Direct finite element computation of non-linear modal coupling coefficients for reduced-order shell models
Summary (2 min read)
1 Introduction
- Thin shells vibrating at large amplitude can exhibit complex dynamics.
- In conjunction with a numerical continuation method [3,4], or perturbation analytical methods [5], one is able to obtain complete bifurcation diagrams for free vibrations and forced responses of thin structures.
- Unfortunately, this strategy is restricted to simple geometries, for which ad-hoc functional bases made of simple – often analytical – functions can be constructed, ensuring convergence for a small number of modes.
- From the computation of the residual, and via algebraic manipulations, nonlinear coupling coefficients can be evaluated.
- The authors propose and implement a direct method with application to FE shells discretized with general shell elements of the MITC family (Mixed Interpolation of Tensorial Components) [18].
2 Direct computation of nonlinear stiffness
- This section gives the analytical and implementation details for the direct computation of the nonlinear coupling coefficients describing the geometrical nonlinearity of the shell.
- Other basis functions could have been used for the Galerkin projection, such as POD modes (Proper Orthogonal Decomposition, see for instance [20– 22]).
- As is well-known, shell finite elements suffer from very serious numerical pathologies when directly discretizing standard kinematical assumptions – namely, when using so-called “displacements-based elements”– see [18] and references therein.
- Therefore, the authors substitute for the above displacement-based strain vector the modified expression ˜ e = [err ess ers esr Irerz Irezr Isesz Isezs] t, Ir and Is denoting the interpolation operators, and likewise for the linear variations.
- Similar results up to machine precision have been obtained, hence validating the direct numerical computation of non-linear coefficients.
3 Application : a hyperbolic paraboloid panel
- The general methodology given in the previous section is now applied with a practical test case given by a shallow hyperbolic paraboloid panel.
- The first truncation T1 shown in the insert of Fig. 6 contains only the fundamental mode, and displays a hardening behavior.
- The transverse modes, with the lowest eigenfrequencies, are given for p=1 to 22 in Table 2.
- In the case of forced and damped vibrations, the amplitude responses predicted by the ROM and shown in Fig. 7 can be compared to direct simulations on the full FE model, so as to ascertain the quality of the ROM and its ability to predict the correct amplitude values.
- This might be ascribed to the fact that the 3:1 resonance is activated when the nonlinear frequencies share the perfect 3:1 relationship, and the complete model may feature a slight difference in the evolution of the higher frequencies versus vibration amplitude.
4 Conclusion
- A direct method has been presented for computing the nonlinear stiffness in geometric nonlinear vibration of thin shells discretized by finite elements.
- These formulas have been implemented in a FE code within the framework of MITC shell elements.
- The method gives fast and reliable computations for these coefficients, the accuracy of which has been validated by comparison with an indirect method.
- The computation of these coefficients can be used for deriving reducedorder models of great accuracy, allowing for precise prediction of nonlinear vibration characteristics of shell models.
- Such studies will also allow a more extensive assessment of the criterion used in the present paper to select the important modes in the ROM, which can lead to an automatized selection procedure.
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Citations
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Cites background from "Direct finite element computation o..."
...We also restrict our attention to geometrically nonlinear structures, for which fnl(x) is a quadratic and cubic polynomial function of x [22, 21, 23]....
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Cites methods from "Direct finite element computation o..."
...Application of these methods to finite-element (FE) structures with geometric distributed nonlinearity, remains however scarce [26,32,33]....
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...elasticity, it implies that only quadratic and cubic polynomial terms have to be taken into account [3,57,32,33], they are expressed thanks to the terms G(X,X) and H(X,X,X), using a functional notation for the quadratic and cubic terms with coefficients gathered in third-order tensor G and fourth-order tensor H....
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References
8,068 citations
"Direct finite element computation o..." refers methods in this paper
...We consider general shell elements [23,18], hence discrete displacements are defined according to an isoparametric strategy as...
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Additional excerpts
...time-discretization following a Störmer-Verlet (or leap-frog) scheme [40]....
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"Direct finite element computation o..." refers background in this paper
...Instead, each strain component in the (r, s, z) coordinate system is re-interpolated within every element, according to a specific rule based on given points – called the “tying points” – at which the strains are exactly calculated [18,24]....
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Related Papers (5)
Frequently Asked Questions (20)
Q2. What future works have the authors mentioned in the paper "Direct finite element computation of non-linear modal coupling coefficients for reduced-order shell models" ?
