The post-processed Galerkin method applied to non-linear shell vibrations

We present the results of a computational study of the post-processed Galerkin methods put forward by Garcia-Archilla et al. applied to the non-linear von Karman equations governing the dynamic response of a thin cylindrical panel periodically forced by a transverse point load. We spatially discretize the shell using finite differences to produce a large system of ordinary differential equations (ODEs). By analogy with spectral non-linear Galerkin methods we split this large system into a 'slowly' contracting subsystem and a 'quickly' contracting subsystem. We then compare the accuracy and efficiency of (i) ignoring the dynamics of the 'quick' system (analogous to a traditional spectral Galerkin truncation and sometimes referred to as 'subspace dynamics' in the finite element community when applied to numerical eigenvectors), (ii) slaving the dynamics of the quick system to the slow system during numerical integration (analogous to a non-linear Galerkin method), and (iii) ignoring the influence of the dynamics of the quick system on the evolution of the slow system until we require some output, when we 'lift' the variables from the slow system to the quick using the same slaving rule as in (ii). This corresponds to the post-processing of Garcia-Archilla et al. We find that method (iii) produces essentially the same accuracy as method (ii) but requires only the computational power of method (i) and is thus more efficient than either. In contrast with spectral methods, this type of finite-difference technique can be applied to irregularly shaped domains. We feel that post-processing of this form is a valuable method that can be implemented in computational schemes for a wide variety of partial differential equations (PDEs) of practical importance.

Modelling non-linearities in a MEMS square wave oscillator

The modelling of the non-linear behaviour of MEMS oscillators is of interest to understand the effects of non-linearities on start-up, limit cycle behaviour and performance metrics such as output frequency and phase noise. This paper proposes an approach to integrate the non-linear modelling of the resonator, transducer and sustaining amplifier in a single numerical modelling environment so that their combined effects may be investigated simultaneously. The paper validates the proposed electrical model of the resonator through open-loop frequency response measurements on an electrically addressed flexural silicon MEMS resonator driven to large motional amplitudes. A square wave oscillator is constructed by embedding the same resonator as the primary frequency determining element. Measurements of output power and output frequency of the square wave oscillator as a function of resonator bias and driving voltage are consistent with model predictions ensuring that the model captures the essential non-linear behaviour of the resonator and the sustaining amplifier in a single mathematical equation. © 2012 IEEE.

Non linear constitutive models for lattice materials

We use a computational homogenisation approach to derive a non linear constitutive model for lattice materials. A representative volume element (RVE) of the lattice is modelled by means of discrete structural elements, and macroscopic stress-strain relationships are numerically evaluated after applying appropriate periodic boundary conditions to the RVE. The influence of the choice of the RVE on the predictions of the model is discussed. The model has been used for the analysis of the hexagonal and the triangulated lattices subjected to large strains. The fidelity of the model has been demonstrated by analysing a plate with a central hole under prescribed in plane compressive and tensile loads, and then comparing the results from the discrete and the homogenised models. © 2013 Elsevier Ltd.

Structure and non-linear optical properties of beta-barium borate

The non-linear optical (NLO) properties of crystalline beta-BaB2O4 (beta-barium borate, BBO) have been investigated from the chemical bond viewpoint. The contributions of each type of chemical bond to the total NLO coefficient have been quantitatively determined. The calculations indicate that the true space group of BBO is R3 rather than R3c.

Comparison of non-linear optical susceptibilities of KNbO3 and LiNbO3

From the chemical bond viewpoint, second-order non-linear optical (NLO) tenser coefficients of KNbO3 and LiNbO3 crystals have been calculated. By using the bond-valence theory of complex crystals and the modified bond-charge model, we were able to determine contributions of each type of constituent chemical bond to the total second-order NLO susceptibility. The tenser values thus calculated are in good agreement with experimental data. From the comparison of NLO tenser coefficients of these two crystals, we found that the major NLO contributors are KO12 groups and LiO6 octahedra not the distorted NbO6 octahedra. The difference between their NLO properties arises from their different structural characters, and the high coordination number of constituent elements in KNbO3 makes its valence electrons become more delocalised compared with those of LiNbO3. (C) 1997 Elsevier Science Ltd. All rights reserved.