Stochastic model reduction for robust dynamical characterization of structures with random parameters

Ghienne, Martin; Blanzé, Claude; Laurent, Luc

doi:10.1016/j.crme.2017.09.006

Cited by 6 publications

(16 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In situations where the operator derivatives are not available, which is the case when using with black-box FEM solvers, numerical differentiation using finite differences can be used, especially if the operator is a linear function of the parameter. Such approach has been used for instance in [47].…”

Section: Implementation Detailsmentioning

confidence: 99%

A high order continuation method to locate exceptional points and to compute Puiseux series with applications to acoustic waveguides

Nennig

Perrey‐Debain

2020

Journal of Computational Physics

View full text Add to dashboard Cite

A numerical algorithm is proposed to explore in a systematic way the trajectories of the eigenvalues of non-Hermitian matrices in the parametric space and exploit this in order to find the locations of defective eigenvalues in the complex plane. These non-Hermitian degeneracies also called exceptional points (EP) have raised considerable attention in the scientific community as these can have a great impact in a variety of physical problems.The method requires the computation of successive derivatives of two selected eigenvalues with respect to the parameter so that, after recombination, regular functions can be constructed. This algebraic manipulation permits the localization of exceptional points (EP), using standard root-finding algorithms and the computation of the associated Puiseux series up to an arbitrary order. This representation, which is associated with the topological structure of Riemann surfaces allows to efficiently approximate the selected pair in a certain neighbourhood of the EP.Practical applications dealing with guided acoustic waves propagating in straight ducts with absorbing walls and in periodic guiding structures are given to illustrate the versatility of the proposed method and its ability to handle large size matrices arising from finite element discretization techniques. The fact that EPs are associated with optimal dissipative treatments in the sense that they should provide best modal attenuation is also discussed.

show abstract

Section: Implementation Detailsmentioning

confidence: 99%

A high order continuation method to locate exceptional points and to compute Puiseux series with applications to acoustic waveguides

Nennig

Perrey‐Debain

2020

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…If the operator is a linear function of the parameter, only the first derivative is requiered in the recurence formula (7) and a non-intrusive version of the method can be achieve. Similar approach has been used for instance in [25].…”

Section: Proposed Algorithmmentioning

confidence: 99%

“…This constitutes the major limitation of perturbation methods and is emphasized by the lack of intrinsic quantitative criterion to a priori define a "small variability" of a parameter and thus the validity domain of these methods. To tackle random parameter with larger variability, it is possible to carefully choose the expansion point [9] or to consider a first order Taylor expansion of the structure eigenvectors to estimate random eigenfrequencies [25]. This yields a non-intrusive and scalable approach to approximate the random eigenvalues with a better accuracy than the classical second order perturbation technique while reducing the computational costs.…”

Section: Introductionmentioning

confidence: 99%

Beyond the limitations of perturbation methods for real random eigenvalue problems using Exceptional Points and analytic continuation

Ghienne

Nennig

2020

Journal of Sound and Vibration

Self Cite

View full text Add to dashboard Cite

A numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods. The radius of convergence of eigenvalue perturbation methods, based on Taylor series, is known to decrease when eigenvalues are getting closer to each other. This phenomenon, knwon as veering in structural dynamics, is a direct consequence of the existence of branch point singularity in the complex plane of the varying parameters where some eigenvalues are defective. When this degeneracy, referred to as Exceptional Point (EP), is close to the real axis, the veering becomes stronger.The main idea of the proposed approach is to combined a pair of eigenvalues to remove this singularity. To do so, two analytic auxiliary functions are introduced and are computed through high order derivatives of the eigenvalue pair with respect to the parameter. This yields a new robust eigenvalue reconstruction scheme which is compared to Taylor and Puiseux series. In all cases, theoretical bounds are established and all approximations are compared numerically on a three degrees of freedom toy model. This system illustrate the ability of the method to handle the vibrations of a structure with a randomly varying parameter. Computationally efficient, the proposed algorithm could also be relevant for actual numerical models of large size, arising from other applications involving parametric eigenvalue problems, e.g., waveguides, rotating machinery or instability problems such as squeal or flutter.

show abstract

“…Sparse Polynomial Chaos method has been proposed by Blatman and Sudret [78] to decrease the computational time in a nonintrusive manner, nevertheless PC based methods remain complex to implement. In the context of this work, we choose to use a non-intrusive method referred to as the Stochastic Model Reduction (SMR) approach [27]. This approach benefits from simple implementation as perturbation methods and allows to consider random variables not limited to small variations.…”

Section: State Of Artmentioning

confidence: 99%

“…Finally, the characteristics of the parameters of this stochastic model of joint 2 are identified using an approximation of the maximum likelihood principle. A straightforward non-intrusive approach developed in previous work [27,28] and referred to as the Stochastic Model Reduction (SMR) approach is used to simulate the vibrational behaviour of the structure. The efficiency of this approach allows to integrate it in the identification procedure of a stochastic model of joint.…”

Section: Introductionmentioning

confidence: 99%

Robust characterization of the vibrational behaviour of light assembled structures with random parameters

Ghienne

Laurent

Blanzé

2020

Mechanical Systems and Signal Processing

Self Cite

View full text Add to dashboard Cite

Predictive simulations of assembled structures raises some computational challenges for usual industrial structures. It is indeed necessary to deal with non-linear phenomena, such as contact or friction, and to consider sources of uncertainties mainly responsible for the deviation between the effective behaviour of the structure and results from deterministic simulations. Each of these two features of assembled structures requires intensive computational resources and the significant number of joints of actual structures increases even more the computational cost. In order to address such problems, this work aims to provide a pragmatic approach to robustly characterize the vibrational behaviour of light assembled structures considering the variability of the parameters of the joints. All the uncertainties of the whole problem are assumed to be associated to the joint interfaces. First, a nominal model of a joint consistent with the studied structure is identified using experimental data. Then a stochastic model of a joint is defined by considering the mechanical design variables of the nominal model as random variables. Based on an approximation of the maximum likelihood principle, the parameters of this stochastic model of joint are identified. A straightforward non-intrusive strategy referred to as the Stochastic Model Reduction (SMR) approach is used in the identification procedure of the stochastic model of the joint. The ease of implementation and the efficiency of the SMR approach justify this choice. The proposed approach is applied to an academic structure which is representative of light assembled structures: a three beam bolted frame. A stochastic model of joint characterized by its stiffness is thus implemented for each elementary joint of this frame. It allows us to characterize the variability of the vibrational behaviour of the frame taking into account the variability of each joint. Finally the comparison of the predicted stochastic response with the experimental behaviour of the structure shows that the proposed approach allows to tackle the variability of the actual structure. The proposed approach has been developed in the purpose of industrial applications. Based on a pragmatic modeling of the joint, a non-intrusive numerical tool and a stochastic modeling, the whole strategy could be indeed easily implemented on more complex industrial assemblies.

show abstract

Stochastic model reduction for robust dynamical characterization of structures with random parameters

Cited by 6 publications

References 44 publications

A high order continuation method to locate exceptional points and to compute Puiseux series with applications to acoustic waveguides

A high order continuation method to locate exceptional points and to compute Puiseux series with applications to acoustic waveguides

Beyond the limitations of perturbation methods for real random eigenvalue problems using Exceptional Points and analytic continuation

Robust characterization of the vibrational behaviour of light assembled structures with random parameters

Contact Info

Product

Resources

About