TOWARDS DEEP AND SIMPLE UNDERSTANDING OF THE TRANSCENDENTAL EIGENPROBLEM OF STRUCTURAL VIBRATIONS

When using exact methods for undamped free vibration problems the generalized linear eigenvalue problem (K−ω²M) D=0 of approximate methods, e.g., finite elements, is replaced by the transcendental eigenvalue problem K (ω) D=0. Here ω is the circular frequency; D is the displacement amplitude vector; M and K are the mass and static stiffness matrices; and K (ω) is the dynamic stiffness matrix, with coefficients which include trigonometric and hyperbolic functions involving ω and mass because elements (for example, bars or beams) are analyzed exactly by solving their governing differential equations. The natural frequencies of this transcendental eigenvalue problem are generally found by the Wittrick-Williams algorithm which gives the number of natural frequencies below ω_t, a trial value of ω, as ∑J_m+s{K (ω_t)} wheres {} denotes the readily computed sign count property of K (ω) and the summation is over the clamped-clamped natural frequencies of all elements of the structure. Understanding the alternative solution forms of the transcendental eigenvalue problem is important both to accelerate convergence to natural frequencies, e.g., by plotting ∣K (ω)∣, and to improve the mode calculations, which lack the complete reliability of natural frequencies obtained by using the Wittrick-Williams algorithm. The three solution forms are: ∣K (ω)∣=0; D=0 with ∣K (ω)∣→∞; and ∣K (ω)∣≠0 with D≠0. The literature covers the first two forms thoroughly but the third form has been almost totally ignored. Therefore, it is now examined thoroughly, principally by analytical studies of simple bar structures and also by confirmatory numerical results for a rigidly jointed plane frame. Although structures are unlikely to have exactly the properties giving this form, it needs to be understood, particularly because ill-conditioning can occur for structures approximating those for which it occurs.     

On the dynamic conductivity for an electron-impurity system

The correlation function formula for the dynamic conductivity of a system of non-interacting electrons in the field of impurities is analyzed in terms of proper connected diagrams. By selecting those diagrams appropriate in the region of weak coupling and low impurity concentration, a set of coupled equations for the energy broadening γ (ω, ε, n_s) and the energy shift Δ(ω, ε, n_s) is derived, where both γ and Δ depend on the frequency ω of a probing field, the energy ε of the electron, and the concentration, n_g, of impurities. With the assumption of a finite range potential, these equations are solved. It is found that γ (ω, n_s) is smaller than that extrapolated value which the conventional expression γ₀ for the low-concentration collision frequency would predict, in the entire region studied, that the difference γ₀-γ becomes appreciable when the ratio of the average time between scatterings, τ_c, to the average duration of a scattering, τ_d, is 100 or less, that γ (ω, n_s) decreases monotonically from its static value γ (0, n_s), and becomes vanishingly small in the region ω≈1/τ_d, and that in the static limit (ω=0), γ=γ₀[1?(2/π) (γ₀τ_d)+…], that the energy shift Δ is positive, and increases from 0 and reach a peak of magnitude γ₀ as ω is raised from 0. By using the γ and Δ obtained, the dynamic conductivity σ(ω, n_s) for degenerated electrons is calculated. The deviation, σ-σ₀, from the conventional expression σ₀=(?i) (n_ee²/M) [ω-iΓ₀]^?1, (n_e]=number density of electrons), for 0°K, is appreciable when the ratio τ_c/τ_d is 100 or less. The field-term correction, which arises from the modification of the scattering due to the probing field, is found to be negligible in the entire region studied.     

Off-shell effects in the coherent π0 photoproduction off nuclei

The strong sensitivity of coherent π⁰ photoproduction to the choice of the reaction energy ω in the elementary t_πγ (ω)-matrix is demonstrated. The best agreement of the DWIA results with the experimental data is achieved when ω is chosen as an eigenvalue of the free relativistic hamiltonian of the πN system. This is in agreement with the consequences of the relativistic potential theory.     

