Modified Galerkin Method Research Articles

The free vibration of free-clamped fluid-coupled coaxial cylindrical shells.

The linear free vibration of free-clamped coaxial cylinders partially filled with incompressible, inviscid liquid in the annular gap is investigated theoretically on the basis of the Donnell-type equations for cylinders and the velocity potential theory for liquid motion. The problem is solved by the modified Galerkin method. The initial axisymmetric deformation of the shell due to the static liquid pressure as well as the boundary condition on the free liquid surface are fully taken into consideration. It is found that the static liquid pressure and the liquid surface condition have a significant effect on the natural frequency, and that the interactive effect of the coaxial cylinders becomes small and the mode shape changes with an increase in the wave number and the annular gap.

Vibrations of plates of arbitrary shape by a modified Galerkin method

This paper deals with a modification of Galerkin's method which allows for the optimization of the fundamental eigenvalue when a single co-ordinate function is used. Use of the conformal mapping method allows for the consideration of two dimensional domains of complicated boundary shape.

The dynamic stability of liquid-filled cylindrical shells under vertical excitation.

Theoretical solutions are presented for the dynamic stability of liquid-filled cylindrical shells under vertical excitation. For the motion of the shell, the modified Galerkin method is applied to the dynamic version of the Donnell basic equation ; while, for the motion of the liquid, the velocity potential of the liquid is used so as to satisfy the boundary conditions. The equation of motions coupled with both the shell and the liquid are derived from a type of coupled Mathieu's equations. The instability boundaries at which parametric resonance occurs are determined using Hsu's method. The validity of the theoretical analysis was confirmed through precise experiments using the polyester test cylinder and water. As a result, it was found that a principal instability resonance and a combination instability resonance of the sum type of two natural vibrations, each of which has the same circumferential wave number and different axial mode of vibration, are likely to occur.

Natural convection in a vertical porous enclosure with internal heat generation

A theoretical study is performed on steady natural convection inside a rectangular porous enclosure with uniform internal heat generation and cooling from the side walls. For low Rayleigh numbers, the problem is tackled by regular expansion in the Rayleigh number and the resulting Poisson equations are sequentially solved by the modified Galerkin method. For large Rayleigh numbers, the boundary layer equations valid near the vertical walls along with the core region equations are examined using the Gill linearization method. The analytical predictions agree well with the numerical solutions based on the power-law difference representation of the full governing equations.

Finite Element Analysis of Diffusion and Flow Problems : 3rd Report, The Analysis of Transient, Two-Dimensional, Incompressible Viscous Flow by Modified Galerkin Method

Finite Element Analysis of Diffusion and Flow Problems : 3rd Report, The Analysis of Transient, Two-Dimensional, Incompressible Viscous Flow by Modified Galerkin Method

Acoustic transmission in non-uniform ducts with mean flow, part I: The method of weighted residuals

3n this and a companion paper the problem of transmission of sound through non-uniform ducts carrying a high speed subsonic compressible flow is approached using the method of weighted residuals (MWR) in the form of a modified Galerkin method and the finite element method (FEM). The intent of the investigation is to carry out a careful evaluation of these methods in this computationally difficult problem. To this end both MWR and FEM have been limited to in-core computer implementations to generate useful results with relatively modest computational requirements. This paper (Part I) details the MWR formulation and presents numerical results establishing the degree of accuracy of MWR as compared to exact eigenvalue calculations and approximate one dimensional transmission calculations. The comparison of MWR and FEM results is carried out in the companion paper (Part II).

Acoustic transmission in non-uniform ducts with mean flow, part II: The finite element method

This second paper in a two part series describes the implementation of the finite element method for the solution of the problem of acoustic transmission through a non-uniform duct carrying a high speed subsonic compressible flow. A finite element scheme based on both the Galerkin method and the residual least squares method and with eight noded isoparametric elements is described. Multi-modal propagation is investigated by coupling of the solution in the duct non-uniform section to modal expansions in uniform sections. The accuracy of the finite element results for both the eigenvalue and transmission problems is assessed by comparison with exact solutions and with results from the method of weighted residuals in the form of a modified Galerkin method as introduced in Part I of this pair of papers. The results of calculations show that modal interactions, particularly in transmitted modes, become increasingly important with increasing duct flow Mach number. Power transmission coefficient calculations for the geometries studied reveal no indication of a linear basis for the phenomenon of subsonic acoustic choking.

