Hyperspherical approach to three-electron atomic systems

A computational procedure for carrying out calculations on three-electron atomic systems in hyperspherical coordinates is presented. By adopting an adiabatic approximation, the solution of the Schrödinger equation at a fixed hyperradius is reduced to the solution of a set of coupled partial differential equations involving the two hyperangles. The derivation of such a set of coupled equations is explicitly illustrated.     

An extension of TOPSIS for group decision making

An extension of TOPSIS (technique for order performance by similarity to ideal solution), a multi-attribute decision making (MADM) technique, to a group decision environment is investigated. TOPSIS is a practical and useful technique for ranking and selection of a number of externally determined alternatives through distance measures. To get a broad view of the techniques used, we provide a few options for the operations, such as normalization, distance measures and mean operators, at each of the corresponding steps of TOPSIS. In addition, the preferences of more than one decision maker are internally aggregated into the TOPSIS procedure. Unlike in previous developments, our group preferences are aggregated within the procedure. The proposed model is indeed a unified process and it will be readily applicable to many real-world decision making situations without increasing the computational burden. In the final part, the effects of external aggregation and internal aggregation of group preferences for TOPSIS with different computational combinations are compared using examples. The results have demonstrated our model to be both robust and efficient.     

The Ansatz Method for Analyzing Schrödinger’s Equation with Three Anharmonic Potentials in D Dimensions

In this letter, by applying a suitable ansatz to the wave functions, the solutions of the D-dimensional radial Schrödinger equation with some anharmonic potentials are obtained.     

Distribution of Particles Which Produces a “Smart” Material

If A _q(β, α, k) is the scattering amplitude, corresponding to a potential , where D⊂ℝ³ is a bounded domain, and is the incident plane wave, then we call the radiation pattern the function , where the unit vector α, the incident direction, is fixed, β is the unit vector in the direction of the scattered wave, and k>0, the wavenumber, is fixed. It is shown that any function , where S ² is the unit sphere in ℝ³, can be approximated with any desired accuracy by a radiation pattern: , where ∊ >0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ∊, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles D _m⊂ D, 1≤ m≤ M, distributed in an a priori given bounded domain D⊂ℝ³. The geometrical shape of a small particle D _m is arbitrary, the boundary S _m of D _m is Lipschitz uniformly with respect to m. The wave number k and the direction α of the incident upon D plane wave are fixed. It is shown that a suitable distribution of the above particles in D can produce the scattering amplitude , at a fixed k>0, arbitrarily close in the norm of L ²(S ²× S ²) to an arbitrary given scattering amplitude f(α ', α), corresponding to a real-valued potential q∊ L ²(D), i.e., corresponding to an arbitrary refraction coefficient in D. MSC: 35J05, 35J10, 70F10, 74J25, 81U40, 81V05, 35R30. PACS: 03.04.Kf.     

Non-linear flexural oscillations of a partially tapered annular plate

The free finite amplitude axisymmetric oscillations of an isotropic annular plate with partially tapered thickness are investigated. The time variable is eliminated by a Ritz-Kantorovich averaging method. The von Karman plate equations are then reduced to two non-linear ordinary differential equations, which form a non-linear eigenvalue problem. Solutions to the problem are obtained by utilizing a direct computational method. The results reveal the effects of large amplitude upon the dynamic responses. Also, an annulus of constant thickness, which has the same boundary conditions and the same volume as the partially tapered one, is investigated. Their results, which may shed light on the optimal design of annular plates, are compared.     

Effect of Vibrations on Pore Fluid Distribution in Porous Media

Understanding the role of shuttle vibrations in pore fluid distribution is an essential task in the exploration of plant growth in root modules aboard space flights. Results from experimental investigations are reported in this paper on the distribution of immiscible fluid phases in glass beads under vibrations. Hexadecane, a petroleum compound immiscible with and lighter than water, was used in the experiments. The higher freezing point of Hexadecane (18 °C) allowed the solidification of the entrapped blobs in the presence of water in porous media, so that their size distribution can be obtained. van Genuchten function, commonly used to express moisture retention curves, is found to be an adequate fit for blob size distribution at residual saturation. The effect of vibrations on the fate (mobilization, stranding, or breakup) of a solitary ganglion in porous media was studied using a network model. A mobility criterion considering viscous, gravity, and capillary forces was developed to determine the fate of a solitary ganglion in a porous medium. It is concluded that the effect of vibrations is to increase the likelihood of breakup and mobilization of blobs entrapped in porous media at residual saturation. The pore fluid distributions after vibrations are less uniform than those before vibrations.     

Modal analysis of measurements from a large-scale VIV model test of a riser in linearly sheared flow

Large-scale model testing of a tensioned steel riser in well-defined sheared current was performed at Hanøytangen outside Bergen, Norway in 1997. The length of the model was 90 m and the diameter was 3 cm. The aim of the present work is to look into this information and try to improve the understanding of vortex-induced vibrations (VIV) for cases with very high order of responding modes, and in particular to study if and under which circumstances the riser motions would be single-mode or multi-mode. The measurement system consisted of 29 biaxial gauges for bending moment. The signals are processed to yield curvature and displacement and further to identify modes of vibration. A modal approach is used successfully employing a combination of signal filtering and least-squares fitting of precalculated mode-shapes. As a part of the modal analysis, it is demonstrated that the equally spaced instrumentation limited the maximum mode number to be extracted to be equal to the number of instrumentation locations. This imposed a constraint on the analysis of in-line (IL) vibration, which occurs at higher frequencies and involves higher modes than cross-flow (CF). The analysis has shown that in general the riser response was irregular (i.e. broad-banded) and that the degree of irregularity increases with the flow speed. In some tests distinct spectral peaks could be seen, corresponding to a dominating mode. No occurrences of single-mode (lock-in) were seen. The IL response is more broad-banded than the CF response and contains higher frequencies. The average value of the displacement r.m.s over the length of the riser is computed to indicate the magnitude of VIV motion during one test. In the CF direction the average displacement is typically 1/4 of the diameter, almost independent of the flow speed. For the IL direction the values are in the range 0.05–0.08 of the diameter. The peak frequency taken from the spectra of the CF displacement at riser midpoint show approximately to be equal to the Strouhal frequency. The peak frequency in IL direction was typically twice the Strouhal frequency.     

Mode II fracture-parameter estimates for concrete from beam specimens

At the present time, work on determination of fracture parameters for concrete subjected solely to mode I deformation states has advanced to the point where standard methods of testing and data evaluation have been proposed. Thus, many researchers are now considering the dual problems of (1) determination of fracture parameters for mode II deformation and (2) the need for this information (or put another way—the possible application of this information). In fact, the physical testing arrangements for mode II testing with no influence of mode I are not obvious; compromises must be made which will lead to approximations—perhaps fairly gross—in the estimates of mode II fracture parameters. Herein are presented experimental and analytical results for fracture parameters for beams loaded in antisymmetric four-point bending with a single-edge starter notch located in a region of high-shear and low-bending moment. This study complements another, presented elsewhere, in which beams in three-point bending were used with notches located off center. From this study, which included tests on 18 beam specimens, the following conclusions are reached. The beams must be tested in strain control. Crack initiation started in mode II but quickly changed to a mixed mode (although finite-element calculations indicate mode I is dominant). The fracture energy associated with the onset of unstable crack propagation is much larger than normally obtained for mode I—this is thought to be due to extensive aggregate interlock forces not present in normal bending or direct tension tests. Probably this type of testing arrangement is not suitable for determination of mode II fracture parameters because propagation does not occur under mode II conditions.