Accounting for Quasi-Static Coupling in Nonlinear Dynamic Reduced-Order Models

Engineering structures are often designed using detailed finite element (FE) models. Although these models can capture nonlinear effects, performing nonlinear dynamic analysis using FE models is often prohibitively computationally expensive. Nonlinear reduced-order modeling provides a means of capturing the principal dynamics of an FE model in a smaller, computationally cheaper reduced-order model (ROM). One challenge in formulating nonlinear ROMs is the strong coupling between low- and high-frequency modes, a feature we term quasi-static coupling. An example of this is the coupling between bending and axial modes of beams. Some methods for formulating ROMs require that these high-frequency modes are included in the ROM, thus increasing its size and adding computational expense. Other methods can implicitly capture the effects of the high-frequency modes within the retained low-frequency modes; however, the resulting ROMs are normally sensitive to the scaling used to calibrate them, which may introduce errors. In this paper, quasi-static coupling is first investigated using a simple oscillator with nonlinearities up to the cubic order. ROMs typically include quadratic and cubic nonlinear terms, however here it is demonstrated mathematically that the ROM describing the oscillator requires higher-order nonlinear terms to capture the modal coupling. Novel ROMs, with high-order nonlinear terms, are then shown to be more accurate, and significantly more robust to scaling, than standard ROMs developed using existing approaches. The robustness of these novel ROMs is further demonstrated using a clamped–clamped beam, modeled using commercial FE software.

X-ray diffraction data as a source of the vibrational free-energy contribution in polymorphic systems

In this contribution we attempt to answer a general question: can X-ray diffraction data combined with theoretical computations be a source of information about the thermodynamic properties of a given system? Newly collected sets of high-quality multi-temperature single-crystal X-ray diffraction data and complementary periodic DFT calculations of vibrational frequencies and normal mode vectors at the Γ point on the yellow and white polymorphs of dimethyl 3,6-dichloro-2,5-dihydroxyterephthalate are combined using two different approaches, aiming to obtain thermodynamic properties for the two compounds. The first approach uses low-frequency normal modes extracted from multi-temperature X-ray diffraction data (normal coordinate analysis), while the other uses DFT-calculated low-frequency normal mode in the refinement of the same data (normal mode refinement). Thermodynamic data from the literature [Yang et al. (1989), Acta Cryst. B45, 312–323] and new periodic ab initio DFT supercell calculations are used as a reference point. Both approaches tested in this work capture the most essential features of the systems: the polymorphs are enantiotropically related, with the yellow form being the thermodynamically stable system at low temperature, and the white form at higher temperatures. However, the inferred phase transition temperature varies between different approaches. Thanks to the application of unconventional methods of X-ray data refinement and analysis, it was additionally found that, in the case of the yellow polymorph, anharmonicity is an important issue. By discussing contributions from low- and high-frequency modes to the vibrational entropy and enthalpy, the importance of high-frequency modes is highlighted. The analysis shows that larger anisotropic displacement parameters are not always related to the polymorph with the higher vibrational entropy contribution.

Modified Elastic Dynamic Analysis (EDA) for Seismic Demand on In-Span Hinge Shear Keys in Multi-Frame Bridges

Long concrete box-girder bridges are typically constructed in multiple frames that are separated by in-span hinges. Shear keys, located at the in-span hinges, help preserve the transverse integrity of the bridge frames. To date, no reliable method other than nonlinear time history analysis exists to estimate the seismic force demands on in-span shear keys. Methods such as pushover and elastic dynamic analysis (EDA) do not provide accurate estimations. In this study, a rational and reliable analysis method was developed for obtaining the seismic demand on in-span shear keys of multi-frame bridges. A large number of time history analyses were performed on two- to five-frame bridge models with single- and two-column bents that were designed according to California Department of Transportation (Caltrans) Seismic Design Criteria. The results show that high-frequency modes of vibration of the superstructure significantly contribute to the shear key force demands. These modes may also cause transverse yielding in columns. It was established that a modified EDA method may be used to approximate the shear key force demands. In the proposed modified method, the modal forces are reduced separately by the corresponding modal displacement ductility before performing modal combination. This method accounts for the nonlinear response under high-frequency modes of vibration.

The improved Ginzburg-Landau technique

We discuss an innovative method for the description of inhomogeneous phases designed to improve the standard Ginzburg-Landau expansion. The method is characterized by two key ingredients. The first one is a moving average of the order parameter designed to account for the long-wavelength modulations of the condensate. The second one is a sum of the high frequency modes, to improve the description of the phase transition to the restored phase. The method is applied to compare the free energies of 1D and 2D inhomogeneous structures arising in the chirally symmetric broken phase.

A note on the necessity of filtering mechanism for polynomial observability of time-discrete wave equation

The problem of uniform polynomial observability was recently analyzed. It is shown that, when the continuous model is uniformly polynomially observable, it is sufficient to filter initial data to derive uniform polynomial observability inequalities for suitable time-discretization schemes. In this note, we prove that a filtering mechanism of high frequency modes is necessary to obtain uniform polynomial observability. More precisely, we give a counterexample which proves that this latter fails without filtering the initial data for time semi-discrete approximations of the wave equation.