Global Galerkin Research Articles

Kinetic Theory of Plasma Waves - Part I: Introduction

The kinetic description of linear waves in plasmas is succinctly presented, with emphasis on applications to high-frequency (hf) wave heating and current drive. The Maxwell-Vlasov system of equations is introduced. Its two-timescale analysis yields the linearized Vlasov and the quasilinear Fokker-Planck equations. The standard guiding centre and Hamiltonian formalisms are presented. Two formulations of the hf plasma wave equation are given: as a partial differential equation to hold at each position, and as a global Galerkin (’variational’) form.

Implementation of arbitrary inner product in the global Galerkin method for incompressible Navier–Stokes equations

The global Galerkin or weighted residuals method applied to the incompressible Navier–Stokes equations is considered. The basis functions are assumed to be divergence-free and satisfy all the boundary conditions. The method is formulated for an arbitrary inner product, so that the pressure cannot be eliminated by Galerkin projections on a divergence-free basis. A proposed straightforward procedure for the elimination of the pressure reduces the problem to an ODE system without algebraic constraints. To illustrate the applicability and the robustness of the numerical approach and to show that numerical solutions with unit and non-unit weight functions yield similar results the driving lid cavity and natural convection benchmark problems are solved using the unit and Chebyshev weight functions. Further implications of the proposed Galerkin formulation are discussed.

Kinetic Theory of Plasma Waves - Part I: Introduction

The kinetic description of linear waves in plasmas is succinctly presented, with emphasis on applications to high-frequency (hf) wave heating and current drive. The Maxwell-Vlasov system of equations is introduced. Its two-timescale analysis yields the linearized Vlasov and the quasilinear Fokker-Planck equations. The standard guiding centre and Hamiltonian formalisms are presented. Two formulations of the hf plasma wave equation are given: as a partial differential equation to hold at each position, and as a global Galerkin (’variational’) form.

The numerical solution of exterior neumann problem for Helmholtz's equation via modified green's functions approach

In the 1970s, modified Green's function approach for solving the Helmholtz equation was proposed by Jones and Ursell and in the 1980s was clarified by Kleinman, Roach and Kress. To this date there are no numerical results available for this approach. In this paper, a global Galerkin method is used to numerically solve the exterior Neumann problem for the Helmholtz equation in three dimensions based on Jones' modified integral equation approach. Jones approach directly leads to an integral equation which only involves weakly singular operators, thus is a good alternative for solving the exterior Neumann problem. Theoretical and computational details of the method are presented.

Stability and slightly supercritical oscillatory regimes of natural convection in a 8:1 cavity: solution of the benchmark problem by a global Galerkin method

AbstractThe global Galerkin method is applied to the benchmark problem that considers an oscillatory regime of convection of air in a tall two‐dimensional rectangular cavity. The three most unstable modes of the linearized system of the Boussinesq equations are studied. The converged values of the critical Rayleigh numbers together with the corresponding oscillation frequencies are calculated for each mode. The oscillatory flow regimes corresponding to each of the three modes are approximated asymptotically. No direct time integration is applied. Good agreement with the previously published results obtained by solution of the time‐dependent Boussinesq equations is reported. Copyright © 2004 John Wiley & Sons, Ltd.

The numerical solution of the exterior dirichlet problem for Helmholtz's equation via modified green's functions approach

In the 1970s, a modified Green's function approach for solving the Helmholtz equation was proposed by Jones and Ursell, and in the 1980s was clarified by Kleinman, Roach and Kress. But to date, there are no numerical results available for this approach. In this paper, a global Galerkin method is used to numerically solve the exterior Dirichlet problem for the Helmholtz equation in three dimensions based on Jones' modified integral equation approach. Theoretical and computational details of the method are presented.

Kinetic Theory of Plasma Waves - Part I: Introduction

The kinetic description of linear waves in plasmas is succinctly presented, with emphasis on applications to high-frequency (hf) wave heating and current drive. The Maxwell-Vlasov system of equations is introduced. Its two-timescale analysis yields the linearized Vlasov and the quasilinear Fokker-Planck equations. The standard guiding centre and Hamiltonian formalisms are presented. Two formulations of the hf plasma wave equation are given: as a partial differential equation to hold at each position, and as a global Galerkin (‘variational’) form.

