Abstract: The problem of nonlinear oscillations of a two-layer liquid completely filling a limited volume is considered. Using two basic asymmetric harmonics excited in two mutually perpendicular planes, ordinary differential equations of nonlinear oscillations of the interface of a two-layer liquid are investigated. In this paper, hydrodynamic coefficients of linear and nonlinear problems in integral relations were determined. As a result, the instability regions of forced oscillations of a two-layered liquid in a cylindrical tank occurring in the plane of action of the disturbing force are constructed, as well as the dynamic instability regions of the parametric resonance for different ratios of densities of the upper and lower liquids depending on the amplitudes of liquids from the excitations frequencies. Steady-state regimes of fluid motion were found in the regions of dynamic instability of the initial oscillation form. The Bubnov-Galerkin method is used to construct instability regions for approximate solution of nonlinear differential equations.
Abstract: A method for determining the stress distribution of a rectangular plate subjected to two pairs of arbitrarily distributed in-plane edge shear loads is proposed, and the free vibration and buckling of such a rectangular plate are investigated in this work. The method utilizes two stress functions to synthesize the stress-resultant field of the plate with each of the stress functions satisfying the biharmonic compatibility equation. The sum of stress-resultant fields due to these two stress functions satisfies the boundary conditions at the edges of the plate, from which these two stress functions are determined. Then, the free vibration and buckling of the rectangular plate are investigated by the Galerkin method. Numerical results obtained by this work are compared with those appeared in the literature, and good agreements are observed.
Abstract: In this paper, longitudinal vibration of a micro-beam in micro-scale fluid media has been investigated. The proposed mathematical model for this study is made up of a micro-beam and a micro-plate at its free end. An AC voltage is applied to the pair of piezoelectric layers on the upper and lower surfaces of the micro-beam in order to actuate it longitudinally. The whole structure is bounded between two fixed plates on its upper and lower surfaces. The micro-gap between the structure and the fixed plates is filled with fluid. Fluids behave differently in micro-scale than macro, so the fluid field in the gap has been modeled based on micro-polar theory. The coupled governing equations of motion of the micro-beam and the micro-scale fluid field have been derived. Due to having non-homogenous boundary conditions, derived equations have been transformed to an enhanced form with homogenous boundary conditions. Using Galerkin-based reduced order model, the enhanced equations have been discretized over the beam and fluid domains and solve simultaneously in order to obtain force response of the micro-beam. Effects of micro-polar parameters of the fluid as characteristic length scale, coupling parameter and surface parameter on the response of the micro-beam have been studied.
Abstract: Pull-in instability is a nonlinear and crucial effect that is important for the design of microelectromechanical system devices. In this paper, the appropriate electrostatic voltage range is determined by measuring fluid flow pressure via micro pressure sensor based microbeam. The microbeam deflection contains two parts, the static and perturbation deflection of static. The second order equation regarding the equivalent stiffness, mass and damping matrices based on Galerkin method is introduced to predict pull-in instability due to the external voltage. Also the reduced order method is used for solving the second order nonlinear equation of motion. Furthermore, in the present study, the micro capacitive pressure sensor is designed for measuring special fluid flow pressure range. The results show that the measurable pressure range can be optimized, regarding damping field and external voltage.
Abstract: Electrical current measurement is a suitable method for the performance determination of electrical devices. There are two contact and noncontact methods in this measuring process. Contact method has some disadvantages like having direct connection with wire which may endamage the system. Thus, in this paper, a bimorph piezoelectric cantilever beam which has a permanent magnet on its free end is used to measure electrical current in a noncontact way. In mathematical modeling, based on Galerkin method, the governing equation of the cantilever beam is solved, and the equation presenting the relation between applied force and beam’s output voltage is presented. Magnetic force resulting from current carrying wire is considered as the external excitation force of the system. The results are compared with other references in order to demonstrate the accuracy of the mathematical model. Finally, the effects of geometric parameters on the output voltage and natural frequency are presented.
Abstract: We present and analyze reliable numerical techniques
for simulating complex flow and transport phenomena related to
natural gas transportation in pipelines. Such kind of problems
are of high interest in the field of petroleum and environmental
engineering. Modeling and understanding natural gas flow and
transformation processes during transportation is important for the
sake of physical realism and the design and operation of pipeline
systems. In our approach a two fluid flow model based on a system
of coupled hyperbolic conservation laws is considered for describing
natural gas flow undergoing hydratization. The accurate numerical
approximation of two-phase gas flow remains subject of strong
interest in the scientific community. Such hyperbolic problems are
characterized by solutions with steep gradients or discontinuities, and
their approximation by standard finite element techniques typically
gives rise to spurious oscillations and numerical artefacts. Recently,
stabilized and discontinuous Galerkin finite element techniques
have attracted researchers’ interest. They are highly adapted to the
hyperbolic nature of our two-phase flow model. In the presentation
a streamline upwind Petrov-Galerkin approach and a discontinuous
Galerkin finite element method for the numerical approximation of
our flow model of two coupled systems of Euler equations are
presented. Then the efficiency and reliability of stabilized continuous
and discontinous finite element methods for the approximation is
carefully analyzed and the potential of the either classes of numerical
schemes is investigated. In particular, standard benchmark problems
of two-phase flow like the shock tube problem are used for the
comparative numerical study.
