Andrea Opreni

Bio: Andrea Opreni is an academic researcher from Polytechnic University of Milan. The author has contributed to research in topics: Finite element method & Nonlinear system. The author has an hindex of 3, co-authored 11 publications receiving 33 citations.

Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance

Abstract: Dimensionality 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.

Nonlinear Response of PZT-Actuated Resonant Micromirrors

Abstract: In this work, we address the modelling of piezo-micromirrors with large opening angles. We focus on various sources of nonlinearites induced by the interaction with the surrounding fluid and by occurrence of geometric large transformations. Specific attention is also devoted to the piezoelectric coupling coefficient. Piezoelectric thin films have gained attention as key materials for the actuation of micro devices since they provide high drive forces, enabling devices with higher performances compared to electrostatic actuation. However they also induce material nonlinearities that cannot be neglected. The model is validated using experimental data obtained for various excitation levels. [2020-0229]

Analysis of the Nonlinear Response of Piezo-Micromirrors with the Harmonic Balance Method

Abstract: In this work, we address the simulation and testing of MEMS micromirrors with hardening and softening behaviour excited with patches of piezoelectric materials. The forces exerted by the piezoelectric patches are modelled by means of the theory of ferroelectrics developed by Landau–Devonshire and are based on the experimentally measured polarisation hysteresis loops. The large rotations experienced by the mirrors also induce geometrical nonlinearities in the formulation up to cubic order. The solution of the proposed model is performed by discretising the device geometry using the Finite Element Method, and the resulting large system of coupled differential equations is solved by means of the Harmonic Balance Method. Numerical results were validated with experimental data collected on the devices.

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.

Abstract: 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 expressed in the physical basis, so that the technique is directly applicable to 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. The same development is performed on the reduced-order dynamics which is computed at generic order following the 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 the invariant manifold of the first mode shows a folding point at large amplitudes which is not connected to an internal resonance. This exemplifies the failure of the graph style due to the folding point, whereas the normal form style is able to pass over the folding. Finally, A MEMS micromirror undergoing large rotations is used to show the importance of using high-order expansions on an industrial example.

High-order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to generic forcing terms and parametrically excited systems

Abstract: The direct parametrisation method for invariant manifolds is used for model order reduction of forced-damped mechanical structures subjected to geometric nonlinearities. Nonlinear mappings are introduced, allowing one to pass from the degrees of freedom of the finite-element model to the normal coordinates. Arbitrary orders of expansions are considered for the unknown mappings and the reduced dynamics, which are then solved sequentially through the homological equations for both autonomous and time-dependent terms. It is emphasised that the two problems share a similar structure, which can be used for an efficient implementation of the non-autonomous added terms. Special emphasis is also put on the new resonance conditions arising due to the presence of the external forcing frequencies, which allow predicting phenomena such as parametric excitation and isolas formation. The method is then applied to structures of academic and industrial interest. First, the large amplitude vibrations of a forced-damped cantilever beam are studied. This example highlights that high-order non-autonomous terms are compulsory to correctly estimate the maximum vibration amplitude experienced by the structure. The birth of isolated solutions is also illustrated on this example. The cantilever is then used to show how quadratic coupling creates conditions for the excitation of the parametric instability, and that this feature is correctly embedded in the reduction process. A shallow arch excited with multi-modal forcing is then studied to detail different forcing effects. Finally, the approach is validated on a structure of industrial relevance, i.e. a comb-driven micro-electro-mechanical resonator. The accuracy and computational performance reported suggest that the proposed methodology can accurately predict the nonlinear dynamic response of a large class of nonlinear vibratory systems.

The parameterization method for invariant manifolds

Abstract: We describe the parameterization method and we illustrate its application in two cases to obtain two results on invariant manifolds.

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

Abstract: 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 based on direct normal form: application to large finite element MEMS structures featuring internal resonance

Abstract: Dimensionality 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.

How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

Abstract: Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.