An adaptive procedure based on background cells for meshless methods

In practical implementations, meshless methods using moving least-squares (MLS) approximation and global Galerkin formulation require a background mesh for domain integration. An adaptive procedure based on background mesh is developed for meshless methods using MLS. It comprises a cell energy error estimate and a local domain refinement technique. The error estimate differs from conventional pointwise approaches in that it evaluates error based on individual cells instead of points. For each cell, a computed cell energy and a reference cell energy are generated based on a same stress field by using two different integration schemes; the difference between the two energy values is used as the basic measure of error estimation. The advantage of this strategy is that it requires only one stress field and no reference field needs to be furnished. This has significance, as the stress field generated by meshless methods is already very smooth and traditional error estimates based on stress smoothing techniques for finite element methods are not applicable. To achieve high efficiency in domain refinement, a local technique based on the Delaunay algorithm is implemented. In this technique, each node is assigned a scaling factor to control local nodal density; refinement of the neighborhood of a node is accomplished simply by adjusting its scaling factor. The numerical experiments in this paper show that the proposed adaptive procedure is simple, effective and efficient. The limitations of the approach are also discussed.

Three-dimensional instability of a two-layer Dean flow

Stability of a two-layer Dean flow in a cylindrical annulus with respect to three-dimensional perturbations is studied by a global Galerkin method. It is shown that for large inner radius of the annulus (i) the instability becomes three-dimensional if one of the fluid layers is thin, (ii) its onset is not affected by possible small deformations of the interface, and (iii) multiple three-dimensional flow states are expected in a slightly supercritical flow regime. Stability diagrams and patterns of the three-dimensional perturbations are reported. It is concluded that even when the axisymmetric perturbation is the most dangerous, the resulting supercritical flow is expected to be three-dimensional. Possible multiplicity of supercritical three-dimensional states is predicted. The basis functions of the global Galerkin method are constructed so as to satisfy analytically the boundary conditions on no-slip walls and at the liquid–liquid interface. A modification of the numerical approach, accounting for small deformations of the interface which is subject to the action of the capillary force, is proposed. The results are of potential importance for development of novel bioseparators employing Dean vortices for enhancement of mass transfer of a passive scalar (say, a protein) through the interface. The developed numerical approach can be used for stability analysis in other two-fluid systems.

Three-dimensional instability of axisymmetric flow in a rotating lid–cylinder enclosure

The axisymmetry-breaking three-dimensional instability of the axisymmetric flow between a rotating lid and a stationary cylinder is analysed. The flow is governed by two parameters – the Reynolds number Re and the aspect ratio γ (=height/radius). Published experimental results indicate that in different ranges of γ axisymmetric or non-axisymmetric instabilities can be observed. Previous analyses considered only axisymmetric instability. The present analysis is devoted to the linear stability of the basic axisymmetric flow with respect to the non-axisymmetric perturbations. After the linearization the stability problem separates into a family of quasi-axisymmetric subproblems for discrete values of the azimuthal wavenumber k. The computations are done using the global Galerkin method. The stability analysis is carried out at various densely distributed values of γ in the range 1 < γ < 3.5. It is shown that the axisymmetric perturbations are dominant in the range 1.63 < γ < 2.76. Outside this range, for γ < 1.63 and for γ > 2.76, the instability is three-dimensional and sets in with k = 2 and k = 3 or 4, respectively. The azimuthal periodicity, patterns, characteristic frequencies and phase velocities of the dominant perturbations are discussed.

Kinetic Theory of Plasma Waves - Part I: Introduction

The kinetic description of linear waves in plasmas is succinctly presented, with emphasis on applications to high-frequency (hf) wave heating and current drive. The Maxwell-Vlasov system of equations is introduced. Its two-timescale analysis yields the linearized Vlasov and the quasilinear Fokker-Planck equations. The standard guiding centre and Hamiltonian formalisms are presented. Two formulations of the hf plasma wave equation are given: as a partial differential equation to hold at each position, and as a global Galerkin (‘variational’) form.

Stability of multiple steady states of convection in laterally heated cavities

A parametric study of multiple steady states, their stability, onset of oscillatory instability, and some supercritical unsteady regimes of convective flow of a Boussinesq fluid in laterally heated rectangular cavities is presented. Cavities with four no-slip boundaries, isothermal vertical and perfectly insulated horizontal boundaries are considered. Four distinct branches of steady-state flows are found for this configuration. A complete study of stability of each branch is performed for the aspect ratio A (length/height) of the cavity varying continuously from 1 to 11 and for two fixed values of the Prandtl number: Pr = 0 and Pr = 0.015. The results are represented as stability diagrams showing the critical parameters (critical Grashof number and the frequency at the onset of the oscillatory instability) corresponding to transitions from steady to oscillatory states, appearance of multi-roll states, merging of multiple states and backwards transitions from multi-roll to single-roll states. For better comparison with the existing experimental data, an additional stability study for varying Prandtl number (0.015 [les ] Pr [les ] 0.03) and fixed value of the aspect ratio A = 4 was carried out. It was shown that the dependence of the critical Grashof number on the aspect ratio and the Prandtl number is very complicated and a very detailed parametric study is required to reproduce it correctly. Comparison with the available experimental data for A = 4 shows that the results of a two-dimensional stability analysis are in good agreement with the experimental results if the width ratio (width/height) of the experimental container is sufficiently large. The study is carried out numerically with the use of two independent numerical approaches based on the global Galerkin and finite-volume methods.

