Spline-based Rayleigh-Ritz methods for the approximation of the natural modes of vibration for flexible beams with tip bodies

Rayleigh-Ritz methods for the approximation of the natural modes for a class of vibration problems involving flexible beams with tip bodies using subspaces of piecewise polynomial spline functions are developed. An abstract operator theoretic formulation of the eigenvalue problem is derived and spectral properties investigated. The existing theory for spline-based Rayleigh-Ritz methods applied to elliptic differential operators and the approximation properties of interpolatory splines are useed to argue convergence and establish rates of convergence. An example and numerical results are discussed

Thermal stresses in a spherical pressure vessel having temperature-dependent, transversely isotropic, elastic properties

Rayleigh-Ritz and modified Rayleigh-Ritz procedures are used to construct approximate solutions for the response of a thick-walled sphere to uniform pressure loads and an arbitrary radial temperature distribution. The thermoelastic properties of the sphere are assumed to be transversely isotropic and nonhomogeneous; variations in the elastic stiffness and thermal expansion coefficients are taken to be an arbitrary function of the radial coordinate and temperature. Numerical examples are presented which illustrate the effect of the temperature-dependence upon the thermal stress field. A comparison of the approximate solutions with a finite element analysis indicates that Ritz methods offer a simple, efficient, and relatively accurate approach to the problem

Rayleigh-Ritz variation method and connected-moments polynomial approach

We show that the connected-moments polynomial approach proposed recently is equivalent to the well known Rayleigh-Ritz variation method in the Krylov space. We compare the latter with one of the original connected-moments methods by means of a numerical test on an anharmonic oscillato

Accurate approximations to eigenpairs using the harmonic Rayleigh Ritz Method

The problem in this paper is to construct accurate approximations from a subspace to eigenpairs for symmetric matrices using the harmonic Rayleigh-Ritz method. Morgan introduced this concept in [14] as an alternative forRayleigh-Ritz in large scale iterative methods for computing interior eigenpairs. The focus rests on the choice and in uence of the shift and error estimation. We also give a discussion of the dierences and similarities with the rened Ritz approach for symmetric matrices. Using some numerical experiments we compare dierent conditions for selecting appropriate harmonic Ritz vectors

Fluctuations in the Irreversible Decay of Turbulent Energy

A fluctuation law of the energy in freely-decaying, homogeneous and isotropic turbulence is derived within standard closure hypotheses for 3D incompressible flow. In particular, a fluctuation-dissipation relation is derived which relates the strength of a stochastic backscatter term in the energy decay equation to the mean of the energy dissipation rate. The theory is based on the so-called ``effective action'' of the energy history and illustrates a Rayleigh-Ritz method recently developed to evaluate the effective action approximately within probability density-function (PDF) closures. These effective actions generalize the Onsager-Machlup action of nonequilibrium statistical mechanics to turbulent flow. They yield detailed, concrete predictions for fluctuations, such as multi-time correlation functions of arbitrary order, which cannot be obtained by direct PDF methods. They also characterize the mean histories by a variational principle.Comment: 26 pages, Latex Version 2.09, plus seceq.sty, a stylefile for sequential numbering of equations by section. This version includes new discussion of the physical interpretation of the formal Rayleigh-Ritz approximation. The title is also change

Parallel eigensolvers in plane-wave Density Functional Theory

We consider the problem of parallelizing electronic structure computations in plane-wave Density Functional Theory. Because of the limited scalability of Fourier transforms, parallelism has to be found at the eigensolver level. We show how a recently proposed algorithm based on Chebyshev polynomials can scale into the tens of thousands of processors, outperforming block conjugate gradient algorithms for large computations

Buckling of a sublaminate in a quasi-isotropic composite laminate

The buckling of an elliptic delamination embedded near the surface of a thick quasi-isotropic laminate was predicted. The thickness of the delaminated ply group (the sublaminate) was assumed to be small compared to the total laminate thickness. Finite-element and Rayleigh-Ritz methods were used for the analyses. The Rayleigh-Ritz method was found to be simple, inexpensive, and accurate, except for highly anisotropic delaminated regions. Effects of delamination shape and orientation, material anisotropy, and layup on buckling strains were examined. Results show that: (1) the stress state around the delaminated region is biaxial, which may lead to buckling when the laminate is loaded in tension; (2) buckling strains for multi-directional fiber sublaminates generally are bounded by those for the 0 deg and 90 deg unidirectional sublaminates; and (3) the direction of elongation of the sublaminate that has the lowest buckling strain correlates with the delamination growth direction