Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions

Functionally graded materials(FGMs)are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties.This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method.Under the assumptions of the Euler-Bernoulli beam theory,the governing differential equations of an individual FGM beam are derived with Hamilton’s principle and decoupled via the separation-of-variable approach.Then,the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method(TMM).Two models,i.e.,a three-level FGM stepped beam and a five-level FGM stepped beam,are considered,and their natural frequencies and mode shapes are presented.To demonstrate the validity of the method in this paper,the simulation results by ABAQUS are also given.On this basis,the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out.The results show the accuracy and efficiency of the TMM.

Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method

The two-dimensional nonlinear shallow water equations in the presence of Coriolis force and bottom topography are solved numerically using the fractional steps method. The fractional steps method consists of splitting the multi-dimensional matrix inversion problem into an equivalent one dimensional problem which is successively integrated in every direction along the characteristics using the Riemann invariant associated with the cubic spline interpolation. The height and the velocity field of the shallow water equations over irregular bottom are discretized on a fixed Eulerian grid and time-stepped using the fractional steps method. Effects of the Coriolis force and the bottom topography for particular initial flows on the velocity components and the free surface elevation have been studied and the results are plotted.

3D anatomy and flow dynamics of net-depositional cyclic steps on the world’s largest submarine fan:a joint 3D seismic and numerical approach

Supercritical flows are ubiquitous in natural environments;however,there is rare 3D anatomy of their deposits.This study uses high-quality 3D seismic datasets from the world’s largest submarine fan,Bengal Fan,to interpret 3D architectures and flow processes of Pliocene undulating bedforms that were related to supercritical flows.Bengal undulating bedforms as documented in this study were developed in unconfined settings,and are seismically imaged as strike-elongated,crescentic bedforms in plan view and as rhythmically undulating,upstream migrating,erosive,discontinuous reflections in section view.Their lee sides are overall 3 to 4 times steeper(0.280 to 1.19°in slope)and 3 to 4 times shorter(117 to 419 m in length)than their stoss flanks and were ascribed to faster(high flow velocities of 2.70 to 3.98 m/s)supercritical flows(Froude numbers of 1.53 to 2.27).Their stoss sides,in contrast,are overall 3 to 4 times gentler(0.120 to 0.270 in slope)and 3 to 4 times longer(410 to 1139 m in length)than their lee flanks and were related to slower(low velocities of 2.35 to 3.05 m/s)subcritical flows(Froude numbers of 0.58 to 0.97).Bengal wave-like features were,thus,created by supercriticalto-subcritical flow transformations through internal hydraulic jumps(i.e.,cyclic steps).They have crests that are positive relative to the surrounding region of the seafloor,suggesting the predominant deposition of draping sediments associated with net-depositional cyclic steps.Turbidity currents forming Bengal wave-like features were,thus,dominated by deposition,resulting in net-depositional cyclic steps.Sandy deposits associated with Bengal net-depositional cyclic steps are imaged themselves as closely spaced,strike-elongated high RMS-attribute patches,thereby showing closely spaced,long and linear,strike-elongated distribution patterns.

Direct observation of bunching of elementary steps on protein crystals under forced flow conditions

Bunching of elementary steps by solution flow is still not yet clarified for protein crystals. Hence, in this study, we observed elementary steps on crystal surfaces of model protein hen egg-white lysozyme （HEWL） under forced flow conditions, by our advanced optical microscopy. We found that in the case of a HEWL solution of 99.99% purity, forced flow changed bunched steps into elementary ones （debunching） on tetragonal HEWL crystals. In contrast, in the case of a HEWL solution of 98.5% purity, forced flow significantly induced bunching of elementary steps. These results indicate that in the case of HEWL crystals, the mass transfer of impurities is more significantly enhanced by forced solution flow than that of solute HEWL molecules. We also showed that forced flow induced the incorporation of microcrystals into a mother crystal and the subsequent formation of screw dislocations and spiral growth hillocks.

Weak and Strong Convergence of Self Adaptive Inertial Subgradient Extragradient Algorithms for Solving Variational Inequality Problems

Many solutions of variational inequalities have been proposed,among which the subgradient extragradient method has obvious advantages.Two different algorithms are given for solving variational inequality problem in this paper.The problem we study is defined in a real Hilbert space and has L-Lipschitz and pseudomonotone condition.Two new algorithms adopt inertial technology and non-monotonic step size rule,and their convergence can still be proved when the value of L is not given in advance.Finally,some numerical results are designed to demonstrate the computational efficiency of our two new algorithms.