Scattering From a Large Planar Slotted Waveguide Array Antenna

ABSTRACT The problem of scattering from a large planar slotted waveguide array antenna is investigated. The structural scattering and slot scattering are solved independently through the application of the equivalence principle. The former is calculated by a high-frequency method, such as the physical optics (PO) and the shooting and bouncing ray (SBR) method combined with edge diffraction. The latter is solved using the method of moments (MoM) in conjunction with global sinusoidal basis functions and Galerkin's testing technique. Special attention is paid to the evaluation of the admittance matrix elements, which involve the internal and external mutual coupling. Numerical results are presented to demonstrate the effects of a variety of factors on the radar cross section (RCS) of a slot array, and these include the number of basis functions in the MoM solution, the external mutual coupling between the slots, the waveguide terminations, and the host object. Also given are the RCS space distribution and the time responses of slot arrays to illustrate their unique scattering characteristics. The RCS of a complicated real slot array is given finally to demonstrate the capability of the numerical method as well as the distinct features of its RCS pattern.

S matrix of a broad wall coupler between dissimilar rectangular waveguides

The paper describes a method of moment analysis of a four-port junction between two dissimilar rectangular waveguides coupled through either a longitudinal slot or a transverse slot on the common broad wall. The Green's function approach has been used to obtain the scattered field inside the primary and the secondary waveguide for both cases. The coupled integro-differential equations, arising from the continuity of the tangential component of the magnetic field at both the interfaces of the slot, are transformed into matrix equation using entire domain sinusoidal basis functions in conjunction with global Galerkin's technique. The results obtained using the present method have been compared with the results available in the literature. The theoretical results on the magnitude and phase of the scattering parameters have been compared with the experimental results for the case of a longitudinal slot coupler with moderate offset from the centerline. The use of global sinusoidal basis functions gives a much faster rate of convergence compared to the use of subsectional basis functions. The equivalent loading of a single slot derived in the paper can be effectively utilized to design a multiaperture coupler. >

A Pseudospectral Three-Dimensional Boundary Integral Method Applied to a Nonlinear Model Problem from Finite Elasticity

This paper concerns the numerical analysis of the pseudospectral boundary integral method for the solution of linear boundary value problems for the vector Laplace equation in a three-dimensional (3D) region formed when a small ball is removed from inside the unit ball $B \subset R^3 $. These problems arise as Newton iterates for a certain nonlinear model problem deriving from finite elasticity, which exhibits a mechanism under which a deformed body undergoes a degenerate form of cavitation. The unit ball represents an elastic body, with the small ball representing a “core region,” which is mapped to a cavity by the (possibly large) deformation. This deformation satisfies the (vector) Laplace equation in the body minus the core region, together with a Dirichlet condition on the outer boundary and a nonlinear Neumann condition on the inner boundary. Newton’s method yields a sequence of linear problems that are reformulated as a $6 \times 6$ coupled system of boundary integral equations. The authors introduce a pseudospectral (discrete global Galerkin) method for this system, using the spherical harmonics as basis functions. By extending the known approximation theory for spherical polynomials it is proved that this method converges faster than any power of $\frac{1} {n}$ when the number of degrees of freedom is $O((n + 1)^2 )$. As well as providing a convergence theory for this boundary value problem, the paper also contains a number of new results on spherical polynomial approximation and on the convergence of the pseudospectral method for classical integral equations of 3D potential theory on spheres.

FLOW OF NEWTONIAN FLUIDS THROUGH SINUSOID ALLY CONSTRICTED TUBES. NUMERICAL AND EXPERIMENTAL RESULTS

Numerical and experimental predictions of pressure drops in the flow of Newtonian fluids through sinusoidally constricted tubes (SCT) are carried out. The numerical evaluations analyzed in this work are obtained from the following methods: Geometric Iteration (GJM), Geometric Iteration with First and Second Upwinds (GIM1, GIM2), Successive Over Relaxation by Line (SORBLM), Global Galerkin Spectral Method (GGSM), Collocation (CM) and Dufort-Frankel (DFM). The GIM1, GIM2 and SORBLM are applied to SCT and explained in this work. The other methods have been previously reported in the literature with the same purpose. Experiments are accomplished for constrictions of approximately 40,60 and 80% of the average tube diameter and results compare well with numerical predictions of the steady flow. It is concluded that special attention should be given to evaluations of the friction factor ƒ for Reynolds numbers Re between the onsets of flow separation and turbulence due to flow instability. Finally, the SCT as a model for porous media (PM) is discussed.

