Non-linear vibrations and frequency response analysis of piezoelectrically driven microcantilevers

Microcantilevers have recently received widespread attentions due to their extreme applicability and versatility in both biological and non-biological applications. Along this line, this paper undertakes the non-linear vibrations of a piezoelectrically driven microcantilever beam as a common configuration in many scanning probe microscopy and nanomechanical cantilever biosensor systems. A part of the microcantilever beam surface is covered by a piezoelectric layer (typically ZnO), which acts both as an actuator and sensor. The bending vibrations of the microcantilever beam are studied considering the inextensibility condition and the coupling between electrical and mechanical properties in the piezoelectric materials. The non-linear terms appear in the form of quadratic expression due to presence of piezoelectric layer, and cubic form due to geometrical non-linearities. The Galerkin approximation is then utilized to discretize the equations of motion. In addition, the method of multiple scales is applied to arrive at the closed form solution for the fundamental natural frequency of the system. An experimental setup consisting of a commercial piezoelectric microcantilever attached on the stand of a state-of-the-art microsystem analyzer for non-contact vibration measurement is utilized to verify the theoretical developments. It is found that the experimental results and theoretical findings are in good agreement, which demonstrates that the non-linear modeling framework could provide a better dynamic representation of the microcantilever than the previous linear models. Due to microscale nature of the system, excitation amplitude plays an important role since even a small change in the amplitude of excitation can lead to significant vibrations and frequency shift.     

New studying of Lattice Boltzmann method for two-phase driven in porous media

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Determining the role of the underlying orbital‐dependence of PBE0‐DH and PBE‐QIDH double‐hybrid density functionals

We study the orbital‐dependence of three (parameter‐free) double‐hybrid density functionals, namely the PBE0‐DH, the PBE‐QIDH models, and the SOS1‐PBE‐QIDH spin‐opposite‐scaled variant of the latter. To do it, we feed all their energy terms with different sets of orbitals obtained previously from self‐consistent density functional theory calculations using several exchange‐correlation functionals (e.g., PBE, PBE0, PBEH&H), or directly with HF‐PBE orbitals, to see their effect on selected datasets for atomization and reaction energies, the latter proned to marked self‐interaction errors. We find that the PBE‐QIDH double‐hybrid model shows a great consistency, as the best results are always obtained for the set of orbitals corresponding to its hybrid scheme, which prompts us to recommend this model without any other fitting or reparameterization. © 2017 Wiley Periodicals, Inc.     

Modeling Variable Density Flow and Solute Transport in Porous Medium: 2. Re‐Evaluation of the Salt Dome Flow Problem

Case 5, Level 1 of the international HYDROCOIN groundwater flow modeling project is an example of idealized flow over a salt dome. The groundwater flow is strongly coupled to solute transport since density variations in this example are large (20%).Several independent teams simulated this problem using different models. Results obtained by different codes can be contradictory. We develop a new numerical model based on the mixed hybrid finite elements approximation for flow, which provides a good approximation of the velocity, and the discontinuous finite elements approximation to solve the advection equation, which gives a good approximation of concentration even when the dispersion tensor is very small. We use the new numerical model to simulate the salt dome flow problem.In this paper we study the effect of molecular diffusion and we compare linear and nonlinear dispersion equations. We show the importance of the discretization of the boundary condition on the extent of recirculation and the final salt distribution. We study also the salt dome flow problem with a more realistic dispersion (very small dispersion tensor). Our results are different to prior works with regard to the magnitude of recirculation and the final concentration distribution. In all cases, we obtain recirculation in the lower part of the domain, even for only dispersive fluxes at the boundary. When the dispersion tensor becomes very small, the magnitude of recirculation is small. Swept forward displacement could be reproduced by using finite difference method to compute the dispersive fluxes instead of mixed hybrid finite elements.     

Coupled Groundwater Flow and Transport in Porous Media. A Conservative or Non-conservative Form?