The real part of Kp forward scattering with rising total cross sections

Forward dispersion relations have been applied to Kp elastic scattering with the assumption that the total cross sections continue to rise with primary energy ω like log ω and log² ω respectively. The results depend on the way of approach with ω of the two total cross sections. In this connection, we further discuss the consequences of a generalized version of the Pomeranchuk theorem when total cross sections are rising like log^β ω. The ratio α₊ of real and imaginary parts of the forward elastic scattering amplitude for K⁺ p is seen to change sign, and becomes positive at some high energy (e.g. at about 400 GeV in the log² case). If the total cross section difference behaves like 1/ω^b, then α_ will remain positive at all (high) energies. However, if the cross sections approach each other as slowly as allowed by the generalized Pomeranchuk theorem, then also α_ will change sign at some very high energy.     

The Tikhonov α-regularization method in the dielectric spectroscopy of ferroelectrics

A method is suggested for calculating the relaxation oscillator spectrum using the experimental frequency dependence of the real part of the complex permittivity ε′(ω). A logarithmic transformation of the argument in the Fröhlich equation makes it possible to solve the Tikhonov equations for the spectrum of ε′(ω). The broadening of the relaxation oscillator spectrum due to experimental errors is reduced by 20 to 30%, since the accuracy of measurement of the real part ε′(ω) is usually 5 to 10 times higher than that for the imaginary part ε″(ω).     

An h-adaptive finite element solver for the calculations of the electronic structures

In this paper, a framework of using h-adaptive finite element method for the Kohn–Sham equation on the tetrahedron mesh is presented. The Kohn–Sham equation is discretized by the finite element method, and the h-adaptive technique is adopted to optimize the accuracy and the efficiency of the algorithm. The locally optimal block preconditioned conjugate gradient method is employed for solving the generalized eigenvalue problem, and an algebraic multigrid preconditioner is used to accelerate the solver. A variety of numerical experiments demonstrate the effectiveness of our algorithm for both the all-electron and the pseudo-potential calculations.     

Inverse Monte Carlo solutions for radiative transfer in inhomogeneous media

The inverse Monte Carlo method is used to construct solutions for three radiative transfer inverse problems in which the single scatter albedo, ω, varies within the medium and the scattering is isotropic. The first problem concerns a half space whose single scatter albedo varies exponentially according to ω = ω₀e^?τ/s, where τ is the optical depth, s is known and we seek ω₀; the second problem concerns a two-region slab for which we seek ω for each region. The procedure is also used to construct an approximate solution for a finite, plane-parallel medium whose single scatter albedo varies exponentially by considering the medium to be composed of five regions, each with constant single scatter albedo.     

A GROUP THEORETIC APPROACH TO THE LINEAR FREE VIBRATION ANALYSIS OF SHELLS WITH DIHEDRAL SYMMETRY

This paper deals with a group theoretic approach to the finite element analysis of linear free vibrations of shells with dihedral symmetry. Examples of such shell structures are cylindrical shells, conical shells, shells with circumferential stiffeners, corrugated shells, spherical shells, etc. The group theoretic approach is used to exploit the inherent symmetry in the problem. For vibration analysis, the group theoretic results give the correct symmetry-adapted basis for the displacement field. The stiffness matrix K and the mass matrix M are identically block diagonalized in this basis. The generalized linear eigenvalue problem of free vibration gets split into independent subproblems due to this block diagonalization. The Simo element is used in the finite element formulation of the shell equilibrium equations. Numerical results for natural frequencies and natural modes of vibration of several dihedral shell structures are presented. The results are shown to be in very good agreement with those reported in the literature. The computational advantages and physical insights due to the group theoretic approach are also discussed.     

Optical constants and dielectric function of DNA's nucleotides in UV range

We calculated frequency-dependent real n(ω) and imaginary k(ω) parts of the complex refractive index for adenine A, guanine G, cytosine C and thymine T in ultraviolet range using Kramers-Kroning analysis of their absorption spectra. The frequency-dependent real ε′(ω) and imaginary ε″(ω) parts of the dielectric function for the corresponding nucleotides were restored. We calculated also the dependence of the dielectric function on the imaginary frequency ε(iω) for the corresponding nucleotides.     

