
The first methodology consists in making a calculation of transitory response per under-structuring.
The equation checked by the complete structure is then projected on the basis of substructure.
precision of this method thus is directly determined by the extent of these bases. These
last can be enriched without leading to prohibitory calculating times because the substructures
are, in theory, relatively reduced sizes. At all events, it is difficult to estimate the effect of
modal truncation with the only knowledge of the modes of the bases of the substructures. In addition, them
bases of projection of the substructures are made up of modes which all are not orthogonal
between them (clean modes and static deformations). Matrices of generalized mass and rigidity
constitutive of the final transitory problem are thus not-diagonals. All in all, their width of
tape can be given starting from the number of static deformations of the bases of projection of
substructures [R4.06.02]. Duration of integration in the operator of transitory calculation
DYNA_TRAN_MODAL will be thus all the more long as there will be degrees of freedom of interface. Of other
leaves, the step of acceptable time of integration maximum by the diagram of integration explicit is
determined starting from the maximum frequency of the base of projection. In the case of a calculation
transient by under-structuring, this frequency results, in theory, of the static modes of which them
diagonal terms are high in the matrix of rigidity generalized and weak in the matrix of
mass generalized. Consequently, the step of time of integration cannot be a priori given.
The experiment shows that it is very weak, taking into consideration Eigen frequency of the bases of under
structures and that the use of the diagram of integration to step of adaptive time of DYNA_TRAN_MODAL
is very advantageous.
The second methodology consists in making a transitory calculation on the basis of modal structure
supplements obtained by under-structuring. It is known that the stage consisting in calculating the clean modes
structure can be expensive in term of calculating time. This is all the more true when one
consider nonlinear forces because the base of projection must then be sufficiently extended for
to represent the dynamics of the system well. In addition, the modal base on which is calculated
transitory answer is of size lower than that determined by the clean vectors of under
structures (clean modes and static deformations). It thus does not constitute a generating system.
Double projection thus amounts introducing a cut-off frequency. One must thus expect it
that this method is less precise than the preceding one. However, the calculation of the clean modes
allows to estimate the effect of modal truncation. In addition, it can make it possible to validate the models of
under-stuctures if one has experimental results. Finally, essential interest of this
method is that the matrices of mass and rigidity used in transitory calculation are
diagonals. Numerical integration is thus very fast.
To conclude, it is noted that transitory calculation by under-structuring is identified with the methods
direct of transitory calculation. One does not have access to modal information and the matrices are not
diagonals. In this case, one can say that the vectors of the bases of projection of the substructures
play the same part that the functions of form of the finite elements. Transitory calculation on basis
modal calculated by under-structuring is identified, as for him, with the methods of modal recombination
traditional.
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Code_Aster ®
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Titrate:
Transitory response by traditional dynamic under-structuring
Date:
17/10/95
Author (S):
C. VARE
Key:
R4.06.04-A
Page:
12/16
5
Implementation in Code_Aster
5.1
Study of the substructures separately
If one wishes to introduce a damping of Rayleigh, the parameters E and E of this
damping are defined, by operator DEFI_MATERIAU [U4.23.01].
The processing of the substructures are identical to the case of modal calculation [R4.06.02] and harmonic
[R4.06.03]. The dynamic clean modes are calculated with the operators: MODE_ITER_SIMULT
[U4.52.02] or MODE_ITER_INV [U4.52.01]. The conditions with the interfaces of connection are applied
with operator AFFE_CHAR_MECA [U4.25.01].
Operator DEFI_INTERF_DYNA [U4.55.03] allows to define the interfaces of connection of
substructure. Operator DEFI_BASE_MODALE [U4.55.04] allows to calculate the base of projection
substructure supplements (recopy of the clean modes and calculation of the static deformations).
Operator MACR_ELEM_DYNA [U4.55.05] calculates the generalized matrices of stiffness, mass and
possibly of damping of the substructure, as well as the matrices of connection.
The damping of Rayleigh is taken into account by supplementing operand MATR_AMOR.
Damping proportional is introduced by operand AMOR_REDUIT.
The transitory loading is defined, on the level of the substructure, by the operators
AFFE_CHAR_MECA [U4.25.01] (application of the force on the grid), CALC_VECT_ELEM [U4.41.02]
(calculation of the associated elementary vectors) and ASSE_VECTEUR [U4.42.03] (assembly of the vector of
loading on the grid of the substructure).
The operator AFFE_CHAM_NO [U4.26.01] who allows to affect a field on the nodes of a model
allows to describe the field of initial displacement or/and the initial field speed of the substructure.
5.2
Assembly of the generalized model
As in the case of modal calculation [R4.06.02] and harmonic [R4.06.03], the model of the structure
supplements is defined by operator DEFI_MODELE_GENE [U4.55.06]. Its classification is carried out by
operator NUME_DDL_GENE [U4.55.07]. Matrices of mass, stiffness and possibly
of damping generalized of the structure supplements are assembled according to this
classification with operator ASSE_MATR_GENE [U4.55.08].
The loadings are projected on the basis of substructure to which they are applied, then
assembled starting from classification resulting from NUME_DDL_GENE [U4.55.07] by the operator
ASSE_VECT_GENE [U4.55.09].
Initial generalized displacements and initial speeds generalized for each substructure,
are calculated by operator ASSE_VECT_GENE [U4.55.09]. This operator also realizes
assembly of these vectors according to classification resulting from NUME_DDL_GENE [U4.55.07].
In the case of a transitory calculation projected on the “bases” of the substructures, the matrices and vectors
assembled generalized obtained with resulting from this stage are directly used for calculation
transient. In the case of a calculation on the basis of complete modal structure calculated by
under-structuring, it is necessary to carry out specific operations which are presented at the § 5.3.
Handbook of Référence
R4.06 booklet: Under-structuring
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Code_Aster ®
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Titrate:
Transitory response by traditional dynamic under-structuring
Date:
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Author (S):
C. VARE
Key:
R4.06.04-A
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5.3
Calculation of the modal base of the complete structure and projection
This chapter is specific to transitory calculation on modal basis calculated by under-structuring.
The modal base of the complete structure is calculated with the traditional operators of Code_Aster:
MODE_ITER_SIMULT [U4.52.02] or MODE_ITER_INV [U4.52.01]. One defines a classification of
final problem generalized with operator NUME_DDL_GENE [U4.55.07]. Matrices of mass, of
stiffness and possibly of damping generalized is projected on the basis of clean mode
structure with operator PROJ_MATR_BASE [U4.55.01]. Generalized vectors corresponding
with the external loadings are projected on the basis of clean mode of the structure with
operator PROJ_VECT_BASE [U4.55.02].
5.4
Resolution and restitution about physical base
The calculation of the transitory response of the complete structure is carried out by the operator
DYNA_TRAN_MODAL [U4.54.03].
The restitution of the results on physical basis utilizes operator REST_BASE_PHYS [U4.64.01];
it is identical to the case of modal calculation [R4.06.02] and harmonic [R4.06.03]. One can use
operator DEFI_SQUELETTE [U4.75.01] to create a grid “skeleton”. Coarser than it
grid of calculation, it makes it possible to reduce the durations of the graphic processing.
Handbook of Référence
R4.06 booklet: Under-structuring
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Code_Aster ®
Version
3
Titrate:
Transitory response by traditional dynamic under-structuring
Date:
17/10/95
Author (S):
C. VARE
Key:
R4.06.04-A
Page:
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6 Conclusion
We presented, in this report/ratio, work completed to introduce, in Code_Aster, it
calculation of transitory response linear per dynamic under-structuring. The methods which were
chosen consist, for the first of them, to project the transitory equations on the “bases”
of each substructure, made up of dynamic clean modes and static deformations and
for the second, to calculate the clean modes of the structure supplements by under-structuring and with y
to project the transitory equations.
We begin with a talk from the theoretical bases on which are based the first method
of transitory under-structuring to lead to the matric formulation of the final problem. In
private individual, an original processing of the equation of continuity of displacements to return the matrix
of invertible mass and to ensure an optimal stability of the algorithm of integration, led us
to modify the shape of the matrices of mass and damping of the transitory problem.
For the second method, the essential difficulty consists in restoring the results obtained in
co-ordinates generalized on the physical basis. Indeed, it is necessary to take account of the double projection:
on the basis of modal structure supplements on the one hand, and on the basis of substructure of other
leaves.
The developments carried out resulted in modifications of the operators
DYNA_TRAN_MODAL [U4.54.03] and REST_BASE_PHYS [U4.64.01]. Their syntax was modified very little,
so that their use is identical during a calculation by under-structuring and of a calculation
direct by modal recombination.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-61/95/072/A