Abstract: The motion of an axially moving beam with rotating prismatic joint with a tip mass on the end is analyzed to investigate the nonlinear vibration and dynamic stability of the beam. The beam is moving with a harmonic axially and rotating velocity about a constant mean velocity. A time-dependent partial differential equation and boundary conditions with the aid of the Hamilton principle are derived to describe the beam lateral deflection. After the partial differential equation is discretized by the Galerkin method, the method of multiple scales is applied to obtain analytical solutions. Frequency response curves are plotted for the super harmonic resonances of the first and the second modes. The effects of non-linear term and mean velocity are investigated on the steady state response of the axially moving beam. The results are validated with numerical simulations.
Abstract: The present study investigates the effect of inclusions on the shape and size of crack tip plastic zones in engineering materials subjected to static loads by employing the element free Galerkin method (EFGM). The modeling of the discontinuities produced by cracks and inclusions becomes independent of the grid chosen for analysis. The standard displacement approximation is modified by adding additional enrichment functions, which introduce the effects of different discontinuities into the formulation. The level set method has been used to represent different discontinuities present in the domain. The effect of inclusions on the extent of crack tip plastic zones is investigated by solving some numerical problems by the EFGM.
Abstract: In this work, functionally graded materials (FGMs), subjected to loading, which varies with time has been studied. The material properties of FGM are changing through the thickness of material as power law distribution. The conical shells have been chosen for this study so in the first step capability equations for FGM have been obtained. With Galerkin method, these equations have been replaced with time dependant differential equations with variable coefficient. These equations have solved for different initial conditions with variation methods. Important parameters in loading conditions are semi-vertex angle, external pressure and material properties. Results validation has been done by comparison between with those in previous studies of other researchers.
Abstract: A nonlinear model of the mathematical fluid dynamics which describes the motion of an incompressible viscous rotating fluid in a homogeneous gravitational field is considered. The model is a generalization of the known Navier-Stokes system with the addition of the Coriolis parameter and the equations for changeable density. An explicit algorithm for the solution is constructed, and the proof of the existence and uniqueness theorems for the strong solution of the nonlinear problem is given. For the linear case, the localization and the structure of the spectrum of inner waves are also investigated.
Abstract: This article presents a numerical method to find the
heat flux in an inhomogeneous inverse heat conduction problem with
linear boundary conditions and an extra specification at the terminal.
The method is based upon applying the satisfier function along with
the Ritz-Galerkin technique to reduce the approximate solution of the
inverse problem to the solution of a system of algebraic equations.
The instability of the problem is resolved by taking advantage of
the Landweber’s iterations as an admissible regularization strategy.
In computations, we find the stable and low-cost results which
demonstrate the efficiency of the technique.
Abstract: This paper develops a meshless approach, called Element Free Galerkin (EFG) method, which is based on the weak form Moving Least Squares (MLS) of the partial differential governing equations and employs the interpolation to construct the meshless shape functions. The variation weak form is used in the EFG where the trial and test functions are approximated bye the MLS approximation. Since the shape functions constructed by this discretization have the weight function property based on the randomly distributed points, the essential boundary conditions can be implemented easily. The local weak form of the partial differential governing equations is obtained by the weighted residual method within the simple local quadrature domain. The spline function with high continuity is used as the weight function. The presently developed EFG method is a truly meshless method, as it does not require the mesh, either for the construction of the shape functions, or for the integration of the local weak form. Several numerical examples of two-dimensional static structural analysis are presented to illustrate the performance of the present EFG method. They show that the EFG method is highly efficient for the implementation and highly accurate for the computation. The present method is used to analyze the static deflection of beams and plate hole
Abstract: In this paper, the influence of van der Waals, as well as electrostatic forces on the structural behavior of MEMS and NEMS actuators, has been investigated using of a Euler-Bernoulli beam continuous model. In the proposed nonlinear model, the electrostatic fringing-fields and the mid-plane stretching (geometric nonlinearity) effects have been considered. The nonlinear integro-differential equation governing the static structural behavior of the actuator has been derived. An original Galerkin-based reduced-order model has been developed to avoid problems arising from the nonlinearities in the differential equation. The obtained reduced-order model equations have been solved numerically using the Newton-Raphson method. The basic design parameters such as the pull-in parameters (voltage and deflection at pull-in), as well as the detachment length due to the van der Waals force of some investigated micro- and nano-actuators have been calculated. The obtained numerical results have been compared with some other existing methods (finite-elements method and finite-difference method) and the comparison showed good agreement among all assumed numerical techniques.
