Code_Aster ®
Version
7.4
Titrate:
Calculation of sensitivity in dynamics


Date:
02/05/05
Author (S):
H. ANDRIAMBOLOLONA, Key S. CAMBIER
:
R4.03.04-B Page
: 1/26

Organization (S): EDF-R & D/AMA
Handbook of Référence
R4.03 booklet: Analyze sensitivity
Document: R4.03.04
Calculation of sensitivity in dynamics

Summary

Obtaining the sensitivities is important in dynamics where small disturbances of the parameters can
to have a great influence on the sizes of interest. The objective of the document is to clarify the calculation of
sensitivities (i.e derived) of the principal sizes: modes and Eigen frequencies, dynamic responses in
frequential and temporal, into linear and nonlinear). For each size, the “derived problem”,
i.e. the problem whose derivative is solution, is written (in variational and/or matric form).
numerical algorithms of calculation of these derivative are then detailed and discussed. All establishments
corresponding data processing in Code_Aster were not carried out. A table thus gives the current state
developments.
Derivation compared to the parameters of the matrices of mass, stiffness, damping, the loading
and also the particular case of derivation compared to a variation of field is presented.
calculations by the finite element method, which is specific to the derivative, are clarified on the level of
elementary calculations.
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Calculation of sensitivity in dynamics


Date:
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Author (S):
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:
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Contents

1 Introduction ............................................................................................................................................ 3
2 Derived from dynamic responses ....................................................................................................... 4
2.1 Standard problem .......................................................................................................................... 4
2.2 Derived from a harmonic dynamic response ........................................................................... 5
2.2.1 Direct method ..................................................................................................................... 5
2.2.2 Simple modal method ......................................................................................................... 5
2.2.3 Modal method doubles ......................................................................................................... 6
2.2.4 Establishment by the finite element method .................................................................... 7
2.3 Derived from a temporal dynamic response into linear .............................................................. 8
2.3.1 Direct method ..................................................................................................................... 8
2.3.2 Modal method .................................................................................................................... 9
2.3.3 Establishment by the finite element method .................................................................. 10
2.4 Derived from a temporal dynamic response into nonlinear ..................................................... 10
3 Derived from the frequencies and clean modes from vibration .................................................................... 11
3.1 Standard problem ........................................................................................................................ 11
3.2 Derived from the Eigen frequencies ................................................................................................. 12
3.2.1 The case of the eigenvalues simple ..................................................................................... 12
3.2.2 The case of the eigenvalues multiple ................................................................................... 12
3.2.3 Establishment by the finite element method .................................................................. 15
3.3 Derived from the clean modes of vibration .................................................................................... 16
3.3.1 The case of the eigenvalues simple ..................................................................................... 16
3.3.2 The case of the eigenvalues multiple ................................................................................... 17
3.3.3 Establishment by the finite element method .................................................................. 18
4 Derived compared to a variation from field ............................................................................... 18
4.1 Variational formulation of the standard problem ......................................................................... 18
4.2 Description of the variation of field and form of derivation .............................................. 19
4.3 Derivation of the variational formulation .................................................................................... 20
4.3.1 Derivation of the elastic operator of stiffness ..................................................................... 20
4.3.2 Derivation of the geometrical operator of stiffness ............................................................... 21
4.3.3 Derivation of the operator of mass ..................................................................................... 22
4.3.4 Derivation of the variational formulation of the problem of modal analysis ......................... 23
4.3.5 Obtaining derived from the Eigen frequencies and the clean modes ............................. 23
4.3.6 Establishment by the finite element method .................................................................. 23
5 effective Establishment in Code_Aster .............................................................................................. 24
6 Bibliography ........................................................................................................................................ 25
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Calculation of sensitivity in dynamics


Date:
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Author (S):
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:
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1 Introduction

In a mechanical context of calculation on a model of structure, one understands by “sensitivities” them
derived from a size result of the model compared to parameters of model input of the aforesaid. In
dynamics, of small disturbances of the parameters can have a great influence on
dynamic and vibratory sizes [bib1]. The calculation of the sensitivities proves thus paramount, because it
allows to carry out analyzes of influences of the parameters, analyzes probabilistic, of
identification of parameters (or retiming), optimization…

This documentation develops obtaining the sensitivities of the dynamic sizes compared to
mechanical parameters, in the boundary conditions, with the loading, and a variation of field.
The dynamic sizes treated are the dynamic responses in displacement, the frequencies
clean and clean modes. Sensitivities of the other dynamic sizes (forced,
efforts….) which is in general calculated by postprocessing can result some a posteriori by
ad hoc calculations.

The general method of calculation retained is a method of exact direct differentiation (by
opposition to the methods of the associated state, the methods of finished differences or other methods
semi-analytical (the calculation of derived by an semi-analytical method consists in calculating them
derived from the elementary matrices by finished differences and then to calculate them analytically
derived from the sizes results. This method is easier to establish than the analytical method
complete and was retained in Nastran or Abaqus for example (cf [bib2]))).

This method has as principal advantage of being exact. One avoids the problems thus of
convergence of the algorithms of optimization based on the gradient or the problems of precise details of
finished differences (when the results result from an iterative algorithm, the finished differences can
to be particularly vague). It is also more general than the method of the assistant state where
a functional calculus to be derived must be specified before calculation (it thus makes it possible for example to obtain
directly fields derived).

This method can be very economic in calculating times if the algorithms of resolution of
derived problems - i.e. the problems whose derivative are solutions - are well chosen and if
the programming is neat. In the most favorable cases, the calculating time of a derivative
represent that a very small percentage of the calculating time of the standard size.

Its principal disadvantage is the extent of the data-processing developments necessary to sound
establishment (more especially as the architecture of Code_Aster does not make it possible to use the differentiation
automatic).

The document attempts to present the various “derived problems”. They are written in form
variational and/or matric according to the need. These derived problems often have the very maid
properties. For example, in linear dynamics, the first members of the systems to be solved are
preserved. In nonlinear dynamics, the derived problem is a linear problem.

The methods of resolution which meet a priori more the needs for the users of Code_Aster
are presented. In order to minimize the calculating time, the calculation algorithm of derived must exploit
as much as possible intermediate results of the resolution of the standard problem. In addition,
the effectiveness of certain algorithms depends on the properties of the matrices of the problems to solve, of
a number of parameters per report/ratio for which it is necessary to derive, and of the required precision.

