Compact model for a MEM perforation cell with viscous, spring, and inertial forces

A compact model for calculating damping, inertial, and spring forces in a perforated squeeze-film damper is reported. The repetitive pressure patterns around each perforation are utilized by analyzing the visco-acoustic wave transmission around the hole in a cylindrical volume, called perforation cell. The model is needed in applications where the acoustic wavelength of the oscillation is comparable with the dimensions of the perforation cell. The model is constructed of acoustic impedance two-ports. A novel model is derived for the air gap region, and a published two-port model is used for the hole. The impedances for these two-ports are derived from the low reduced frequency model that is equivalent with linearized, harmonic Navier–Stokes equations for acoustic wave propagation in thin channels. This model considers also the transition from the isothermal conditions at low frequencies to the adiabatic ones at high frequencies. The dimensions of MEMS structures are considered using slip conditions for velocities and temperatures. Also, an easy-to-use simplified model for frequencies where the squeeze number and the Reynolds numbers are below unity is derived. The analytical compact model is verified with FEM simulations using a harmonic solver for linearized Navier–Stokes equations with slip boundary conditions in a wide range of perforation ratios. The maximum relative error in the damping coefficient in the simulated cases was 20% upto the first resonant frequency.     

Experimental validation of compact damping models of perforated MEMS devices

Measured damping coefficients of six different perforated micromechanical test structures are compared with damping coefficients given by published compact models. The motion of the perforated plates is almost translational, the surface shape is rectangular, and the perforation is uniform validating the assumptions made for compact models. In the structures, the perforation ratio varies from 24 to 59%. The study of the structure shows that the compressibility and inertia do not contribute to the damping at the frequencies used (130–220 kHz). The damping coefficients given by all four compact models underestimate the measured damping coefficient by approximately 20%. The reasons for this underestimation are discussed by studying the various flow components in the models.     

Fluid Effects in Vibrating Micromachined Structures

Squeeze film damping and hydrodynamic lift for a micromechanical perforated proof mass are calculated and measured. This paper has resulted in closed-form expressions that can be used to design accelerometers, tuning-fork gyroscopes (TFGs), and other micromechanical devices. The fluid damping and lift are determined using finite-element analyses of the normalized and linearized governing equations where the boundary condition of the pressure relief holes is derived using pipe flow analysis. The rarefaction of gas is incorporated in the governing equations based on slip flow condition. As a further check, a one-dimensional (1-D) network model is developed to account for the boundary condition of the holes on a tilted proof mass. Both closed-form and numerical solutions are compared against experimental data over a range of pressures.hfillhbox[1221]     

An artificial neural network-based multiscale method for hybrid atomistic-continuum simulations

This paper presents an artificial neural network-based multiscale method for coupling continuum and molecular simulations. Molecular dynamics modelling is employed as a local “high resolution” refinement of computational data required by the continuum computational fluid dynamics solver. The coupling between atomistic and continuum simulations is obtained by an artificial neural network (ANN) methodology. The ANN aims to optimise the transfer of information through minimisation of (1) the computational cost by avoiding repetitive atomistic simulations of nearly identical states, and (2) the fluctuation strength of the atomistic outputs that are fed back to the continuum solver. Results are presented for prototype flows such as the isothermal Couette flow with slip boundary conditions and the slip Couette flow with heat transfer.     

Comparative numerical study of FEM methods solving gas damping in perforated MEMS devices

Three different numerical strategies are presented for the estimation of the damping force acting on perforated movable MEMS dampers. Results from the 2D Perforated Profile Reynolds (PPR) method and the simplified 2D ANSYS method are compared with accurate full 3D flow simulations. Altogether, 32 different topologies are compared varying, e.g., the dimensions of the square damper and the square holes, and the number of holes. The case of uniform perforation and perpendicular motion is studied. Oscillation in the low frequency regime is assumed, that is, the compressibility and inertia of the gas are ignored in the study. While the PPR method is in good agreement with the 3D simulations, the forces given by the ANSYS method were considerably smaller. The reasons for this are studied, and a compact expression to explain the small forces is derived.     