A smoothed projection method for singular, nonlinear Volterra equations

A modified Galerkin method previously used to approximate the solution of nonlinear Volterra integral equations of the second kind with smooth kernels is generalized to include such equations with singular, monotone kernels of convolution type. Several singular kernel approximations are considered, including positive convolution operators and integral splines. The main results relating to the original integral equation supply error estimates resulting from using the kernel approximations and an approximating system of ordinary differential equations.

Finite Element Analysis of Diffusion and Flow problems : 1st Report, A proposal of Modified Galerkin Method for One-Dimensional Non-Steady Diffusion Problems

Finite Element Analysis of Diffusion and Flow problems : 1st Report, A proposal of Modified Galerkin Method for One-Dimensional Non-Steady Diffusion Problems

A finite element method for transmission in non-uniform ducts without flow: Comparison with the method of weighted residuals

A finite element method is developed for the study of transmission of sound in non-uniform ducts without flow. The formulation is based on a weighted residual approach and eight noded isoparametric elements are used. Two computational schemes are described, one based on the Helmholtz equation obtained by combining the basic conservation equations and one based on the conservation equations themselves. The latter case is considered because in future extensions to problems involving mean flow a single governing equation is not readily obtainable except for irrotational flows. Both two-dimensional and circular duct geometries are considered. Comparisons are made with a Method of Weighted Residuals in the form of a Modified Galerkin Method in the two-dimensional case to assess both accuracy and computational efficiency. It is found that the finite element method produces results for transmission and reflection coefficients nearly identical to those from the Galerkin approach. Used to its best advantage the finite element method is of comparable efficiency.

A method of weighted residuals with trigonometric basis functions for sound transmission in circular ducts

A modified Galerkin method is employed to formulate the problem of transmission of sound in a uniform circular duct. The basis functions used are trigonometric and are derived from the equivalent problem in a two-dimensional duct. In the case when flow is present basis functions from the case without flow are used. It is found that the success of the method is dependent on the choice of basis functions within the class of functions discussed, particularly when flow is present. Results for axial wave numbers are generally of accuracy similar to the results achievable by a similar approach in the two-dimensional case.

Numerical solution of the stability of hydromagnetic couette flow with hall currents

In the present problem, we have studied numerically the effects of Hall currents and the magnetic Prandtl number P m on the stability of hydromagnetie Couette flow. In obtaining a numerical solution of the problem by Runge-Kutta-Merson method (here after will be called as RKM method), we have used the approximate solution found by modified Galerkin method as the starting solution. The accuracy of the method is tested by comparing the numerical solution with the exact solution of Chandrasekhar [1]. It is found that even though Hall currents are destabilizing for small and large P m , they tend to stabilize for certain intermediate values.

Asymptotic stability of a catalyst particle

The catalyst asymptotic stability problem is studied by means of several new methods that allow accurate solutions to be calculated where other methods have given qualitatively erroneous results. The underlying eigenvalue problem is considered in three limiting situations Le = ∞, 1 and 0. These are solved first to give expansion functions for the full eigenproblem at an arbitrary value of Le. A modified Galerkin method based on the two sets of eigenfunctions for the Le = 1 problem even in the lowest order approximation able to solve the full problem for all Le. Perturbation methods developed from the solutions at the limiting Le-values may if properly handled be extremely useful, e.g. for calculating the stability limit. Finally approximate values of the higher eigenvalues are found by simple formulas based on the steady state solution. Thus the whole eigenvalue spectrum can be studied in detail and certain properties of the spectrum that should appear also with other rate expressions are described.

An Analysis of Some Galerkin Schemes for the Solution of Nonlinear Time-Dependent Problems

