Random matrix theory for modeling uncertainties in computational mechanics

TLDR: In this paper, a nonparametric probabilistic approach of random uncertainties is presented for linear dynamical systems and for nonlinear dynamical system constituted of a linear part with additional localized nonlinearities.

About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2005-04-08 and is currently open access. It has received 269 citations till now. The article focuses on the topics: Linear dynamical system & Uncertainty quantification.

Figures

Fig. 2. Graphs of probability density functions p Λ̃j for j=1,...n corresponding to the order statistics of the random eigenvalues of random matrix [Gn].

Fig. 12. Graphs of functions r 7→δA (horizontal thin solid line), r 7→m δ̂0r (thin solid line) and r 7→m δ̂r (thick solid line).

Fig. 13. Graphs of functions r 7→σ δ̂0r (thin solid line) and r 7→σ δ̂r (thick solid line).

Fig. 8. Random normalized response spectrum in dB versus frequency in Hertz (log scale). Deterministic response of the mean model (mid irregular thin solid line). Mean value of the model with random uncertainties (mid smoothed thin solid line). Lower and upper envelopes of the confidence region (lower and upper thick solid lines). Extreme value statistics (lower and upper thin solid lines).

Fig. 7. Convergence in the nonlinear case. Graphs of functions ns 7→log10{Conv(ns,n)} for n=10,20 and 30 (three upper thin solid lines, for n=50 (lower thin solid line) and for n=100 (lower thick solid line).

Fig. 11. Shock Response Spectrum at observation point 2 in plate 1 with a point force excitation applied in plate 2.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Random matrix theory for modeling uncertainties in computational mechanics" ?

This paper deals with data uncertainties and model uncertainties issues in computational mechanics. The first part is devoted to randommatrix theory for which the authors summarize previous published results and for which two new ensembles of random matrices useful for the nonparametric models are introduced. In a second part, the nonparametric probabilistic approach of random uncertainties is presented for linear dynamical systems and for nonlinear dynamical systems constituted of a linear part with additional localized nonlinearities. 

Q2. What is the nonlinear evolution problem defined by Eq. (95)?

For given generalized mass, damping and stiffness matrices , the nonlinear evolution problem defined by Eq. (95) is solved by using the Newmark implicit step-bystep integration scheme and an additional numerical iteration procedure for solving the nonlinear algebraic equations at each time step. 

Q3. How many DOFs are in the mean finite element model of each substructure?

The mean finite element model of each substructure (plate 1, plate 2 and complex joint) is represented by a uniform finite element model of 4 nodes thin plates elements with 6222 DOFs for plate 1, 8052 DOFs for plate 2 and 2745 DOFs for the complex joint, e.g. a total amount of more than 16000 DOFs. 

Q4. what is the generalized mass, damping and stiffness matrices of the mean?

The generalized mass, damping and stiffness matrices [ Mn], [ Dn] and [ Kn] are positive-definite symmetric (n× n) real matrices such that [ Mn]αβ = µα δαβ , [ Dn]αβ =< [! ] β, α > and [ Kn]αβ = µα ω 2 α δαβ , in which, generally, [ Dn] is a full matrix. 

Q5. What is the reason for the increase in model uncertainties?

For such a complex system: (i) If an additional smaller spatial scale is introduced in the predictive model for reducing model uncertainties, then data uncertainties increase due to the increasing of the number of parameters. 

Q6. What is the definition of the mean reduced model of the nonlinear dynamic system?

the mean reduced model of the nonlinear dynamic system is written as the projection yn of y on Hn can be written as yn(t) = [ Φn] q n(t) in which the vector qn(t) ∈ #n of the generalized coordinates verifies the mean nonlinear differential equation,[ Mn] q̈ n(t) + [ Dn] q̇ n(t) + [ Kn] q n(t) + FnNL(q n(t), q̇n(t)) = Fn(t) , ∀t ≥ 0 , (88)where, for all q and p in #n,FnNL(q, p) = [ Φn] T fNL([ Φn] q, [ Φn] p) . (89)The principle of construction of the nonparametric probabilistic approach of random uncertainties for the linear and nonlinear dynamic systems whose mean finite element model is defined byC. Soize - Computer Methods in Applied Mechanics and Engineering (CMAME) (accepted in March 2004) 25Eq. (87), is given in Section 3. 

Q7. what is the convergence of the iteration algorithm for the construction of matrix?

The convergence of the iteration algorithm for the construction of matrix [Un] is analyzed in computing ε (j) = ‖ [Bn] − [B(j)n ] ‖ / ‖ [Bn] ‖ as a function of the iteration number j. 

Q8. What is the m-valued vector of the random reduced matrices?

If all these m DOFs were measured, then the m-valued vector Yexp(ω, θα) could be measured for each realization S(θα) of structure S. In point of fact, only a small number of DOFs can be measured and this is the mout -valued vector Zexp(ω, θα) introduced in Eq. (100) with m ≫ mout. 

Q9. what is the probability density function of each random variable?

The joint probability density function pΛ1,...,Λn(λ1, . . . , λn) with respect to dλ1 . . . dλn of random variables Λ1, . . . , Λn is written [28] aspΛ1,...,Λn(λ1, . . . , λn) = ![0,+∞[(λ1) × . . .× ![0,+∞[(λn) × C × (λ1 × . . . × λn)(n+1) (1−δ2) 2δ2× {Πα<β |λβ − λα|} e− (n+1) 2δ2 (λ1+...+λn) , (26)C. Soize - Computer Methods in Applied Mechanics and Engineering (CMAME) (accepted in March 2004) 10in whichC is a constant of normalization defined by the equation ∫ +∞ 0 . . . ∫ +∞ 0 pΛ1,...,Λn(λ1, . . . , λn) dλ1 . . . dλn = 1. Presently, the authors are interested in the probability density function of each random eigenvalue for the order statistics. 

Q10. What is the generalized eigenvalue problem associated with the mean finite element model?

The generalized eigenvalue problem associated with the mean mass and stiffness matrices of the mean finite element model is written as [" ] = λ [ ] . 

Q11. What is the first ensemble of random matrices used for modeling random uncertainties?

The second ensemble, that the authors call the pseudo-inverse ensemble of random matrices can be used for modeling random uncertainties in the coupling operator between an elastic solid and an acoustic fluid for structural-acoustic systems. 

Q12. What is the generalized mass, damping and stiffness of the mean reduced model?

It consists in substituting the generalized mass, damping and stiffness matrices of the mean reduced model (see Eq. (88)) by random matrices [Mn], [Dn] and [Kn]. 

Q13. What is the problem with the parametric probabilistic model of data uncertainties?

The problem is then to introduce a nonparametric probabilistic approach of data and model uncertainties allowing the mean-square error defined by Eq. (81) to be reduced. 

Q14. What is the first ensemble useful for modeling random uncertainties?

For instance, the first one is useful for modeling uncertainties of the mass operator of a dynamical system for which the spatial distribution of the mass is uncertain but for which the total mass is given.