Code_Aster ®
Version
3
Titrate:
Transitory response by traditional dynamic under-structuring
Date:
17/10/95
Author (S):
C. VARE
Key:
R4.06.04-A
Page:
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7 Bibliography
[1]
C. VARE: “Code of Mécanique Aster - Manuel of reference: Methods of analysis” Clé:
R4.06.02 “Modal Calculation by traditional and cyclic dynamic under-structuring”
[2]
C. VARE: “Code of Mécanique Aster - Manuel of reference: Methods of analysis” Clé:
R4.06.03 “Harmonic Response by traditional dynamic under-structuring”
[3]
P. RICHARD: “Methods of cyclic under-structuring in finite elements” Rapport EDF
HP-61/91.156
[4]
P. RICHARD: “Methods of under-structuring in Code_Aster” - Rapport EDF
HP-61/92.149
[5]
R. ROY, J. CRAIG & Mr. C. BAMPTON: “Coupling off Substructures for Dynamic Analysis” -
AIAA Journal, (July 1968), Vol. 6, N° 7, p. 1313-1319.
[6]
R.H. Mac Neal: “A hybrid method off component mode synthesis” Computers and Structures,
(1971), vol. 1, p. 581-601.
[7]
C. VARE: “Schedule of conditions of the implementation of linear transitory calculation by
dynamic under-structuring in Code_Aster " - Rapport D.E.R. HP-61/94.135/B
[8]
C. VARE: “User's documentations and Validation of the operators of transitory calculation by
under-structuring " - Rapport D.E.R. HP-61/94.208/A
[9]
J. PELLET: “Code of Mécanique Aster - Manuel of reference: Finite elements in Aster
- Key: R3.03.01 “Dualisation of the boundary conditions”
Handbook of Référence
R4.06 booklet: Under-structuring
HP-61/95/072/A

Code_Aster ®
Version
3
Titrate:
Transitory response by traditional dynamic under-structuring
Date:
17/10/95
Author (S):
C. VARE
Key:
R4.06.04-A
Page:
16/16
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R4.06 booklet: Under-structuring
HP-61/95/072/A

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