Numerical modelling of industrial microwave heating

The numerical modelling of electromagnetic waves has been the focus of many research areas in the past. Some specific applications of electromagnetic wave scattering are in the fields of Microwave Heating and Radar Communication Systems. The equations that govern the fundamental behaviour of electromagnetic wave propagation in waveguides and cavities are the Maxwell's equations. In the literature, a number of methods have been employed to solve these equations. Of these methods, the classical Finite-Difference Time-Domain scheme, which uses a staggered time and space discretisation, is the most well known and widely used. However, it is complicated to implement this method on an irregular computational domain using an unstructured mesh. In this work, a coupled method is introduced for the solution of Maxwell's equations. It is proposed that the free-space component of the solution is computed in the time domain, whilst the load is resolved using the frequency dependent electric field Helmholtz equation. This methodology results in a timefrequency domain hybrid scheme. For the Helmholtz equation, boundary conditions are generated from the time dependent free-space solutions. The boundary information is mapped into the frequency domain using the Discrete Fourier Transform. The solution for the electric field components is obtained by solving a sparse-complex system of linear equations. The hybrid method has been tested for both waveguide and cavity configurations. Numerical tests performed on waveguides and cavities for inhomogeneous lossy materials highlight the accuracy and computational efficiency of the newly proposed hybrid computational electromagnetic strategy.

Numerical modelling of load bearing steel stud walls under fire conditions

Fire design is an essential part of the overall design procedure of structural steel members and systems. Conventionally, increased fire rating is provided simply by adding more plasterboards to Light gauge Steel Frame (LSF) stud walls, which is inefficient. However, recently Kolarkar & Mahendran (2008) developed a new composite wall panel system, where the insulation was located externally between the plasterboards on both sides of the steel wall frame. Numerical and experimental studies were undertaken to investigate the structural and fire performance of LSF walls using the new composite panels under axial compression. This paper presents the details of the numerical studies of the new LSF walls and the results. It also includes brief details of the experimental studies. Experimental and numerical results were compared for the purpose of validating the developed numerical model. The paper also describes the structural and fire performance of the new LSF wall system in comparison to traditional wall systems using cavity insulation.

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.

Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term

Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term (TSFFPE-NST), which involve the Caputo time fractional derivative (CTFD) of order α ∈ (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order μ ∈ (1, 2). Approximating the CTFD and RSFD using the L1-algorithm and shifted Grunwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

Computationally efficient numerical methods for time- and space-fractional Fokker–Planck equations

Fractional Fokker–Planck equations have been used to model several physical situations that present anomalous diffusion. In this paper, a class of time- and space-fractional Fokker–Planck equations (TSFFPE), which involve the Riemann–Liouville time-fractional derivative of order 1-α (α(0, 1)) and the Riesz space-fractional derivative (RSFD) of order μ(1, 2), are considered. The solution of TSFFPE is important for describing the competition between subdiffusion and Lévy flights. However, effective numerical methods for solving TSFFPE are still in their infancy. We present three computationally efficient numerical methods to deal with the RSFD, and approximate the Riemann–Liouville time-fractional derivative using the Grünwald method. The TSFFPE is then transformed into a system of ordinary differential equations (ODE), which is solved by the fractional implicit trapezoidal method (FITM). Finally, numerical results are given to demonstrate the effectiveness of these methods. These techniques can also be applied to solve other types of fractional partial differential equations.

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.

Numerical evaluation of pollutant dispersion at a toll plaza based on system dynamics and Computational Fluid Dynamics models

The health of tollbooth workers is seriously threatened by long-term exposure to polluted air from vehicle exhausts. Using traffic data collected at a toll plaza, vehicle movements were simulated by a system dynamics model with different traffic volumes and toll collection procedures. This allowed the average travel time of vehicles to be calculated. A three-dimension Computational Fluid Dynamics (CFD) model was used with a k–ε turbulence model to simulate pollutant dispersion at the toll plaza for different traffic volumes and toll collection procedures. It was shown that pollutant concentration around tollbooths increases as traffic volume increases. Whether traffic volume is low or high (1500 vehicles/h or 2500 vehicles/h), pollutant concentration decreases if electronic toll collection (ETC) is adopted. In addition, pollutant concentration around tollbooths decreases as the proportion of ETC-equipped vehicles increases. However, if the proportion of ETC-equipped vehicles is very low and the traffic volume is not heavy, then pollutant concentration increases as the number of ETC lanes increases.

Executive function effects and numerical development in children : behavioural and ERP evidence from a numerical Stroop paradigm

Most research on numerical development in children is behavioural, focusing on accuracy and response time in different problem formats. However, Temple and Posner (1998) used ERPs and the numerical distance task with 5-year-olds to show that the development of numerical representations is difficult to disentangle from the development of the executive components of response organization and execution. Here we use the numerical Stroop paradigm (NSP) and ERPs to study possible executive interference in numerical processing tasks in 6–8-year-old children. In the NSP, the numerical magnitude of the digits is task-relevant and the physical size of the digits is task-irrelevant. We show that younger children are highly susceptible to interference from irrelevant physical information such as digit size, but that access to the numerical representation is almost as fast in young children as in adults. We argue that the developmental trajectories for executive function and numerical processing may act together to determine numerical development in young children.

Atomistic numerical investigation of single-crystal copper nanowire with surface defects

Based on the embedded atom method (EAM), a molecular dynamics (MD) simulation is performed to study the single-crystal copper nanowire with surface defects through tension. The tension simulations for nanowire without defect are first carried out under different temperatures, strain rates and time steps and then surface defect effects for nanowire are investigated. The stress-strain curves obtained by the MD simulations of various strain rates show a rate below 1 x 10(9) s-1 will exert less effect on the yield strength and yield point, and the Young's modulus is independent of strain rate. a time step below 5 fs is recommend for the atomic model during the MD simulation. It is observed that high temperature leads to low Young's modulus, as well as the yield strength. The surface defects on nanowires are systematically studied in considering different defect orientations. It is found that the surface defect serves as a dislocation source, and the yield strength shows 34.20% decresse with 45 degree surface defect. Both yield strength and yield point are significantly influenced by the surface defects, except the Young's modulus.

Numerical solution to the Saffman-Taylor finger problem with kinetic undercooling regularisation

The Saffman-Taylor finger problem is to predict the shape and,in particular, width of a finger of fluid travelling in a Hele-Shaw cell filled with a different, more viscous fluid. In experiments the width is dependent on the speed of propagation of the finger, tending to half the total cell width as the speed increases. To predict this result mathematically, nonlinear effects on the fluid interface must be considered; usually surface tension is included for this purpose. This makes the mathematical problem suffciently diffcult that asymptotic or numerical methods must be used. In this paper we adapt numerical methods used to solve the Saffman-Taylor finger problem with surface tension to instead include the effect of kinetic undercooling, a regularisation effect important in Stefan melting-freezing problems, for which Hele-Shaw flow serves as a leading order approximation when the specific heat of a substance is much smaller than its latent heat. We find the existence of a solution branch where the finger width tends to zero as the propagation speed increases, disagreeing with some aspects of the asymptotic analysis of the same problem. We also find a second solution branch, supporting the idea of a countably infinite number of branches as with the surface tension problem.