Maximum entropy approach for modeling random uncertainties in transient elastodynamics.

TL;DR: 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.

Summary

Introduction

• A new approach is presented for analyzing random uncertainties in dynamical systems.
• 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).
• That wewill 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).the authors.
• In addition, it should be noted that Eqs. (65)-(68) have been deduced from the convergence analysis carried out.
• Section III is devoted to construction of the nonparametric model of random uncertainties for the reduced matrix model.

A. Definition of the mean boundary value problem

• The authors 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.
• B. Variational formulation of the mean boundary value problem Below, Eqs. (14) and (17) will be used to prove the convergence properties of the stochastic transient response.
• From the usual reference books (see for instance Ref. 33), the authors deduce that the problem defined by Eq. (12) with Eq. (4) has a unique solution t 7→ u(t) with values in , that they refer to as the transient response of the mean model of the structure submitted to impulsive loads.
• Note that underlined quantities refer to the mean mechanical model.

B. 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}.
• Let t 7→ u(t) from !+ into be the unique solution of Eq. (12) with the initial conditions defined by Eq. (4) and let un(t) be the projection of u(t) on n (in structural dynamics, this corresponds to the usual mode-superposition method).

A. Principle of construction

• It should be noted that if the boundary value problem defined by Eqs. (1)-(3) corresponds to an exact mechanical model of the structure under consideration, there are no uncertainties in the model which is then sure.
• Soc. Am. 10 C. Soize - Revised version - February 2001 allows the transient response to be predicted with good accuracy.
• This means that the variational formulation of the mean boundary value problem does not constitute available information for constructing the nonparametric model of random uncertainties.

B. Random reduced matrix model

• Using the construction principle presented in Section III.A, the random reduced matrix model J. Acoust.
• Soc. Am. 11 C. Soize - Revised version - February 2001 associated with the mean reduced matrix model introduced in Section II.
• It should be noted that the mathematical property related to the positiveness of the random matrices is absolutely fundamental and is required so that the second-order differential equation in time corresponds effectively to a dynamical system.

D. Available information for the construction of the nonparametric model

• The authors have to define the available information which is useful for constructing the probabilistic model.
• The basic available informations are the mean reduced matrix model, the positive-definiteness of the random generalized matrices and the existence of second-order moments of inverses of these random generalized matrices.
• The mean reduced matrix model is constituted of mean generalized mass, damping and stiffness matrices [ Mn], [ Dn] and [ Kn] defined in Section II.B and which belong to !+n ( ).
• Random generalized mass, damping and stiffness matrices [Mn], [Dn] and J. Acoust.
• This construction is presented in Section IV.

IV. PROBABILITY MODEL FOR SYMMETRIC POSITIVE-DEFINITE REAL RANDOM

• In a part of this section, the authors recall the main results established in Refs. 28 and 29 concerning the construction of a probability model for random matrices with values in +n (!) using the entropy optimization principle which allows only the available information to be used.
• It should be noted that the results obtained and presented below differ from the known results concerning Gaussian and circular ensembles for random matrices (orthogonal, symplectic, unitary and antisymmetric Hermitian ensembles) which have been extensively studied in the literature (see for instance Refs 38 to 43).
• In another part of this section, the authors complete the construction given in Refs. 28 and 29 in order to obtain a consistent probabilistic model which allows the convergence properties to J. Acoust.
• Soc. Am. 13 C. Soize - Revised version - February 2001 be studied when dimension n approaches infinity.
• In particular, the authors give an explicit expression of parameter λAn as a function of scalar parameter δA which is independent of dimension n of random matrix [An] and which allows the dispersion of random matrix [An] to be given.

C. Probability model using the maximum entropy principle

• When λAn is an integer, the probability distribution defined by Eq. (50) or (51) is a Wishart distribution47,48.
• If λAn is not an integer, the probability distribution defined by Eq. (50) or (51) is not a Wishart distribution.
• D. Dispersion parameter δA of random matrix [An] Since [An] is a positive-definite real matrix, there is an upper triangular matrix [LAn ] in n(!) such that [An] = [LAn ] T [LAn ] , (56) which corresponds to the Cholesky factorization of matrix [An].
• From convergence considerations when n → +∞ (see Section VI) and J. Acoust.
• The following results, which allow a procedure for the Monte Carlo simulation of random matrix [An] to be defined, are proved28,29: (1) Random variables {[LAn ]jj′ , j ≤ j′} are independent.

