Closed-form solutions for the time-variant spectral characteristics of non-stationary random processes

Spectral characteristics are important quantities in describing stationary and non-stationary random processes. In this paper, the spectral characteristics for complex-valued random processes are evaluated and closed-form solutions for the time-variant statistics of the response of linear single-degree-of-freedom (SDOF) and both classically and non-classically damped multi-degree-of-freedom (MDOF) systems subjected to modulated Gaussian colored noise are obtained. The time-variant central frequency and bandwidth parameter of the response processes of linear SDOF and MDOF elastic systems subjected to Gaussian colored noise excitation are computed exactly in closed-form. These quantities are useful in problems which require the use of complex modal analysis, such as random vibrations of non-classically damped MDOF linear structures, and in structural reliability applications. Monte Carlo simulation has been used to confirm the validity of the proposed solutions.     

Designs in Three and Four Components For Mixtures Models With Inverse Terms

Scheffé's mixture models are given and the D-optimality of various designs for these models is discussed. Approximate D-optimal (measure) and D_n -optimal (exact) designs are given for use with mixture models with inverse terms in three and four components.     

Prediction of the energy dissipation rate in ductile crack propagation

In this paper, energy dissipation rate D vs. Δa curves in ductile fracture are predicted using a ‘conversion’ between loads, load‐point displacements and crack lengths predicted by NLEFM and those found in real ELPL propagation. The NLEFM/ELPL link was recently discovered for the DCB testpiece, and we believe it applies to other cracked geometries. The predictions for D agree with experimental results. The model permits a crack tip toughness R(Δa) which rises from J_c and saturates out when (if) steady state propagation is reached after a transient stage in which all tunnelling, crack tip necking and shear lip formation is established. J_R is always greater than the crack tip R(Δa) and continues to rise even after R(Δa) levels off. The analysis is capable of predicting the usual D vs. Δa curves in the literature which have high initial values and fall monotonically to a plateau at large Δa. It also predicts that D curves for CCT testpieces should be higher than those for SENB/CT, as found in practice. The possibility that D curves at some intermediate Δa may dip to a minimum below the levelled‐off value at large Δa is predicted and confirmed by experiment. Recently reported D curves that have smaller initial D than the D‐values after extensive propagation can also be predicted. The testpiece geometry and crack tip R(Δa) conditions required to produce these different‐shaped D vs. Δa curves are established and confirmed by comparison with experiment. The energy dissipation rate D vs. Δa is not a transferable property as it depends on geometry. The material characteristic R(Δa) may be the ‘transferable property’ for scaling problems in ELPL fracture. How it can be deduced from D vs. Δa curves (and by implication, J_R vs. Δa curves) is established.     

A Synthetic Control Chart for Monitoring Process Variability

The control chart based on Downton's estimator (D chart) has recently been introduced in the literature for monitoring the process variability. The D chart is found to be equally efficient to the S chart in terms of detecting shifts in process variability. In this paper, salient features of D chart and the conforming run length chart are combined to produce synthetic D chart. The average run length performance of the synthetic D chart is investigated using simulation study and is compared with the originally proposed D chart and some other procedures proposed in the literature. It is found that it has an improved performance in comparison with the traditional control charts for process variability. Copyright © 2013 John Wiley & Sons, Ltd.     

Pattern scaling in axial segregation

Granular tumbled frequently segregate by grain size along the axis of partially-filled, horizontal, rotating tubes. When segregation approaches saturation at the surface, a well-defined pattern of bands with wavelength λ emerges. The long-term dynamics of the pattern involves a much slower coarsening process. We characterized the initial saturated wavelength λ as a function of the diameter of the tube D for a filling fraction of 30 %, for 1.9 cm ≤D≤11.5 cm. We also studied the initial growth-rate of the bands as D varies. We find that λ/ D is not constant, but rather increases rapidly for small D. The growth-rate of bands decreases with smaller D and segregation is suppressed completely for sufficiently small D. These relatively simple features are not captured by any of the existing models of axial segregation.     