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Code_Aster ®
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Titrate:
Calculation of sensitivity in dynamics


Date:
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2
Derived from dynamic responses

2.1 Problem
standard

In the frequential field, equations of linearized dynamics and the dualisation of the conditions
to the limits lead to the following matric equation (it is the discretization by the method of
finite elements of the variational formulation which makes it possible to obtain this equation. During
document, the variational formulations and with the method of the elements are clarified when
necessary):

(- 2M () p+ iC () p+K () p) X (,) p = F (,) p
éq
2.1-1
where:
·
X (, p) RN

(N =n+2) is, for the pulsation, the vector made up of N ddl physical
structures discretized and of the 2 multipliers of Lagrange [R3.03.01],
·
p is a scalar parameter of mechanical property, loading or of geometry which
intervenes directly explicitly in the matrices of mass, stiffness or
of damping, or in the loading. Let us notice that that can be a parameter
of geometry for the elements of structure (for example a thickness of hull) but not
a geometrical parameter of the grid for which the field of integration is modified and
for which thus it is necessary to return to a variational formulation of the problem and to establish one
parameter setting of the field (cf [§4]),
·
M (p), C (p) and K (p) are the real symmetrical matrices of mass, damping and
of stiffness “generalized” (thus not necessarily definite positive),
·
F (, p) is the vector of the external forces.

Code_Aster solves a more general equation for the systems fluid-structures [R5.05.03].
The writing of the derived problem and its resolution are not basically modified by it. For more
of clearness and concision, one thus limits oneself in the continuation of this Doc. to the equation [éq 2.1-1].

In Code_Aster, the resolution of [éq 2.1-1] is carried out by the direct methods multifrontale and
“LDLT” [R6.02.02]. These two methods proceed in three stages:

·
renumerotation of the unknown factors,
·
factorization of the matrix,
·
gone up descent/(resolution of two triangular systems).

If several linear systems, of the same matrix, are to be solved, only them
gone up descents/are to be carried out several times. The same if several of the same matrices
structure are to be factorized, the renumerotation of the unknown factors will not be to remake (this phase
preliminary has a considerable cost, even if its relative weight in calculating times decreases with
cut matrices to be factorized). These two properties are very important and will allow
to calculate the derivative at numerical lower cost.
Handbook of Référence
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Code_Aster ®
Version
7.4
Titrate:
Calculation of sensitivity in dynamics


Date:
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:
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: 5/26


2.2
Derived from a harmonic dynamic response

2.2.1 Method
direct

The derivation of [éq 2.1-1] gives (after handling for exhiber a “good” second member):

(
X ()
F ()
M
C K
- 2 + I +)
=
- - 2
M
C K
+ I
+
X ()

éq
2.2.1-1
p
p

p
p
p
X ()
The resolution of the linear system [éq 2.2.1-1] gives the size

sought.
p

One can observe that the system [éq 2.2.1-1] has the same matrix coefficient as [éq 2.1-1], and
a second different member but whom one can obtain without inversion starting from elementary calculations.
This second member is called in the literature “pseudo-load” giving to the direct method it
name also of “method of the pseudo-loads”.

There are as many resolutions of [éq 2.2.1-1] that parameters by report/ratio to which one derives. For
to carry out these resolutions by minimizing the calculating time, one uses the properties of the methods
direct multifrontale and “LDLT” recalled in [§2.1]. The renumerotation of the unknown factors and
factorization of the matrix are thus carried out only once for all the resolutions.

Also let us notice that the majority of the terms of the second member are null. Indeed, if p is one
F ()
nodal force only remainder the term
, if p is a density only remainder the term
p
2 M X (), etc… Cette notice valid residence subsequently Doc., for all them
p
terms containing of the derivative.

2.2.2 Simple modal method

To accelerate the resolutions of [éq 2.1-1] frequency per frequency, it is usual to project
the operator on a truncated modal basis [] = [,

L
1
2,
,
L L] where I is the i-ième vector
clean (cf [§3.1]) and with L<n [R5.06.01]. The new system to be solved becomes:

[(] T
2
T
L (- M + iC + K) [] L) () = [] L F () éq
2.2.2-1
with the projection of X () on the incomplete modal basis which is written X = []
L
L ().
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:
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The derivation of [éq 2.2.2-1] gives:

T
[(
2



] T -
M +
C + K
=
F
+
F
L (



I
) []) () [] L
L
() [] T ()
p

p
L

p


- [

2 M
C K

] T -
+ I


+
L
[]


éq
2.2.2-2

p

p

p
L



[] T

L
+
(2
T

- M + iC + K) [] +
- 2M + C +
L
[] L (
I
K) [] L (
)
p

p


()
The resolution of [éq 2.2.2-2] provides

. One obtains then the derivative of projection on the basis
p
XL
modal of X (), i.e. by:
p

X
[
]
(
L
L
)
=
() + []


L

éq
2.2.2-3
p
p
p

The remark made for the direct method is valid for the simple modal method: the pre ones
processing of the matrix coefficient in the linear system [éq 2.2.2-1] (factorization
primarily) can advantageously be re-used for the resolution of [éq 2.2.2-2].

In this method, only truncation carried out is that of the projection of X () on the basis
modal incomplete. The calculation of the derivative is then carried out without another approximation. In
counterpart, the resolution of [éq 2.2.2-2] and the calculation of [éq 2.2.2-3] require calculation as a preliminary
[
]
of
L, for example by one of the methods presented at the chapter [§3.3].
p

2.2.3 Double modal method

X ()
In the same way that X () is projected on truncated modal basis,

can be projected on
p
this same base. Applying this projection to [éq 2.2.1-1], one obtains:

[(] T
2
µ
L (- M + iC + K) [] L) () =
éq
2.2.3-1
= [] T F
() -
2

L
[] T M

-
+ I C

K

+
X
L
()




p

p
p
p

X ()
with projection on the basis of modal
µ

who is written []
.
p
L ()
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In practice, in [éq 2.2.3-1], X () is known only after the first modal projection. Method
by double projection thus consists in carrying out two projections successively, and thus writing
(substitution of X () by XL in [éq 2.2.3-1]):

[(] T - 2 +
+


=
L (M
iC K) [] L) ()
= [] T F
()

2 M

C

K

-
-
+
+
L
[] T
I
L
X



L ()
p


p

p

p

éq 2.2.3-2

X ()
It should well be seen that the resolution of [éq 2.2.3-2] gives an approximation of

after one
p
double truncation. The first truncation is usual truncation carried out to obtain
X ()
projection of X (). The second truncation is of comparable nature but concerns

.
p

X ()
Techniques of acceleration of the convergence of the series giving

were proposed in
p
literature with [bib3]. The user of Code_Aster will use the simple modal method preferentially
to obtain more precision in the results.