Engineering MEMS Resonators With Low Thermoelastic Damping

This paper presents two approaches to analyzing and calculating thermoelastic damping in micromechanical resonators. The first approach solves the fully coupled thermomechanical equations that capture the physics of thermoelastic damping in both two and three dimensions for arbitrary structures. The second approach uses the eigenvalues and eigenvectors of the uncoupled thermal and mechanical dynamics equations to calculate damping. We demonstrate the use of the latter approach to identify the thermal modes that contribute most to damping, and present an example that illustrates how this information may be used to design devices with higher quality factors. Both approaches are numerically implemented using a finite-element solver (Comsol Multiphysics). We calculate damping in typical micromechanical resonator structures using Comsol Multiphysics and compare the results with experimental data reported in literature for these devices     

A two-port model for wave propagation along a long circular microchannel

A compact model for oscillatory flow in a long microchannel with a circular cross-section is derived from the linearised Navier–Stokes equations. The resulting two-port model includes the effects of viscosity due to rarefied gas in the slip flow regime, inertia, compressibility and losses due to heat exchange. Both an acoustic impedance T network and an acoustic admittance Π network are presented for implementation in system level and circuit simulation tools. Also, reduced T and Π networks with constant component values are given to be used in the low frequency region. They are useful in time domain simulations, too. To verify the analytical model, simulations with a harmonic finite element solver for acoustic viscous flow are performed for microchannels exploiting the axisymmetry. The simulation results with both open and closed outlet conditions are compared with the two-port model with excellent agreement. Contribution of the slip conditions and the accuracy of the simple model are demonstrated.     

孔的形状和排列方式对厚孔板微结构压膜空气阻尼的影响分析

本文对微结构上孔的形状和排列方式对压膜空气阻尼的影响进行了理论和模拟分析.理论研究表明对于不同厚度、不同排列方式下的孔单元阵列,若孔的总面积和孔单元面积均为常数,当孔数增加到某一值时有最小阻尼力,并用FEM工具ANSYS证明了该结论的正确性.结果还表明孔数对恒定尺寸微结构空气阻尼的影响随着结构厚度和孔数的增加而变得更加明显.分析结果对比表明在同样的尺寸条件下,孔方形排列微结构的空气阻尼小于孔蜂窝式排列微结构的空气阻尼,该现象随着孔单元面积的增加变得越明显,但是随着孔单元接近微结构的边界,阻尼之间的差距减小.研究结果可以用在高精度MEMS器件如MEMS地震检波器、MEMS光开关和MEMS红外光传感器等的优化设计中去.     

Static bending of perforated nanobeams including surface energy and microstructure effects

This article aims to present comprehensive model and analytical solution to investigate the static bending behavior of regularly squared cutout perforated thin/thick nanobeams incorporating the coupled effect of the microstructure and surface energy for the first time. The perforation influence is considered to be deriving equivalent geometrical and material characteristics. The modified couple stress theory is adopted to incorporate the microstructure effect while the modified Gurtin–Murdoch surface elasticity model is employed to incorporate the surface stress effect in perforated nanobeams. A variational formulation based on minimization of the total potential energy principle is employed to derive the equilibrium equations of perforated nanobeams based on both Euler–Bernoulli and Timoshenko beams theories are developed to investigate the associated effect of the shear deformation due to perforation process. Additionally, Poisson’s effect is also incorporated. Analytical closed-form for the non-classical bending profiles as well as the rotational displacement are developed for both beam theories considering the simultaneous effect of both couple stress and surface stress for both uniformly distributed and concentrated loading patterns. The verification of the developed model is verified and compared with previous works, and an excellent agreement is obtained. The applicability of the developed model is demonstrated and applied to study and analyze the nonclassical bending behavior of regularly squared perforated simply supported beams under different loading conditions. Additionally, effects of the perforation configuration parameters, beam size as well as beam aspect ratio on the bending behavior of perforated beams in the presence of microstructure and surface stress effects are also investigated and analyzed. The obtained results reveal that both couple stress and surface stress significantly affect the bending behavior of regularly squared cutout perforated beam structures. Results obtained are supportive for the design, analysis and manufacturing of perforated NEMS applications.

     

A numerical molecular dynamics approach for squeeze-film damping of perforated MEMS structures in the free molecular regime

Accurate determination of the squeeze-film damping in rare air is crucial for the design of high-Q MEMS devices. In the past, for the MEMS structures with no perforations, there have been two approaches to treating the squeeze-film damping in rare air: the approach based on the continuum assumption and the approach using molecular dynamics (MD) method. The amount of squeeze-film damping can be controlled by providing perforations in microstructures. To model perforation effects on squeeze-film damping, many methods have been proposed. However, almost all the previous methods are based on the continuum assumption. Only one paper focuses on analytical modeling of squeeze-film damping of a perforated microplate using the MD method. Hutcherson and Ye (J Micromech Microeng 14:1726–1733, 2004) developed a novel MD method to model the squeeze-film damping in free molecular regime. The method possesses high computational efficiency. However, their work is valid only for non-perforated rectangular microplate. This paper presents a numerical MD approach for calculating the squeeze-film damping of a perforated rectangular plate and a perforated circular plate in free molecular regime. In Hutcherson and Ye’s work, the microplate is non-perforated. After each collision with the non-perforated plate, all the molecules are reflected to the substrate. In this paper, the plate is perforated. For the molecules in the air gap striking the surface of the perforated microplate, some of the molecules are reflected to the substrate. The rest leave the air gap through the perforations. This paper is an extension of the work done by Hutcherson and Ye (J Micromech Microeng 14:1726–1733, 2004). The accuracy of the present numerical MD approach is verified by comparing its results with the experimental results available in the literature and the finite element method results.     