Response surface optimization for a nonlinearly constrained irregular experimental design space

Finding optimum conditions for process factors in an engineering optimization problem with response surface functions requires structured data collection using experimental design. When the experimental design space is constrained owing to external factors, its design space may form an asymmetrical and irregular shape and thus standard experimental design methods become ineffective. Computer-generated optimal designs, such as D-optimal designs, provide alternatives. While several iterative exchange algorithms for D-optimal designs are available for a linearly constrained irregular design space, it has not been clearly understood how D-optimal design points need to be generated when the design space is nonlinearly constrained and how response surface models are optimized. This article proposes an algorithm for generating the D-optimal design points that satisfy both feasibility and optimality conditions by using piecewise linear functions on the design space. The D-optimality-based response surface design models are proposed and optimization procedures are then analysed.     

Edge distance effects on residual stress distribution around a cold expanded hole in Al 2024 alloy

Cold expansion process is a well-known technique for improving the fatigue life of aerospace structures by introducing a compressive residual stress around the fastener holes. However, there are concerns about the residual stress distribution around those holes which are located near the free edges of structure. The purpose of this study is to investigate the influence of edge distance ratio (e/D) on the residual stress distribution around a cold expanded hole in Al 2024 alloy. A two-dimensional finite element simulation was carried out with various degrees of cold expansion and various values of e/D. It was found that for edge distance ratios less than e/D = 3, there are considerable effects on the residual stress profile. Also, the dependency of residual stress distribution on the degree of expansion was reduced significantly for small e/Ds. The results revealed that the bulging of the plate free edge increases with reduction of e/D but in worse cases the maximum bulging at the plate free edge was lower than 3% of the hole radius. The weight function method was then used for determining stress intensity factors for a crack emanating from a cold expanded hole.     

Directional nature of Goodman–Kruskal gamma and some consequences: identity of Goodman–Kruskal gamma and Somers delta,and their connection to Jonckheere–Terpstra test statistic

Although usually taken as a symmetric measure, G is shown to be a directional coefficient of association. The direction in G is not related to rows or columns of the cross-table nor the identity of the variables to be a predictor or a criterion variable but, instead, to the number of categories in the scales. Under the conditions where there are no tied pairs in the dataset, G equals Somers’ D so directed that the variable with a wider scale (X) explains the response pattern in the variable with a narrower scale (g), that is, D(g│X). Hence, G = G(g│X) = D(g│X) but G ≠ D(X│g) and G ≠ D(symmetric). If there are tied pairs, the estimates by G = G(g│X) are more liberal in comparison with those by D(g│X). Algebraic relation of G and D with Jonckheere–Terpstra test statistic (JT) is derived. Because of the connection to JT, G = G(g│X) and D = D(g│X) indicate the proportion of logically ordered test-takers in the item after they are ordered by the score. It is strongly recommendable that gamma should not be used as a symmetric measure, and it should be used directionally only when willing to explain the behaviour of a variable with a narrower scale by the variable with a wider scale. This fits well with the measurement modelling settings.

     

An analysis of grain-growth data in duplex materials on static annealing and during superplastic deformation

An analysis of data on grain growth in several duplex materials subjected to superplastic deformation (SPD) and to static annealing (SA) is presented. Also, for comparison, data of a single-phase alloy and a particle-strengthened one were considered. The theoretical equationD ⁿ -D _o ⁿ =Kt was employed, whereD andD _o are the instantaneous and initial average grain size, respectively,t is the deformation or annealing time,n is the kinetic exponent andK is the rate constant. When analysing the selectedD-t data sets, the fact thatD _o values were generally not small enough to be negligible in the above equation was taken into account. It is concluded: (i) for a given duplex allay, the coarsening mechanisms acting during SA and SPD are different, the kinetics being enhanced by concurrent deformation, and (ii) whatever the alloy, the value forn for grain growth under SPD is near to 2, (as for normal grain growth in single-phase materials).     

Sequentially Generated Second Order Quasi D-Optimal Designs for Experiments With Mixture and Process Variables

Second order designs for experiments with mixture and process variables are proposed. They are constructed on the basis of continuous D-optimal designs by use of a three-stage procedure for sequentially generating optimal designs. The determinants of the information matrices of the designs obtained are very near to those of continuous D-optimal designs. Tables of discrete quasi D-optimal designs for q + r ≤ 7 are given, where q is the number of mixture components and r is the number of process variables. The experimenter can choose the number of trials N within the interval k ≤ N ≤ min(2k, k + 20), where k is the number of model coefficients. An application of the proposed designs in an investigation of truck tire properties is given.     

