Investigation on the choice of boundary conditions and shape functions for flexible multi-body system

The objective of this investigation is to examine the correctness and efficiency of the choice of boundary conditions when using assumed mode approach to simulate flexible multi-body systems. The displacement field due to deformation is approximated by the Rayleigh-Ritz assumed modes in floating frame of reference (FFR) formulation. The deformations obtained by the absolute nodal coordinate (ANC) formulation which are transformed by two sets of reference coordinates are introduced as a criterion to verify the accuracy of the simulation results by using the FFR formulation. The relationship between the deformations obtained from different boundary conditions is revealed. Numerical simulation examples demonstrate that the assumed modes with cantilevered-free, simply-supported and freefree boundary conditions without inclusion of rigid body modes are suitable for simulation of flexible multi-body system with large over all motion, and the same physical deformation can be obtained using those mode functions, differ only by a coordinate transformation. It is also shown that when using mode shapes with statically indeterminate boundary conditions, significant error may occur. Furthermore, the slider crank mechanism with rigid crank is accurate enough for investigating boundary condition problem of flexible multi-body system, which cost significant less simulating time.     

Exact solutions to one-dimensional transient response of incompressible fluid-saturated single-layer porous media

Based on the Biot theory of porous media,the exact solutions to onedimensional transient response of incompressible saturated single-layer porous media under four types of boundary conditions are developed.In the procedure,a relation between the solid displacement u and the relative displacement w is derived,and the well-posed initial conditions and boundary conditions are proposed.The derivation of the solution for one type of boundary condition is then illustrated in detail.The exact solutions for the other three types of boundary conditions are given directly.The propagation of the compressional wave is investigated through numerical examples.It is verified that only one type of compressional wave exists in the incompressible saturated porous media.     

A numerical approach based on the meshless collocation method in elastodynamics

In this paper, a collocation technique with the modified equilibrium on line method （ELM） for imposition of Neumann （natural） boundary conditions is presented for solving the two-dimensional problems of linear elastic body vibrations. In the modified ELM, equilibrium over the lines on the natural boundary is satisfied as Neumann boundary condition equations. In other words, the natural boundary conditions are satisfied naturally by using the weak formulation. The performance of the modified version of the ELM is studied for collocation methods based on two different ways to construct meshless shape functions： moving least squares approximation and radial basis point interpolation. Numerical examples of two-dimensional free and forced vibration analyses show that by using the modified ELM, more stable and accurate results would be obtained in comparison with the direct collocation method.     

High-order boundary conditions for the problems of Laplace equation in infinite region and their application

The high-order boundary conditions for the problems of Laplace equation in infiniteregion have been developed.The improvement in accuracy for numerical solution isachieved by imposing the high-order boundary conditions on the exterior boundary of areduced finite region in which the numerical method is used.So both the computing effortsand the required storage in computer are reduced.The numerical examples show that thelst-order boundary condition approaches to the exact boundary condition and it is clearlysuperior to the traditional boundary condition and the2nd-order boundary condition.     

VIBRATION AND STABILITY OF VERTICAL UPWARD-FLUID-CONVEYING PIPE IMMERSED IN RIGID CYLINDRICAL CHANNEL

A theoretical model is developed for the vibration and stability of a vertical pipe subjected concurrently to two dependent axial flows. The external fluid, after exiting the outer annular region between the pipe and a rigid cylindrical channel, is conveyed upwards inside the pipe. This configuration thus resembles of a pipe that aspirating fluid. The equation of planar mo- tion is solved by means of the differential quadrature method （DQM）. Calculations are conducted for a slender drill-string-like and a bench-top-size system, for different confinement conditions of the outer annular channel. It is shown that the vibrations of these two systems are closely related to the degree of confinement of the outer annular channel. For a drill-string-like system with narrow annuli, buckling instability may occur in the second and third modes. For a bench-top-size system, however, both buckling and flutter may occur in the lowest three modes. The form of instability depends on the annuli size.     

Evaluation of the effect of wall boundary conditions on numerical simulations of circulating fluidized beds

A computational fluid dynamics （CFD） modeling of the gas-solids two-phase flow in a circulating fluidized bed （CFB） riser is carried out. The Eularian-Eularian method with the kinetic theory of granular flow is used to solve the gas-solids two-phase flow in the CFB riser. The wall boundary condition of the riser is defined based on the Johnson and Jackson wall boundary theory （Johnson ＆ Jackson, 1987） with specularity coefficient and particle-wall restitution coefficient.The numerical results show that these two coefficients in the wall boundary condition play a major role in the predicted solids lateral velocity, which affects the solid particle distribution in the CFB riser. And the effect of each of the two coefficients on the solids distribution also depends on the other one. The generality of the CFD model is further validated under different operatin~ conditions of the CFB riser.     

