Galerkin Basis Research Articles

On solenoidal high-degree polynomial approximations to solutions of the stationary Navier–Stokes equations

The cell discretization algorithm, a nonconforming extension of the finite element method, is used to obtain approximations to the Navier–Stokes equations for viscous steady-state incompressible flow. An implementation using polynomial bases is described that permits the use of approximations that exactly satisfy the solenoidal requirement. A Galerkin basis is constructed and, starting with the solution of the associated Stokes problem, Newton's method can be used to generate an approximate solution. A unique solution exists if the viscosity is sufficiently large; here an a posteriori computation is described that tests if uniqueness is likely to hold and gives an error estimate showing closeness of the approximation. Tests of the algorithm applied to problems in polygonal domains are described where approximations are 10th degree polynomials.

Low-dimensional dynamical system model for observed coherent structures in ocean satellite data

The dynamics of coherent structures present in real-world environmental data is analyzed. The method developed in this paper combines the power of the Proper Orthogonal Decomposition (POD) technique to identify these coherent structures in experimental data sets, and its optimality in providing Galerkin basis for projecting and reducing complex dynamical models. The POD basis used is the one obtained from the experimental data. We apply the procedure to analyze coherent structures in an oceanic setting, the ones arising from instabilities of the Algerian current, in the western Mediterranean Sea. Data are from satellite altimetry providing Sea Surface Height, and the model is a two-layer quasigeostrophic system. A four-dimensional dynamical system that correctly describes the observed coherent structures (moving eddies) is obtained. Finally, a bifurcation analysis is performed on the reduced model.

Local adaptive Galerkin bases

Local adaptive Galerkin bases

On numerical stability analysis of double-diffusive convection in confined enclosures

The onset of thermosolutal convection and finite-amplitude flows, due to vertical gradients of heat and solute, in a horizontal rectangular enclosure are investigated analytically and numerically. Dirichlet or Neumann boundary conditions for temperature and solute concentration are applied to the two horizontal walls of the enclosure, while the two vertical ones are assumed impermeable and insulated. The cases of stress-free and non-slip horizontal boundaries are considered. The governing equations are solved numerically using a finite element method. To study the linear stability of the quiescent state and of the fully developed flows, a reliable numerical technique is implemented on the basis of Galerkin and finite element methods. The thresholds for finite-amplitude, oscillatory and monotonic convection instabilities are determined explicitly in terms of the governing parameters. In the diffusive mode (solute is stabilizing) it is demonstrated that overstability and subcritical convection may set in at a Rayleigh number well below the threshold of monotonic instability, when the thermal to solutal diffusivity ratio is greater than unity. In an infinite layer with rigid boundaries, the wavelength at the onset of overstability was found to be a function of the governing parameters. Analytical solutions, for finite-amplitude convection, are derived on the basis of a weak nonlinear perturbation theory for general cases and on the basis of the parallel flow approximation for a shallow enclosure subject to Neumann boundary conditions. The stability of the parallel flow solution is studied and the threshold for Hopf bifurcation is determined. For a relatively large aspect ratio enclosure, the numerical solution indicates horizontally travelling waves developing near the threshold of the oscillatory convection. Multiple confined steady and unsteady states are found to coexist. Finally, note that all the numerical solutions presented in this paper were found to be stable.

On The Properties of a Chaotic Attractor Observed in an Inhomogenous Magnetised Plasma

The equations governing the non-linear dynamics of low frequency, long wave-length electromagnetic fields in a non-uniform magnetised plasma are discussed. This pair of non-linear partial differential equations turns out to be non-integrable. A Galerkin basis type expansion is then used to deduce six coupled non-linear ordinary differential equations. These equations are then analysed from the viewpoint of stability and chaos. The system is treated numerically using the techniques of Power spectrum. Lyapunov exponent and phase space analysis. The various regions of the attractor are quantified using the concept of local divergence rate and the corresponding time averages. The predictability property of the various sections is also analysed by studying the probability density functions of such time averages. Furthermore, the phase spatial variation of the predictability is studied using the Poincaré section for different sections of incoming and outgoing orbits separately. The local divergence rate of these different sections are also computed to detail the behaviour of the attractor. The chaotic scenario of the attractor is controlled using (a) adaptive control method (b) application of a short pulse (c) application of a nonlinear pulse on the Poincaré section.

A multilevel method applied in the nonhomogeneous direction of the channel flow problem

In this paper, we apply a spectral multilevel method in the nonhomogeneous direction of a channel. The spectral tau method being not well suited to separate the scales, we use a Galerkin basis in the wall normal direction. Then we can separate the scales, as in the periodic case, from the spectral decomposition of the velocity field. In this way, the quantities associated with the small and large scales verify the no slip boundary conditions. Then, we resolve the large and the small scale equations, simplifying the computation of the small scales. Indeed, we use a quasi-static approximation to compute the small scales. As for the interaction terms, we apply a quasi-static approximation to estimate the modulus, the phase being updated, at each time step, in function of the large scales. To validate the method proposed, we have done two simulations of the channel with the multilevel method. They correspond to two different choices of the total number of modes and of the coarse cut-off level for the multilevel method in the wall normal direction. The results obtained are compared with the results stemming from direct numerical simulations (DNS): one fine DNS (fine resolution) and one low DNS (coarse resolution).