A priori error estimates are derived for several Galerkin procedures. For a nonlinear parabolic initial-boundary value problem, optimal rate L2-error estimates are derived for the forward and central difference Laplace modified Galerkin methods and the forward and central difference Laplace modified alternating direction Galerkin methods. For a nonlinear second order hyperbolic initial-boundary value problem, optimal rate L2-error estimates are derived for the Laplace modified Galerkin method and the Laplace modified alternating direction Galerkin method. 1. Introduction. In this paper we derive a priori error estimates for several Galerkin procedures. In ?3 we consider a nonlinear parabolic initial-boundary value problem, and we derive error estimates for the forward and central difference Laplace modified Galerkin methods and the forward and central difference Laplace modified alternating direction Galerkin methods of (5); these methods were formulated in (5), but in the central difference methods proofs were given only for the case in which the elliptic part was linear. In ?4 we consider a nonlinear second order hyperbolic initial-boundary value problem, and we derive error estimates for the Laplace modified Galerkin method and the Laplace modified alternating direction Galerkin method of (5); these methods are the obvious extensions of the methods of (5) to the nonlinear case. In ?2 we present certain notation, assumptions, and known results which are common to the development of ? 3 and 4. Section 5 contains some computational results.

A numerical study of rotating annulus flows using a modified Galerkin method

Physical phenomena fundamental to rotating, baroclinically driven flows are studied with reference to results of numerical simulation of rotating annulus flows, using a modified Galerkin Model. Both local and global effects of sources, sinks, and transports of heat and momentum are discussed. A convenient ‘energy exchange” diagram reveals detailed information that is used to analyze nonlinear equilibration and amplitude vacillation of quasi-geostrophic baroclinic eddies. Transient inertial oscillations, sidewall boundary layers, and internal boundary layers are also discussed. A detailed study of symmetric flows is made, eleven of which are tested numerically for stability with respect to three-dimensional disturbances of a given zonal wave number. Two of the four unstable cases are integrated to a numerical steady state with finite-amplitude, quasi-geostrophic baroclinic waves. With the ‘rigid-lid’ geometry assumed, the average zonal velocity is zero, resulting in zero phase velocity of the waves. The structure of the thermal wave is nearly coherent in the vertical. These numerical results are consistent with laboratory observations. The eddy flow is quasi-geostrophic except in horizontal boundary layers, where the flow is driven toward low pressure. A small cross-isotherm advection is sufficient to maintain the temperature wave against diffusion and vertical advection. The eddy flow adjusts spontaneously toward the form of the fastest growing or slowest decaying disturbance representable by the truncated space resolution. The eddy flow feeds energy into the mean zonal flow in ‘barotropic-type’ interactions reflected mainly by the familiar ‘tilted trough’. During equilibration, the eddy flow alters the mean zonal flow in such a way that eddy energy sources are reduced relative to energy sinks. However, this adjustment is small compared to the change of total flow, which reflects a relatively large change of eddy amplitude. This suggests that small errors in the mean zonal flow representation can lead to relatively large errors in total flow representation. In most flows studied the kinetic energy dissipation is concentrated in thin boundary layers. In spite of this thinness, the basically laminar character of these dissipative boundary layers allows accurate and economical numerical simulation through the use of characteristic functions, which is a natural refinernent of the basic Galerkin method used. (In this prototype study, only “moderately characteristic’ functions are used, thus sacrificing numerical economy while simplifying the programming.) Similarly, the generation of potential energy, which is transformed into the kinetic energy of the flow, is accurately simulated. In most cases studied, this generation is also concentrated in thin boundary layers where thermal energy is extracted from cold fluid and added to warm fluid.

Recherche numérique d'oscillations composées du type additif dans un système oscillant non-linéaire amorti à deux degrés de liberté

The combination tones of summed type are studied in a non-linear damped vibratory system of two degrees of freedom with spring characteristics of second and third order and excited by two external harmonic forces of different frequencies. A modified Galerkin method is presented, based on the replacement of some integrals by their Simpson approximations instead of using the formulae given by M. Urabe. For some systems this modified method has the advantage to ameliorate the precision of the Galerkin approximations and to diminish considerably the error of these approximations. The existence and stability of an exact periodic solution is discussed and the error bound of the Galerkin approxi¬ mations is given. The modified Galerkin method is compared to other approximation methods for two numerical examples.

Large amplitude vibrations of plates on elastic foundations

Simplified non-linear, uncoupled differential equations for the dynamic case of an elastic plate resting on an elastic foundation, vibrating with large amplitudes are presented by making the assumption, due to Berger [5], that the strain energy corresponding to the second invariant of the middle surface strains can be neglected. A modified Galerkin method is used to derive a non-linear differential equation of which the solution is given in terms of elliptic functions. Numerical results are presented for a simply supported rectangular plate and simply supported and clamped circular plates performing large amplitude vibrations.