An Algorithm for the Construction of “D-Optimal” Experimental Designs

This paper presenm the algorithm “DETMAX” whose purpose is to construct experimental designs that are “D-optimal.” These are designs for which the determinant of X'X is maximum, where X is the “matrix of independent variables” in the usual linear model y = Xβ + ε. Although the algorithm does not guarantee D-optimality, it has performed well in many cases where D-optimal designs are known. Five examples are given, illustrating the use of DETMAX to construct designs “from scratch” and to augment existing data. A FORTRAN listing is available on request.     

Three-dimensional analysis of modulated photoreflectance in a silicon wafer

The effects of probe and pumping beam size and modulation frequency on photoreflectance were investigated for a silicon wafer by considering one- and three-dimensional generation and propagation of thermal and plasma waves,PR _1D andPR _3D. The magnitude ofPR _1D decreased as the inverse square of the effective beam radius and that ofPR _3D was 100 times smaller thanPR _1D at 0.1 m effective beam radius and decreased with the effective beam radius. The phase shift ofPR _1D was nearly constant at 225°, whereas that ofPR _3D increased with the effective beam radius from 0° to 225°. The magnitude and phase ofPR _3D become the same as those ofPR _1D by satisfying the equivalence conditions, where the probe and pumping beam radii are larger than the thermal and plasma wavelengths, when the effective beam radius was larger than 112 m.PR _1D decreased with modulation frequency as ^–1/2, whereas the magnitude ofPR _3D was nearly constant and 100 times smaller than that ofPR _1D at 1 kHz modulation frequency. ThePR _1D phase varied from 180° to 225°, but that of thePR _3D increased from 0° to that ofPR _1D with increase of the modulation frequency. As the modulation frequency increased, the magnitude and phase ofPR _3D approached to those ofPR _1D by approaching the equivalence conditions, owing to the decrease of the thermal and plasma wavelengths. The good agreements in the modulation frequency dependence of the magnitude and phase ofPR _3D with those measured, justified the three-dimensional analysis of the photoreflectance.     

Bridge Designs for Modeling Systems With Low Noise

For deterministic computer simulations, Gaussian process models are a standard procedure for fitting data. These models can be used only when the study design avoids having replicated points. This characteristic is also desirable for one-dimensional projections of the design, since it may happen that one of the design factors has a strongly nonlinear effect on the response. Latin hypercube designs have uniform one-dimensional projections, but are not efficient for fitting low-order polynomials when there is a small error variance. D-optimal designs are very efficient for polynomial fitting but have substantial replication in projections. We propose a new class of designs that bridge the gap between D-optimal designs and D-optimal Latin hypercube designs. These designs guarantee a minimum distance between points in any one-dimensional projection allowing for the fit of either polynomial or Gaussian process models. Subject to this constraint they are D-optimal for a prespecified model.     

Superflow and the depairing critical current for spin singlet-triplet states of superfluid Fermi liquid

The unitary states of superfluid Fermi liquid with singlet D and triplet P type of pairing are investigated in the framework of the weak coupling approximation. The superflow pair-breaking critical current is calculated at zero temperature and in the Ginzburg-Landau region for various values of the respective strengths of singlet and triplet components of the pairing interaction. The dependence of the mass superflow on the a ₁ (F ₁ ^s ) Landau amplitude is determined. The mixed singlet-triplet states BS (a mixture of BW anisotropic and D-wave states) and 2DS (a mixture of 2D planar and D-wave states) are found to be stable for some region of the superfluid velocity.     

Scanning Kerr Microscopy of the Spin Hall Effect in <Emphasis Type="Italic">n</Emphasis>-Doped GaAs with Various Doping Concentration

We investigated the doping concentration (N _D) dependence of the extrinsic spin Hall effect (SHE) in n-doped GaAs with N _D raging from 3×10¹⁶ cm⁻³ to 5×10¹⁷ cm⁻³. By using scanning Kerr microscopy (SKM) measurements, we observed the Kerr rotation signal due to the spin accumulation near the channel edges in all the samples with different N _D. Moreover, the position and in-plane magnetic field dependence of the Kerr rotation signal are found to vary with N _D. We analyzed the N _D dependence of the spin Hall conductivity by taking account of the N _D-dependent spin lifetime based on the typical drift-diffusion model.     