Geometry of Local Adaptive Galerkin Bases

The local adaptive Galerkin bases for large-dimensional dynamical systems, whose long-time behavior is confined to a finite-dimensional manifold, are optimal bases chosen by a local version of a singular decomposition analysis. These bases are picked out by choosing directions of maximum bending of the manifold restricted to a ball of radius {epsilon} . We show their geometrical meaning by analyzing the eigenvalues of a certain self-adjoint operator. The eigenvalues scale according to the information they carry, the ones that scale as {epsilon}{sup 2} have a common factor that depends only on the dimension of the manifold, the ones that scale as {epsilon}{sup 4} give the different curvatures of the manifold, the ones that scale as {epsilon}{sup 6} give the third invariants, as the torsion for curves, and so on. In this way we obtain a decomposition of phase space into orthogonal spaces E {sub m} , where E{sub m} is spanned by the eigenvectors whose corresponding eigenvalues scale as {epsilon}{sup m} . This decomposition is analogous to the Frenet frames for curves. We also discover a practical way to compute the dimension and local structure of the invariant manifold.

Identification of Unknown Physical Parameters of Cantilevered Beam Using its Vibration Data.

In this paper, an efficient computer algorithm is proposed for identifying the unknown spatially invariant/varying parameter values such as flexural stiffness and both air and structural damping coefficients of a class of mechanically flexible cantilevered beams, using dynamic data measured on its vibration in a non-destructive manner. First, the mathematical model of the cantilevered beam is given by an Euler-Bernoulli type partial differential equation and the identification problem is formulated as one of the inverse problems by minimizing the quadratic error functional between mathematical model and measurement data with respect to model parameters. Then, the identification algorithm is developed by introducing finite approximations both to the modeled partial differential equation and the unknown parameter profiles using Galerkin method and basis functions. Finally, the efficacy of the proposed identification algorithm is examined for two kinds of beams-one is spatially uniform and the other is nonuniform in its shape by simulations and experiments.

LQR Control of Thin Shell Dynamics: Formulation and Numerical Implementation

A PDE-based feedback control method for thin cylindrical shells with surface-mounted piezoceramic actuators is presented. Donnell-Mushtari equations modified to incorporate both passive and active piezoceramic patch contributions are used to model the system dynamics. The well-posedness of this model and the associated LQR problem with an unbounded input operator are established through analytic semigroup theory. The model is discretized using a Galerkin expansion and basis functions constructed from Fourier polynomials tensored with cubic splines, and convergence criteria for the associated approximate LQR problem are established. The effectiveness of the method for attenuating the coupled longitudinal, circumferential and transverse shell displacements is illustrated through a set of numerical examples.

On the cost of controlling unstable systems: The case of boundary controls

We discuss the cost of controlling parabolic equations of the formyt + δ2y +kδy = 0 in a bounded smooth domain Ώ ofℝd by means of a boundary control. More precisely, we are interested in the cost of controlling from zero initial state to a given final state (in a suitable approximate sense) at timeT > 0 and in the behavior of this cost ask → ∞. We introduce finite-dimensional Galerkin approximations and prove that they are exactly controllable. Moreover, we also prove that the cost of controlling converges exponentially to zero ask → ∞. This proves, roughly speaking, that when the system becomes more unstable it is easier to control. The system under consideration does not admit a variational formulation. Thus, in order to introduce its Galerkin approximation, we first approximate it by means of a singular perturbation. We also develop a method for the construction of special Galerkin bases well adapted to the control problem.

Contrôlabilité exacte des approximations de Galerkin des équations de Navier-Stokes

We consider Navier-Stokes equations in a smooth domain of ℝ n, n = 2, 3, with Dirichlet boundary conditions. We introduce the classical Galerkin approximation and study its exact controllability when the control acts on an open non-empty subset of the domain. Under suitable assumptions on the elements of the Galerkin basis, we prove that this finite-dimensional system is exactly controllable. The proof combines HUM together with a classical fixed point argument, and relies on the cancellation properties of the non-linearity appearing in the Navier-Stokes equations.

A wavelet-Galerkin method for solving population balance equations

A new wavelet-Galerkin method is developed to solve the population balance equations which arise in the description of particle-size distribution of a continuous, mixed-suspension, mixed-product removal crystallizer with taking account of the effect of particle breakage. The class of Daubechies wavelets, which is both compactly supported and orthonormal, is adopted as the Galerkin bases. Some elegant results concerned with the exact evaluation of functions on wavelets and their derivatives and integrals are derived. These results along with the 2-scale relation which defines the wavelet bases make the Galerkin method feasible for the solution of population balance equations containing a scaled argument.

