Abstract
Non-intrusive methods have been used since two decades to derive reduced-order models for geometrically nonlinear structures, with a particular emphasis on the so-called STiffness Evaluation Procedure (STEP), relying on the static application of prescribed displacements in a finite-element context. We show that a particularly slow convergence of the modal expansion is observed when applying the method with 3D elements, because of nonlinear couplings occurring with very high frequency modes involving 3D thickness deformations. Focusing on the case of flat structures, we first show by computing all the modes of the structure that a converged solution can be exhibited by using either static condensation or normal form theory. We then show that static modal derivatives provide the same solution with fewer calculations. Finally, we propose a modified STEP, where the prescribed displacements are imposed solely on specific degrees of freedom of the structure, and show that this adjustment also provides efficiently a converged solution.
Highlights
Nonlinear effects appear generally in thin structures such as beams, plates and shells, when the amplitude of the vibration is of the order of the thickness [26,37]
When the structure under study is discretized with the finite element (FE) method, the problem of deriving accurate reduced-order models (ROM) is more stringent since the user cannot rely on a partial differential equations (PDE) in order to unfold an ad hoc mathematical method for Computational Mechanics (2020) 66:1293–1319 building the ROM
In parallel to the static condensation emphasised here, one can use the reduction formulae given by the normal form approach, restricting the motion to a single Nonlinear Normal Mode (NNM) [42,46,47]
Summary
Nonlinear effects appear generally in thin structures such as beams, plates and shells, when the amplitude of the vibration is of the order of the thickness [26,37]. The objective of this paper is to diagnose properly the issues one can encounter when applying the STEP with a modal basis to a structure discretized with 3D finite elements and present methods to overcome the problems. It is shown that a nonlinear coupling of bending modes with very high frequency modes involving deformations in the thickness of the structures occurs, and called thickness modes Those modes are the result of 3D deformation effects that are not present when using beam or plate models. This unexpected coupling is different from the traditional bending-longitudinal coupling and the numerical examples show that they are of prime importance to achieve a converged solution.
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