Generating efficient dynamical models for microelectromechanicalsystems from a few finite-element simulation runs

In this paper, we demonstrate how efficient low-order dynamical models for micromechanical devices can be constructed using data from a few runs of fully meshed but slow numerical models such as those created by the finite-element method (FEM). These reduced-order macromodels are generated by extracting global basis functions from the fully meshed model runs in order to parameterize solutions with far fewer degrees of freedom. The macromodels may be used for subsequent simulations of the time-dependent behavior of nonlinear devices in order to rapidly explore the design space of the device. As an example, the method is used to capture the behavior of a pressure sensor based on the pull-in time of an electrostatically actuated microbeam, including the effects of squeeze-film damping due to ambient air under the beam. Results show that the reduced-order model decreases simulation time by at least a factor of 37 with less than 2% error. More complicated simulation problems show significantly higher speedup factors. The simulations also show good agreement with experimental data     

A comparative study of analytical squeeze film damping models in rigid rectangular perforated MEMS structures with experimental results

Several analytical models exist for evaluating squeeze film damping in rigid rectangular perforated MEMS structures. These models vary in their treatment of losses through perforations and squeezed film, in their assumptions of compressibility, rarefaction and inertia, and their treatment of various second order corrections. We present a model that improves upon our previously reported work by incorporating more accurate losses through holes proposed by Veijola and treating boundary cells and interior cell differently as proposed by Mohite et al. We benchmark all these models against experimental results obtained for a typical perforated MEMS structure with geometric parameters (e.g., perforation geometry, air gap, plate thickness) that fall well within the acceptable range of parameters for these models (with the sole exception of Blech’s model that does not include perforations but is included for historical reasons). We compare the results and discuss the sources of errors. We show that the proposed model gives the best result by predicting the damping constant within 10% of the experimental value. We study the validity of the proposed model over the entire range of perforation ratios (PR) by comparing its results with numerically computed results from 3D Navier-Stokes equation. These results are also compared with other analytical models. The proposed model shows considerably better results than other models, especially for large values of PR.     

Analytical model of squeeze film air damping of perforated plates in the free molecular regime

In this paper, an analytical model of squeeze film damping (SQFD) of perforated plates in the free molecular regime is developed, which is based on: (1) the modification of the perforated energy transfer model (P-ETM) (Li and Hu, J Micromech Microeng 21:025006, 2011) by giving the probability of molecules entering the gap through holes; (2) the application of Sumali’s formula (J Micromech Microeng 17:2231–2240, 2007) to relate to the Monte Carlo model (MC) (Hutcherson and Ye, J Micromech Microeng 14:1726–1733, 2004) quantitatively. The analytical model can model the perforation effect on SQFD of plates of various hole sizes. Compared with experiment data and numerical models, the analytical model is proved to be accurate, easy to operate. The effect of gap distance on SQFD of perforated plate in the free molecular regime is discussed. Due to perforation effect, as gap distance increases, the damping constant of non-perforated plate decreases faster than that of perforated plate of the same size.

     

A two-level computational graph method for the adjoint of a finite volume based compressible unsteady flow solver

The adjoint method is a useful tool for finding gradients of design objectives with respect to system parameters for fluid dynamics simulations. But the utility of this method is hampered by the difficulty in writing an efficient implementation for the adjoint flow solver, especially one that scales to thousands of cores. This paper demonstrates a Python library, called adFVM, that can be used to construct an explicit unsteady flow solver and derive the corresponding discrete adjoint flow solver using automatic differentiation (AD). The library uses a two-level computational graph method for representing the structure of both solvers. The library translates this structure into a sequence of optimized kernels, significantly reducing its execution time and memory footprint. Kernels can be generated for heterogeneous architectures including distributed memory, shared memory and accelerator based systems. The library is used to write a finite volume based compressible flow solver. A wall clock time comparison between different flow solvers and adjoint flow solvers built using this library and state of the art graph based AD libraries is presented on a turbomachinery flow problem. Performance analysis of the flow solvers is carried out for CPUs and GPUs. Results of strong and weak scaling of the flow solver and its adjoint are demonstrated on subsonic flow in a periodic box.     