Local adaptive Galerkin bases for large-dimensional dynamical systems

The authors suggest and develop a method for following the dynamics of systems whose long-time behaviour is confined to an attractor or invariant manifold A of potentially large dimension. The idea is to embed A in a set of local coverings. The dynamics of the phase point P on A in each local ball is then approximated by the dynamics of its projections into the local tangent space. Optimal coordinates in each local patch are chosen by a local version of a singular value decomposition (SVD) analysis which picks out the principal axes of inertia of a data set. Because the basis is continually updated, it is natural to call the procedure an adaptive basis method. The advantages of the method are the following. (i) The choice of the local coordinate system in the local tangent space of A is dictated by the dynamics of the system being investigated and can therefore reflect the importance of natural nonlinear structures which occur locally but which could not be used as part of a global basis. (ii) The number of important or active local degrees of freedom is clearly defined by the algorithm and will usually be much lower than the number of coordinates in the local embedding space and certainly considerably fewer than the number which would be required to provide a global embedding of A. (iii) While the local coordinates indicate which nonlinear structures are important there, the transition matrices which glue the coordinate patches together carry information about the global geometry of A. (iv). The method also suggests a useful algorithm for the numerical integration of complicated spatially extended equation systems, by first using crude integration schemes to generate data from which optimal local and sometimes global Galerkin bases are chosen.

Application of the Finite Element Method to Rotary‐Wing Aeroelasticity

A finite element method for the spatial discretization of the dynamic equations of equilibrium governing rotary-wing aeroelastic problems is presented. Formulation of the finite element equations is based on weighted Galerkin residuals. This Galerkin finite element method reduces algebraic manipulative labor significantly, when compared to the application of the global Galerkin method in similar problems. The coupled flap-lag aeroelastic stability boundaries of hingeless helicopter rotor blades in hover are calculated. The linearized dynamic equations are reduced to the standard eigenvalue problem from which the aeroelastic stability boundaries are obtained. The convergence properties of the Galerkin finite element method are studied numerically by refining the discretization process. Results indicate that four or five elements suffice to capture the dynamics of the blade with the same accuracy as the global Galerkin method.

General problems in the theory of machines and mechanisms: Address of the President of IFToMM Before the 13th ASME Mechanisms Conference, New York, 8 October 1974

This paper investigates the effect of hydrodynamic thrust bearings on the nonlinear vibrations and the bifurcations occurring in rotor/bearing systems. In order to examine the influence of thrust bearings, run-up simulations may be carried out. To be able to perform such run-up calculations, a computationally efficient thrust bearing model is mandatory. Direct discretization of the Reynolds equation for thrust bearings by means of a Finite Element or Finite Difference approach entails rather large simulation times, since in every time-integration step a discretized model of the Reynolds equation has to be solved simultaneously with the rotor model. Implementation of such a coupled rotor/bearing model may be accomplished by a co-simulation approach. Such an approach prevents, however, a thorough analysis of the rotor/bearing system based on extensive parameter studies.A major point of this work is the derivation of a very time-efficient but rather precise model for transient simulations of rotors with hydrodynamic thrust bearings. The presented model makes use of a global Galerkin approach, where the pressure field is approximated by global trial functions. For the considered problem, an analytical evaluation of the relevant integrals is possible. As a consequence, the system of equations of the discretized bearing model is obtained symbolically. In combination with a proper decomposition of the governing system matrix, a numerically efficient implementation can be achieved.Using run-up simulations with the proposed model, the effect of thrust bearings on the bifurcations points as well as on the amplitudes and frequencies of the subsynchronous rotor oscillations is investigated. Especially, the influence of the magnitude of the axial force, the geometry of the thrust bearing and the oil parameters is examined. It is shown that the thrust bearing exerts a large influence on the nonlinear rotor oscillations, especially to those related with the conical mode of the rotor. A comparison between a full co-simulation approach and a reduced Galerkin implementation is carried out. It is shown that a speed-up of 10–15 times may be obtained with the Galerkin model compared to the co-simulation model under the same accuracy.