半导体短程势深能级在浅能区的波函数行为

本文对短程缺陷势深能级在能量趋向禁带边缘时波函数的行为作了研究。当深能级能量与禁带边能量差ε很小时,在一定的能量和对称性条件满足下,波函数在k空间向极值收缩而在实空间扩展。本文发展了一种方法可有效地用于ε很小时的波函数数值计算。对于靠近导带的A₁态波函数,对于出Si,当ε小于2meV时,开始显著地在k空间收缩;对于GaAs,直至ε=0.1meV还未发生显著收缩。发生显著收缩对应的ε值与有效质量以及缺陷态和能带边波函数对称性匹配度有灵敏的关系。对于靠近价带顶的A₁态,由于与价带顶对称性不匹配,不发生k空间收缩现象。 关键词：     

Self-consistent medium polarization in RPA

The standard random-phase approximation for finite systems is extended by including the effect of the exchange of the RPA phonons in the residual interaction selfconsistently. It is shown that this particle-hole interaction is strongly energy dependent due to the presence of poles corresponding to 2p2h (and more complex) excitations. The RPA eigenvalue problem with this energy-dependent residual interaction also provides solutions for these predominantly 2p2h-like states. In addition a modified normalization condition is obtained.This new scheme is applied to ⁵⁶Ni (⁵⁶Co) in a large (up to 7ħω) configuration space using a residual interaction of G-matrix type. It is shown that the lowest 2⁺ eigenvalue, which in the standard RPA becomes imaginary, is stabilized when the selfconsistent screening is taken into account. Another feature observed is the splitting of the M1 strength as an example of 1p1h and 2p2h mixing.     

Effect of the hexagonal warping on the dynamical conductivity of surface states in a topological insulator

We report on the effect of hexagonal warping on the dynamical conductivity of the surface states of a topological insulator in the presence of nonmagnetic impurities. It is found that the photon energy dependent conductivities are determined by a polarization-function-liked term,  Π₂ (q,ω), which contains a velocity term corresponding to the difference of group velocities between the two states due to an electron-impurity scattering. This is different from the conductivity of 2-dimentional electron systems where the conductivity depends on the inverse imaginary part of the dielectric function Im [1/κ(q,ω)]. We present both the real part and imaginary part of the polarization function with different warping strength. It is found that the warping strength can both enhance single particle excitations (SPEs) and suppress the screening effect of electrons. As a result the inverse scattering time is enhanced by up to about two orders of magnitudes. The real part of the longitudinal conductivity of the intra-band process is analog to the case with a conductivity of σ ~ μδ(ω). The broadening of the spectrum in the low energy is not only determined by chemical potential, but also dependent on the warping strength. At higher frequency, the real part of conductivity shows a jump at the threshold photon energy of μ, where the inter-band contribution takes over.     

Frequency Dependence of the Dielectric Loss Angle in Disordered Semiconductors in the Terahertz Frequency Range

Frequency dependence of the real part of the conductivity σ₁(ω) in the region of the transition from almost linear (s < 1) to quadratic (s ≈ 2) can indicate a change in the conduction mechanism (the transition from the variable-range to the fixed-range hopping with increasing frequency); in this case, the sharpness of the change in the slope of the frequency characteristic is related to the dependence of the preexponential factor of the resonance integral on the intercenter distance in the pair. The frequency dependence of the imaginary part of the conductivity σ₂(ω) has no kink in the vicinity of the transition frequency ωcr, remaining almost linear. A large dielectric loss angle |cotγ| = |σ₂|/σ₁ can indicate that the imaginary part of the conductivity at ω < ω_cr is defined by the larger zero-phonon contribution in σ₂^res the region of weak variation in the loss angle γ(ω), which significantly exceeds the relaxation contribution σ₂^res.     

EFFECT OF ELASTIC FOUNDATION ON ASYMMETRIC VIBRATION OF POLAR ORTHOTROPIC LINEARLY TAPERED CIRCULAR PLATES

Asymmetric vibrations of polar orthotropic circular plates of linearly varying thickness resting on an elastic foundation of Winkler type are discussed on the basis of the classical plate theory. Ritz method has been employed to obtain the natural frequencies of vibration using the functions based upon the static deflection of polar orthotropic plates, which has faster rate of convergence as compared to the polynomial co-ordinate functions. Frequency parameter of the plate with elastically restrained edge conditions are presented for various values of taper parameter, rigidity ratio and foundation parameter. A comparison of the results with those available in the literature obtained by finite element method, receptance method and polynomial co-ordinate functions shows an excellent agreement.     