2.2.4 Establishment by the finite element method

For the calculation of the second members of [éq 2.2.1-1], [éq 2.2.2-2] and [éq 2.2.3-2], several choices are
possible according to whether the matric products are calculated on an elementary level or not.

For reasons of performance, it was selected to carry out the matric products of the second member
of [éq 2.2.1-1] on an elementary level. In Code_Aster, the second member of [éq 2.2.1-1] is thus
calculated directly by the assembly of the following elementary matrices:

F
(G)
2
E
T
E
E

E
T
E
E
- G
((G)) (G) (G) X + gi
(Ca (G)) (G) (G) X
p

p
p
+ (E

G (G))T
E
(H (G))(E
E
(G) X)
p
éq 2.2.4-1
where:

·
same diagrams of integration (same functions of forms, points of Gauss,…) who have
allowed to obtain the standard linear system are used,
·
G are the parametric co-ordinates of the point of Gauss of weight G,
·
(G) is the matrix of the functions of form of the element E,
·
Be (G) is the matrix connecting the deformations to the nodal variables for the element E.
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Thus, the assembled vector - 2 M
C
K
+ I
+
X ()


is obtained without calculation and
p
p
p
MR. K
C
the assembly preconditions of the matrices,
and
. This procedure is thus economic in
p
p
p
assembly and storage of assembled matrices. It has on the other hand the disadvantage of
to require the calculation of the elementary matrices derived at each frequency since the product by
X () is carried out on an elementary level and thus prevents the storage of these matrices
elementary derived (at least with the architecture of Code_Aster).

The second members of [éq 2.2.2-2] and [éq 2.2.3-2] will result them from a shandy between operations
elementary and matric operations.

Once again, one can notice that many terms of [éq 2.2.4-1] are null. For example if
p is Young the modulus of certain elements, only remainder the term
E
T
((
E
E
G))
(H ()) () X (
G
) for these elements and [éq 2.2.4-1] is null everywhere

G (
G
G
p
)
elsewhere.

2.3
Derived from a temporal dynamic response into linear

One limits oneself in this chapter and the following if one does not derive compared to
parameters intervening under the initial conditions. This case raises theoretical difficulties and
practical out of the field of this Doc.

2.3.1 Method
direct

Let us leave the differential equation of dynamics in matric form:

M (p) X
& (T, p) + C (p) X& (T, p) + K (T, p) X (T, p) = F (T, p)
éq
2.3.1-1
where:
·
X (T) RN
(N = n+2) is, at the moment T, the vector made up of N ddl physical of
structures discretized and of the 2 multipliers of Lagrange [R3.03.01],
·
F (T, p) is the vector of the external forces, with, to calculate the response in transient
linear in Code_Aster [U4.53.02], the loading which must be written in the form
F = (T F
) (X
I
I
).
I

X
The differential equation governing the derivative compared to p of the vector displacement X, Y =, is
p
obtained while deriving directly [éq 2.3.1-1] compared to p. Après change about
derivation compared to time and p (the sufficient regularity is supposed), and after rearrangement
terms, one obtains:

M (p) Y& (T, p) + (
C p) Y& (T, p) + K (p) Y (T, p) =
F
(T, p)
M
(p)



éq 2.3.1-2
-
X& (
C
K
T, p)
(p)
-
X& (T, p)
(p)
-
X (T, p)
p

p

p

p

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The vector of displacements and its derivatives are obtained while solving successively [éq 2.3.1-1]
then [éq 2.3.1-2]. For that, one uses a diagram of numerical integration (implicit or clarifies in
Code_Aster). On the same principle as that of the derivation of a harmonic answer, one exploits
intermediate results of the resolution of the standard problem in order to minimize calculations related to
resolution of the derived problems.

Let us specify this in the case of an implicit scheme, the identical remaining principle in the case of one
explicit diagram: For the standard problem, the vectors accelerations and speeds, with the step
from integration tn+1, are obtained by the linear approximations of the following form:

&X
= L (X
, X,
n+1
1
n+1
N &
X,
N &X)
N
éq
2.3.1-3
&X
= L (X
, X,
n+1
2
n+1
N &
X,
N &X)
N éq
2.3.1-4

One solves the standard problem then, i.e one obtains Xn+1, by the substitution of [éq 2.3.1-3] and
[éq 2.3.1-4] in [éq 2.3.1-1]. This gives a linear system to solve.

The idea for the problems derived is to use the same diagram of integration:

&Y
= L (Y
, Y,
+1
1
+
&Y, &Y)
N
N 1
N
N
N
éq
2.3.1-5
&Y
= L (Y
, Y,
+1
2
+
&Y, &Y)
N
N 1
N
N
N éq
2.3.1-6

Thus, factorization used to solve the system can be also used to obtain Yn+1.
Indeed, since the matrices coefficient of [éq 2.3.1-1] and [éq 2.3.1-2] are identical, and if one uses
the same diagram of integration for the calculation of Xn+1 and Yn+1, then the matrix coefficient is
even in the two linear systems to solve after substitution of [éq 2.3.1-3] and [éq 2.3.1-4]
in [éq 2.3.1-1] and of [éq 2.3.1-5] and [éq 2.3.1-6] in [éq 2.3.1-2].

The calculating time of derived can then be negligible compared to the calculating time associated with
standard problem. It should however be noted that, in the case of a resolution by explicit diagram,
each calculation of derived can be as expensive as standard calculation. Indeed, in this case,
the main part of time CPU for the standard problem is associated at the stage of calculation of the second member
with each step of time; and it is also necessary to carry out this stage for the derivative (calculation of the second
member of [éq 2.3.1-2]).