Abstract: This paper investigates the parametric stability of an
axially moving web subjected to non-uniform in-plane edge
excitations on two opposite, simply-supported edges. The web is
modeled as a viscoelastic plate whose constitutive relation obeys the
Kelvin-Voigt model, and the in-plane edge excitations are expressed
as the sum of a static tension and a periodical perturbation. Due to the
in-plane edge excitations, the moving plate may bring about
parametric instability under certain situations. First, the in-plane
stresses of the plate due to the non-uniform edge excitations are
determined by solving the in-plane forced vibration problem. Then,
the dependence on the spatial coordinates in the equation of transverse
motion is eliminated by the generalized Galerkin method, which
results in a set of discretized system equations in time. Finally, the
method of multiple scales is utilized to solve the set of system
equations analytically if the periodical perturbation of the in-plane
edge excitations is much smaller as compared with the static tension of
the plate, from which the stability boundaries of the moving plate are
obtained. Numerical results reveal that only combination resonances
of the summed-type appear under the in-plane edge excitations
considered in this work.
Abstract: New physical insights into the nonlinear Lorenz
equations related to flow resistance is discussed in this work. The
chaotic dynamics related to Lorenz equations has been studied in
many papers, which is due to the sensitivity of Lorenz equations to
initial conditions and parameter uncertainties. However, the physical
implication arising from Lorenz equations about convectional motion
attracts little attention in the relevant literature. Therefore, as a first
step to understand the related fluid mechanics of convectional motion,
this paper derives the Lorenz equations again with different forced
conditions in the model. Simulation work of the modified Lorenz
equations without the viscosity or buoyancy force is discussed. The
time-domain simulation results may imply that the states of the
Lorenz equations are related to certain flow speed and flow resistance.
The flow speed of the underlying fluid system increases as the flow
resistance reduces. This observation would be helpful to analyze the
coupling effects of different fluid parameters in a convectional model
in future work.
Abstract: In this paper, a numerical algorithm using a coupled Galerkin-Differential Quadrature (DQ) method is proposed for the solution of dam-reservoir interaction problem. The governing differential equation of motion of the dam structure is discretized by the Galerkin method and the DQM is used to discretize the fluid domain. The resulting systems of ordinary differential equations are then solved by the Newmark time integration scheme. The mixed scheme combines the simplicity of the Galerkin method and high accuracy and efficiency of the DQ method. Its accuracy and efficiency are demonstrated by comparing the calculated results with those of the existing literature. It is shown that highly accurate results can be obtained using a small number of Galerkin terms and DQM sampling points. The technique presented in this investigation is general and can be used to solve various fluid-structure interaction problems.
Abstract: The effect of internal heat generation is applied to the Rayleigh-Benard convection in a horizontal micropolar fluid layer. The bounding surfaces of the liquids are considered to be rigid-free, rigid-rigid and free-free with the combination of isothermal on the spin-vanishing boundaries. A linear stability analysis is used and the Galerkin method is employed to find the critical stability parameters numerically. It is shown that the critical Rayleigh number decreases as the value of internal heat generation increase and hence destabilize the system.
Abstract: This paper deals with a high-order accurate Runge
Kutta Discontinuous Galerkin (RKDG) method for the numerical
solution of the wave equation, which is one of the simple case of a
linear hyperbolic partial differential equation. Nodal DG method is
used for a finite element space discretization in 'x' by discontinuous
approximations. This method combines mainly two key ideas which
are based on the finite volume and finite element methods. The
physics of wave propagation being accounted for by means of
Riemann problems and accuracy is obtained by means of high-order
polynomial approximations within the elements. High order accurate
Low Storage Explicit Runge Kutta (LSERK) method is used for
temporal discretization in 't' that allows the method to be nonlinearly
stable regardless of its accuracy. The resulting RKDG
methods are stable and high-order accurate. The L1 ,L2 and L∞ error
norm analysis shows that the scheme is highly accurate and effective.
Hence, the method is well suited to achieve high order accurate
solution for the scalar wave equation and other hyperbolic equations.
Abstract: In this paper, a numerical solution based on sinc
functions is used for finding the solution of boundary value problems
which arise from the problems of calculus of variations. This
approximation reduce the problems to an explicit system of algebraic
equations. Some numerical examples are also given to illustrate the
accuracy and applicability of the presented method.
Abstract: The Wavelet-Galerkin finite element method for
solving the one-dimensional heat equation is presented in this work.
Two types of basis functions which are the Lagrange and multi-level
wavelet bases are employed to derive the full form of matrix system.
We consider both linear and quadratic bases in the Galerkin method.
Time derivative is approximated by polynomial time basis that
provides easily extend the order of approximation in time space. Our
numerical results show that the rate of convergences for the linear
Lagrange and the linear wavelet bases are the same and in order 2
while the rate of convergences for the quadratic Lagrange and the
quadratic wavelet bases are approximately in order 4. It also reveals
that the wavelet basis provides an easy treatment to improve
numerical resolutions that can be done by increasing just its desired
levels in the multilevel construction process.