Reduced-order modeling of geometrically nonlinear rotating structures using the direct parametrisation of invariant manifolds

The direct parametrisation method for invariant manifolds is a nonlinear reduction technique which derives nonlinear mappings and reduced-order dynamics that describe the evolution of dynamical systems along a low-dimensional invariant-based span of the phase space. It can be directly applied to finite element problems. When the development is performed using an arbitrary order asymptotic expansion, it provides an efficient reduced-order modeling strategy for geometrically nonlinear structures. It is here applied to the case of rotating structures featuring centrifugal effect. A rotating cantilever beam with large amplitude vibrations is first selected in order to highlight the main features of the method. Numerical results show that the method provides accurate reduced-order models (ROMs) for any rotation speed and vibration amplitude of interest with a single master mode, thus offering remarkable reduction in the computational burden. The hardening/softening transition of the fundamental flexural mode with increasing rotation speed is then investigated in detail and a ROM parametrised with respect to rotation speed and forcing frequencies is detailed. The method is then applied to a twisted plate model representative of a fan blade, showing how the technique can handle more complex structures. Hardening/softening transition is also investigated as well as interpolation of ROMs, highlighting the efficacy of the method

Model Order Reduction based on Direct Normal Form: Application to Large Finite Element MEMS Structures Featuring Internal Resonance

International audienceDimensionality reduction in mechanical vibratory systems poses challenges for distributed structures including geometric nonlinearities, mainly because of the lack of invariance of the linear subspaces. A reduction method based on direct normal form computation for large finite element (FE) models is here detailed. The main advantage resides in operating directly from the physical space, hence avoiding the computation of the complete eigenfunctions spectrum. Explicit solutions are given, thus enabling a fully non-intrusive version of the reduction method. The reduced dynamics is obtained from the normal form of the geometrically nonlinear mechanical problem, free of non-resonant monomials, and truncated to the selected master coordinates, thus making a direct link with the parametrisation of invariant manifolds. The method is fully expressed with a complex-valued formalism by detailing the homological equations in a systematic manner, and the link with real-valued expressions is established. A special emphasis is put on the treatment of second-order internal resonances and the specific case of a 1:2 resonance is made explicit. Finally, applications to large-scale models of Micro-Electro-Mechanical structures featuring 1:2 and 1:3 resonances are reported, along with considerations on computational efficiency

Bronchial hyperreactivity and spirometric impairment in polysensitized patients with allergic rhinitis

<p>Abstract</p> <p>Background</p> <p>We previously demonstrated in a group of patients with perennial allergic rhinitis alone impairment of spirometric parameters and high percentage of subjects with bronchial hyperreactivity (BHR). The present study aimed at evaluating a group of polysensitized subjects suffering from allergic rhinitis alone to investigate the presence of spirometric impairment and BHR during the pollen season.</p> <p>Methods</p> <p>One hundred rhinitics sensitized both to pollen and perennial allergens were evaluated during the pollen season. Spirometry and methacholine bronchial challenge were performed.</p> <p>Results</p> <p>Six rhinitics showed impaired values of FEV1 without referred symptoms of asthma. FEF 25–75 values were impaired in 28 rhinitics. Sixty-six patients showed positive methacholine bronchial challenge. FEF 25–75 values were impaired only in BHR positive patients (p < 0.001). A significant difference was observed both for FEV1 (p < 0.05) and FEF 25–75 (p < 0.001) considering BHR severity.</p> <p>Conclusions</p> <p>This study evidences that an impairment of spirometric parameters may be observed in polysensitized patients with allergic rhinitis alone during the pollen season. A high percentage of these patients had BHR. A close relationship between upper and lower airways is confirmed.</p

Model order reduction of nonlinear piezoelectric microstructures through direct parametrisation of invariant manifolds

International audiencePiezoelectric actuation represents the most effective out-of-plane actuation mechanism for resonant microstructures as scanning micromirrors and piezoelectric ultrasonic transducers. Nevertheless, predicting the dynamic response of such devices at their design stage is often impractical since numerical computation of periodic orbits from finite element systems is computational demanding. In this work, we propose a model order reduction strategy based on the direct parametrisation for invariant manifolds tailored for nonlinear piezoelectric structures. The innovative aspect of the method is the introduction of nonlinear terms that arise due to piezoelectric coupling in the reduction procedure

Asymptotic Computation of Invariant Manifolds of large Finite Element structures with Geometric Nonlinearities

International audienceIn this contribution we present a method to directly compute asymptotic expansion of invariant manifolds of large finite element models from physical coordinates and their reduced order dynamics on the manifold. We show the accuracy of the reduction method on selected models, exhibiting large rotations and internal resonances. The results obtained with the reduction compared to full-order harmonic balance simulations show that the proposed methodology can reproduce extremely accurately the dynamics of the original systems with a very low computational cost

High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point

International audienceAbstract This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems in oscillatory form expressed in the physical basis, so that the technique is directly applicable to mechanical problems discretised by the finite element method. Two nonlinear mappings, respectively related to displacement and velocity, are introduced, and the link between the two is made explicit at arbitrary order of expansion, under the assumption that the damping matrix is diagonalised by the conservative linear eigenvectors. The same development is performed on the reduced-order dynamics which is computed at generic order following different styles of parametrisation. More specifically, three different styles are introduced and commented: the graph style, the complex normal form style and the real normal form style. These developments allow making better connections with earlier works using these parametrisation methods. The technique is then applied to three different examples. A clamped-clamped arch with increasing curvature is first used to show an example of a system with a softening behaviour turning to hardening at larger amplitudes, which can be replicated with a single mode reduction. Secondly, the case of a cantilever beam is investigated. It is shown that invariant manifold of the first mode shows a folding point at large amplitudes. This exemplifies the failure of the graph style due to the folding point on a real structure, whereas the normal form style is able to pass over the folding. Finally, a MEMS (Micro Electro Mechanical System) micromirror undergoing large rotations is used to show the importance of using high-order expansions on an industrial example

A numerical package for model order reduction of large dimensional finite element systems of nonlinear vibrating structures based on invariant manifold theory

International audienceDimensionality reduction through parametrisation of the system motion along a low dimensional invariant-based span of the phase space represents the most efficient technique for deriving reduced order models (ROM) of structures vibrating with large amplitudes. In this work we present the first release of an efficient software for deriving reduced models of structures based on the Direct Parametrisation of Invariant Manifolds (DPIM). The package exploits an algorithmic implementation of the method tailored for mechanical systems, hence achieving low memory consumption and unprecedented speed. Examples of large scale systems of industrial interest are shown and comparisons with experimental data and full order numerical simulations are reported

Model order reduction for geometrically nonlinear beams featuring internal resonance and centrifugal effect

[ENOC] Conférence internationale sur la dynamique non linéaire en ingénierieInternational audienceThe direct parametrisation of invariant manifold is used for model order reduction of large amplitude vibrations of clampedclamped and rotating cantilever beams. A particular emphasis is set on the computation of the backbone curve in case of internal resonance. For the clamped beam, the 1:5 resonance between first and third mode occuring at large amplitude, is reproduced with the model. For the rotating cantilever, a Campbell diagram is first used to detect the appearance of a 1:5 resonance, which is then computed with the reduction method