Nonparametric Stochastic Modeling of Structures with Uncertain Boundary Conditions/Coupling Between Substructures

Abstract

The focus of this investigation is on the formulation and validation of a novel approach for the inclusion of uncertainty in the modeling of the boundary conditions of linear structures and of the coupling between linear substructures. This work is particularly relevant to complex structures assembled from simpler substructures as in aerospace applications. First, a mean structural dynamic model that includes boundary condition/coupling flexibility is obtained using classical substructuring concepts. The application of the nonparametric stochastic modeling approach to this mean model is next described and thus permits the consideration of both model and parameter uncertainty.

Uncertainty Quantification

“…We thus present an approach [141] that allows for taking into account uncertain boundary conditions by using the nonparametric probabilistic approach of model uncertainties. In addition, certain boundary conditions that are designed can have a complex behavior and cannot correctly be modeled and consequently, some modeling errors are systematically introduced in the mean computational model.…”

“…The method presented hereinafter can directly be extended to the uncertain coupling between substructures (see [141]). − A part of the boundary of the structure is not perfectly clamped, but corresponds to an uncertain flexible boundary with unknown and nonhomogeneous elastic properties.…”

“…13 Mean (left figure) and coefficient of variation in percent (right figure) of E bc (k, δ) corresponding to the response at the first resonance associated with the first elastic modes, as a function of k and δ [Figure from[141]]. …”

Nonparametric Probabilistic Approach of Model Uncertainties Introduced by a Projection-Based Nonlinear Reduced-Order Model

“…The second step consists in constructing a SROM by substituting the deterministic matrices of the ROM (such as the mass, the damping, and the stiffness reduced matrices) with random matrices for which the probability distributions are constructed using the Maximum Entropy (MaxEnt) principle [40,41,42] under the constraints defined by available information assocaited with algebraic properties (such as lower bounds, positiveness, integrability of the inverse, etc) and statistical information (such as the mean value equals to the nominal values, etc), and for which advanced algorithms have been developed for the high dimensions [39,43,44]. This approach has been extended to different ensembles of random matrices (see [38,45]), static boundary value problems [46], and has been experimentally validated and applied in many areas, including: dynamics of composite structures [47] and viscoelastic structures [48,49], dynamic substructuring techniques [50,51,52,53,54], vibroacoustic systems [48,55,56,57], soil-structure interactions and earthquake engineering [58,59,60], robust design and optimization [61,62], to name only a few. More recently, this nonparametric probabilistic approach has been extended in structural dynamics to nonlinear geometrical effects [63,64].…”

Random Matrix Models and Nonparametric Method for Uncertainty Quantification

“…the nonparametric stochastic modeling of uncertain structures with uncertain boundary conditions/coupling between substructures [79];…”