Maximum entropy approach for modeling random uncertainties in transient elastodynamics.
Summary (4 min read)
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.
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Citations
577 citations
Cites background or methods from "Maximum entropy approach for modeli..."
...Finally, the methods presented herein can be extended in a straightforward manner to situations where the Hermitian range space of the transformations is replaced with more general spaces such as the space of matrices or a Banach space [20, 21]....
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...However, in the context of modeling physical systems that exhibit uncertainty either in their behavior or in their environment, probability theory is often used as a framework for modeling the uncertainty [14, 10, 20, 21]....
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338 citations
269 citations
Cites background or methods from "Maximum entropy approach for modeli..."
...(2) The second ensemble SE of random matrices, herein called the the positive-definite ensemble, has been constructed in [28,29], simultaneously with SG....
[...]
...(1) The first ensemble SG of random matrices, herein called the the normalized positive-definite ensemble, has recently been constructed (see [28,29]) in the context of the development of a new approach for modeling random uncertainties in dynamical systems with a nonparametric approach....
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...First, for readability of the paper, the results concerning the main ensemble introduced in [28,29] are summarized....
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...Probability density function p[Gn]([Gn]) is then written ([28,29]) as p[Gn]([Gn]) = " n (!)([Gn])×CGn × ( det [Gn] )(n+1) (1−δ2) 2δ2 × exp { − + 1) 2δ2 tr [Gn] } , (12)...
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...[29], it is proved that the dispersion of the probability model is fixed by giving parameter δ which has to be independent of n and which has to be such that 0 < δ < √ (n + 1)(n + 5)−1 ....
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226 citations
Cites background or result from "Maximum entropy approach for modeli..."
...Positive-definite ensemble SE of random matrices The second ensemble SE of random matrices, called the the positive-definite ensemble, has been constructed in [14,15], simultaneously with SG....
[...]
...[15], it is proved that the dispersion parameter of the probability model has to be independent of n and has to be such that...
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...[15] or [23]) that the family {Qn(ω) , ω ∈ B} of random variables verifying Eq....
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...[14,15,25]) and will be summarized in Section 5....
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...[15]) that the sequence of second-order stochastic fields {V (x, ω), x ∈ I, ω ∈ B}n≥1 converges to a second-order stochastic field {V (x, ω), x ∈ I, ω ∈ B} when dimension n goes to infinity for the norm defined by Eq....
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179 citations
Cites background or methods from "Maximum entropy approach for modeli..."
...From Refs....
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...[74,75,81], we introduced the ensemble SG of the random matrices [Gn], defined on a probability space (Θ, T , P )with values in!+n ( ), whose probability distribution was constructed by using the entropy optimization principle [65,34,36,37]...
[...]
...In Refs....
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...[74,75], it can directly be deduced that this probability distribution on !+n ( ) is defined by a probability density function [Gn] 7→ p[Gn(x)]([Gn]) from !+n ( ) into , with respect to the measure d̃Gn on !Sn( ) defined by d̃Gn = 2 n(n−1)/4 Π1≤i≤j≤n d[Gn]ij , (39)...
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...However, in Refs....
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References
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Frequently Asked Questions (15)
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.