2.3.2 Method
modal

Two modal methods simple and double for calculation of derived from a dynamic response
frequential ([§2.2.2] and [§2.2.3]) are easily transposable with the calculation of the dynamic response in
temporal. One obtains a linear system of the same type as [éq 2.3.1-2] with second members
different according to whether one uses the simple or double modal method. In the case, method
modal double, one obtains:

[] T
^
^
^

&
+
&
+

=
L M [] L (p) Y (T, p)
[] T
L
(
C p) [] L Y (T, p) [] L K (p) [] L Y (T, p)
[
] T F
(T, p)
M

C

K

-
&
-
&
-
L
[] T
(p)
L
XL (T, p) [] T
(p)
L
XL (T, p) [] T
(p)
L
XL (T, p)
p

p

p

p

éq 2.3.2-1
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The response in generalized co-ordinates is obtained by solving this system by a diagram
of numerical integration of the same type as [éq 2.3.1-3] and [éq 2.3.1-4]. And in the same way, them
factorizations used for the calculation of the answer can be used for the calculation of the derivative.

2.3.3 Establishment by the finite element method

In a way similar to the preceding cases [§2.2.4], the second member of [éq 2.3.1-2] can be calculated
MR. K
C
after assembly of the matrices,
and
, or in a more powerful way directly to one
p
p
p
elementary level.
The temporal integration of X to the step tn+1 is carried out the temporal integration before of
its derivative Y. With the step of integration tn+1 of Y, terms X (T
, p)
n+1
, &
X (T
, p)
n+1
, and &X (T
, p)
n+1

are thus known (the last two terms are calculable by [éq 2.3.1-3] and [éq 2.3.1-4] by
example).
M (p)
C (p)
K (p)
With the step of integration tn+1 of Y,
X
& (T
, p) +
X
n+1
& (T
, p) +
X (T
, p)
p
p
n+1
p
n+1

result thus from the assembly of the following elementary matrices (cf [§3.2.3] for the others
notations):




(()) E
NR () T NR () E
X & (T
) +
(C ()) E
NR () T NR () X& (, T
)
G
G
G
G
N 1
+
G
has
G
G
G
G
N 1
+
p
p

éq
2.3.3-1
+
E
B () T
(E
H ()) E
B () T E
X & (T
)
G
G
G
G
N 1
+
p

2.4
Derived from a temporal dynamic response into nonlinear

Let us consider the class of the problems of nonlinear dynamics of which the equation discretized of
movement is form:

M (p) X
& (T, p) + C (p) X& (T, p) + [
G X (T, p), p] = F (T, p) éq
2.4-1

where
[
G X (T, p), p] is a function (nonlinear) X (T,)
p and of p. Elle represents the forces
interns of the system and all the other forces which are dependant on X (T).

[
G X (T, p), p]
Noting the tangent matrix K (X (T, p), p) =

, the equation of derived compared to p
X
vector displacement is obtained:

M (p) Y & T
(, p) + C (p) Y & T
(, p) + K (X T
(, p), p) Y T
(, p) =
F
(T, p)
M
(p)
p

T, p, p
-

éq
2.4-2
X& (T, p)
(
C)
-
X& (T, p)
G (X (
))
-
p

p

p

p

X fixed
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G (p)
With share the term
, all the terms derived from [éq 2.4-2] are calculated in a traditional way
p X fixed
G (p)
(i.e by assembly of the matrices or elementary vectors derived). The term
can
p X fixed
to require a particular processing. Indeed, in [éq 2.4-1], dependence between G and the value of
X (T,)
p is explicit at the moment T; however G also depends on the history in plasticity. This
G (p)
dependence complicates calculations clearly of
that we will not approach here.
p X fixed

What is remarkable, it is that the derived problems [éq 2.4-2] are linear problems then
that the standard problem is nonlinear. This property allows a calculation of really derived
very economic compared to the cost calculation of the standard problem. This is all the more true as,
for the resolution of [éq 2.4-1] and [éq 2.4-2], factorizations of matrices used for the calculation of
vector displacement can be re-used for the calculation of the derived vector; as it is the case
in the case of a linear standard problem. Indeed, the resolution of [éq 2.4-1] by an algorithm of
Newton type requires the resolution of a linear system whose matrix coefficient is the same one as
in [éq 2.4-2]. Once the algorithm of the Newton type converged to obtain X (T
)
n+1, the vector
gradient Y (T
)
n+1 can thus be obtained directly not a simple resolution of two systems
triangular without convergence of an iterative algorithm.

3
Derived from the frequencies and clean modes from vibration

3.1 Problem
standard

The problem with the eigenvalues (problem known as quadratic) associated [éq 2.1-1] or [éq 2.3.1-1]
is written:

((p) 2M (p) + (p) C (p) +K (p))(p) = 0
éq
3.1-1

where (p) and (p) are complex values and clean vectors.

In the continuation, to gain in compactness of the equations, we will omit the dependence of the sizes
compared to the parameter p and/or time T, except when the clarification of this
dependence will be able to avoid ambiguities.

The reduction of the quadratic problem [éq 3.1-1] in a problem generalized equivalent [R5.01.02]
give the system of size NR × NR = (N
2) × (N
2) according to:

- K O
C M


=

éq
3.1-2
O
M
MR. O
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One can standardize the clean vectors so that they check:


T
- K 0
J
J



=




éq
3.1-3
0
M

J


J



J

T
C M
J
J

=




1
éq
3.1-4
M
0



J

J

3.2
Derived from the Eigen frequencies

The multiple eigenvalues are not differentiable with the direction running of derived (i.e within the meaning of
Fréchet). We must thus distinguish the case from the simple eigenvalues of the case of the eigenvalues
multiples for which some precautions must be taken.

3.2.1 The case of the simple eigenvalues

The derivation of the equation [éq 3.1-1] written for the jième mode compared to a scalar parameter p
give:

(


J
J

M
C K
2M +
2
J
jC + K)
= - (2 M + C)
-
+
+


J
J
J


J

J éq 3.2.1-1
p
p
p
p
p

While multiplying on the left per T
J two members of [éq 3.2.1-1] and while using [éq 3.1-1] and
[éq 3.1-4], the derivative of the eigenvalue can be obtained from:

J




T
= - 2 M
C
K
+
+


J
J


J

J
éq
3.2.1-2
p
p
p
p

The equation [éq 3.2.1-2] above requires the calculation of the clean vector J associated with J. One cannot
thus not to exempt calculation of the clean vectors for the calculation of derived from eigenvalues.

