Geometrically nonlinear shell finite element based on the geometry of a planar curve

An engineering approach for constructing a curved triangular finite element of a thin shell is considered. The approach is based on the assumption that the triangle sides are planar nearly circular curves before and after deformation. A geometrically nonlinear formulation of a triangular finite element of a thin Kirchhoff–Love shell is given. The predictive capabilities of the element are tested using benchmark problems of nonlinear deformation of elastic plates and shells.     

Efficient use of Jacobi rotations for orbital optimization and localization

Summary Quantum chemical orbital optimizations can be accomplished by Newton-type iterations, whereall orbitals are improved ateach step; or by a succession of Jacobi rotations, where onlytwo orbitals are improved at any one step. In both schemes, the iterative updating of the four-index two-electron integrals requires a large computational effort. We show that the four-index transformation due to a Jacobi rotation can be simplified to such a degree that the successive execution of the four-index transformations ofN(N–1)/2 different Jacobi rotations requires no greater computational effort than that required by theone full orthogonal transformation which is the product of allN(N–1)/2 Jacobi rotations. The four-index updating has therefore no bearing on the relative merit of the Newton approach versus the Jacobi approach. The Jacobi approach has, however, an advantage if the optimization of each Jacobi rotation angle is simple and if the effectiveness of the individual Jacobi rotations can be assessed without the execution of four-index transformations. For, in that case, all ineffectual rotations are easily deleted from the iterative sequence. Whether convergence can be guaranteed for one or the other approach is also relevant. Our conclusions are illustrated by application to the problem of intrinsic orbital localization where the succession of Jacobi rotations is the more effective strategy.Dedicated to Professor Per Olov LöwdinAmes Laboratory is operated for the U.S. Department of Energy by Iowa State University under contract No. W-7405-Eng-82. This work was supported by the Division of Chemical Sciences, Office of Basic Energy Sciences     

Statistical Problems Related to Irrational Rotations

Let be random variables as functions of β in the probability space [0,1) with the Lebesgue measure, where is considered to be an unknown parameter which we want to estimate from the observation ξ :=ξ₁, ξ₂...ξ _m . Let an observation ξ be given, which is a finite Sturmian sequence. We determine the likelihood function P _α(ξ) as a function of parameter α, and obtain the maximum likelihood estimator as the relative frequency of 1s in a minimal cycle of ξ, where a factor η of ξ is called a minimal cycle if ξ is a factor of η^∞ and η has the minimum length among them. We also obtain a minimum sufficient statistics. The sample mean (ξ₁ + ξ₂ + ... + ξ _m )/m which is an unbiased estimator of α is not admissible if m=6 or m ≥ 8 since it is not based on the minimum sufficient statistics.     

General Wigner Rotations in D Dimensions

We construct general Wigner rotations for both massive and massless particles in D-dimensional spacetime.We work out the explicit expressions of these Wigner rotations for arbitrary Lorentz transformations. We study the relation between the electromagnetic gauge invariance and the non-uniqueness of Wigner rotation.     

Dissipative soft modes in viscous nematodynamics

The paper analyses a possible occurrence of soft and semi-soft viscous modes in slow (low Reynolds number) flows of uniaxially anisotropic nematic liquids as described by the five parametric Leslie-Ericksen-Parodi (LEP) constitutive equations (CEs). As in the similar elastic case, the soft viscous modes theoretically cause no resistance to flow, nullifying the corresponding components of the viscous part of the total stress tensor, and do not contribute to the dissipation. That is why these modes can also be called dissipative soft modes. In some flows, these dissipative soft modes may cause the effect of nematic superfluidity. As in the theories of nematic elastic solids, this effect is caused by a marginal thermodynamic stability. The analysis is simplified in a specific local, rotating orthogonal coordinate system whose one axis is directed along the director. We demonstrate that depending on closeness of material parameters to the marginal stability conditions, LEP CEs describe the entire variety of soft, semi-soft and harder behaviors of nematic viscous liquids. When the only shearing dissipative modes are soft, the viscous part of stress tensor is symmetric, and LEP CE for stress, scaled with isotropic viscosity is reduced to a one-parametric, stress-strain rate anisotropic relation. When additionally the elongation dissipative mode is also soft, this scaled relation has no additional parameters and shows that the dissipation is always less than that in isotropic phase. Simple shearing and simple elongation flows illustrate these possible effects.     

Measurement of in-plane motions and rotations using a simple electronic speckle pattern interferometer

We describe the construction and operation of a simple electronic speckle pattern interferometer which is sensitive to in-plane motions and rotations. The interferometer is extremely simple and easy to use. It uses a commercial digital still camera for image acquisition, and a personal computer for image storage and analysis. The interferometer was used to measure very small in-plane rotations of a rough surface, and the results were found to be in good agreement with the expected values. We propose to use this system as an instrument for the measurement of small-angle rotations.     

The structure of hydantoins in solution and in the solid state

The results of NMR spectroscopic and X-ray crystallographic studies are critically discussed with respect to the structure of hydantoins, their tautomerism, and their acidity. The imide NH proton of the preferred, nearly planar 2,4-imidazolidine-dione tautomer proved to be more acidic than the corresponding amide NH proton. Phenyl substituents at the ring nitrogen atoms and at C-5 are twisted from the plane of the hydantoin ring; in case ofortho substituents restricted rotation about the N-aryl bond was found and the barrier to rotation determined by dynamic NMR spectroscopy. For 5-benzyl substituents, afolded conformation of the two rings, due to intramolecular interactions, was found and for 5-exo-methylene substituted hydantoins the relevant E/Z isomerism at theexo-cyclic C, C double bond was studied. In addition, the¹H and¹³C chemical shifts of the hydantoins proved to excellently indicate the electronic distribution along the hydantoin ring moiety. Finally, the mass spectrometric fragmentation of the hydantoins is critically discussed.Dedicated to Prof. Rolf Borsdorf on the occasion of his 65th birthday.     

Discarding or downweighting high-noise variables in factor analytic models

This work examines the factor analysis of matrices where the proportion of signal and noise is very different in different columns (variables). Such matrices often occur when measuring elemental concentrations in environmental samples. In the strongest variables, the error level may be a few percent. For the weakest variables, the data may consist almost entirely of noise. This paper demonstrates that the proper scaling of weak variables is critical. It is found that if a few weak variables are scaled to too high a weight in the analysis, the errors in computed factors would grow, possibly obscuring the weakest factor(s) by the increased noise level. The mathematical explanation of this phenomenon is explored by means of Givens rotations. It is shown that the customary form of principal component analysis (PCA), based on autoscaling the original data, is generally very ineffective because the scaling of weak variables becomes much too high. Practical advice is given for dealing with noisy data in both PCA and positive matrix factorization (PMF).