Hybrid Elements for Modelling Squeeze Film Effects Coupled with Structural Interactions in Vibratory MEMS Devices

We present a hybrid finite element based methodology to solve the coupled fluid structure problem of squeeze film effects in vibratory MEMS devices, such as gyroscopes, RF switches, and 2D resonators. The aforementioned devices often have a thin plate like structure vibrating normally to a fixed substrate, and are generally not perfectly vacuum packed. This results in a thin air film being trapped between the vibrating plate and the fixed substrate which behaves like a squeeze film offering both stiffness and damping. For accurate modelling of such devices the squeeze film effects must be incorporated. Extensive literature is available on squeeze film modelling, however only a few studies address the coupled fluid elasticity problem. The majority of the studies that account for the plate elasticity coupled with the fluid equation, either use approximate mode shapes for the plate or use iterative solution strategies. In an earlier work we presented a single step coupled methodology using only one type of displacement based element to solve the coupled problem. The displacement based finite element models suffer from locking issues when it comes to modelling very thin structures with the lateral dimensions much larger than the plate thickness as is typical in MEMS devices with squeeze film effects. In this work we present another coupled formulation where we have used hybrid elements to model the structural domain. The numerical results show a huge improvement in convergence and accuracy with coarse hybrid mesh as compared to displacement based formulations. We further compare our numerical results with experimental data from literature and find them to be in good accordance.

Note on modelling directionality in piezoresistivity

When computing the change in electrical resistivity of a piezoresistive cubic material embedded in a deforming structure, the piezoresistive and the stress tensors should be in the same coordinate system. While the stress tensor is usually calculated in a coordinate system aligned with the principal axes of a regular structure, the specified piezoresistive coefficients may not be in that coordinate system. For instance, piezoresistive coefficients are usually given in an orthogonal cartesian coordinate system aligned with the <100> crystallographic directions and designers sometimes deliberately orient a crystallographic direction other than <100> along the principal directions of the structure to increase the gauge factor. In such structures, it is advantageous to calculate the piezoresistivity tensor in the coordinate system along which the stress tensors are known rather than the other way around. This is because the transformation of stress will have to be done at every point in the structure but piezoresistivity tensor needs to be transformed only once. Here, using tensor transformation relations, we show how to calculate the piezoresistive tensor along any arbitrary Cartesian coordinate system from the piezoresistive coefficients for the <100> coordinate system. Some of the software packages that simulate the piezoresistive effect do not have interfaces for calculation of the entire piezoresistive tensor for arbitrary directions. This warrants additional work for the user because not considering the complete piezoresisitive tensor can lead to large errors. This is illustrated with an example where the error is as high as 33%. Additionally, for elastic analysis, we used hybrid finite element formulation that estimates stresses more accurately than displacement-based formulation. Therefore, as shown in an example where the change in resistance can be calculated analytically, the percentage error of our piezoresistive program is an order of magnitude lower relative to displacement-based finite element method.

Synthesis and vesicle formation from hybrid bolaphile/amphiphile ion-pairs. Evidence of membrane property modulation by molecular design

Four new hybrid (bolaphile/amphiphile) ion-pairs were synthesized. Electron microscopy indicated that each of these forms bilayer membranes upon dispersion in aqueous media. Membrane properties have also been examined by differential scanning calorimetry, microcalorimetry, temperature-dependent fluorescence anisotropy measurements, and UV-vis spectroscopy. The T-m values for the vesicular 1, 2, 3, 4, and 5 were 38, 12, 85, 31.3, and 41.6 degrees C, respectively. Interestingly the T-m values for 1 and 3 were found to depend on their concentration. The entrapment of small solute and the release capability have also been examined to demonstrate that these bilayers form enclosed vesicles. X-ray diffraction of the cast films has been performed to understand the nature and the thickness of these membrane organizations. The membrane widths ranged from 33 to 47 Angstrom. Finally, the above observations have been analyzed in light of the results obtained from molecular modeling studies. Thus we have demonstrated that membrane properties can be modulated by simple structural changes at the amphiphile level. It was shown that by judicious incorporation of central, isomeric, disubstituted aromatic units as structural anchors into different bolaphiles, one can modulate the properties of the resulting vesicles.

Hybrid stiff-string-polynomial basis functions for vibration analysis of high speed rotating beams

A new finite element is developed for free vibration analysis of high speed rotating beams using basis functions which use a linear combination of the solution of the governing static differential equation of a stiff-string and a cubic polynomial. These new shape functions depend on rotation speed and element position along the beam and account for the centrifugal stiffening effect. The natural frequencies predicted by the proposed element are compared with an element with stiff-string, cubic polynomial and quintic polynomial shape functions. It is found that the new element exhibits superior convergence compared to the other basis functions.

