Maximum entropy approach for modeling random uncertainties in transient elastodynamics.

TLDR: An explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to Monte Carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads.

Abstract: A new approach is presented for analyzing random uncertainties in dynamical systems. This approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics. The information used does not require the description of the local parameters of the mechanical model. The probability model is deduced from the use of the entropy optimization principle, whose available information is constituted of the algebraic properties related to the generalized mass, damping, and stiffness matrices which have to be positive-definite symmetric matrices, and the knowledge of these matrices for the mean reduced matrix model. An explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to Monte Carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads. The fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed. Finally, an example is presented.

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Maximum entropy approach for modeling random
uncertainties in transient elastodynamics
Christian Soize
To cite this version:
Christian Soize. Maximum entropy approach for modeling random uncertainties in transient elasto-
dynamics. Journal of the Acoustical Society of America, Acoustical Society of America, 2001, 109 (5),
pp.1979-1996. �10.1121/1.1360716�. �hal-00686287�

MAXIMUM ENTROPY APPROACH FOR MODELING RANDOM
UNCERTAINTIES IN TRANSIENT ELASTODYNAMICS
Christian Soize
Structural Dynamics and Coupled Systems Department, ONERA, BP 72, 92322 Chatillon Cedex,
France
ABSTRACT
A new approach is presented for analyzing random uncertainties in dynamical systems. This
approach consists in modeling random uncertainties by a nonparametric model allowing transient
responses of mechanical systems submitted to impulsive loads to be predicted in the context of
linear structural dynamics. The information used does not require the description of the local
parameters of the mechanical model. The probability model is deduced from the use of the entropy
optimization principle whose available information is constituted of the algebraic properties related
to the generalized mass, damping and stiffness matrices which have to be positive-definite symmetric
matrices, and the knowledge of these matrices for the mean reduced matrix model. An explicit
construction and representation of the probability model have been obtained and are very well suited
to algebraic calculus and to Monte Carlo numerical simulation in order to compute the transient
responses of structures submitted to impulsive loads. The fundamental properties related to the
convergence of the stochastic solution with respect to the dimension of the random reduced matrix
model is analyzed. Finally, an example is presented.
PACS numbers: 43.40
Keywords: Random uncertainties; dynamical systems, structural dynamics; structural acoustics;
transient response; impulsive load; entropy optimization principle
INTRODUCTION
This paper deals with predicting the transient responses of structures submitted to impulsive loads in
linear structural dynamics. The theory presented below can be extended without any difficulties to
structural-acoustic problems such as a structure coupled with an internal acoustic cavity. In general,
this kind of prediction is relatively difficult because the structural models have to be adapted to
J. Acoust. Soc. Am. 1 C. Soize - Revised version - February 2001

large, medium and small vibrational wavelengths which correspond to the low-, medium- and
high-frequency ranges.
Here, we are interested in the case where the impulsive load under consideration has an energy
which is almost entirely distributed over a broad low-frequency band and for which prediction of
the impulsive load response can be obtained with a reduced matrix model constructed using the
generalized coordinates of the mode-superposition method associated with the structural modes
corresponding to the n lowest eigenfrequencies of the structure. It should be noted that, for a
complex structure, only a numerical approximation of the first structural modes can be calculated
using a large finite element model of the structure. The low-frequency case considered in this paper
is important for many applications, and details concerning such a case can be found in the literature
on structural dynamics and vibrations (see Refs. 1 to 8).
Under the above assumptions and for a complex structure, dimension n of the reduced matrix model
generally has to be high (several dozen or hundred structural modes may be necessary to predict
transient responses). However, it is known that the higher the eigenfrequency of a structural mode,
the lower its accuracy because the uncertainties in the model increase (in linear structural dynamics
and vibrations, the effects of uncertainties on the model increase with the frequency and it should
be kept in mind that the mechanical model and the finite element model of a complex structure tend
to be less reliable in predicting the higher structural modes). This is why random uncertainties in
the mechanical model have to be taken into account. This is a fundamental problem in structural
dynamics and in structural acoustics when the mechanical model has to be adapted to predict a
transient response for which not only the low-frequency band is mainly concerned, but also the
upper part of this low-frequency band and maybe the medium-frequency-band have to be taken into
account.
Random uncertainties in linear structural dynamics and structural acoustics are usually modeled
using parametric models. This means that 1) the uncertain parameters (scalars, vectors or fields)
occurring in the boundary value problem (geometrical parameters; boundary conditions; mass
density; mechanical parameters of constitutive equations; structural complexity, interface and
junction modeling, etc.) have to be identified; 2) appropriate probabilistic models of these uncertain
parameters have to be constructed, and 3) functions mapping the domains of uncertain parameters
into the mass, damping and stiffness operators have to be constructed. Concerning details related
to such a parametric approach, we refer the reader to Refs. 9 to 15 for general developments, to
J. Acoust. Soc. Am. 2 C. Soize - Revised version - February 2001

