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Open accessJournal ArticleDOI: 10.3390/VIBRATION4010014

Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures

04 Mar 2021-Vol. 4, Iss: 1, pp 175-204
Abstract: The aim of this contribution is to present numerical comparisons of model-order reduction methods for geometrically nonlinear structures in the general framework of finite element (FE) procedures. Three different methods are compared: the implicit condensation and expansion (ICE), the quadratic manifold computed from modal derivatives (MD), and the direct normal form (DNF) procedure, the latter expressing the reduced dynamics in an invariant-based span of the phase space. The methods are first presented in order to underline their common points and differences, highlighting in particular that ICE and MD use reduction subspaces that are not invariant. A simple analytical example is then used in order to analyze how the different treatments of quadratic nonlinearities by the three methods can affect the predictions. Finally, three beam examples are used to emphasize the ability of the methods to handle curvature (on a curved beam), 1:1 internal resonance (on a clamped-clamped beam with two polarizations), and inertia nonlinearity (on a cantilever beam).

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Topics: Beam (structure) (55%), Finite element method (54%), Invariant (mathematics) (53%) ... show more

11 results found

Open accessJournal ArticleDOI: 10.1016/J.CMA.2021.113957
Alessandra Vizzaccaro1, Yichang Shen2, Loic Salles1, Jiří Blahoš1  +1 moreInstitutions (2)
Abstract: The direct computation of the third-order normal form for a geometrically nonlinear structure discretised with the finite element (FE) method, is detailed. The procedure allows to define a nonlinear mapping in order to derive accurate reduced-order models (ROM) relying on invariant manifold theory. The proposed reduction strategy is direct and simulation free , in the sense that it allows to pass from physical coordinates (FE nodes) to normal coordinates, describing the dynamics in an invariant-based span of the phase space. The number of master modes for the ROM is not a priori limited since a complete change of coordinate is proposed. The underlying theory ensures the quality of the predictions thanks to the invariance property of the reduced subspace , together with their curvatures in phase space that accounts for the non-resonant nonlinear couplings. The method is applied to a beam discretised with 3D elements and shows its ability in recovering internal resonance at high energy. Then a fan blade model is investigated and the correct prediction given by the ROMs are assessed and discussed. A method is proposed to approximate an aggregate value for the damping, that takes into account the damping coefficients of all the slave modes, and also using the Rayleigh damping model as input. Frequency–response curves for the beam and the blades are then exhibited, showing the accuracy of the proposed method.

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Topics: Finite element method (55%), Nonlinear system (55%), Normal coordinates (53%) ... show more

17 Citations

Open accessJournal ArticleDOI: 10.1007/S11071-021-06641-7
18 Mar 2021-Nonlinear Dynamics
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.

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Topics: Model order reduction (56%), Finite element method (53%), Computation (52%) ... show more

9 Citations

Open accessJournal ArticleDOI: 10.1007/S11071-021-06693-9
01 Jul 2021-Nonlinear Dynamics
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.

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Topics: Invariant manifold (65%), Invariant (mathematics) (57%), Nonlinear system (54%) ... show more

7 Citations

Open accessJournal ArticleDOI: 10.1007/S11071-021-06496-Y
22 May 2021-Nonlinear Dynamics
Abstract: We present an enhanced version of the parametric nonlinear reduced-order model for shape imperfections in structural dynamics we studied in a previous work. In this model, the total displacement is split between the one due to the presence of a shape defect and the one due to the motion of the structure. This allows to expand the two fields independently using different bases. The defected geometry is described by some user-defined displacement fields which can be embedded in the strain formulation. This way, a polynomial function of both the defect field and actual displacement field provides the nonlinear internal elastic forces. The latter can be thus expressed using tensors, and owning the reduction in size of the model given by a Galerkin projection, high simulation speedups can be achieved. We show that the adopted deformation framework, exploiting Neumann expansion in the definition of the strains, leads to better accuracy as compared to the previous work. Two numerical examples of a clamped beam and a MEMS gyroscope finally demonstrate the benefits of the method in terms of speed and increased accuracy.