Viscous falling film instability around a vertical moving cylinder

The present work discusses both the linear and nonlinear stability conditions of a viscous falling film down the outer surface of a solid vertical cylinder which moves in the direction of its axis with a constant velocity.After studying the linear conditions,a generalized nonlinear kinematic model is then derived to present the physical system.Applying the boundary conditions,analytical solutions are obtained using the long-wave perturbation method.In the first step,the normal mode method is used to characterize the linear behaviors.In the second step,the nonlinear film flow model is solved by using the method of multiple scales,to obtain Ginzburg-Landau equation.The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem,and the results are displayed in many plots showing the stability criteria in various parameter planes.     

Free vibrations of beam-mass-spring systems: analytical analysis with numerical confirmation

Free vibrations of a beam-mass-spring system with different boundary conditions are analyzed both analytically and numerically.In the analytical analysis,the system is divided into three subsystems and the effects of the spring and the point mass are considered as internal boundary conditions between any two neighboring subsystems.The partial differential equations governing the motion of the subsystems and internal boundary conditions are then solved using the method of separation of variables.In the numerical analysis,the whole system is considered as a single system and the effects of the spring and point mass are introduced using the Dirac delta function.The Galerkin method is then employed to discretize the equation of motion and the resulting set of ordinary differential equations are solved via eigenvalue analysis.Analytical and numerical results are shown to be in very good agreement.     

Viscous fluid damping in a laterally oscillating finger of a comb-drive micro-resonator based on micro-polar fluid theory

Viscous damping is a dominant source of energy dissipation in laterally oscillating micro-structures. In micro-resonators in which the characteristic dimensions are compa-rable to the dimensions of the fluid molecules, the assumption of the continuum fluid theory is no longer justified and the use of micro-polar fluid theory is indispensable. In this paper a mathematical model was presented in order to predict the viscous fluid damping in a laterally oscillating finger of a micro-resonator considering micro-polar fluid theory. The coupled governing partial differential equations of motion for the vibration of the finger and the micro-polar fluid field have been derived. Considering spin and no-spin boundary conditions, the related shape functions for the fluid field were presented. The obtained governing differential equations with time varying boundary conditions have been trans-formed to an enhanced form with homogenous boundary conditions and have been discretized using a Galerkin-based reduced order model. The effects of physical properties of the micro-polar fluid and geometrical parameters of the oscillat-ing structure on the damping ratio of the system have been investigated.     

Symmetries of boundary layer equations of power-law fluids of second grade

A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.     

ANALYSIS ON ACOUSTICAL SCATTERING BY A CRACKED ELASTIC STRUCTURE

The acoustical scattering by a cracked elastic structure is studied. The mixed method of boundary element and fractal finite element is adopted to solve the cracked structure-acoustic coupling problem. The fractal two-level finite element method is employed for the cracked structure, which can reduce the degree of freedoms (DOFs) greatly, and the boundary element method is used for the exterior acoustic field which can automatically satisfy Sommerfeld‘s radiation condition. Numerical examples show that the resonance frequency is lower with the crack‘s depth increase, and that the effect on the acoustical field by the crack is particularly pronounced in the vicinity of the crack tip. This mixed method of boundary element and finite element is effective in solving the scattering problem by a cracked structure.     

A note on a boundary condition for spouted beds

The boundary condition, zero solids pressure at the top of a particle bed of maximum spoutable height, Hm, is shown to eliminate any resort to empiricism in the derivation of the fluid velocity in the annulus of a spouted bed for which both viscous and inertial effects are taken into account. The same boundary condition fails when applied to a spouted bed for which the bed height H 〈 Hm, especially when H 〈 0.8Hm.     

High-order numerical methods of fractional-order Stokes’ first problem for heated generalized second grade fluid

The high-order implicit finite difference schemes for solving the fractionalorder Stokes’ first problem for a heated generalized second grade fluid with the Dirichlet boundary condition and the initial condition are given. The stability, solvability, and convergence of the numerical scheme are discussed via the Fourier analysis and the matrix analysis methods. An improved implicit scheme is also obtained. Finally, two numerical examples are given to demonstrate the effectiveness of the mentioned schemes.     

A NUMERICAL METHOD FOR SIMULATING NONLINEAR FLUID-RIGID STRUCTURE INTERACTION PROBLEMS

A numerical method for simulating nonlinear fluid-rigid structure interaction problems is developed. The structure is assumed to undergo large rigid body motions and the fluid flow is governed by nonlinear, viscous or non-viscous, field equations with nonlinear boundary conditions applied to the free surface and fluid-solid interaction interfaces. An Arbitrary-Lagrangian-Eulerian (ALE) mesh system is used to construct the numerical model. A multi-block numerical scheme of study is adopted allowing for the relative motion between moving overset grids, which are independent of one another. This provides a convenient method to overcome the difficulties in matching fluid meshes with large solid motions. Nonlinear numerical equations describing nonlinear fluid-solid interaction dynamics are derived through a numerical discretization scheme of study. A coupling iteration process is used to solve these numerical equations. Numerical examples are presented to demonstrate applications of the model developed.     

