Code_Aster ®
Version
6.4
Titrate:
Extrapolation of measurements on a digital model in dynamics
Date:
11/02/03
Author (S):
S. AUDEBERT, Key H. ANDRIAMBOLOLONA
:
R7.20.02-B Page
: 1/8
Organization (S): EDF-R & D/AMA
Handbook of Référence
R7.20 booklet: Projection of results or measurements
Document: R7.20.02
Extrapolation of measurements on a digital model
in dynamics
Summary:
A step of extrapolation of experimental results of measurement in dynamics (displacements, speeds,
accelerations, strains, stresses) on a digital model is presented. Based on one
representation of the structure on a basis of projection chosen beforehand, it consists of the resolution
opposite problem defined by the identification of the generalized co-ordinates relating to the base of projection.
The resolution suggested uses a minimization, within the meaning of least squares, by using the decomposition LU
or decomposition in singular values (SVD), of a functional calculus possibly regularized via
addition of a criterion of proximity of a solution known a priori. In the case of a temporal identification,
an explicit formulation of information a priori is proposed.
Handbook of Référence
R7.20 booklet: Projection of results or measurements
HT-66/02/004/A
Code_Aster ®
Version
6.4
Titrate:
Extrapolation of measurements on a digital model in dynamics
Date:
11/02/03
Author (S):
S. AUDEBERT, Key H. ANDRIAMBOLOLONA
:
R7.20.02-B Page
: 2/8
Count
matters
1 Problems ........................................................................................................................................ 3
2 Notations ................................................................................................................................................ 3
3 Step of extrapolation ...................................................................................................................... 4
4 Relations between the physical sizes and the generalized sizes .............................................. 4
5 Calculation of the modal contributions .......................................................................................................... 5
5.1 Formulation of the problem reverses .................................................................................................... 5
5.2 Determination of a quasi-solution ................................................................................................. 5
5.3 Determination of a regularized opposite solution ........................................................................... 6
5.3.1 Principles of the methods of regularization ............................................................................. 6
5.3.2 Choice of information a priori ................................................................................................. 7
6 Implementation in Code_Aster ....................................................................................................... 8
7 Bibliography ........................................................................................................................................... 8
Handbook of Référence
R7.20 booklet: Projection of results or measurements
HT-66/02/004/A
Code_Aster ®
Version
6.4
Titrate:
Extrapolation of measurements on a digital model in dynamics
Date:
11/02/03
Author (S):
S. AUDEBERT, Key H. ANDRIAMBOLOLONA
:
R7.20.02-B Page
: 3/8
1 Problems
One wishes to estimate numerically, the behavior in any point of a structure from
measurements taken in some points of the structure. Taking into account the costs and constraints
of accessibility, experimental measurements in a limited number and are generally located in
places which inevitably are not requested. Thus, in dynamics, the knowledge of the zones
of stress and local value concentration of constraints is crucial to check the behavior
mechanics of the hardware. One is then brought to extrapolate results of measurement located, on
the whole of the numerical grid of the structure.
The step of extrapolation suggested is based on a representation of the structure on a basis
of judiciously selected projection (clean modes, static answer,…). It consists of
determination of the generalized co-ordinates relating to this base of projection. The resolution
proposed uses a minimization, within the meaning of least squares, by LU decomposition or in
singular values (SVD), of a functional calculus possibly regularized via the addition of a criterion
of proximity of a solution known a priori. In the case of a temporal identification, a formulation
explicit of information a priori is proposed.