It also paves the way for further fine numerical studies of bifurcation diagrams for shells of arbitrary complex geometry. Such studies will also allow a more extensive assessment of the criterion used in the present paper to select the important modes in the ROM, which can lead to an automatized selection procedure.
Q3. What is the key to a complete bifurcation diagram?
In order to have a significant understanding of the possible nonlinear behaviors of a given structure, the computation of a complete bifurcation diagram is key, as it gives access to all the solution branches (stable and unstable) under variations of some selected control parameters.
Q4. What is the method for obtaining complete bifurcation diagrams?
In conjunction with a numerical continuation method [3,4], or perturbation analytical methods [5], one is able to obtain complete bifurcation diagrams for free vibrations and forced responses of thin structures.
Q5. What is the method used for calculating the nonlinear coupling coefficients?
Analytical expressions of the nonlinear coupling coefficients have been expressed from the internal elastic energy by using a modal expansion for the displacement.
Q6. What is the first physical effect that the reduced-order model must capture?
The first physical effect that the reduced-order model must mandatorily capture is the coupling between bending and membrane motions.
Q7. How is the ROM used to predict the response of a thin shell?
A converged ROM including 20 linear modes has been shown to predict with excellent accuracy the resonant response of a HP panel in the vicinity of its fundamental frequency.
Q8. What is the way to compute the convergence of the reduced-order model?
As peculiar nonlinear phenomena will be exhibited, a very good accuracy on all the numerical values is needed so as to ensure a good convergence of the reduced-order model.
Q9. What is the effect of the internal resonance tongue on the global dynamics of the forceddamped system?
An internal resonance tongue has been found with mode 22, albeit it occurs in a very narrow interval, so that its effect on the global dynamics of the forceddamped system is likely to be negligible.
Q10. Why do the authors use linear eigenmodes as a projection basis?
the authors specifically select linear eigenmodes as a projection basis for the following two reasons:– it allows for a decoupling between linear components, – eigenmodes are easily computable in any standard FE code.
Q11. What is the way to compute the bifurcation diagram?
The reduced-order model may be used for either direct time integration, or, as shown next, for computing by a continuation method the bifurcation diagram in the vicinity of a given dynamical state.
Q12. What is the effect of the absence of stable periodic orbit?
This absence of stable periodic orbit is consistent with previous studies on the transition to turbulence for thin plates and shells, see e.g. [34–36], where it has been found that from vibration amplitudes of 2 to 4h (depending on the structure considered), no stable periodic solutions exist anymore, so that the dynamical solution is at least quasi-periodic.
Q13. What is the effect of the internal resonance tongue on the frequency response in the forced-damped?
it has been emphasized in [36] that when the internal resonance occurs in a narrow interval as here observed, its influence on the frequency response in the forced-damped system is negligible.
Q14. What is the first one in increasing frequency order that shows a strong coupling with the fundamental?
Concerning the M-modes, the mode labeled 224 is the first one in increasing frequency order that shows a strong coupling with the fundamental mode and thus will be key in the selection procedure explained in the next section.
Q15. What is the effect of adding the modes in order to reproduce the nonlinear vibrating behavior?
One observes that the hardening behavior is less pronounced, which reveals the effect of adding those modes in order to correctly reproduce the nonlinear vibrating behavior of the panel.
Q16. How long does the amplitude limit of the HP panel converge?
The amplitude limit in the case presented here of the HP panel is found to converge to a value of 2.3h, as shown by the last two truncations which are superimposed, T6 and T7.
Q17. Why do the authors need to deal with nonlinear tensors?
This is because the authors need to deal with non-symmetric tensors for the computations of the nonlinear stiffness coefficients in (14)-(15), since the tensor ∇tΦi.
Q18. What is the amplitude of the transverse displacement at the center of the ROM?
For the full FE model, the maximum amplitude of the transverse displacement at the center is shown, whereas for the ROM the continuation software AUTO gives the maximum amplitude of each coordinate Xp, without their relative phases, so that one is not able to reconstruct precisely the complete transverse displacement by adding all modal coordinates according to Eq. (10).
Q19. How do the authors compute the coefficients gpij and h p ijk?
In order to compute the coefficients gpij and h p ijk consistently with the finite element procedure, the authors thus need to modify the components of the second-order tensors ∇tΦi.
Q20. What is the convergence with the number of high-frequency modes?
The convergence with this number of high-frequency modes is shown in the insert of Fig. 6, where only the fundamental B-mode is retained with an increasing number of M-modes.