3.2.2 The case of the multiple eigenvalues

In a clean subspace whose associated eigenvalue is multiple of command m, any combination
linear of vectors is clean vector. One cannot however apply the formula directly
[éq 3.2.1-2] with any of these vectors, because the clean subspace can be divided into m
distinct clean subspaces when a parameter p varies.
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On the theoretical level, the multiple eigenvalues are not differentiable as a function with
several variables, only of the derivative directional exist (this can pose problems of
convergence with many algorithms of optimization which are based on the differentiability of
function objectifies as a function with several variables). As example, let us consider the values
clean of the following matrix:

4 + 2y X

2
2
1 = 4 + y +
X + y
With =
,

X
4

2
2
2 = 4 + y -
X + y

For (X, y) = (,
0)
0, the eigenvalue of A are multiple. And, one a:

1



= - 1
1
= 1
2
= 1
2
= - 1

,
,
,
.
X



(0 -, 0)
X (0+ 0,)
X (0 -, 0)
X (0+, 0)

It is possible “to connect”, 1 for X < 0 with 2 for X > 0, and “to connect” 2 for X < 0
with 1 for X > 0 (let us take for example y = 0. Then, “the smallest eigenvalue of A” (2 - X)
is not differentiable in 0, and “the largest either” (2 + X). On the other hand, “the eigenvalue of A
corresponding to the clean vector (,
1) T
1 “(2 + X) and “the eigenvalue of A corresponding to
clean vector (,
1 -) T
1 “(2 - X) is differentiable into 0). If one does not consider independently
the two Eigen frequencies of A, one could thus write/X = ±1. (One can proceed of
even with the derivative in y and to write/y = 0 or 2). However, eigenvalues of A
are thus not differentiable as functions with two variables (i.e one cannot


to write D =
D X +
D y

), and have only derivative directional. For of
X
y
to convince, it is enough to notice that:

(cos, sin)
(cos, sin) - (,
0)
0
= lim
= sin ± 1





=

0
0

On the plan practices, for the calculation of these derivative directional, the choice of the clean vector J
associated J in [éq 3.2.1-2] is ambiguous when the eigenvalue is multiple. This choice thus must
to be specified.
In the case of an eigenvalue m of command of multiplicity m, the problem with the eigenvalues
[éq 3.1-1] can rewrite itself in the following matric form:

[
M] 2 + [
C] + K [
m
m
m
m
m] = 0
éq
3.2.2-1

with m = m Im and [] = [,

m
i+1
i+2,
,
L i+m] where the i+k are the clean vectors
associated Mr.
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Note:

In any rigor and in the general case, the matrices M and K not being definite positive (with
cause dualisation of the boundary conditions), it does not exist inevitably m vectors
clean generating a clean subspace of dimension equalizes with the multiplicity of the value
clean. This point is never approached in the literature. By convenience, we will not approach
either this pathological point which becomes delicate when it is about calculation of derivative.

The conditions of orthogonality of the clean vectors are written with the matric notations
above:

[] T (2 m
M +)
C [
m
m] = Im éq
3.2.2-2

Let us note [X] the m vectors clean of the same multiple subspace clean which become m
clean vectors [X (p)] m subspaces two to two distinct when when a parameter p
vary. When p tightens towards its face value Po, [X (p)] tends towards [X] and the m eigenvalues
distinct tighten towards the eigenvalue multiple Mr. Dans the literature, such vectors clean are
called clean vectors “adjacent”. It should be noted that these vectors depend on the parameter which varies
i.e. parameter by report/ratio to which one derives. In other words, if parameter is changed, it
is then necessary to change adjacent vectors.
The adjacent vectors [X] can be connected to [m] by an orthogonal transformation T ([X]
and [m] two bases of the same clean subspace constitute):

[]
X = [m] T
éq
3.2.2-3
with TTT = Im.
m
The procedure consists in finding the transformation T for then deducing some [X] and. It is it
p
that we develop below.

The adjacent vectors check the problem with the eigenvalues [éq 3.2.2-1]:

[
MR. X] 2
m + [
C X] + K
m
[X] = 0
éq
3.2.2-4
The derivation of [éq 3.2.2-4] compared to p gives:

(
X


M
C K
2M +
m
2
m
mC + K) [] = - (2mM +)
C [X]
-
+
+
[X]


m


m


p
p
p
p
p
éq 3.2.2-5
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T
While multiplying [éq 3.2.2-5] on the left by [m], using [éq 3.2.2-2] and substituent []
X = [m] T, one
the new problem with the eigenvalues obtains:


DT = T
m


éq
3.2.2-6
p
T
M
C K
with D = [] 2
+
+

m
m
[
m
m]


.
p
p
p

m
The resolution of the problem to the eigenvalue [éq 3.2.2-6] gives the derivative of the eigenvalues
p
and stamps it orthogonal transformation T.
It should be stressed that there will be a subproblem with the eigenvalues of size m to be solved for each
eigenvalue of multiplicity m to be derived and for each parameter.

3.2.3 Establishment by the finite element method

In a traditional way, several choices are possible for the calculation of the second member of [éq 3.2.1-2] and
matrix D of [éq 3.2.2-6] according to whether the matric products are calculated on an elementary level
or not.
The calculation of the second member of [éq 3.2.1-2] on an elementary level is a priori most powerful.




T 2 M
C
K
+
+



can be written directly on an elementary level by the scalar:
p
p
p


T

T
2
((
E
G
G))((G)) ((G)) +
(C (
G
has
G))((G)) ((G))
p
p
éq
3.2.3-1
T
+
E
E
E
G (B () (
G
G))
(H (G) () B () (
G
G))
p
In particular, a procedure not to start the calculation of each elementary term of



[éq 3.2.3-1] that if quantities
()
E
(H ())
(C ())

,
and
are nonnull allows
p
p
G
p has G
to minimize time CPU.