2 Notations
Q, q&, Q &: vectors of displacements, speeds and accelerations in the physical reference mark
, &, &: vectors of displacements, speeds and accelerations generalized
: stamp formed of the basic vectors of projection (displacements)
: stamp formed of the basic vectors of projection (deformations)
: stamp formed of the basic vectors of projection (forced)
: stamp basic vectors (displacements), restricted with the measured degrees of freedom
: stamp basic vectors (deformations), restricted with the measured degrees of freedom
: stamp basic vectors (forced), restricted with the measured degrees of freedom
I: stamp identity
Nnum: a number of basic vectors of projection, Nexp: a number of degrees of freedom measured
T: variable time: variable pulsation
TF: transform of Fourier, TF - 1: opposite transform of Fourier
A+: pseudo-opposite of matrix A
Handbook of Référence
R7.20 booklet: Projection of results or measurements
HT-66/02/004/A
Code_Aster ®
Version
6.4
Titrate:
Extrapolation of measurements on a digital model in dynamics
Date:
11/02/03
Author (S):
S. AUDEBERT, Key H. ANDRIAMBOLOLONA
:
R7.20.02-B Page
: 4/8
3 Step
of extrapolation
The step which one wishes to set up in order to extrapolate of the vibratory results of measurement
on a digital model breaks up into 4 stages [bib1]:
Experimental measurement of a physical size
Numerical calculation of
(displacement, speed, acceleration,
Nnum first
strain or stress) in NR
clean modes of the structure
exp points
Calculation of the relative generalized co-ordinates
at the base of projection (according to time
or of the frequency) by minimization
Restitution of the results in physical base
This step is based on the concept of projection of a field in a space of dimension
lower than the dimension of the space of the digital model and then extrapolation on the space of
digital model. The fact of projecting the field in a space of reduced size generates
necessarily a loss of information during extrapolation. One thus sees here the importance of the choice of
the base of projection. This base can made of modal answer and/or static answer. One
suppose here that the digital model is linear.
4
Relations between the physical sizes and the sizes
generalized
It is supposed that the intrinsic behavior of the structure is represented in a generated space
by the NR num basic vectors of projection. The transformation of Rayleigh-Ritz establishes the relation
between the degrees of freedom of the structure in the physical reference mark and its generalized co-ordinates:
Q =
In this formulation, the matrix of the basic vectors contains all space information;
generalized co-ordinates, as for them, depend:
· time, in the case of a calculation of temporal answer: Q (M, T) = (M) (T)
· pulsation, in the case of a harmonic calculation of answer: Q (M,) = (M) ()
Handbook of Référence
R7.20 booklet: Projection of results or measurements
HT-66/02/004/A
Code_Aster ®
Version
6.4
Titrate:
Extrapolation of measurements on a digital model in dynamics
Date:
11/02/03
Author (S):
S. AUDEBERT, Key H. ANDRIAMBOLOLONA
:
R7.20.02-B Page
: 5/8
One can thus deduce from them very simply the following relations:
Harmonic answer
Temporal answer
Displacement
Q (M,) = (M) ()
Q (M, T) = (M) (T)
Speed
& (
Q M,) = J (M) ()
& (
Q M, T) = (M) & (T)
Acceleration
& (
Q M,) = - 2 (M) ()
& (
Q M, T) = (M) & (T)
Deformations
(M,) = (M)
()
(M, T) = (M)
(T)
Constraints
(M,) = (M)
()
(M, T) = (M)
(T)
All these formulations thus present an equivalent form: in the continuation of the document, us
will treat primarily the case of temporal displacement, but the results obtained are
transposable with all the other sizes: speed, acceleration, strain and stress.
In the same way, the relations established according to time are applicable in the spectral field:
TF (Q (M, T)) = (M) TF ((T)) = (M) ()
TF - 1 (Q (M,)) = (M) TF - 1 (()) = (M) (T)
5
Calculation of the generalized co-ordinates
5.1
Formulation of the problem
The calculation of the generalized co-ordinates is carried out on the matrix of displacements
(respectively speeds, accelerations, strains, stresses) restricted with the degrees of freedom
measured, by resolution of the matric system:
qexp =
Dimensions of the matrix “to be reversed” are (NR
, NR
exp
num).
It is seen here that the calculation of the generalized co-ordinates is carried out in a restricted space:
dimension of the space generated by the basic vectors is lower than the dimension of the model
numerical, one exploits only information with the measured degrees of freedom.