A new formulation for the modeling of density coupled flow and transport in porous media is presented. This formulation is based on the development of the mass balance equation by using the conservative form. The system of equations obtained by coupling the flow and transport equations using a state equation is solved by a combination of the mixed hybrid finite element method (MHFEM) and the discontinuous finite element method (DFEM). The former is applied in order to solve the flow equation and the dispersive part of the transport equation, whilst the latter is used to solve the advective part of the transport equation. Although the advantages of the MHFEM are known (efficiency calculation of velocity field and continuity of fluxes from one element to an adjacent one), its application in a classical development form (volumetric fluxes as unknowns) leads to the non-conservative version of the mass balance equation. The associated matrix of the system of equations obtained by hybridization is positive definite but non-symmetrical. By using a new approach (mass fluxes as unknowns) the conservative form of the continuity equation is preserved and the associated matrix of the system of equations obtained by hybridization becomes symmetrical. When applied to Elder's problem involving a strong density contrast, this new approach, with a lower calculation cost, leads to similar or identical results to those found in the specialized literature. The comparison between the conservative and non-conservative formulations solved with the same MHFEM and DFEM combination emphasizes the rigor and the pertinence of this new approach. Furthermore, we show the existence of a limit refinement defining the stability of the numerical solution for Elder's problem.     

磁驱动高速飞片实验样品设计

利用理论与数值模拟相结合的方法对磁驱动飞片实验样品的尺寸设计进行了分析探讨,给出了横 向尺寸设计的估算公式、长度设计原则、影响厚度设计的主要因素等。     

A multigrid approach for steady state laminar viscous flows

In this paper, we use the laminar viscous flow in a lid‐driven cavity as an example to describe and verify a numerical scheme for non‐linear partial differential equations. The proposed scheme combines a new analytical method for strongly non‐linear problems, namely the homotopy analysis method, with the multigrid techniques. A family of formulas at different orders is given. At the lowest order, the current approach is the same as the traditional multigrid methods. However, our high‐order scheme needs a fewer number of iterations and less CPU time than the classical ones. Copyright © 2001 John Wiley & Sons, Ltd.     

Free surface Stokes flows obstructed by multiple obstacles

Gravity‐driven Stokes flow down an inclined plane over and around multiple obstacles is considered. The flow problem is formulated in terms of a boundary integral equation and solved using the boundary element method. A Hermitian radial basis function (RBF) is used for the interpolation of the free surface, generation of the unit normal and curvature, and to prescribe the far‐field conditions. For flow over an obstacle, hemispheres are taken. For flow around an obstacle, circular cylinders are modelled and the contact angle condition on the obstacle/free surface intersection specified using the RBF formulation. Explicit profiles are produced for flow over and around two obstacles placed in various locations relative to one another. Interaction due to two obstacles is given by comparisons made with the profiles for flow over and around individual obstacles. In general, when the obstacles are separated by a sufficiently large distance the flow profiles are identical to a single obstacle analysis. For flow over and around two obstacles in‐line with the incident flow, effects of the governing parameters are examined, with variations in plane inclination angle, Bond number, obstacle size, and in the case of obstacles intersecting the free surface, static contact angle is considered. Finally flows over and around three obstacles are modelled. Copyright © 2009 John Wiley & Sons, Ltd.     

A new positive‐definite regularization of incompressible Navier–Stokes equations discretized with Q1/P0 finite element

A new regularization method is proposed for the Galerkin approximation of the incompressible Navier–Stokes equations with Q1/P0 element, by newly introducing a square‐type linear form into the variational divergence‐free constraint regularized with the global pressure jump (GPJ) method. The addition of the square‐type linear form is intended to eliminate the hydrostatic pressure mode appearing in confined flows, and to make the discretized matrix positive definite and then non‐singular without the pressure pegging trick. Effects of the free parameters for the regularization on the solutions are numerically examined with a 2‐D driven cavity flow problem. Furthermore, the convergences in the conjugate gradient iteration for the solution of the pressure Poisson equation are compared among the mixed method, the GPJ method and the present method for both leaky and non‐leaky 3‐D driven cavity flows. Finally, the non‐leaky 3‐D cavity flows at different Re numbers are solved to compare with the literature data and to demonstrate the accuracy of the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.     

间接驱动的冲击波特性实验

报道在星光激光器上进行的X光辐射驱动(间接驱动)和激光直接驱动的冲击波实验。实验观察到了铝背冲击波发光信息。结果表明,间接驱动比直接驱动的冲击波平面性好。