In a way similar to the case of the simple eigenvalues, the matrix D of the system [éq 3.2.3-1] can
to be calculated directly on an elementary level or after effective assembly of the matrices
MR. K
C
,
and
.
p
p
p
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3.3
Derived from the clean modes of vibration

3.3.1 The case of the simple eigenvalues

One cannot directly obtain the derivative of the clean vectors by solving the linear system
[éq 3.2.1-1] because it is a noninvertible system since 2M +
J
J C + K




is singular (by
definition of J).
Many methods were proposed in the literature to solve or circumvent this
difficulty. Among them, two types of methods emergent: methods of projection on basis
modal incomplete [bib4], [bib5] and algebraic methods [bib6], each one of them with theirs
alternatives. These methods are presented in [bib1]. The algebraic methods are methods
exact which, moreover, does not require the other clean vectors to calculate the derivative of one
clean vector given. One presents only one algebraic method here consisting in adding one
equation with [éq 3.2.1-1] to obtain a regular and symmetrical system to solve [bib6].

The rewriting of the condition of orthonormality [éq 3.1-4] gives:

T (2 jM +)
C
J
J = 1
éq
3.3.1-1

The derivation of this equation compared to the parameter p leads to:


1

MR. C
T (
J
T
J
2


jM +)
C
+
2
M +
+

J
J
J
J = 0

éq
3.3.1-2
p
2
p
p p

The equations [éq 3.2.1-1] and [éq 3.3.1-2] can be rewritten together:

(2

M + C +

J
J
K) (
2 M +
J
C)

J

J


p

T
J (
2 M +
J
C)

=


0



0



2 M
C
- (
2 M +
K
J
C)
J




-
+
+


J

J
J

J
p

p
p
p


1



T
J
MR. C
- 2
M +
+



J
J


J
2
p
p
p






éq 3.3.1-3

The resolution of the system [éq 3.3.1-3] is the key idea of the algebraic method. It can be shown that
J
this system is always regular. The derivative of the clean vectors can be obtained
p
directly by solving this system of dimensions (
N +)
1 × (
N +)
1 with a factorization LDLT
for example.
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The algebraic method preserves the structure in tape and the symmetry of the matrices. Moreover, for
to calculate the derivative of a clean vector, only the knowledge of this clean vector and value
clean corresponding is necessary. Indeed, [éq 3.3.1-3] does not utilize the other vectors
clean that that to derive.

However, the algebraic method requires the resolution of a system of size (
N +)
1 × (
N +)
1
different for each vector suitable to derive and each parameter.

3.3.2 The case of the multiple eigenvalues

[
X]
Taking again the notations of the paragraph [§3.2.2], the derivative of the clean vectors “adjacent”
p
[X] = [m] T are solutions of the equation [éq 3.2.2-5].

The matrix (2M +
m
mC + K) being singular, one cannot solve [éq 3.2.2-5] directly.
[
X]
The idea, still, is to add an additional equation for.
p

The condition of standardization of [X] is written:

T (2 jM +)
C
J
J = 1
éq
3.3.2-1

The derivative of [éq 3.3.2-1] gives the sought additional equation:

[
X
M

1
C
X] T (2 M
T
T
m
T
m
+) []
C
= [
- X]
[X] - [X]
[
MR. X]
- [X]
[X]
m

éq 3.3.2-2
p
p
p
2
p

[éq 3.2.2-5] and [éq 3.3.2-2] form the following system to solve:

(2

M + C + K
m
m
) (2 M+C
m
) [X] [X]



p

[X] T (
2
M + C
m
)

=
0



0



2




- (
2
M + C
m
) []

m -
M
X

+
C

+ K
m
m
[X]





p

p
p
p

T M
T
m 1 T C
- [X]
[X] -
m
[X] M [X]
- [X]
[X]

p
p
2
p

éq 3.3.2-3

The system [éq 3.3.2-3] to solve is of dimension (
N +)
m × (
N +)
Mr. Pour to calculate the derivative of one
clean vector, the knowledge of the clean vectors associated the eigenvalue is necessary.

As for the case of the nonmultiple eigenvalues, the structure in tape and the symmetry of
matrices are preserved and it can be shown that the system to be solved is always regular. Of
more, the system must be solved for each vector suitable to derive and each parameter.
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3.3.3 Establishment by the finite element method

In the equations [éq 3.3.1-3] and [éq 3.3.2-3], the second members can be calculated either in
MR. K
C
assembling the matrices initially,
and
then by carrying out matric operations, that is to say
p
p
p
by directly calculating the various terms on an elementary level.





A direct calculation at the elementary level being more powerful, 2 M
C
K
+
+



, by
p
p
p
example, will be calculated on an elementary level by (cf [§3.2.3] for the notations):



2
((
E
)) (T
) () +
(C (
E
)) (T
) (
G
)

G
G
G
G
p

has
G
G
G
p
éq
3.3.3-1
T
+
E
E
E
G ((G))
(H (G) () () (
G
G))
p

4
Derived compared to a variation from field

The matric equation [éq 3.1-1] uses an explicit dependence of the matrices of mass,
of damping, and stiffness compared to the variable parameter p. Lorsqu' it acts to derive by
report/ratio with a variation of field, one cannot in the general case write this dependence
directly. In this case, obtaining derived from the dynamic sizes requires a work
precondition. It is necessary on the one hand to describe the variation of field in a mathematical form which one can
to handle, and in addition to return to the variational formulation of the equations of dynamics that
one derives directly.

This chapter discusses these various items. One gives in particular the derivative of the various operators
of mass, stiffness necessary to the calculations of derived from all the dynamic sizes. One
limit then with the calculation of derived from the Eigen frequencies and clean vectors. One limits oneself to
conservative problem and thus with the real clean modes.

4.1
Variational formulation of the standard problem

One considers an elastic solid on a field, subjected to surface static loadings
F and voluminal F. In variational form, the problem of balance is written:

To find U
V0
:
that

such
éq
4.1-1
With (U): (v)
D = f.v
D + F.v
D
v



V0
F
where A is the tensor of elasticity, (U) the linearized traditional tensor of the deformations and V0 space
3
of Sobolev of (H1 ()) displacements kinematically acceptable.
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The clean modes around this balance are obtained by the resolution of the following spectral problem:

To find


real

clean

frequency



> 0
W

mode



and
V
:
that

such

no one

not
éq
4.1-2
With (W): (v)
D + W (U)
2
: v
D -
w.v


D = 0
v



V0

(notation: if A and B are two tensors of command 2, A B
is the contracted product of A and B)

where is a parameter of possible amplification of the solution of [éq 4.1-1] for the loadings
statics above, is the tensor of the constraints given by (U) =

With (U) and masses it
voluminal of material considered.