5.2
Determination of a quasi-solution
For the resolution of the opposite problem, 3 cases can arise:
·
NR
= NR
exp
num: the number of measured degrees of freedom is equal to the number of vectors of
base projection which one wishes to identify the generalized co-ordinates.
In this case, there is a single solution with the problem of inversion: = - 1qexp
·
NR
> NR
exp
num: the number of measured degrees of freedom is higher than the number of
basic vectors of projection of the digital model which one wishes to identify them
co-ordinates generalize.
In this case, there is not exact solution with the problem of inversion. A quasi-solution
can however be defined, which minimizes the distance
: qexp -. The formula
+
= [T] Tq then provides the solution (single) within the meaning of least squares. In
exp
+
this expression, the matrix [T] T indicates the opposite matrix generalized of.
Handbook of Référence
R7.20 booklet: Projection of results or measurements
HT-66/02/004/A
Code_Aster ®
Version
6.4
Titrate:
Extrapolation of measurements on a digital model in dynamics
Date:
11/02/03
Author (S):
S. AUDEBERT, Key H. ANDRIAMBOLOLONA
:
R7.20.02-B Page
: 6/8
calculation of pseudo-opposite can be carried out by using the decomposition LU or
decomposition in singular values (SVD).
·
NR
< NR
exp
num: the number of measured degrees of freedom is lower than the number of vectors
basic of projection which one wishes to identify the generalized co-ordinates (what
corresponds to the case more running).
In this case, there is an infinity of solutions to the problem of inversion and the objective is of
to determine an acceptable solution by introducing an additional condition (standard
minimal of the solution or application of methods known as “of regularization”).
5.3
Determination of a regularized opposite solution
5.3.1 Principles of the methods of regularization
The goal of the methods of regularization [bib4], [bib5] is to propose an approximate and stable solution
with respect to the variations of the data input. One does not seek any more to solve the equation of
minimization resulting from the formulation: qexp =, but to determine an approximate solution (or
regularized) answering two requirements:
· it satisfies a condition of proximity: one seeks such as qexp - num <,
· it answers additional a condition a priori known as “information”.
The methods of regularization thus consist in supplementing the statement of the problem by introducing one
information to extract a priori, in the family of the solutions which are compatible with the data
experimental, that which corresponds best to the problem. This is done while amalgamating in a criterion
single a measurement of the fidelity of the solution compared to the experimental data and a measurement
of its fidelity to information a priori [bib2].
An approach which can be easily implemented in finished dimension is the regularization by
optimization. To bring closer the method of regularization of Tikhonov [bib3], it consists with
to consider a solution a priori priori problem of minimization and to seek the solution of
approximate system nearest to this solution. One then seeks to minimize the functional calculus
following:
2
2
qexp - + has - priori
Parameter A determines the affected weight with information a priori.
The solution of the equation of minimization is given by:
- 1
= [T + have] (Tqexp +
a priori)
or, while revealing explicitly the variation compared to the solution a priori:
- 1
=
T
T
priori + [+ have] (qexp - priori)
If priori = 0 is posed, this formulation consists in seeking the solution known as of “standard
minimal “(or Tikhonov of command 0).
The regularizing addition of the term related to the matrix have has as a role to shift the spectrum of T so
to ensure the stage of matric inversion. This step of calculation thus makes it possible to implement
a procedure of calculation conditioned better, which softens the effects of the noise and which provides a solution
physically acceptable.
In addition, the choice of the values of the matrix have results from a compromise between the stability of
required solution and the confidence which one can grant to the solution a priori.
Handbook of Référence
R7.20 booklet: Projection of results or measurements
HT-66/02/004/A
Code_Aster ®
Version
6.4
Titrate:
Extrapolation of measurements on a digital model in dynamics
Date:
11/02/03
Author (S):
S. AUDEBERT, Key H. ANDRIAMBOLOLONA
:
R7.20.02-B Page
: 7/8
5.3.2 Choice of information a priori
In the case of the methods of regularization, the choice of information a priori constitutes a stage-key
who determines the representativeness of the final results. This choice can be based on a knowledge
physics of the solution or on a knowledge of its evolution according to the parameter selected.