Refs. 16 to 21 for general aspects related to stochastic finite elements and to Refs. 22 to 27 for
other aspects related to this kind of parametric models of random uncertainties in the context of
developments written in stochastic dynamics and parametric stochastic excitations.
In this paper we present a new approach, that we will call a nonparametric approach, for constructing
a model of random uncertainties in linear structural dynamics in order to predict the transient
response of complex structures submitted to impulsive loads (as indicated above, this approach
can be directly extended to structural-acoustic problems). This nonparametric model of random
uncertainties does not require identifying the uncertain parameters in the boundary value problem as
described above for the parametric approach but is based on the use of recent research (see Refs. 28
and 29) in which the construction of a probability model for symmetric positive-definite real random
matrices using the entropy optimization principle has been introduced and developed. These results
will allow the direct construction of a probabilistic model of the reduced matrix model deduced
from the variational formulation of the boundary value problem to be obtained, for which the only
information used in this construction is the available information constituted of the mean reduced
matrix model, the existence of second-order moments of inverses of the random matrices and some
algebraic properties relative to the positive-definiteness of these random matrices. It should be
noted that these properties have to be taken into account in order to obtain a mechanical system with
random uncertainties, which models a dynamical system. For instance if there are uncertainties on
the generalized mass matrix, the probability distribution has to be such that this random matrix be
positive definite. If not, the probability model would be wrong because the generalized mass matrix
of any dynamical system has to be positive definite.
In Refs. 28 and 29, we presented the calculation of the matrix-valued frequency response functions
for discretized linear dynamical systems with random uncertainties. Unfortunately, convergence
results were not obtained yet and consequently, a parameter of the probability model were not
clearly defined for a designer. In this paper, an explicit construction of the probabilistic reduced
matrix model of finite dimension n is given and its convergence is studied as n approaches infinity.
In such a probabilistic theory, it seems absolutely fundamental to prove the convergence. It is not
self-evident that convergence properties exist in such a construction. In addition, it should be noted
that Eqs. (65)-(68) have been deduced from the convergence analysis carried out. Thanks to this new
analysis presented in this paper, we have obtained a new consistent and coherent theory in which all
the parameters are clearly defined. In Section I, the mean boundary value problem is introduced and
J. Acoust. Soc. Am. 3 C. Soize - Revised version - February 2001

its variational formulation is given in order to construct the mean reduced matrix model, which is
carried out in Section II, using the mode-superposition method. Section III is devoted to construction
of the nonparametric model of random uncertainties for the reduced matrix model. In this section,
we introduce the available information which is directly used for constructing the probabilistic
model of random uncertainties. In Section IV, we give a summary of the main results established
in Refs. 28 and 29 concerning the probability model for symmetric positive-definite real random
matrices and we complete this construction in order to obtain a consistent probabilistic model useful
for studying convergence as dimension n approaches infinity. The nonparametric model of random
uncertainties for the reduced matrix model constructed using Sections III and IV, is presented in
Section V. The convergence properties of this nonparametric model of random uncertainties as
dimension n approaches infinity are given in Section VI. The convergence properties prove that the
construction proposed is consistent. Finally, an example is presented in Section VII.
I. MEAN BOUNDARY VALUE PROBLEM FOR MEAN TRANSIENT RESPONSE AND
ITS VARIATIONAL FORMULATION
A. Definition of the mean boundary value problem
We consider the linear transient response of a three-dimensional damped fixed structure around
a static equilibrium configuration considered as a natural state without prestresses, submitted to
an impulsive load. The mean mechanical model is described by the following mean boundary
value problem. Let Ω be the bounded open domain of
3
occupied by the mean structure at
static equilibrium and made of viscoelastic material without memory. Let ∂Ω = Γ
0
∪ Γ be the
boundary such that Γ
0
∩ Γ = ∅ and let n be its outward unit normal. Let u = (u
1
, u
2
, u
3
) be
the displacement field at each point x = (x
1
, x
2
, x
3
) in Cartesian coordinates. On part Γ
0
of the
boundary, the structure is fixed (u = 0) while on part Γ it is free. There are external prescribed
impulsive volumetric and surface force fields applied to Ω and Γ, written as {g
vol
(x, t), t ≥ 0} and
{g
surf
(x, t), t ≥ 0} respectively. Let T be a positive real number. The mean transient response
{u(x, t), x ∈ Ω, t ∈ [0, T ]} is the solution of the following mean boundary value problem:
ρ ¨u
i
−
∂σ
ij
∂x
j
= g
vol,i
in Ω , t ∈ [0, T ] , (1)
σ
ij
n
j
= g
surf,i
on Γ , t ∈ [0, T ] , (2)
u
i
= 0 on Γ
0
, t ∈ [0, T ] , (3)
J. Acoust. Soc. Am. 4 C. Soize - Revised version - February 2001

Frequently asked questions

Q1. What are the contributions in "Maximum entropy approach for modeling random uncertainties in transient elastodynamics" ?