This problem can be rewritten by using the following traditional multilinear operators:

·
elastic operator of stiffness has (W, v) =
With (W): (v) D


·
geometrical operator of stiffness B (U, W, v) =
W
(U):vd




=
(U) W


ik
L I,
kl (v) D

·
operator of mass C (W, v) =
W. vd




in the form:

To find


reality



> 0 and W V0
:
that

such

no one

not

has (W, v) + B
(U, W, v) - 2
C (W v
,) = 0 v V
éq
4.1-3

0


C (W W
,) = 1

4.2
Description of the variation of field and form of derivation

One considers now that the field is likely to change form, by a variation
of its edge (not included/understood the edge where are applied imposed displacements). To describe this
variation, one uses a bijective transformation making correspond the area of reference (that
that one nets) with the field modified (representation known as Lagrangienne of variation of field,
cf [R7.02.04]).

The edge of the field (
) is controlled by the scalar (such as (
)
0 =), according to the field of
vectors.

The derivative of a field of vector X compared to a variation of field can be now written
like the Lagrangian derivative, which one will note with a point (cf [R4.03.01]):


&
X
D
X
X =
=
+ X
éq
4.2-1
D

=0
=0
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And, the derivative of the gradient of X is written:

D ()
X
dX
=
-
X
éq
4.2-2
D
D

The derivation of the operators of mass and stiffnesses [§4.1] requires the derivation of sizes
integrals whose field of integration depends on the scalar. These integral sizes are
form:

I (U ()
=
(E ((U))
()
D éq
4.2-3

where (
U) is the solution of the problem of balance [éq 4.1-1] on the field (
).

This integral for derived (theorem of Reynolds):



dI

(
of ((
U)
)

(U) = (E (U))div +
D éq
4.2-4
D

D
=0



=0
The derivative intervening in the intégrande is expressed:

(
of (U ()
) E D (U ()

=
.

D

D
with

D (U ()

()

T
T
=
- U - U éq
4.2-5
D
D

Note:

Implicitly, in the continuation the derivative will be taken in = 0.

4.3
Derivation of the variational formulation

4.3.1 Derivation of the elastic operator of stiffness

Using the symmetry of the tensor of elasticity, has (W, v) is also written:

has (W, v) =
With (
W): (
v) D

éq
4.3.1-1

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The derivation of this expression and the use of [éq 4.2-4] give:

D (

dA W
D v
has (W, v))
()
()
= A (
W): (
v) div +
: (
v) + A (
W):
D éq
4.3.1-2
D

D
D

With (
W)
(
W)
v
By [éq 4.2-5], and knowing
= A
= 0

and
, [éq 4.3.1-2] becomes:

D (
W
has (W, v))

=

With (W): (v)
D
div +
With
: (v) -
With (W): (v)





D

D
éq
4.3.1-3
- A (W): (v
) D

that one can rewrite, by using amongst other things again the symmetry of the tensor of elasticity:

D (
W
has (W, v)) =
D
has (
, v) + a^ (W, v)
D
D
éq
4.3.1.- 4
with a^ (W, v) =
(
With (W): (v) div - (W)
: v - (W): v
)


D


4.3.2 Derivation of the geometrical operator of stiffness

In the same way, one obtains:

D

dw
B U
(, W, v) = (W U
(): v) div
U
(): v
W
U
(): v


+

-








D

D



+ W
With
: v - W
With U
: v - W U
(): v
D






D

éq 4.3.2-1

that one can rewrite:

D
dw
B (U, W, v) = B (
, W, v) + B (U,
, v) + b$ (U, W, v) éq
4.3.2-2
D
D
D
with:

b^ U
(, W, v) =
(
(W U
(): v) div - W U
(): v - W
With U
:

v



- W U
(): v
)
D
éq 4.3.2-3
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Code_Aster ®
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7.4
Titrate:
Calculation of sensitivity in dynamics


Date:
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:
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In [éq 4.3.2-2], one sees that the calculation of derived from the geometrical operator of stiffness requires
au préalable the derivative D of the solution of the problem of prestressed [éq 4.1-1].
This problem of prestressing, which is thus to derive, is written:

To find U V such as: has (,
U v) = F. D
v + F.D
v = F (v) v

V


0 éq
4.3.2-4
F
The derivation of this variational equation gives:


D
has
, v =
F (v) - a$ (,
U v)

éq
4.3.2-5
D
D
with, by supposing that the forces F, F are independent of and:

D F (v) = (f.v.(div) + (F.) .v) d+
F.v div -
.n + F. .v



((, N) ()) dS
D
F
éq 4.3.2-6

(The formula of derivation of the surface integral is given by proposal 4 of appendix 2 of
[R4.03.01]).
The resolution of the linear system associated [éq 4.3.2-5] giving D has the same matrix of
rigidity that the problem of standard prestressing, only the second member changes. Code_Aster solves
this problem in deformations plane and 2D-axisymmetric, cf [R4.03.01].

4.3.3 Derivation of the operator of mass

The derivation of the operator of mass gives:

D (

dw
C,
v W) =
v.w
+. (
v.w) +

div
v.
D
D

D

that one can rewrite:

D (
W
C (W, v)) = D
C (
, v) + c^ (W, v)
D
D
éq
4.3.3-1
with c^ (,
v W) =
(v.w

div +. (v.w)


D

Handbook of Référence
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A

Code_Aster ®
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7.4
Titrate:
Calculation of sensitivity in dynamics


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:
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4.3.4 Derivation of the variational formulation of the problem of modal analysis

Derivation in = 0 of the problem [éq 4.1-3] of calculation of the clean modes on (
), brought back on
(
)
0 =, the parameter of initial loading being fixed, gives now:

To find

reality



&

W

and

& V
:

that

such

has (W & v,) + ^a (W v,) + (B (&, uw v,) + B (U W & v,) + ^
,
B (,
U W v
,))

éq
4.3.4-1

-
(2 &c
(W v
,) + (C (W & v
,) + ^c (W v
,) = 0v V
2c (W & W
,) + ^c (W W
,)

= 0

4.3.5 Obtaining derived from the Eigen frequencies and the clean modes

In a preoccupation with a simplification, we suppose here who the eigenvalue of the mode which one drift is
simple. The case of a multiple eigenvalue results from the adaptation of the developments from
paragraphs [§3.2.2] and [§3.3.2].