Bismuth deformation potentials calculated from SdH periods under static strain and from magnetoacoustic attenuation

We describe the non-parabolicity of the electron dispersion in bismuth by the Lax model, which replaces the energy Eby E(1+E/E_G), E_G being the L-point energy gap. It is assumed that the effect of small strains can be accounted for solely by small changes of the electron and hole Fermi energies, dE_F = D_jke_jk,where D_jk and e_jk denote deformation potentials and strains. With this assumption we show that the deformation potentials come out the same whether the dispersion relation is non-parabolic or parabolic. This finding we use in a re-evaluation of the deformation potentials obtained from SdH-measurements under static strain. We further make a mass data correction of deformation potentials obtained from magnetoacoustic attenuation. The two sets of values so obtained are in excellent agreement. This allows us to improve the accuracy, and we recommend to use the following values (unit eV): for electrons: D₁₁ = 2.74 ± 0.50, D₂₂ = –7.38 ± 0.56, D₃₃ = 2.17 ±0.25, D₂₃ = –1.85 ± 0.44and for holes: D₁₁ = –1.06 ± 0.27, D₃₃ = –1.06 ± 0.09     

Molecular dynamics simulation on diffusion of 13 kinds of small molecules in polyethylene terephthalate

Understanding the diffusion of migrants in polyethylene terephthalate (PET) and calculating the diffusion coefficients are very important for migration research. In this study, the diffusion coefficients of 13 kinds of small molecules with molecular weights ranging from 32 to 339 g/mol in amorphous PET are calculated based on molecular dynamics (MD) simulation. By comparison of diffusion coefficients simulated by MD simulation techniques, predicted by the Piringer model and by experiments, the accuracy of the Piringer model and MD simulation techniques for the estimation of diffusion coefficients of migrants in PET is evaluated. The MD simulation shows that D_simu is very close to D_exp, within one order of magnitude of the experimental diffusion coefficients except for a few samples. The possible reasons for the differences among D_simu, D_pred and D_exp are analysed from the molecular weight and temperature. The results show that the Piringer‐model‐predicted values at high temperatures overestimate significantly higher than that at lower temperatures. The activation energy is calculated by the Arrhenius equation, which shows the relationship between diffusion coefficient and temperature. It is shown that the MD simulation yields acceptable activation energy. The study suggests that MD simulation may be a useful approach to calculate the diffusion coefficients of small molecules in PET. Copyright © 2010 John Wiley & Sons, Ltd.     

Separation in D‐optimal experimental designs for the logistic regression model

The D‐optimality criterion is often used in computer‐generated experimental designs when the response of interest is binary, such as when the attribute of interest can be categorized as pass or fail. The majority of methods in the generation of D‐optimal designs focus on logistic regression as the base model for relating a set of experimental factors with the binary response. Despite the advances in computational algorithms for calculating D‐optimal designs for the logistic regression model, very few have acknowledged the problem of separation, a phenomenon where the responses are perfectly separable by a hyperplane in the design space. Separation causes one or more parameters of the logistic regression model to be inestimable via maximum likelihood estimation. The objective of this paper is to investigate the tendency of computer‐generated, nonsequential D‐optimal designs to yield separation in small‐sample experimental data. Sets of local D‐optimal and Bayesian D‐optimal designs with different run (sample) sizes are generated for several “ground truth” logistic regression models. A Monte Carlo simulation methodology is then used to estimate the probability of separation for each design. Results of the simulation study confirm that separation occurs frequently in small‐sample data and that separation is more likely to occur when the ground truth model has interaction and quadratic terms. Finally, the paper illustrates that different designs with identical run sizes created from the same model can have significantly different chances of encountering separation.     

The Fictitious Time Integration Method to Solve the Space- and Time-Fractional Burgers Equations

We propose a simple numerical scheme for solving the space- and time-fractional derivative Burgers equations: D_t^αu + εuu_x = vu_xx + ηD_x^βu, 0 < α, β ≤ 1, and u_t + D_*^β(D_*^1-βu)²/2 = vu_xx, 0 < β ≤ 1. The time-fractional derivative D_t^αu and space-fractional derivative D_x^βu are defined in the Caputo sense, while D_*^βu is the Riemann-Liouville space-fractional derivative. A fictitious time τ is used to transform the dependent variable u(x,t) into a new one by (1+τ)^γu(x,t) =: v(x,t,τ), where 0 < γ ≤ 1 is a parameter, such that the original equation is written as a new functional-differential type partial differential equation in the space of (x,t,τ). When the group-preserving scheme is used to integrate these equations under a semi-discretization of u(x,t,τ) at the spatial-temporal grid points, we can achieve rather accurate solutions.