B. Construction of the stochastic transient response

• Below, the authors present a formulation which is adapted to Monte Carlo numerical simulation.
• 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).
• Since matrices [Mn], [Dn] and [Kn] are full matrices (not diagonal) as samplings of random matrices [Mn], [Dn] and [Kn], Eq. (84) is not used but secondorder differential Eq. (81) is solved directly using an unconditionally stable implicit step-by-step integration method (such as the Newmark integration scheme2) with initial conditions defined by Eq. (82).
• 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.
• It should be noted that for many applications, integer n is sufficiently high that λM , λD and λK can be considered as positive integers without introducing any significant limitation in the model.

VI. CONVERGENCE PROPERTIES AS THE DIMENSION APPROACHES INFINITY

• A. Introduction of norms useful for the convergence properties.
• As above, all the random variables are defined on probability space (A, T , P ).

B. Prerequisite to the construction of basic inequalities

• For ω fixed inA, the norm of matrix [GAn(ω)].
• The authors have numerically verified Eq. (97) using a Monte Carlo numerical simulation based on Eqs. (95)-(96), Section IV.E and the usual estimator of the second-order moment of random variable 1/Σ̃1.
• These numerical results confirm Eq. (97) which is mathematically proved.

C. Basic inequalities derived from the random energy equation

• Be the vector spaces of all the second-order random variables defined on probability space (A, T , P ) with values in ! and " respectively.
• And that the sequence of functions {t 7→ respectively.

A. Definition of the mean model

• This plate is simply supported on 3 edges and free on the fourth edge corresponding to x2 = 0 (see Figure 2).
• The spectral problem related to the mean reduced matrix model is analyzed using the finite element method.
• All the finite elements are the same and each one is a 4-node square plate element.
• It can be seen in Figure 4 that the main part of the energy of impulse function e is distributed over the [150 , 250].
• It is assumed that the damping rate ξ of the mean model is 0.001 for frequencies around 200 Hz.

B. Transient response of the mean model

• The transient response of the mean model is calculated by solving the evolution problem defined by Eqs. (26)-(27) using an unconditionally stable implicit step-by-step integration method (Newmark integration scheme) with a time step size ∆t = 1/4000 s.
• This time-step corresponds to 10 timesteps per period for the structural mode of the mean model whose eigenfrequency is ν61 = 402.24 Hz.
• From Figure 5, it can be deduced that the transient response of the mean model is reasonably converged when n ≥ 40.
• Figure 6 shows the graph of function J. Acoust.
• For n = 40, the corresponding value of the dynamic magnification factor is bn = 1.85.

C. Transient response of the model with random uncertainties

• Therefore, for the convergence analysis with respect to dimension n of the reduced matrix model with random uncertainties, the authors have to consider n ≥ n0 = 4.
• The dispersions of the generalized mass, damping and stiffness random matrices of the reduced matrix model with random uncertainties, are controlled by parameters δM , δD and δK introduced in Section V.A, which have to verify the constraints defined by Eq. (76), 0 < δM , δD, δK < 0.7453 .
• The authors are interested in the random response ratio Rn(t) defined by Eq. (35) and the random dynamic magnification factor Bn = maxt≥0 Rn(t) defined by Eq. (36).
• The transient response of the structure with random uncertainties is calculated using the Monte Carlo numerical simulation method.
• This value has to be compared to the value for the mean model which is 1.85.

VIII. CONCLUSIONS

• The authors have presented a new approach allowing the random uncertainties to be modeled by a nonparametric model for prediction of transient responses to impulsive loads in linear structural dynamics.
• The nonparametric approach presented is useful when the number of uncertain parameters is high or when the probabilistic model is difficult to construct for the set of parameters considered.
• Soc. Am. 28 C. Soize - Revised version - February 2001 description of the local parameters of the mechanical model.
• This convergence analysis carried out has allowed the consistency of the theory proposed to be proved and the parameters of the probability distribution of the random generalized matrices to be clearly defined.

Full text

HAL Id: hal-00686287
https://hal-upec-upem.archives-ouvertes.fr/hal-00686287
Submitted on 10 Apr 2012
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
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.