Modified genetic algorithm to design arbitrary response filters for rectangular waveguides

In this study, we present an improved genetic algorithm (GA) to design user‐specified filters housed in a waveguide of arbitrary cross‐sectional dimensions. An edge‐based finite element method (FEM) is employed as the forward solver for the problem. Additionally, the structures generated via the GA are easily constructed using standard printed circuit board fabrication techniques. Two of the major improvements to the GA are: (1) a technique for enhancing the GA's ability to avoid local minima and (2) a fine‐tuning mechanism which allows the GA to more efficiently seek out a minimum once a low error has been obtained. Using the aforementioned techniques, numerical/experimental results are presented for notch, low pass, high pass, and bandpass filters. © 2007 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2007.     

Accelerated numerical simulations of a heaving floating body by coupling a motion solver with a two-phase fluid solver

This paper presents a study on the coupling between a fluid solver and a motion solver to perform fluid–structure interaction (FSI) simulations of floating bodies such as point absorber wave energy converters heaving under wave loading. The two-phase fluid solver with dynamic mesh handling, interDyMFoam, is a part of the Computational Fluid Dynamics (CFD) toolbox OpenFOAM. The incompressible Navier–Stokes (NS) equations are solved together with a conservation equation for the Volume of Fluid (VoF). The motion solver is computing the kinematic body motion induced by the fluid flow. A coupling algorithm is needed between the fluid solver and the motion solver to obtain a converged solution between the hydrodynamic flow field around and the kinematic motion of the body during each time step in the transient simulation. For body geometries with a significant added mass effect, simple coupling algorithms show slow convergence or even instabilities. In this paper, we identify the mechanism for the numerical instability and we derive an accelerated coupling algorithm (based on a Jacobian) to enhance the convergence speed between the fluid and motion solver. Secondly, we illustrate the coupling algorithm by presenting a free decay test of a heaving wave energy converter. Thirdly and most challenging, a water impact test of a free falling wedge with a significant added mass effect is successfully simulated. For both test cases, the numerical results obtained by using the accelerated coupling algorithm are in a very good agreement with the experimental measurements.     

Automatic model building and solving for optimization problems

Automatic modelling is one of the key topics in model base management in DSS. We consider the design and implementation of a system which automatically specifies an optimization model from an abstract form with a given set of data specified by a user and solves it by calling an appropriate solver. We implement our idea as a prototype in Prolog and illustrate its validity by simple examples.     

Numerical and experimental validation of out-of-plane resonance closed formulas for MEMS suspended plates with square holes

A compact analytical model for out-of-plane resonance evaluation is proposed for large diffuse MEMS vibrating plates with squared holes. A closed form expression was developed from a structural reduced order model with equivalent concentrated mass and stiffness parameters. Results were validated through FEM models and experimental modal analysis. Numerical FEM simulations were performed by 1D, 2D and 3D FEM models; experiments on polysilicon test structures were conducted using an interferometric microscope. Specimens were designed ad hoc to highlight the sensitivity of proposed formulas to main structural dimensional parameters of vibrating plates. Experiments and models helped investigate the sensitivity of the proposed reduced linearized model when exposed to electrostatic non-linear coupling effects and fluidic damping coupling. Both numerical results and experiments are in good agreement with analytical results predicted by closed formulas.     

Compact damping models for laterally moving microstructures withgas-rarefaction effects

Compact models for the viscous damping coefficient in narrow air gaps between laterally moving structures are reported. In the first part of the paper, a simple frequency-independent first-order slip-flow approximation for the damping coefficient is derived and compared with a more accurate expression obtained from the linearized Boltzmann equation. The simple approximation is slightly modified and fitted to match the accurate model. The resulting simple approximation has a maximum relative error of less than ±6% at arbitrary Knudsen numbers in viscous, transitional and free molecular regions. In the second part of the paper, dynamic models for the damping force are derived, considering again gas rarefaction, by applying various boundary conditions. The damping admittance of the first-order slip-flow model is implemented also as an electrical equivalent admittance, constructed of RC sections, to allow both frequency and time domain simulations with a circuit simulator. The dependence of the damping admittance on pressure and gap displacement is demonstrated with model simulations. The accuracy and validity range of the model are verified with comparative numerical simulations of the Navier-Stokes equation. Finally, the damping coefficient in a lateral resonator is calculated using the compact model and compared with measured data with good agreement