An experimental study of ω neutral decays

The neutral branching ratios of the ω meson have been determined by measuring the c.m. momentum spectrum of single γ's from ω decay. The sample of 400 ω events after background subtraction yields an upper limit of (ω → ηγ)/(ω → neutrals) < 25% (90% confidence). The fit with ω → π⁰γ only cannot be excluded. However, the best fit is obtained with the ratios (ω →π⁰γ)/(ω → neutrals) = (78±7)% and (ω → π⁰π⁰γ)/(ω → neutrals) = (22 ± 7)%.     

An elastodynamic solution of finite long orthotropic hollow cylinder under torsion impact

The paper presents an analytical method to solve the elastodynamic problem of a finite-length orthotropic hollow cylinder subjected to a torsion impact often occurring in engineering fields. The elastodynamic solution is composed of a quasi-static solution of homogeneous equation satisfied with the non-homogeneous boundary condition and a dynamic solution of non-homogeneous equation satisfied with homogeneous boundary condition. The quasi-static solution can be obtained by directly solving the quasi-static equation satisfied with the non-homogeneous boundary condition. The solution of a non-homogeneous dynamic equation is obtained by means of a finite Hankel transform to a radial variable r, Laplace transform to a time variable t and finite Fourier transform to an axial variable z. Thus, the elastodynamic solution of the finite length of an orthotropic hollow cylinder subjected to a torsion impact is obtained. On the other hand, a dynamic finite element for the same problem is also carried out by applying the ANSYS finite-element analysis system. Comparing the theoretical solution with finite-element solution, it can be found that two kinds of results obtained by making use of two different solving methods are suitably approached. Therefore, it is further concluded that the methods and computing processes of the theoretical solution are effective and accurate.     

Sound radiation from a circular cylinder subjected to elastic collision by a sphere

Sound radiation from a steel cylinder impacted by a steel sphere from the longitudinal or the transverse direction is studied. In order to analyze the vibration of the cylinder, Hertz's theory is incorporated to obtain an approximate value of the contact force. The influence of the impact speed and the slenderness of the cylinder on the radiation of sound waves from the vibrating cylinder is analyzed. An experimental apparatus was constructed and vibrations of the cylinder as well as the acoustic pressure radiated were measured to demonstrate the analytical results. It is shown that, no matter whether dispersion in the wave propagation in the impacted cylinder is significant as in the case of a transverse impact, or not as in the case of a longitudinal impact, a decaying sound pulse is predominantly generated by the rigid motion of the cylinder provided that the product of the contact time T and the fundamental natural frequency of the cylinder ω₁i.e., ω₁T, is larger than 4π, while a periodic sound due to the free vibration of the cylinder is predominant provided that ω₁T is smaller than 4π.     

Analytic structure of ρ meson propagator at finite temperature

We analyse the structure of one-loop self-energy graphs for the ρ meson in real time formulation of finite temperature field theory. We find the discontinuities of these graphs across the unitary and the Landau cuts. These contributions are identified with different sources of medium modification discussed in the literature. We also calculate numerically the imaginary and the real parts of the self-energies and construct the spectral function of the ρ meson, which are compared with an earlier determination. A significant contribution arises from the unitary cut of the π ω loop, that was ignored so far in the literature.     

Optimized computation of the voigt and complex probability functions

Methods for computing the complex probability function w(z) = e^?z2 erfc (?iz), which is related to the Voigt spectrum line profiles, are developed. The basic method is a rational approximation, minimizing the relative error of the imaginary part on the real axis. It is complemented by other methods in order to increase efficiency and to overcome the inevitable failure of any rational approximation near the real axis. The procedures enable one to evaluate both real and imaginary parts of w(z) with high relative accuracy. The methods are simple, as demonstrated by a sample FORTRAN program.