Numerical study of cold-formed steel beam subject to lateral-torsional buckling

The use of cold-formed steel members as structural columns and beams in residential, industrial and commercial buildings has increased significantly in recent times. This study is focused on the use of cold-formed steel sections as flexural members subject to lateral-torsional buckling. For this purpose a finite element model of a simply supported lipped channel beam under uniform bending was developed, validated using available numerical and experimental results, and used in a detailed parametric study. The moment capacity results were then compared with the predictions from the current ambient temperature design rules in the cold-formed steel structures codes of Australia, New Zealand, North America and Europe. European design rules were found to be conservative while Australian and American design rules were unsafe. This paper presents the results of the numerical study, the comparison with the current design rules and the new proposed design rules.

Numerical models to simulate the thermal performance of LSF wall panels

Fire safety of buildings has been recognised as very important by the building industry and the community at large. Gypsum plasterboards are widely used to protect light gauge steel frame (LSF) walls all over the world. Gypsum contains free and chemically bound water in its crystal structure. Plasterboard also contains gypsum (CaSO4.2H2O) and calcium carbonate (CaCO3). The dehydration of gypsum and the decomposition of calcium carbonate absorb heat, and thus are able to protect LSF walls from fires. Kolarkar and Mahendran (2008) developed an innovative composite wall panel system, where the insulation was sandwiched between two plasterboards to improve the thermal and structural performance of LSF wall panels under fire conditions. In order to understand the performance of gypsum plasterboards and LSF wall panels under standard fire conditions, many experiments were conducted in the Fire Research Laboratory of Queensland University of Technology (Kolarkar, 2010). Fire tests were conducted on single, double and triple layers of Type X gypsum plasterboards and load bearing LSF wall panels under standard fire conditions. However, suitable numerical models have not been developed to investigate the thermal performance of LSF walls using the innovative composite panels under standard fire conditions. Continued reliance on expensive and time consuming fire tests is not acceptable. Therefore this research developed suitable numerical models to investigate the thermal performance of both plasterboard assemblies and load bearing LSF wall panels. SAFIR, a finite element program, was used to investigate the thermal performance of gypsum plasterboard assemblies and LSF wall panels under standard fire conditions. Appropriate values of important thermal properties were proposed for plasterboards and insulations based on laboratory tests, literature review and comparisons of finite element analysis results of small scale plasterboard assemblies from this research and corresponding experimental results from Kolarkar (2010). The important thermal properties (thermal conductivity, specific heat capacity and density) of gypsum plasterboard and insulation materials were proposed as functions of temperature and used in the numerical models of load bearing LSF wall panels. Using these thermal properties, the developed finite element models were able to accurately predict the time temperature profiles of plasterboard assemblies while they predicted them reasonably well for load bearing LSF wall systems despite the many complexities that are present in these LSF wall systems under fires. This thesis presents the details of the finite element models of plasterboard assemblies and load bearing LSF wall panels including those with the composite panels developed by Kolarkar and Mahendran (2008). It examines and compares the thermal performance of composite panels developed based on different insulating materials of varying densities and thicknesses based on 11 small scale tests, and makes suitable recommendations for improved fire performance of stud wall panels protected by these composite panels. It also presents the thermal performance data of LSF wall systems and demonstrates the superior performance of LSF wall systems using the composite panels. Using the developed finite element of models of LSF walls, this thesis has proposed new LSF wall systems with increased fire rating. The developed finite element models are particularly useful in comparing the thermal performance of different wall panel systems without time consuming and expensive fire tests.

A study on the encryption model for numerical data

The encryption method is a well established technology for protecting sensitive data. However, once encrypted, the data can no longer be easily queried. The performance of the database depends on how to encrypt the sensitive data. In this paper we review the conventional encryption method which can be partially queried and propose the encryption method for numerical data which can be effectively queried. The proposed system includes the design of the service scenario, and metadata.

Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

A fundamental numerical and theoretical study for the vibrational properties of nanowires

Based on the molecular dynamics (MD) simulation and the classical Euler-Bernoulli beam theory, a fundamental study of the vibrational performance of the Ag nanowire (NW) is carried out. A comprehensive analysis of the quality (Q)-factor, natural frequency, beat vibration, as well as high vibration mode is presented. Two excitation approaches, i.e., velocity excitation and displacement excitation, have been successfully implemented to achieve the vibration of NWs. Upon these two kinds of excitations, consistent results are obtained, i.e., the increase of the initial excitation amplitude will lead to a decrease to the Q-factor, and moderate plastic deformation could increase the first natural frequency. Meanwhile, the beat vibration driven by a single relatively large excitation or two uniform excitations in both two lateral directions is observed. It is concluded that the nonlinear changing trend of external energy magnitude does not necessarily mean a nonconstant Q-factor. In particular, the first order natural frequency of the Ag NW is observed to decrease with the increase of temperature. Furthermore, comparing with the predictions by Euler- Bernoulli beam theory, the MD simulation provides a larger and smaller first vibration frequencies for the clamped-clamped and clamped-free thin Ag NWs, respectively. Additionally, for thin NWs, the first order natural frequency exhibits a parabolic relationship with the excitation magnitudes. The frequencies of the higher vibration modes tend to be low in comparison to Euler-Bernoulli beam theory predictions. A combined initial excitation is proposed which is capable to drive the NW under a multi-mode vibration and arrows the coexistence of all the following low vibration modes. This work sheds lights on the better understanding of the mechanical properties of NWs and benefits the increasing utilities of NWs in diverse nano-electronic devices.