Calculation of stress intensity factors for an interfacial crack between dissimilar anisotropic media, using a hybrid element method and the mutual integral

Using Beom and Atluri's complete eigen-function solutions for stresses and displacements near the tip of an interfacial crack between dissimilar anisotropic media, a hybrid crack tip finite-element is developed. This element, as well as a mutual integral method are used to determine the stress intensity factors for an interfacial crack between dissimilar anisotropic media. The hybrid element has, for its Galerkin basis functions, the eigen-function solutions for stresses and displacements embedded within it. The “mutual integral” approach is based on the application of the path-independent J integral to a linear combination of two solutions: one, the problem to be solved, and the second, an “auxiliary” solution with a known singular solution. A comparison with exact solutions is made to determine the accuracy and efficiency of both the methods in various mixed mode interfacial crack problems. The size of the hybrid element was found to have very little effect on the accuracy of the solution: an acceptable numerical solution can be obtained with a very coarse mesh by using a larger hybrid element. An equivalent domain integral method is used in the application of the “mutual” integral instead of the line integral method. It is shown that the calculated mutual integral is domain independent. Therefore, the mutual integral can be evaluated far away from the crack-tip where the finite element solution is more accurate. In addition, numerical examples are given to determine the stress intensity factors for a delamination crack in composite lap joints and at plate-stiffener interfaces.

Low-dimensional representations of early transition in rotating fluid flow

Low-dimensional representations of the axisymmetric Navier-Stokes equations are generated by a Galerkin projection. Proper orthogonal decomposition (POD) techniques based on snapshots generated from a finite-difference algorithm are used. The Reynolds number range is extended by adding displacement vectors to the Galerkin basis. For the fluid flow enclosed in a cylindrical vessel with rotating end cover, the first transition from steady to oscillatory motion is detected as a supercritical Hopf bifurcation. Comparison with the full numerical solution of the Navier-Stokes equations as well as experimental results show excellent agreement.

Buckling of Shear Deformable Antisymmetric Angle-Ply Laminated Cylindrical Panels Under Axial Compression (Design Paper)

This paper deals with the buckling problem of antisymmetric angle-ply laminated circular cylindrical panels subjected to a uniform axial compression. Since a flat plate configuration occurs as a particular case of the cylindrical panel geometry (zero shallow angle parameter), the corresponding flat plate problem is studied as a particular case of the problem considered. The theoretical analysis is based on a nonlinear theory developed in a previous paper (Soldatos, 1992), which accounts for parabolically distributed transverse shear strains through the shell thickness. The linearized differential equations, governing the buckling behavior of a simply supported panel, are solved on the basis of Galerkin’s method. Comparisons of corresponding numerical results, based on both the refined shell theory employed and a classical Love-type shell theory, show the influence of transverse shear deformation on the buckling loads of such laminated composite panels.

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.

Non-linear vibration of circular plates with transverse shear and rotatory inertia

This study is an analytical investigation of large amplitude flexural vibrations of clamped circular plates with stress-free and immovable edges. The effects of transverse shear deformation and rotatory inertia are included in the governing equations. Solutions are formulated on the basis of Galerkin's method and calculated by using the Runge-Kutta numerical procedure. An excellent agreement is found between the present results and those reported earlier for non-linear static and dynamic cases. Numerical results indicate that the effects of transverse shear deformation and rotatory inertia are usually neligible in the non-linear dynamic analysis of circular plates, but can be significant for moderately thick plates with a radius-to-thickness ratio less than 10.

Large amplitude vibration of circular plates including transverse shear and rotatory inertia

This study is an analytical investigation of large amplitude flexural vibration of clamped circular plates with stress-free and immovable edges. The effects of transverse shear deformation and rotatory inertia are included in the governing equations. Solutions are formulated on the basis of Galerkin's method and the Runge-Kutta numerical procedure. An excellent agreement is found between the present results and those reported earlier for nonlinear static and dynamic cases. Numerical results indicate that the effects of transverse shear deformation and rotatory inertia are significant in the nonlinear dynamic analysis of circular plates, particularly for moderately thick plates.

Analysis of a circular microstrip disk antenna with a thick dielectric substrate

The problem of a circular microstrip disk excited by a probe is solved using rigorous analysis. The disk is assumed to have zero thickness, and the current on the probe is taken to be uniform. Using vector Hankel transforms the problem is formulated in terms of vector dual-integral equations, from which the unknown current can be solved for. Due to the singular nature of the current distribution arising from probe excitation, the direct application of Galerkin's basis function expansion method gives a slowly convergent result. Therefore the singular part of the current is removed since the singularity is known a priori. The unknown current to be solved for is then regular and tenable to Galerkin's method of analysis. It is shown that this analysis agrees with the single-mode approximation when the dielectric substrate layer is thin, and that it deviates from the single-mode approximation when the substrate layer is thick. Excellent agreement of both the computed real and imaginary parts of the input impedance with experimental data is noted. The radiation patterns and the current distributions on the disk are also-presented.

Non-linear vibration of anisotropic rectangular plates including shear and rotatory inertia

A non-linear vibration theory which includes the effects of transverse shear deformation and rotatory inertia is formulated for anistropic rectangular elastic plates. Solutions to the governing equations are presented for various boundary conditions on the basis of Galerkin's method and the numerical Runge-Kutta procedure. These equations can readily reduce to the dynamic von Kármán equations of plates. Numerical results indicate significant influence of these effects on the large amplitude vibration behaviour of anisotropic plates.