Nonlinear model reduction for computational vibration analysis of structures with weak geometrical nonlinearity coupled with linear acoustic liquids in the presence of linear sloshing and capillarity

Abstract

International audienceThis paper deals with a novel formulation of a nonlinear reduced-order computational model for analyzing the non-linear vibrations of a linear viscoelastic structure with weak nonlinear geometrical effects, coupled with a linear acoustic liquid with sloshing and capillarity on the free surface. The model proposed is derived from the one used in fluid-structure interaction for linear systems, for which the analysis of the acoustic-sloshing-capillarity phenomena is efficient thanks to the use of a projection on the linear modes of the linear acoustic liquid and on the sloshing modes with capillarity. Concerning the construction of the vector basis for the structure, it is proposed to use a POD approach for the viscoelastic structure with weak nonlinear geometrical effects and taking into account the added mass induced by the liquid. The methodology for constructing the vector bases of the admissible sets and for obtaining the non-linear ROM are detailed. The computational nonlinear ROM that is presented can directly be used for analyzing the vibrations of such fluid-structure systems using commercial finite element softwares for computing the vector bases and the projections

Revisiting the experiment of a free-surface resonance of a liquid in a vibration tank using a nonlinear fluid–structure computational model

“…As shown in (Ohayon and Soize, 2016), the modal bases are computed by solving three different generalized eigenvalue problems: the first one being related to the elastic structure in presence of the liquid, considered as incompressible; the second one being related to the compressible liquid in a rigid container; the third one being related to the free surface in presence gravity field and with surface tension effects. In addition, the computation of the reducedorder bases exhibits major difficulties in terms of computational cost when studying a large-scale fluid-structure model due (i) to the high flexural motion of the thin-walled cylinder and (ii) to the free surface.…”

Novel formulation for the effects of sloshing with capillarity on elastic structures in linear dynamics

“…Because the objective is to detail the formulation and to quantify the role played by the contact line between the free surface of the compressible liquid and the elastic structure in presence of sloshing and capillarity effects, the developments are restricted to the linear case, because the introduction of nonlinearities does not change the analysis and the nature of the difficulties induced by the contact line. The case of linear elastic structure coupled with a nonlinear incompressible fluid can be found in Chu and Kana 1967 [19] and in Dias and Kharif 1999 [20], while the case of a nonlinear elastic structure with a linear compressible fluid can be found in Ohayon and Soize 2016 [21] and in Akkaoui et al 2019 [22] that reinterprets the experimental studies performed by Abramson et al 1966 [23] and 1970 [24].…”

Nonlinear Dynamical Analysis for Coupled Fluid-Structure Systems

“…Let U(t), P(t), and H(t) be the R n S , R n F , and R n H -vectors corresponding to the finite element discretization of the structural displacement, fluid pressure, and free-surface elevation fields. The computational model is then written [3] as,…”

“…is the nonlinear term issued from the large displacements/deformations induced by the geometrical nonlinearities. An adapted numerical nonlinear reduced-order model of order N requiring the numerical computation of the elastic modes of the structure with fluid added mass effect, of the acoustic modes of the fluid, and of the sloshing modes of the free surface [1] is proposed in [3]. Such computation on mid-power computers can be very challenging when large finite element meshes are involved.…”

“…A linear elastic constitutive equation is considered for the structure. It is also assumed that the structure undergoes sufficiently large deformations and large displacements in order to consider the geometrical nonlinear effects [2], but also sufficiently moderate so that the fluid behavior remains linear [3]. A total Lagrangian formulation around a static equilibrium state taken as a reference configuration is used.…”