三步接骨法治疗Sanders Ⅱ型跟骨骨折的有限元研究

目的量化评估三步接骨法治疗Sanders Ⅱ型跟骨骨折效能指标,阐明其科学机制。方法利用健康成年男性踝关节CT数据,构建正常足踝模型与Sanders Ⅱ型跟骨骨折模型,在骨折模型上进行三步接骨法的力学加载,模拟拔伸牵引、提按顶复、端挤捏骨等手法,评估跟骨复位情况并求解不同手法作用下力学的变化。结果建立正常足部模型与Sanders Ⅱ A/B/C型跟骨骨折模型;三步接骨法加载复位后跟骨长、高、宽、Gissane’s角及Bohler’s角得到明显纠正;求解不同手法的力学趋势,发现拔伸牵引法能有效纠正重叠位移,提按顶复法侧重纠正前后侧移位,端挤捏骨法致力纠正内外侧移位。结论三步接骨法通过依次纠正骨折位移,能恢复跟骨解剖结构,有效治疗跟骨SandersⅡ型损伤,具有有效性及科学性。

Application of CLEAR-VOF method to wave and flow simulations

A two-dimensional numerical model based on the Navier-Stokes equations and computational Lagrangian-Eulerian advection remap-volume of fluid （CLEAR-VOF） method was developed to simulate wave and flow problems. The Navier-Stokes equations were discretized with a three-step finite element method that has a third-order accuracy. In the CLEAR-VOF method, the VOF function F was calculated in the Lagrangian manner and allowed the complicated free surface to be accurately captured. The propagation of regular waves and solitary waves over a flat bottom, and shoaling and breaking of solitary waves on two different slopes were simulated with this model, and the numerical results agreed with experimental data and theoretical solutions. A benchmark test of dam-collapse flow was also simulated with an unstructured mesh, and the capability of the present model for wave and flow simulations with unstructured meshes, was verified. The results show that the model is effective for numerical simulation of wave and flow problems with both structured and unstructured meshes.

Physical modelling and scale effects of air-water flows on stepped spillways

During the last three decades, the introduction of new construction materials (e.g. RCC (Roller Compacted Concrete), strengthened gabions) has increased the interest for stepped channels and spillways. However stepped chute hydraulics is not simple, because of different flow regimes and importantly because of very-strong interactions between entrained air and turbu- lence. In this study, new air-water flow measurements were conducted in two large-size stepped chute facilities with two step heights in each facility to study experimental distortion caused by scale effects and the soundness of result extrapolation to pro- totypes. Experimental data included distributions of air concentration, air-water flow velocity, bubble frequency, bubble chord length and air-water flow turbulence intensity. For a Froude similitude, the results implied that scale effects were observed in both facilities, although the geometric scaling ratio was only Lr=2 in each case. The selection of the criterion for scale effects is a critical issue. For example, major differences (i.e. scale effects) were observed in terms of bubble chord sizes and turbulence levels al- though little scale effects were seen in terms of void fraction and velocity distributions. Overall the findings emphasize that physical modelling of stepped chutes based upon a Froude similitude is more sensitive to scale effects than classical smooth-invert chute studies, and this is consistent with basic dimensional analysis developed herein.

Noether＇s theorem and one-step corrections method for holonomic system

In this paper, a new computational method for improving the accuracy of numerically computed solutions is introduced. The computational method is based on the one-step method and conserved quantities of holonomic systems are considered as kinematical constraints in this method.

Air Entrainment and Pressure Fields over Stepped Spillway in Skimming Flow Regime

This paper deals with some aspects of the air entrainment process along the chute of spillway and study of pressure fluctuations. The experimental study has been carried out using stepped spillway model located in the campus of Government College of Engineering, Amravati (India). It is observed that air concentration is increasing with discharge as well as with number of step. Air concentration is increasing along the length of spillway. It is also observed that the bottom mean air concentration increases with step height in the upstream reach of stepped spillway, which is prone to cavitation. The pressure profiles exhibit a wavy pattern down the stepped chute and pressure on each step increases with ratio of critical depth to step height (yc/h).

Synthesis of Codon-optimized Human Interleukin-18 Gene by Combination of Chemical and Enzymatic Method

According to the amino acid sequence and codon preference of E. coli, the human interleukin-18（IL-18） gene was optimized to avoid the rare codons. The total length of the synthesized gene is 571 bp; 18 oligonucleotides, DNA fragments were designed and synthesized by the phosphoramidite four-step chemical method. The whole DNA sequence was synthesized by a one-step total gene synthesis method, and then inserted in pUC18 vector. Five positive clones identified by blue-white colony screening were sent to Shanghai Sangon Biological Engineering Technology and Service Co., Ltd. for sequencing. The sequencing result shows that one clone contained the complete correct gene in all the five positive clones.

Application of Multi-Step Differential Transform Method on Flow of a Second-Grade Fluid over a Stretching or Shrinking Sheet

In this study, a reliable algorithm to develop approximate solutions for the problem of fluid flow over a stretching or shrinking sheet is proposed. It is depicted that the differential transform method (DTM) solutions are only valid for small values of the independent variable. The DTM solutions diverge for some differential equations that extremely have nonlinear behaviors or have boundary-conditions at infinity. For this reason the governing boundary-layer equations are solved by the Multi-step Differential Transform Method (MDTM). The main advantage of this method is that it can be applied directly to nonlinear differential equations without requiring linearization, discretization, or perturbation. It is a semi analytical-numerical technique that formulizes Taylor series in a very different manner. By applying the MDTM the interval of convergence for the series solution is increased. The MDTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. It is predicted that the MDTM can be applied to a wide range of engineering applications.

