An analytical modeling approach for the prediction of the geometric characteristics of five-dimensional (5D) woven composites has been formulated. The model is driven by readily available data including the weaving parameters and constituent material properties. The new model calculates the individual proportions of fiber in each direction, areal density, overall fiber volume fraction, and laminate thickness. This information is useful for the engineer in the design and manufacture of 5D woven composites. In addition the present model outputs the mathematical definition of the 5D woven composite unit cell, which could be implemented as the geometric input for a downstream analytical model that is capable of predicting the elastic stiffness of 5D woven composites. Input parameters have been sourced from existing published work and the subsequent predictions made by the model are compared with the available experimental data on 5D woven composites.

This research presents the development of an analytical model to predict the elastic stiffness performance of orthogonal interlock bound 3D woven composites as a consequence of altering the weaving parameters and constituent material types.The present approach formulates expressions at the micro level with the aim of calculating more representative volume fractions of a group of elements to the layer. The rationale in representing the volume fractions within the unit cell more accurately was to improve the elastic stiffness predictions compared to existing analytical modelling approaches.The models developed in this work show good agreement between experimental data and improvement on existing predicted values by models published in literature. (C) 2010 Elsevier Ltd. All rights reserved.

The three-dimensional (3D) weaving process offers the ability to tailor the mechanical properties via design of the weave architecture. One repeat of the 3D woven fabric is represented by the unit cell. The model accepts basic weaver and material manufacturer data as inputs in order to calculate the geometric characteristics of the 3D woven unit cell. The specific weave architecture manufactured and subsequently modelled had an angle interlock type binding configuration. The modelled result was shown to have a close approximation compared to the experimentally measured values and highlighted the importance of the representation of the binder tow path.

The expected number of real zeros of the polynomial of the form a 0 + a 1 x + a 2 x 2 + ⋯+ a n x n , where a 0 , a 1 , a 2 , …, a n is a sequence of standard Gaussian random variables, is known. For n large it is shown that this expected number in (− ∞ , ∞ ) is asymptotic to (2/ π )log  n . In this paper, we show that this asymptotic value increases significantly to when we consider a polynomial in the form instead. We give the motivation for our choice of polynomial and also obtain some other characteristics for the polynomial, such as the expected number of level crossings or maxima. We note, and present, a small modification to the definition of our polynomial which improves our result from the above asymptotic relation to the equality.

We present a simple formula for the expected number of times that a complex-valued Gaussian stochastic process has a zero imaginary part and the absolute value of its real part is bounded by a constant value M . We show that only some mild conditions on the stochastic process are needed for our formula to remain valid. We further apply this formula to a random algebraic polynomial with complex coefficients. We show how the above expected value in the case of random algebraic polynomials varies for different behaviour of M .

Simulations of deflagration propagation through initially quiescent stoichiometric hydrogen/air mixture inside a large-scale 2.3-m-diameter spherical vessel are performed. A large-eddy simulation (LES) model of premixed combustion is suggested, which is based on a subgrid-scale turbulence model of renormalization group theory, a gradient method for combustion modeling and the dependence of burning velocity on transient pressure and temperature. An unstructured grid with one level of refinement/de-refinement around the flame front area is used. Wrinkled by hydrodynamic instability, the flame front structure has been resolved for the first time in LES of large-scale explosions. The growth of cell sizes with flame radius is reproduced numerically in agreement with theoretical and experimental results. The minimum resolved size of the flame cells is of the order of the simulated flame front thickness, which is about three control volume edges. The fractal dimension of the wrinkled flame front surface grows and reaches a value of 2.15, close to observed in experiments.

There are many known asymptotic estimates for the expected number of real zeros of polynomial Hn(z) = η1 cosh ζz + η2 cosh 2ζz + ⋯ + ηn cosh nζz, where ηj, j = 1, 2, 3, …, n is a sequence of independent random variables. This paper provides the asymptotic formula for the expected density of complex zeros of Hn(z), where ηj = aj + ibj and aj and bj, j = 1, 2, 3, …, n are sequences of independent normally distributed random variables. It is shown that this asymptotic formula for the density of complex zeros remains invariant for other types of polynomials, for instance random trigonometric polynomials, previously studied.

An asymptotic estimate is derived for the expected number of extrema of a polynomial a(0) + a(1)((n)(1))(1/2)x + a(2)((n)(2))(1/2)x(2) + ... + a(n)((n)(n))(1/2)x(n) whose independent normal coefficients possess non-equal non-zero mean values. A result is presented that generalizes in terms of normal processes the analytical device used for construction of similar asymptotic estimates for random polynomials with normal coefficients.

We describe in detail a method to study the asymptotics of the expected density of complex roots of random polynomials with normal coefficients. We present results about random algebraic and hyperbolic polynomials achieved with this method.

In this paper, we obtain an exact formula for the average density of the distribution of complex zeros of a random trigonometric polynomial η0+η1cosθ+η2cos2θ+⋯+ηncosnθ in (0,2π), where the coefficients ηj=aj+ιbj, and {aj}j=1n and {bj}j=1n are sequences of independent normally distributed random variables with mean 0 and variance 1. We also provide the limiting behaviour of the zeros density function as n tends to infinity. The corresponding results for the case of random algebraic polynomials are known.

In this paper we obtain a formula for the average density of the distribution of complex zeros of an algebraic polynomial with random coefficients. The coefficients are assumed independent identical normally distributed random variables with mean μ and variance σ2. The value of the average density for the case of μ=0 and σ2=1 was obtained previously. Some limits of the distribution of the complex zeros are provided using the presented formula.