This paper aims at reviewing nonlinear methods for model order reduction in structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear-based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations. They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then, the specific case of structures discretised with the finite element method is addressed. Implicit condensation, giving rise to a projection onto a stress manifold, and modal derivatives, used in the framework of the quadratic manifold, are first reviewed. Finally, recent developments allowing direct computation of reduced-order models relying on invariant manifolds theory are detailed. Applicative examples are shown and the extension of the methods to deal with further complications are reviewed. Finally, open problems and future directions are highlighted.

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

Direct computation of nonlinear mapping via normal form for reduced-order models of finite element nonlinear structures

Reduced order modeling of blades with geometric nonlinearities and contact interactions

This work is devoted to the robust analysis of the effects of geometric nonlinearities on the nonlinear dynamic behavior of rotating detuned (intentionally mistuned) bladed disks in presence of unintentional mistuning (simply called mistuning). Mistuning induces uncertainties in the computational model, which are taken into account by a probabilistic approach. This paper presents a series of novel results of the dynamic behavior of such rotating bladed disks exhibiting nonlinear geometric effects. The structural responses in the time domain are analyzed in the frequency domain. The frequency analysis exhibits responses outside the frequency band of excitation. The confidence region of the stochastic responses allows the robustness to be analyzed with respect to uncertainties and also allows physical insights to be given concerning the structural sensitivity. The bladed disk structure is made up of 24 blades for which several different detuned patterns are investigated with and without mistuning.

/pdf/robust-dynamic-analysis-of-detuned-mistuned-rotating-bladed-mmh80hefnj.pdf

Robust dynamic analysis of detuned-mistuned rotating bladed disks with geometric nonlinearities

The direct parametrisation method for invariant manifold is a model-order reduction technique that can be applied to nonlinear systems described by PDEs and discretised, for example, with a finite element procedure in order to derive efficient reduced-order models (ROMs). In nonlinear vibrations, it has already been applied to autonomous and non-autonomous problems to propose ROMs that can compute backbone and frequency–response curves of structures with geometric nonlinearity. While previous developments used a first-order expansion to cope with the non-autonomous term, this assumption is here relaxed by proposing a different treatment. The key idea is to enlarge the dimension of the parametrising coordinates with additional entries related to the forcing. A new algorithm is derived with this starting assumption, and as a key consequence, the resonance relationships appearing through the homological equations involve multiple occurrences of the forcing frequency, showing that with this new development, ROMs for systems exhibiting a superharmonic resonance can be derived. The method is implemented and validated on academic test cases involving beams and arches. It is numerically demonstrated that the method generates efficient ROMs for problems involving 3:1 and 2:1 superharmonic resonances, as well as converged results for systems where the first-order truncation on the non-autonomous term showed a clear limitation. 

Direct parametrisation of invariant manifolds for non-autonomous forced systems including superharmonic resonances

The search for ever lighter weight has become a major goal in the aeronautical industry as it has a direct impact on fuel consumption. It also implies the design of increasingly thin structures made of sophisticated and flexible materials. This may result in nonlinear behaviours due to large structural displacements. Stator vanes can be affected by such phenomena, and as they are a critical part of turbojets, it is crucial to predict these behaviours during the design process in order to eliminate them. This paper presents a reduced order modelling process suited for the study of geometric nonlinearities. The method is derived from a classical Component Mode Synthesis with fixed interfaces, in which the reduced nonlinear terms are obtained through a STEP procedure using an adapted basis composed of linear modes completed by modal derivatives. The whole system is solved using a harmonic balance procedure and a classic iterative nonlinear solver. The application is implemented on a schematic stator vane model composed of nonlinear Euler-Bernoulli beams under von Karman assumptions.

/pdf/dynamic-analysis-and-reduction-of-a-cyclic-symmetric-system-4fi8fw692x.pdf

Dynamic Analysis and Reduction of a Cyclic Symmetric System Subjected to Geometric Nonlinearities


 This paper is devoted to the study of the nonlinear harmonic response of an industrial blade model subjected to large geometrical deflection. A reduction procedure is performed on the blade model using the linear normal modes of the structure. Geometrical nonlinear effects are taken into account by considering cubic and quadratic stiffnesses in the dynamical reduced model. Reduced nonlinear stiffness coefficients are computed with the STiffness Evaluation Procedure (STEP) and periodic solutions are sought using the Harmonic Balance Method (HBM) coupled to a pseudo-arclength continuation. Along with the harmonic response, a bifurcation analysis is performed to compute both turning and branching points. Specific attention is paid to the internal resonance phenomenon. 2 to 1 internal resonance occurred during the frequency response analysis close to the first and second modes of the reduced model. Mode coupling phenomena occurred during the harmonic analysis and secondary branches of solutions were obtained from branching point bifurcations.

/pdf/nonlinear-harmonic-analysis-of-a-blade-model-subjected-to-17lotw6llt.pdf

Nonlinear Harmonic Analysis of a Blade Model Subjected to Large Geometrical Deflection and Internal Resonance

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.

/pdf/reduced-order-modeling-of-geometrically-nonlinear-rotating-2kec02gj.pdf

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

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Papers

Dynamic Analysis and Reduction of a Cyclic Symmetric System Subjected to Geometric Nonlinearities

Nonlinear Harmonic Analysis of a Blade Model Subjected to Large Geometrical Deflection and Internal Resonance

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