LIQUID-SOLID COUPLED SYSTEM OF MICROPUMP

This paper employs the integral-averaged method of thickness to approximate the periodical flows in a piezoelectric micropump, with a shallow water equation including nonlinearity and viscous damp presented to characterize the flows in micropump. The finite element method is used to obtain a matrix equation of fluid pressure. The fluid pressure equation is combined with the vibration equation of a silicon diaphragm to construct a liquid-solid coupled equation for reflecting the interaction between solid diaphragm and fluid motion in a micropump. Numerical results of a mode analysis of the coupled system indicate that the natural frequencies of the coupled system are much lower than those of the non-coupled system. The influence of additional mass and viscous damp of fluid on the natural frequencies of the coupled system is more significant as the pump thickness is small. It is found that the vibration shape functions of silicon diaphragm of the coupled system are almost the same as those of the non-coupled system. This paper also gives the first-order amplitude-frequency relationship of the silicon diaphragm, which is necessary for the flow-rate-frequency analysis of a micropump.     

Instability theory of shock wave in a channel

The instability theory of shock wave was extended from the case with an infinitefront to the case of a channel with a rectangular cross section.First,themathematical formulation of the problem was given which included a system ofdisturbed equations and three kinds of boundary conditions.Then,the general solutionsof the equations upstream and downstream were given and each contained fiveconstants to be determined.Thirdly,under one boundary condition and oneassumption,it was proved that all of the disturbances in front of the shock front andone of the two acoustic disturbances behind the shock front should be zero.Theboundary condition was that all of the disturbed physical quantities should approach tozero at infinity.The assumption was that only the unstable shock wave was concernedhere.So it was reasonable to assumeω=iγ,γwas the instability growth rate andwas a positive real number.Another kind of boundary conditions was that the normaldisturbed velocities should be zero at the solid wall of the cha     

New exact solutions for free vibrations of rectangular thin plates by symplectic dual method

The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.     

Generalized expressions for boundary conditions of shells of revolution

For the boundary conditions of shells of revolution, traditionally, four out of the eight quantities which are the four displacements on the middle surface u, v, w and if together with the four corresponding forces, are given. when the generalized displacements on the nodal circles are used as basic unknowns, the number of unknowns on a nodal circle is more than four^[1][2][3][4]. In this case, how to deal with the boundary conditions is still a problem that has not been solved satisfactorily yet. In this paper,the relations between the generalized and nongeneralized quantities of a shell’s edge are derived according to the principle of virtual work. Seven types of common edges are studied and their expressions of boundary conditions in the form of generalized displacements or forces are qiven. The number of expressions for each type of edge may correspond with the number of unknowns used on a nodal circle. Kith these expressions, boundary conditions can be put directly into equations of motion of generalized displacement method so as to solve the generalized displacements. By so doing, the process of transformation and inverse transformation about unknowns in [2] is avoided. Not only is the argument simple and clear, but the calculation work is reduced.Having the set of generalized expressions of boundary conditions, the generalized displacement method of the shell of revolution may be more perfect in theory.     

THE BENDING, STABILITY AND VIBRATIONS OF RECTANGULAR PLATES WITH FREE EDGES ON ELASTIC FOUNDATIONS

This paper discusses the problems of the bending, stabilio, and vibrations of the rectangular plates with free boundaries on elastic foundations, In the present paper we select a flexural function, which satisfies not only all the boundary Conditions of free edges but also the conditions at free corner points, and consequently we obtain a better approximate solution. The energy method is used in this paper.     

Flow patterns and boundary conditions for inlet and outlet conduits of large pump system with low head

The flow patterns in the inlet and outlet conduits have a decisive effect on the safe, stable, and highly efficient operation of the pump in a large pumping station with low head. The numerical simulation of three-dimensional （3D） turbulence flow in conduits is an important method to study the hydraulic performance and conduct an optimum hydraulic design for the conduits. With the analyses of the flow patterns in the inlet and outlet conduits, the boundary conditions of the numerical simulation for them can be determined. The main obtained conclusions are as follows： （i） Under normal operation conditions, there is essentially no pre-swirl flow at the impeller chamber inlet of an axial-flow pump system, based on which the boundary condition at the inlet conduit may be defined. （ii） The circulation at the guide vane outlet of an axial-flow pump system has a great effect on the hydraulic performance of the outlet conduit, and there is optimum circulation for the performance. Therefore, it is strongly suggested to design the guide vane according to the optimum circulation. （iii） The residual circulation at the guide vane outlet needs to be considered for the inlet boundary condition of the outlet conduit, and the value of the circulation may be measured in a specially designed test model.