A new approach is presented for analyzing random uncertainties in dynamical systems. Finally, an example is presented. 

Q2. What have the authors stated for future works in "Maximum entropy approach for modeling random uncertainties in transient elastodynamics" ?

1. Nevertheless, experiments are in progress to study the correlation which could exist between the dispersion of the random responses and parameters δM, δD and δK associated with the random generalized matrices. 

Q3. What are the advantages of parametric approaches?

The parametric approaches existing in literature are very useful when the number of uncertain parameters is small and when the probabilistic model can be constructed for the set of parameters considered. 

Q4. What is the dynamic magnification factor of the mean model?

The dynamic magnification factor increases when the random uncertainties increase, and is greater than the deterministic dynamic amplification factor of the mean model. 

Q5. What is the definition of the variational formulation of the mean boundary value problem?

The variational formulation of the mean boundary value problem defined by Eqs. (1)-(3) consists in finding a function t 7→ u(t) with values in such thatm(ü, v) + d(u̇, v) + k(u, v) = g(v ; t) , ∀v ∈ , ∀t ∈ [0, T ] , (12) with the initial conditions defined by Eq. (4). 

Q6. What is the generalized damping matrix of the mean reduced matrix model?

The generalized damping matrix [ Dn] of the mean reduced matrix model, defined by Eq. (29), is written as [ Dn] = 2 ξ Ωref[ Mn] in which [ Mn] is the generalized mass matrix of the mean reduced matrix model, defined by Eq. (29), and where Ωref = 2 π × 200 rad/s. 

Q7. What is the method for calculating the variance of a random matrices?

In addition, the authors have to calculate multiple integrals in a higher dimension (see Eq. (85)) for which a well suited method consists in using a Monte Carlo calculation with or without variance reduction procedures49−55. 

Q8. What is the eigenfunction of the mean reduced matrix model?

The mean reduced matrix model is obtained using the Ritz-Galerkin projection of the variational formulation of the mean boundary value problem on the subspace n of spanned by the structural modes {1 , . . . , n } of the mean structural model, which correspond to the n lowest eigenfrequen-cies {ω1, . . . , ωn}. 

Q9. What is the convergent model of the Monte Carlo numerical method?

For nS sufficiently high (nS ≥ 300) the Monte Carlo numerical method is reasonably converged and it can be seen that the nonparametric model proposed is convergent with respect to dimension n of the random reduced matrix model (see Section VI.D). 

Q10. What is the way to predict the transient response of a complex structure?

For a complex structure, such a mean boundary value problem defined by Eqs. (1)-(3) is not able to predict the transient response due to impulsive loads whose energy is distributed over a very broad frequency band, i.e. over the low-, medium- and high-frequency ranges (for instance, if there is energy in the medium-frequency range, more advanced probabilistic mechanical models such as the fuzzy structure theory have to be used to take into account the role played by the structural complexity8); the most that this kind of deterministic mean boundary value problem is able to predict is the transient response due to impulsive loads whose energy is mainly distributed over a broad lowfrequency range for which the mean reduced matrix model defined by Eqs. (25)-(27) is suitable andJ. Acoust. 

Q11. what is the hn(t) function of the linear filter associated with second-order?

Fn(τ) dτ , (84)in which t 7→ [hn(t)] is the matrix-valued impulse response function of the linear filter associated with second-order differential Eq. (81). 

Q12. What is the variational form of the themean boundary value problem?

The variational formulation of themean boundary value problem is absolutely necessary to construct the mean reduced matrix model in the general case. 

Q13. How many timesteps per period is the mean reduced matrix model?

This time-step corresponds to 10 timesteps per period for the structural mode of the mean model whose eigenfrequency is ν61 = 402.24 Hz. 

Q14. What is the probability density function of random matrix?

Let us consider ν random matrices [A1n], . . . , [A ν n] with values in +n (!) such that for each j in {1, . . . , ν}, the probability density function of random matrix [Ajn] satisfies Eqs. (47)-(49). 

Q15. What is the way to describe the probability model?

An explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to Monte Carlo numerical simulation.