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• Dwa rodzaje problemów
• Zagadnienie modelowania ( lub zagadnienie
  proste)
• Zagadnienie odwrotne (inwersja)

• Zagadnienie jakosciowego modelowania zrozumienie
  zachodzacego procesu,
• umozliwiajace przewidywanie zachowania sie
  (jakosciowego) systemu.
• Zagadnienie odwrotne (ilosciowe) ilosciowy opis
  obiektu, tzn, ilosciowy opis obiektu
• pozwalajacy na realistyczne (ilosciowe)
  wyznaczenie jego zachowania.

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System fizyczny
Parametry ukladu
Przewidywane mierzalne wielkosci
Parametry ustalone (znane a priori)
Modelowanie d f(m, must)
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Zagadnienie odwrotne
Laczenie informacji (wnioskowanie)
Obserwacja
Teoria
Wiedza a posteriori
Wiedza a priori
Zagadnienie odwrotne
Pomiar posredni (estymacja parametrów)
dobs mest
Modelowanie d f(m, must)
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Zagadnienie odwrotne
Podejscie optymalizacyjne
Przeszukiwanie przestrzeni modeli w celu
znalezienia modelu najlepiej z okreslonego
punktu widzenia odtwarzajacego dane pomiarowe.
Szukanie modelu najlepszego

• Metody niedeterministyczne
• metody Monte Carlo
• Algorytmy genetyczne i ewolucyjne
• Symulowane wyzarzanie
• inne

• Metody deterministyczne
• metody bezgradientowe
• metody gradientowe
• (konieczna analiza wrazliwosci)

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Sensitivity of eigenvalues and eigenvectors
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When the damage identification is based on the
modal data (that is free frequency spectrum and
vibration modes), their variation can be used in
the identification procedure.
The sensitivity derivatives provide the variation
of eigenvalues and eigenvectors with respect to
structural parameters.
Assume that the eigenvalues are distinct and the
structure parameter vector is denoted by s. The
components of s can described both the properties
of structural damage as well as some additional
properties of structure.
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• The state of a discretized model of linear
  elastic structure is described
• by the equation specifying the eigenvalue
  problem

k1,2,n
K and M are the n x n stiffness and mass
matrices,
is the eigenvalue corresponding to the square of
free frequency
is the n-dimensional eigenvector specifying the
amplitude of k-th mode

• For r available eigenvectors and eigenvalues

The n x r modal matrix
k 1,2,r
The r x r spectral matrix

• The eigenvectors are all K-orthogonal
• and Mnormalized

i,j 1, 2,n
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Sensitivity analysis
Differentiating with respect to component s of
vector s equation specifying the eigenvalue
problem and condition of M-normality, the
following set of equations can be obtained
can be one of selected eigenvalues and
eigenvectors.
Premultiplying the first equation by , the
first sensitivity derivative of the eigenvalue
can be obtained
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The sensitivity equations can be then written in
the form
()
and
()
The solution of () and () yields the desired
sensitivties of ? and ?. !!! However, as
is a singular matrix, the direct solution
of () is difficult to obtain and some other
methods should be used !!!
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I. The modal representation of eigenvector
derivative
where ckj are the coefficients to be specified.
Substituting this approximation into
one can obtain
and premultiplying it by
To specify the eigenvector derivative, we have to
know a set of eigenvectors which constitute a
reduced basis of proposed approximation .
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II. The static correction method
Only stiffness matrix K is using in solving
sensitivity equation
Thus
from which one obtains
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III. The combined approximation of modal
representation and static correction method