A class of twostep continuity Runge-Kutta methods for solving singular delay differential equations and its convergence

An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods （TSCRK） is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.

NUMERICAL METHOD FOR THREE-DIMENSIONAL NONLINEAR CONVECTION-DOMINATED PROBLEM OF DYNAMICS OF FLUIDS IN POROUS MEDIA

For the three-dimensional convection-dominated problem of dynamics of fluids in porous media, the second order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Fractional steps techniques are needed to convert a multi-dimensional problem into a series of successive one-dimensional problems. Some techniques, such as calculus of variations, energy method, multiplicative commutation rule of difference operators, decomposition of high order difference operators, and the theory of prior estimates are adopted. Optimal order estimates are derived to determine the error in the second order approximate solution. These methods have already been applied to the numerical simulation of migration-accumulation of oil resources and predicting the consequences of seawater intrusion and protection projects.

INTERACTION OF A SOLITARY WAVE AND A FRONT STEP SIMULATED BY LEVEL SET METHOD

As a new method, the Level Set method had been developed to compute the interface of two-phase flow. The basic mathematical theory and the detailed method to solve the free surface hydrodynamic problem had been investigated. By using the Level Set method, the transformation of a solitary wave over a front step was simulated. The results were in good agreement with laboratory experiments.

A LOW-PRESSURE AND ONE-STEP METHOD FOR THE SYNTHESIS OF ALKYLIDYNETRICOBALT NONACARBONYL CLUSTERS:RCCo_8(CO)_9

Metal clusters RCCo_3(CO)_9(R-H,C1,Br,CH_3,Ph) were prepared in 18.8-57.3% yields from the reaction of cobalt(Ⅱ)salt and RCX_a under mild PTC conditions(latm CO,25℃).The cobalt salt was reduced to Co(CO)_4 in the presence of Na_3S_2O_4.

Effect of Bed Vicinity on Vortex Shedding and Force Coefficients of Fluid Flow on an Offshore Pipeline

The effect of rigid bed proximity on flow parameters and hydrodynamic loads in offshore pipelines exposed to turbulent flow is investigated numerically. The Galerkin finite volume method is employed to solve the unsteady incompressible 2D Navier–Stokes equations. The large eddy simulation turbulence model is solved using the artificial compressibility method and dual time-stepping approach. The proposed algorithm is developed for a wide range of turbulent flows with Reynolds numbers of 9500 to 1.5×10^4.Evaluation of the developed numerical model shows that the proposed technique is capable of properly predicting hydrodynamic forces and simulating the flow pattern. The obtained results show that the lift and drag coefficients are strongly affected by the gap ratio. The mean drag coefficient slightly increases as the gap ratio increases, although the mean lift coefficient rapidly decreases. The vortex shedding suppression happen at the gap ratio of less than 0.2.

Constraint Stabilization and One-Step Correction Method for Dynamical Systems

A computational method of constraint stabilization and correction is introduced. The method is based on the Baumgart＇s one-step method. Constraint conditions are addressed to stabilize and correct the solution. Two examples are given to illustrate the results of the method.

ZnO micro-nano composite hydrophobic film prepared by the three-step method

The hydrophobicity of the lotus leaf is mainly due to its surface micro-nano composite structure. In order to mimic the lotus structure, ZnO micro-nano composite hydrophobic films were prepared via the three-step method. On thin buffer films of SiO2, which were first fabricated on glass substrates by the so,gel dip-coating method, a ZnO seed layer was deposited via RF magnetron sputtering. Then two different ZnO films, micro-nano and micro-only flowerlike structures, were grown by the hydrothermal method. The prepared films have different hydrophobic properties after surface modification. The structures of the obtained ZnO films were characterized using x-ray diffraction and field-emission scanning electron microscopy. A conclusion that a micro-nano composite structure is more beneficial to hydrophobicity than a micro-only structure was obtained through research into the effect of structure on hydrophobic properties.

Study on the wind field and pollutant dispersion in street canyons using a stable numerical method

A stable finite element method for the time dependent Navier-Stokes equations was used for studying the wind flow and pollutant dispersion within street canyons. A three-step fractional method was used to solve the velocity field and the pressure field separately from the governing equations. The Streamline Upwind Petrov-Galerkin(SUPG) method was used to get stable numerical results. Numerical oscillation was minimized and satisfactory results can be obtained for flows at high Reynolds numbers. Simulating the flow over a square cylinder within a wide range of Reynolds numbers validates the wind field model. The Strouhal numbers obtained from the numerical simulation had a good agreement with those obtained from experiment. The wind field model developed in the present study is applied to simulate more complex flow phenomena in street canyons with two different building configurations. The results indicated that the flow at rooftop of buildings might not be assumed parallel to the ground as some numerical modelers did. A counter-clockwise rotating vortex may be found in street canyons with an inflow from the left to right. In addition, increasing building height can increase velocity fluctuations in the street canyon under certain circumstances, which facilitate pollutant dispersion. At high Reynolds numbers, the flow regimes in street canyons do not change with inflow velocity.