Numerical investigation of mechanical properties of nanowires : a review

Nanowires (NWs) have attracted intensive researches owing to the broad applications that arise from their remarkable properties. Over the last decade, immense numerical studies have been conducted for the numerical investigation of mechanical properties of NWs. Among these numerical simulations, the molecular dynamics (MD) plays a key role. Herein we present a brief review on the current state of the MD investigation of nanowires. Emphasis will be placed on the FCC metal NWs, especially the Cu NWs. MD investigations of perfect NWs’ mechanical properties under different deformation conditions including tension, compression, torsion and bending are firstly revisited. Following in succession, the studies for defected NWs including the defects of twin boundaries (TBs) and pre-existing defects are discussed. The different deformation mechanism incurred by the presentation of defects is explored and discussed. This review reveals that the numerical simulation is an important tool to investigate the properties of NWs. However, the substantial gaps between the experimental measurements and MD results suggest the urgent need of multi-scale simulation technique.

Alternating direction implicit numerical method for the space and time fractional Bloch-Torrey equation in 3-D

Recently, some authors have considered a new diffusion model–space and time fractional Bloch-Torrey equation (ST-FBTE). Magin et al. (2008) have derived analytical solutions with fractional order dynamics in space (i.e., _ = 1, β an arbitrary real number, 1 < β ≤ 2) and time (i.e., 0 < α < 1, and β = 2), respectively. Yu et al. (2011) have derived an analytical solution and an effective implicit numerical method for solving ST-FBTEs, and also discussed the stability and convergence of the implicit numerical method. However, due to the computational overheads necessary to perform the simulations for nuclear magnetic resonance (NMR) in three dimensions, they present a study based on a two-dimensional example to confirm their theoretical analysis. Alternating direction implicit (ADI) schemes have been proposed for the numerical simulations of classic differential equations. The ADI schemes will reduce a multidimensional problem to a series of independent one-dimensional problems and are thus computationally efficient. In this paper, we consider the numerical solution of a ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. A fractional alternating direction implicit scheme (FADIS) for the ST-FBTE in 3-D is proposed. Stability and convergence properties of the FADIS are discussed. Finally, some numerical results for ST-FBTE are given.

A new implicit numerical method for the space and time fractional Bloch-Torrey equation

In recent years, it has been found that many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from fractional calculus. Recently, noted a new space and time fractional Bloch-Torrey equation (ST-FBTE) has been proposed (see Magin et al. (2008)), and successfully applied to analyse diffusion images of human brain tissues to provide new insights for further investigations of tissue structures. In this paper, we consider the ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we propose a new effective implicit numerical method (INM) for the STFBTE whereby we discretize the Riesz fractional derivative using a fractional centered difference. Secondly, we prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent, and the order of convergence of the implicit numerical method is ( T2 - α + h2 x + h2 y + h2 z ). Finally, some numerical results are presented to support our theoretical analysis.

Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

Percolation flow problems are discussed in many research fields, such as seepage hydraulics, groundwater hydraulics, groundwater dynamics and fluid dynamics in porous media. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcys law with variable order Riemann-Liouville fractional derivatives, this leads to a new variable-order fractional percolation equation. In this paper, a new two-dimensional variable-order fractional percolation equation is considered. A new implicit numerical method and an alternating direct method for the two-dimensional variable-order fractional model is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of the methods. This technique can be used to simulate a three-dimensional variable-order fractional percolation equation.

The Galerkin finite element approximation of the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. Cable equations with a fractional order temporal derivative have been introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, the fractional cable equation involving two integro-differential operators is considered. The Galerkin finite element approximations of the fractional cable equation are proposed. The main contribution of this work is outlined as follow: • A semi-discrete finite difference approximation in time is proposed. We prove that the scheme is unconditionally stable, and the numerical solution converges to the exact solution with order O(Δt). • A semi-discrete difference scheme for improving the order of convergence for solving the fractional cable equation is proposed, and the numerical solution converges to the exact solution with order O((Δt)2). • Based on the above semi-discrete difference approximations, Galerkin finite element approximations in space for a full discretization are also investigated. • Finally, some numerical results are given to demonstrate the theoretical analysis.

Numerical methods for solving the multi-term time-fractional wave-diffusion equations

In this paper, the multi-term time-fractional wave diffusion equations are considered. The multiterm time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

A second-order accurate numerical approximation for the Riesz space fractional advection-dispersion equation

In this paper, we consider a space Riesz fractional advection-dispersion equation. The equation is obtained from the standard advection-diffusion equation by replacing the ¯rst-order and second-order space derivatives by the Riesz fractional derivatives of order β 1 Є (0; 1) and β2 Є(1; 2], respectively. Riesz fractional advection and dispersion terms are approximated by using two fractional centered difference schemes, respectively. A new weighted Riesz fractional ¯nite difference approximation scheme is proposed. When the weighting factor Ѳ = 1/2, a second- order accurate numerical approximation scheme for the Riesz fractional advection-dispersion equation is obtained. Stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.

Numerical simulation of a fractional mathematical model for epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider numerical simulation of fractional model based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in advection and diffusion terms belong to the intervals (0; 1) or (1; 2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of the Riemann-Liouville and Gr¨unwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.

Numerical analysis of the time variable fractional order mobile-immobile advection-dispersion model

Many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the time variable fractional order mobile-immobile advection-dispersion model. Numerical methods and analyses of stability and convergence for the fractional partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the fractional order mobile immobile advection-dispersion model. In the paper, we use the Coimbra variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation for the equation is proposed and then the stability of the approximation are investigated. As for the convergence of the numerical scheme we only consider a special case, i.e. the time fractional derivative is independent of time variable t. The case where the time fractional derivative depends both the time variable t and the space variable x will be considered in the future work. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.