Fuzzy Logic-Based Approaches in Water Resource System Modelling

Recent research in modelling uncertainty in water resource systems has highlighted the use of fuzzy logic-based approaches. A number of research contributions exist in the literature that deal with uncertainty in water resource systems including fuzziness, subjectivity, imprecision and lack of adequate data. This chapter presents a broad overview of the fuzzy logic-based approaches adopted in addressing uncertainty in water resource systems modelling. Applications of fuzzy rule-based systems and fuzzy optimisation are then discussed. Perspectives on the scope for further research are presented.

Hybrid framework iron(II) phosphate-oxalates

Two inorganic-organic hybrid framework iron phosphate-oxalates, I, [N2C4H12](0.5)[Fe-2(HPO4)(C2O4)(1.5)] and II, [Fe-2(OH2)PO4(C2O4)(0.5)] have been synthesized by hydrothermal means and the structures determined by X-ray crystallography. Crystal Data: compound I, monoclinic, spacegroup = P2(1)/c (No. 14), a=7.569(2) Angstrom, b=7.821(2) Angstrom, c=18.033(4) Angstrom, beta=98.8(1)degrees, V=1055.0(4) Angstrom(3), Z=4, M=382.8, D-calc=2.41 g cm(-3) MoK alpha, R-F=0.02; compound II, monoclinic, spacegroup=P2(1)/c (No. 14), a=10.240(1) b=6.375(3) Angstrom, 9.955(1) Angstrom, beta=117.3(1)degrees, V=577.4(1) Angstrom(3), Z=4, M=268.7, D-calc=3.09 g cm(-3) MoK alpha, R-F=0.03. These materials contain a high proportion of three-coordinated oxygens and [Fe2O9] dimeric units, besides other interesting structural features. The connectivity of Fe2O9 is entirely different in the two materials resulting in the formation of a continuous chain of Fe-O-Fe in II. The phosphate-oxalate containing the amine, I, forms well-defined channels. Magnetic susceptibility measurements show Fen to be in the high-spin state (t(2g)(4)e(g)(2)) in II, and in the intermediate-spin state (t(2g)(5)e(g)(1)) in I.

Hybrid structural control using magnetorheological dampers for base isolated structures

In this paper two nonlinear model based control algorithms have been developed to monitor the magnetorheological (MR) damper voltage. The main advantage of the proposed algorithms is that it is possible to directly monitor the voltage required to control the structural vibration considering the effect of the supplied and commanded voltage dynamics of the damper. The efficiency of the proposed techniques has been shown and compared taking an example of a base isolated three-storey building under a set of seismic excitations. Comparison of the performances with a fuzzy based intelligent control algorithm and a widely used clipped optimal strategy has also been shown.

Energy-momentum conserving algorithm for nonlinear transient analysis within the framework of hybrid elements

This work deals with the formulation and implementation of an energy-momentum conserving algorithm for conducting the nonlinear transient analysis of structures, within the framework of stress-based hybrid elements. Hybrid elements, which are based on a two-field variational formulation, are much less susceptible to locking than conventional displacement-based elements within the static framework. We show that this advantage carries over to the transient case, so that not only are the solutions obtained more accurate, but they are obtained in fewer iterations. We demonstrate the efficacy of the algorithm on a wide range of problems such as ones involving dynamic buckling, complicated three-dimensional motions, et cetera.

Performance modelling of a fault-tolerant real-time multiprocessor using stochastic Petri nets

The fault-tolerant multiprocessor (ftmp) is a bus-based multiprocessor architecture with real-time and fault- tolerance features and is used in critical aerospace applications. A preliminary performance evaluation is of crucial importance in the design of such systems. In this paper, we review stochastic Petri nets (spn) and developspn-based performance models forftmp. These performance models enable efficient computation of important performance measures such as processing power, bus contention, bus utilization, and waiting times.

Influence of time step in the simulation modelling of evapotranspiration

Three simulations of evapotranspiration were done with two values of time step,viz 10 min and one day. Inputs to the model were weather data, including directly measured upward and downward radiation, and soil characteristics. Three soils were used for each simulation. Analysis of the results shows that the time step has a direct influence on the prediction of potential evapotranspiration, but a complex interaction of this effect with the soil moisture characteristic, rate of increase of ground cover and bare soil evaporation determines the actual transpiration predicted. The results indicate that as small a time step as possible should be used in the simulation.

Adsorption isotherms for neutral organic compounds-A hierarchy in modelling: Part II

A two-state model allowing for size disparity between the solvent and the adsorbate is analysed to derive the adsorption isotherm for electrosorption of organic compounds. Explicity, the organic adsorbate is assumed to occupy "n" lattice sites at the interface as compared to "one" by the solvent. The model parameters are the respective permanent and induced dipole moments apart from the nearest neighbour distance. The coulombic interactions due to permanent and induced dipole moments, discreteness of charge effects, and short-range and specific substrate interactions have all been incorporated. The adsorption isotherm is then derived using mean field approximation (MFA) and is found to be more general than the earlier multi-site versions of Bockris and Swinkels, Mohilner et al., and Bennes, as far as the entropy contributions are concerned. The role of electrostatic forces is explicity reflected in the adsorption isotherm via the Gibbs energy of adsorption term which itself is a quadratic function of the electrode charge-density. The approximation implicit in the adsorption isotherm of Mohilner et al. or Bennes is indicated briefly.

Adsorption isotherms for neutral organic compounds-A hierarchy in modelling: Part III

A simple three-state model permitting two different configurational states for the solvent, together with one for the organic adsorbate, is analysed to derive the adsorption isotherm. The implications of this model regarding pseudo-two-state and pseudo-Frumkin adsorption isotherms are indicated. A critique of the earlier theory of Bockris, Devanathan and Müller is presented in brief.

Adsorption isitherms for neutral organic compounds-A hierarchy in modelling: Part IV. Charge of maximum adsorption-Statistical mechanical basis

The charge at which adsorption of orgamc compounds attains a maximum ( \sigma MAX M) at an electrochenucal interface is analysed using several multi-state models in a hierarchical manner The analysis is based on statistical mechamcal results for the following models (A) two-state site parity, (B) two-state muhl-slte, and (C) three-state site parity The coulombic interactions due to permanent and reduced dipole effects (using mean field approximation), electrostatic field effects and specific substrate interactions have been taken into account. The simplest model in the hierarchy (two-state site parity) yields the exphcit dependence of ( \sigma MAX M) on the permanent dipole moment, polarizability of the solvent and the adsorbate, lattice spacing, effective coordination number, etc Other models in the baerarchy bring to hght the influence of the solvent structure and the role of substrate interactions, etc As a result of this approach, the "composition" of oM.x m terms of the fundamental molecular constants becomes clear. With a view to use these molecular results to maxamum advantage, the derived results for ( \sigma MAX M) have been converted into those involving experimentally observable parameters lake Co, C 1, E N, etc Wherever possible, some of the earlier phenomenologlcal relations reported for ( \sigma MAX M), notably by Parsons, Damaskm and Frumkln, and Trasattl, are shown to have a certain molecular basis, vlz a simple two-state sate panty model.As a corollary to the hxerarcbacal modelling, \sigma MAX M and the potential corresponding to at (Emax) are shown to be constants independent of 0max or Corg for all models The lmphcatlon of our analysis f o r OmMa x with respect to that predicted by the generalized surface layer equation (which postulates Om~ and Ema x varlaUon with 0) is discussed in detail Finally we discuss an passing o M. and the electrosorptlon valency an this context.

Adsorption isotherms for neutral organic compounds - A hierarchy in modelling: Part V. Models for the inner layer potential difference in the presence of dipolar adsorption

The relations for the inner layer potential &fference (E) in the presence of adsorbed orgamc molecules are derived for three hterarchlcal models, m terms of molecular constants like permanent &pole moments, polarlzablhtles, etc It is shown how the experimentally observed patterns of the E vs 0 plots (hnear m all ranges of $\sigma^M$, non-linear in one or both regions of o M, etc ) can be understood in a serm-quantltatlve manner from the simplest model in our hierarchy, viz the two-state site panty version Two-state multi-site and three-state (sxte panty) models are also analysed and the slope (3E/80),,M tabulated for these also The results for the Esm-Markov effect are denved for all the models and compared with the earlier result of Parsons. A comparison with the GSL phenomenologlcal equation is presented and its molecular basis, as well as the hmltatlons, is analysed. In partxcular, two-state multa-slte and three-state (site panty) models yield E-o M relations that are more general than the "umfied" GSL equation The posslblhty of vaewlng the compact layer as a "composite medium" with an "effective dlelectnc constant" and obtaimng novel phenomenological descnptions IS also indicated.

Adsorption isotherms-Microscopic modelling: Part II

A lattice formahsm using "spin variables" is employed to analyse multi-state models for the adsorption of neutral dipoles.In particular, a spin-1/2 (two-state) model incorporating permanent and reduced dipole moments of the solvent and the organic adsorbate,substrate interactions, and &screteness of charge effects is analysed The resulting Generalized Islng Hamaltonian is solved under mean field approximation (MFA) in order to derive the adsorption isotherm for organic molecules A few spin-1 (three-state) models are also analysed under MFA to describe the competitive adsorption of multi-state solvent and organic dipoles, and the appropriate equilibrium relations are derived The unification and isomorphism existing at the Hamlltonlan level for several diverse realizations, such as adsorption of ions and solvent/orgamc molecules, is indicated The possibility of analysing phase transitions using this generalized approach is briefly indicated.