has

(,
W w&) + (
B U,
W w&) - 2 (
C,
W w&) = 0
While using
and while taking v = W in [éq 4.3.4-1], one


(C,
W W) = 1
obtains the following expression of derived from the own pulsation:

$a (,
W W) + ((
B &u,
W W) + $b (U,
W W) - 2 $c (,
W W)
& =
éq
4.3.5-1
2

Obtaining [éq 4.3.5-1] is to be brought closer that of [éq 3.2.1-2], except for standardization. In
cases of eigenvalues multiple, the choice v = W condition the calculated directional derivative.
[éq 4.3.4-1] gives the following linear system:

has (&,
W v) + (
B U, &,
W v) - 2 (
C &,
W v) = - a$ (,
W v) - ((
B &u,
W v) + b$ (U,
W v) éq
4.3.5-2
+ (2 & (
C,
W v) + c$ (,
W v))

This system is same form as the system [éq 3.2.1-1]. It must be solved same manner,
for example by using the addition of the equality (
C &,
W W) = - 1 c$ (,
W W).
2

4.3.6 Establishment by the finite element method

In a way similar to the preceding chapters, terms of the second members [éq 4.3.5-1] and
[éq 4.3.5-2] can be calculated, either directly on an elementary level, or after assembly
matrices of mass, stiffness and damping.
In addition, of new procedures of elementary calculations must be developed for
calculations of $a (,
W v) (this term A already was developed in deformations plane and 2D-axisymmetric,
D
cf [R4.03.01]), $
B (U,
W v), $c (,
W v), and
F (v).
D
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Calculation of sensitivity in dynamics


Date:
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:
R4.03.04-B Page
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5
Effective establishment in Code_Aster

The numerical calculation of derived from the clean modes or the dynamic responses proceeds in
two principal stages:

·
stage 1: calculation of the elementary terms constituting the new terms (often in
second member) associated the calculation of the derivative,
·
stage 2: resolution of the derived problem. The calculation of derived from the clean modes requires
the use of specific algorithms different from those used to solve the problem
standard. The calculation of derived from the other sizes, i.e. Eigen frequencies and
the dynamic responses, re-uses the standard algorithm by modifying some terms (in
General second members of the systems to be solved).

Sensitivities of the other dynamic sizes (forced, efforts…) require a third
stage which consists of the derivation of calculations of “postprocessing”. We do not treat these
sizes in this document.

The following table recapitulates the derivative available in Code_Aster. It is updated at the fur and at
measure developments of the derivatives.

Sizes and methods
Operators, and operands private individuals (i.e. Implantée
data necessary to the method of
calculation)
Derived from a dynamic response


harmonic
Direct method
DYNA_LINE_HARM
OUI
Simple modal method
DYNA_LINE_HARM
NON
clean vectors and derived clean vectors
Double modal method
DYNA_LINE_HARM
NON
clean vectors
Derived from a dynamic response


temporal into linear
Direct method
DYNA_LINE_TRAN
OUI
Simple modal method
DYNA_TRAN_MODAL
NON
clean vectors and derived clean vectors
Double modal method
DYNA_TRAN_MODAL
NON
clean vectors
Derived from a dynamic response
standard loading Dirichlet
NON
temporal into nonlinear
DYNA_NON_LINE
for some laws of behavior
OUI
Derived from an Eigen frequency
simple modes
OUI
MODE_ITER_SIMULT, MODE_ITER_INV
multiple modes
NON
Derived from a clean mode
simple modes
OUI
MODE_ITER_SIMULT, MODE_ITER_INV
multiple modes
NON
Derived compared to a variation
* NON
of field
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Calculation of sensitivity in dynamics


Date:
02/05/05
Author (S):
H. ANDRIAMBOLOLONA, Key S. CAMBIER
:
R4.03.04-B Page
: 25/26

6 Bibliography

[1]
R.A. IBRAHIM: Structural dynamics with parameter uncertainties, Applied Mechanics
Reviews, Vol. 40, No 3, pp.309-328, 1987.
[2]
S. CAMBIER: Vibratory sensitivities of sizes. Theory and algorithms for one
establishment in Code_Aster. Note HT-62/01/011.
[3]
Z.Q. Qu: Accurate methods for frequency responses and to their sensitivities off proportionally
damped systems, Computers & Structures, Vol. 79, pp.87-96, 2001.
[4]
Mr. YU, Z.S. LIU, D.J WANG: Modal Comparison off several approximate methods for
computing shape mode derivative, Computers & Structures, Vol. 62, No.2, pp.381-393,
1996.
[5]
S. ADHIKARI, Mr. I. FRISWELL: Not-conservative Eigenderivative analysis off asymmetric
systems, Int. J. Numer. Meth. Engng; Vol. 51, pp. 709-733, 2001.
[6]
I.W. LEE, D.-O. KIM, G. - H. JUNG: Natural frequency and mode shape sensitivities off
damped systems: Leaves I, distinct natural frequencies, Journal off Sound and Vibration 223 (3),
pp. 399-412 (1999). Leaves II, multiple natural frequencies, Journal off Sound and Vibration
223 (3), pp. 313-424, (1999).
[7]
I. VAUTIER: Isoparametric elements [R3.01.00].
[8]
J. PELLET: Dualisation of the boundary conditions [R3.03.01].
[9]
G NICOLAS: Sensitivity of the mechanical thermo fields to a variation of the field
[R4.03.01].
[10]
D. SELIGMANN, R. MICHEL: Algorithms of resolution for the quadratic problem with
eigenvalues [R5.01.02].
[11]
O. BOITEAU: Algorithm of resolution for the generalized problem [R5.01.01].
[12]
G. JACQUART: Methods of RITZ in linear and non-linear dynamics [R5.06.01].
[13]
G. DEBRUYNE: Lagrangienne representation of variation of field [R7.02.04].
[14]
D. GIRARDOT: Operator DYNA_LINE_TRAN [U4.53.02].
Handbook of Référence
R4.03 booklet: Analyze sensitivity
HT-66/05/002/A

Code_Aster ®
Version
7.4
Titrate:
Calculation of sensitivity in dynamics


Date:
02/05/05
Author (S):
H. ANDRIAMBOLOLONA, Key S. CAMBIER
:
R4.03.04-B Page
: 26/26

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