(cf. Sutter et al., 1998)
where the coefficients dkj are obtained by
substituting the above approximation into
sensitivity equation
and premultiplying it by .
Thus, it follows that
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IV. The Nelson method (cf. Nelson,1976)
, we have
.
The normalization condition
is replaced by the condition that the largest
component of eigenvector is equal to one and
other components are properly scaled.
Denoting such components by , and assuming
that , one has .
In the equation specifying the sensitivities
one may then delete the m-th row and m-th column
so the reduced matrix is
not singular and the sensitivity equation can be
solved by standard methods.
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V. The iterative approach to derive the
eigenvector variation
This approach can be constructed in terms of
recursive solutions of consecutive problems using
the same stiffness matrix. One can assume that
the mass matrix does not vary (which is a typical
case in damage identification) and only stiffness
matrix varies with structural parameters.
Thus, one can write
and
For small variation of eigenvalues and
eigenvectors, one can assume
and
where
Thus
and the recursive formula of static eigenvector
correction is obtained in the form
or
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A more accurate recursive formula can be obtained
as follows
Thus, one can obtain the recursive formula
with
The eigenvalue correction can be expressed in
terms of initial eigenvectors, namely
and, similarly as previously, one can have
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Free vibration problem
In order to increase the changes in eigenvalues
with respect to damage some additional support or
attached concentrated masses can be introduced.
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Example Real damaged beam or plate and
their mathematical models described
mostly numerically using FEM

• Control parameters within the beam or plate
  domain
• Additional support (rigid or elastic)
• Additional mass (rigidly or elastically attached
  to the structure)
• Additional load.
• introduced into structure in order to increase
  the sensitivity of vibration frequencies and
  modes with respect to localized damage.
• Damage can be modeled as the change of stiffness
  of some finite elements of structure model
  accompanied with no change of its mass.

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Sensitivity of damage indices in beam for
translating support or mass with respect to
damage location
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Global Indices for lightly damaged beam vs.
moving support (damage in element No. 10)
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Global Indices for severely damaged beam vs.
moving support (damage in elements No. 10 and 20)
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Global Indices for lightly damaged beam (damage
in element No. 10)
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Global Indices for damaged beam vs. moving
mass (damage in elements No. 10 and 20)
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Sensitivity of damage indices in a rectangular
plate

• Consider a rectangular plate with two edges
  built-in and two edges free.
• Two concentrated masses, first one mowing along
  x-line and second one
• mowing along y-line, are attached to the
  plate.
• The plate is divided into 8 x 4 elements, one
  element undergoes damage.

The variations of three first damage indices
with respect to location of masses is considered.
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Variation of DI1 versus masses locations
Variation of DI2 versus masses locations
Variation of DI3 versus masses locations
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Damage identification based on distance norm
dependent on varying structural parameter
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Formulation of identification problem
subject to
where di - identification parameters pk
- structural parameters
More general formulation
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Error /- 0.5
Error /- 1.0
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Error /- 0.5
Error /- 1.0
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No error
Error 0.5
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Identification problem
subjected to
Sensitivity analysis needs the introducing the
adjoint structures described by
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Identification problem (similarly to the static
case)
subjected to
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Identification problem
subjected to
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Following the adjoint sensitivity analysis, the
adjoint structures described by
are introduced and then the optimal mass
distribution is defined by the condition
where µ is the proper Lagrange multipler.
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Control parameters Location of distributed
heat sources or applied temperatures.
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Simple illustrative example One-dimensional
structure subjected to internal heat source
varying linearly.
Real structure S1 fixed k, f0 -
varying Model S1 varying k , f0 - varying
Distance norm
Identification problem
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Particular data
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History of
identification (bisection method for
dI/ds1 0 , dI/dk 0 )   k_max
13.731836 for s1 0.250000, I
0.1591E01 k_max 13.940742 for s1 0.400000,
I 0.1098E01 k_max 13.865089 for s1
0.325000, I 0.4694E00 k_max 13.908579 for
s1 0.362500, I 0.2541E00 k_max
13.888414 for s1 0.343750, I
0.1221E00, k_max 13.898869 for s1
0.353125, I 0.6229E-01 k_max 13.893738 for
s1 0.348437, I 0.3084E-01 k_max
13.896326 for s1 0.350781, I
0.1550E-01 k_max 13.895039 for s1 0.349609,
I 0.7729E-02 k_max 13.895683 for s1
0.350195, I 0.3870E-02 k_max 13.895361 for
s1 0.349902, I 0.1934E-02 k_max
13.895523 for s1 0.350049, I
0.9674E-03 k_max 13.895443 for s1 0.349976,
I 0.4833E-03 k_max 13.895483 for s1
0.350012, I 0.2420E-03 (s1_min.)
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Thanks for your attention !!!