We provide, in the continuation, an example applied to the determination by minimization of one
temporal variable [bib1].
The minimization of a variable according to time can be carried out with each step of time
independently of the step of previous time. The introduction of information a priori allows
however to enrich the functional calculus by supposing a slow evolution by the given variables:
priori (T) = (T - dt)
This assumption is acceptable only when the step of sampling is sufficiently weak. In
effect, the solution at a given moment is approached by (development of Taylor):
(T) = (T - dt) +
dt & (T - dt) + O (dt)
The maximum frequency of response of the structure is determined by the pulsation of the mode of command it
higher max taken in modeling. One thus has:
(T) - (T - dt)
<
dt
max
+ O (dt)
(T - dt)
So that the corrective term is weak (and thus that information a priori constitutes an approximation with
first command of the required solution), the step of sampling must check:
1
dt <<
max
At the initial moment (t=0), since one does not have any information a priori on the solution, it
calculation is carried out by seeking the solution of minimal standard. In order to avoid propagating the error which
in results, it can be necessary to assign a weak confidence to information a priori on
first steps of time (via the parameter) and to exploit the results only from
the moment when one can consider that the errors sufficiently attenuated. If necessary, of
complementary studies will be undertaken in order to determine the optimal parameters of use of
functionality developed in Code_Aster.
In the frequential field, many possibilities are offered to determine information has
priori. They rest is on a physical knowledge of the solution (highlighted
experimental of resonances or forced answers), that is to say on a formulation of displacements
generalized according to the frequency (standard: functions gain), in which case minimization led
finally to characterize the dynamic stresses.
Handbook of Référence
R7.20 booklet: Projection of results or measurements
HT-66/02/004/A
Code_Aster ®
Version
6.4
Titrate:
Extrapolation of measurements on a digital model in dynamics
Date:
11/02/03
Author (S):
S. AUDEBERT, Key H. ANDRIAMBOLOLONA
:
R7.20.02-B Page
: 8/8
6
Implementation in Code_Aster
The base of projection is made up is dynamic modes calculated by the command
MODE_ITER_SIMULT [U4.52.03] stored in a concept of the mode_meca type, are modes
dynamic and of the static modes calculated by command DEFI_BASE_MODALE [U4.64.02]
stored in a concept of the base_modale type.
The phase of calculation of the generalized co-ordinates is treated by command PROJ_MESU_MODAL
[U4.73.01]. The data are gathered there under 4 key words factors.
The data relating to the digital model (projection bases) are gathered under the key word
factor MODELE_CALCUL. The digital model there is specified and projection bases.
The data relating to measurements are gathered under the key word factor MODELE_MESURE. One y
specify in particular the model associated with the structure and the measurement read by command LIRE_RESU.
The possible manual space association of the nodes is given under the key word factor
CORR_MANU.
The data concerning the resolution of the problem reverses are gathered under the key word factor
RESOLUTION. One specifies there the method of decomposition employed (LU, SVD) and the taking into account
of term of regularization.
The restitution of the results in physical base can then be carried out by the command
REST_BASE_PHYS [U4.63.21].
7 Bibliography
[1]
C. VARE: Extrapolation of experimental results of measurement on a digital model
in dynamics. Specification of the developments in Code_Aster. Note EDF/DER
HP-54/98/063/B
[2]
S. AUDEBERT: Comparative evaluation of various methods of inversion. Note EDF/DER
HP-62/93/036
[3]
A. TIKHONOV, V. ARSENINE: Methods of resolution of badly posed problems. ED. Mir -
1976
[4]
Mr. BONNET: Digital processing of problems opposite of source in linear accoustics.
Contract EDF - Convention P55L08/1E5240
[5]
A. TARANTOLA: Opposite problem theory - Methods for dated fitting and model parameter
estimate. Elsevier - 1987
Handbook of Référence
R7.20 booklet: Projection of results or measurements
HT-66/02/004/A
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