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Topics: Displacement field (62%), Galerkin method (53%), Displacement (vector) (53%) ... show more

4 Citations

Journal ArticleDOI: 10.1016/J.COMPSTRUC.2021.106575
Abstract: A reduced-order modelling to predictively simulate the dynamics of piezoelectric structures with geometric nonlinearities is proposed in this paper. A formulation of three-dimensional finite element models with global electric variables per piezoelectric patch, and suitable with any commercial finite element code equipped with geometrically nonlinear and piezoelectric capabilities, is proposed. A modal expansion leads to a reduced model where both nonlinear and electromechanical coupling effects are governed by modal coefficients, identified thanks to a non-intrusive procedure relying on the static application of prescribed displacements. Numerical simulations can be efficiently performed on the reduced modal model, thus defining a convenient procedure to study accurately the nonlinear dynamics of any piezoelectric structure. A particular focus is made on the parametric effect resulting from the combination of geometric nonlinearities and piezoelectricity. Reference results are provided in terms of coefficients of the reduced-order model as well as of dynamic responses, computed for different test cases including realistic structures.

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Topics: Finite element method (52%), Nonlinear system (52%)

3 Citations


69 results found

Open accessBook
01 Aug 1983-
Abstract: Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.- Hyperbolic Sets, Sympolic Dynamics, and Strange Attractors.- Global Bifurcations.- Local Codimension Two Bifurcations of Flows.- Appendix: Suggestions for Further Reading. Postscript Added at Second Printing. Glossary. References. Index.

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Topics: Pitchfork bifurcation (59%), Heteroclinic bifurcation (58%), Dynamical systems theory (57%) ... show more

12,370 Citations

Journal ArticleDOI: 10.1023/A:1020843529530
Gerhard A. Holzapfel1Institutions (1)
01 Jul 2002-Meccanica
Topics: Solid mechanics (66%), Applied mechanics (63%), Computational mechanics (59%) ... show more

2,583 Citations

Open accessBook
28 Dec 2011-
Abstract: 0.- 4.5. The Case ?1 < 0.- 4.6. More Scaling.- 4.7. Completion of the Phase Portraits.- 4.8. Remarks and Exercises.- 4.9. Quadratic Nonlinearities.- 5. Application to a Panel Flutter Problem.- 5.1. Introduction.- 5.2. Reduction to a Second Order Equation.- 5.3. Calculation of Linear Terms.- 5.4. Calculation of the Nonlinear Terms.- 6. Infinite Dimensional Problems.- 6.1. Introduction.- 6.2. Semigroup Theory.- 6.3. Centre Manifolds.- 6.4. Examples.- References.

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1,432 Citations

Open accessBook
01 Aug 2014-
Abstract: Introduction. 1. Nonlinear theories of elasticity of plates and shells 2. Nonlinear theories of doubly curved shells for conventional and advanced materials 3. Introduction to nonlinear dynamics 4. Vibrations of rectangular plates 5. Vibrations of empty and fluid-filled circular cylindrical 6. Reduced order models: proper orthogonal decomposition and nonlinear normal modes 7. Comparison of different shell theories for nonlinear vibrations and stability of circular cylindrical shells 8. Effect of boundary conditions on a large-amplitude vibrations of circular cylindrical shells 9. Vibrations of circular cylindrical panels with different boundary conditions 10. Nonlinear vibrations and stability of doubly-curved shallow-shells: isotropic and laminated materials 11. Meshless discretization of plates and shells of complex shapes by using the R-functions 12. Vibrations of circular plates and rotating disks 13. Nonlinear stability of circular cylindrical shells under static and dynamic axial loads 14. Nonlinear stability and vibrations of circular shells conveying flow 15. Nonlinear supersonic flutter of circular cylindrical shells with imperfections.

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782 Citations

Open accessJournal ArticleDOI: 10.1006/JSVI.1993.1198
Steven W. Shaw1, Christophe Pierre1Institutions (1)
Abstract: A methodology is presented which extends to non-linear systems the concept of normal modes of motion which is well developed for linear systems. The method is constructive for weakly non-linear systems and provides the physical nature of the normal modes along with the non-linear differential equations which govern their dynamics. It also provides the non-linear co-ordinate transformation which relates the original system co-ordinates to the modal co-ordinates. Using this transformation, we demonstrate how an approximate non-linear version of superposition can be employed to reconstruct the overall motion from the individual non-linear modal dynamics. The results presented herein for non-linear systems reduce to modal analysis for the linearized system when non-linearities are neglected, even though the approach is entirely different from the traditional one. The tools employed are from the theory of invariant manifolds for dynamical systems and were inspired by the center manifold reduction technique. In this paper the basic ideas are outlined, a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions.

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Topics: Dynamical systems theory (60%), Modal analysis using FEM (60%), Linear system (56%) ... show more

506 Citations

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