Analysis of steel structures for first- and second-order element (displacement and force) solutions

Finite element method (FEM) relies on an approximate function to fit into a governing equation and minimizes the residual error in the integral sense in order to generate solutions for the boundary value problems (nodal solutions). Because of this FEM does not show simultaneous capacities for accurate displacement and force solutions at node and along an element, especially when under the element loads, which is of much ubiquity. If the displacement and force solutions are strictly confined to an element’s or member’s ends (nodal response), the structural safety along an element (member) is inevitably ignored, which can definitely hinder the design of a structure for both serviceability and ultimate limit states. Although the continuous element deflection and force solutions can be transformed into the discrete nodal solutions by mesh refinement of an element (member), this setback can also hinder the effective and efficient structural assessment as well as the whole-domain accuracy for structural safety of a structure. To this end, this paper presents an effective, robust, applicable and innovative approach to generate accurate nodal and element solutions in both fields of displacement and force, in which the salient and unique features embodies its versatility in applications for the structures to account for the accurate linear and second-order elastic displacement and force solutions along an element continuously as well as at its nodes. The significance of this paper is on shifting the nodal responses (robust global system analysis) into both nodal and element responses (sophisticated element formulation).

Spectral approximations to the fractional integral and derivative

In this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain

Multi-term time-fractional differential equations have been used for describing important physical phenomena. However, studies of the multi-term time-fractional partial differential equations with three kinds of nonhomogeneous boundary conditions are still limited. In this paper, a method of separating variables is used to solve the multi-term time-fractional diffusion-wave equation and the multi-term time-fractional diffusion equation in a finite domain. In the two equations, the time-fractional derivative is defined in the Caputo sense. We discuss and derive the analytical solutions of the two equations with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin conditions, respectively.

Analytical and numerical solutions of the space and time fractional bloch-torrey equation

Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brown-ian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et al., J. Magnetic Resonance, 190 (2008) 255-270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

Some solutions for a compressible isotropic elastic material

In this article we obtain closed-form solutions for the combined inflation and axial shear of an elastic tube in respect of the compressible Isotropic elastic material introduced by Levinson and Burgess. Several other boundary-value problems are also examined, including the bending of a rectangular block and straightening of a cylindrical sector, both coupled with stretching and shearing, and an axially varying twist deformation. Some of the solutions appear in closed form, others are expressible in terms of elliptic functions.

Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by a Caputo fractional derivative, and the second order space derivative by a symmetric fractional derivative. First, a method of separating variables expresses the analytical solution of the TSS-FDE in terms of the Mittag--Leffler function. Second, we propose two numerical methods to approximate the Caputo time fractional derivative: the finite difference method; and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative and the second order space derivative by the symmetric fractional derivative. Firstly, a method of separating variables is used to express the analytical solution of the tss-fde in terms of the Mittag–Leffler function. Secondly, we propose two numerical methods to approximate the Caputo time fractional derivative, namely, the finite difference method and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results are presented to demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Identifying child sexual abuse within school contexts : policy problems and solutions, and a landmark case

As a strategy to identify child sexual abuse, most Australian States and Territories have enacted legislation requiring teachers to report suspected cases. Some Australian State and non-State educational authorities have also created policy-based obligations to report suspected child sexual abuse. Significantly, these can be wider than non-existent or limited legislative duties, and therefore are a crucial element of the effort to identify sexual abuse. Yet, no research has explored the existence and nature of these policy-based duties. The first purpose of this paper is to report the results of a three-State study into policy-based reporting duties in State and non-State schools in Australia. In an extraordinary coincidence, while conducting the study, a case of failure to comply with reporting policy occurred with tragic consequences. This led to a rare example in Australia (and one of only a few worldwide) of a professional being prosecuted for failure to comply with a legislative duty. It also led to disciplinary proceedings against school staff. The second purpose of this paper is to describe this case and connect it with findings from our policy analysis.

Floods, earthquakes and insurance coverage : issues, problems and solutions

The recent floods in Queensland and elsewhere in Australia as well as the recent earthquakes in New Zealand have again given rise to very significant uninsured losses. This article looks at the issue of cover protection against catastrophes such as floods and earthquakes affecting home buildings and contents insurance and the standard cover provisions of the Insurance Contracts Act 1984 (Cth). It points also to the possibility of a national scheme to cover natural disasters including floods.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

Regulating Prostitution: Different Problems, Different Solutions, Same Old Story

This article looks at three main models of intervention that have informed recent policy and practice with people involved in the sex trade. It reveals the inherent contradictions within attempts to both help and punish workers in the existing prostitution strategy.

Numerical solutions of stochastic differential equations – implementation and stability issues

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively.