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.

A hybrid approach for resolving the electromagnetic fields inside a waveguide loaded with a lossy medium

In this work a novel hybrid approach is presented that uses a combination of both time domain and frequency domain solution strategies to predict the power distribution within a lossy medium loaded within a waveguide. The problem of determining the electromagnetic fields evolving within the waveguide and the lossy medium is decoupled into two components, one for computing the fields in the waveguide including a coarse representation of the medium (the exterior problem) and one for a detailed resolution of the lossy medium (the interior problem). A previously documented cell-centred Maxwell’s equations numerical solver can be used to resolve the exterior problem accurately in the time domain. Thereafter the discrete Fourier transform can be applied to the computed field data around the interface of the medium to estimate the frequency domain boundary condition in-formation that is needed for closure of the interior problem. Since only the electric fields are required to compute the power distribution generated within the lossy medium, the interior problem can be resolved efficiently using the Helmholtz equation. A consistent cell-centred finite-volume method is then used to discretise this equation on a fine mesh and the underlying large, sparse, complex matrix system is solved for the required electric field using the iterative Krylov subspace based GMRES iterative solver. It will be shown that the hybrid solution methodology works well when a single frequency is considered in the evaluation of the Helmholtz equation in a single mode waveguide. A restriction of the scheme is that the material needs to be sufficiently lossy, so that any penetrating waves in the material are absorbed.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

Exploring first-year academic achievement through structural equation modelling

The purpose of this research was to develop and test a multicausal model of the individual characteristics associated with academic success in first-year Australian university students. This model comprised the constructs of: previous academic performance, achievement motivation, self-regulatory learning strategies, and personality traits, with end-of-semester grades the dependent variable of interest. The study involved the distribution of a questionnaire, which assessed motivation, self-regulatory learning strategies and personality traits, to 1193 students at the start of their first year at university. Students' academic records were accessed at the end of their first year of study to ascertain their first and second semester grades. This study established that previous high academic performance, use of self-regulatory learning strategies, and being introverted and agreeable, were indicators of academic success in the first semester of university study. Achievement motivation and the personality trait of conscientiousness were indirectly related to first semester grades, through the influence they had on the students' use of self-regulatory learning strategies. First semester grades were predictive of second semester grades. This research provides valuable information for both educators and students about the factors intrinsic to the individual that are associated with successful performance in the first year at university.

Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.

A Petrov-Galerkin method for a singularly perturbed ordinary differential equation with non-smooth data

In this paper, a singularly perturbed ordinary differential equation with non-smooth data is considered. The numerical method is generated by means of a Petrov-Galerkin finite element method with the piecewise-exponential test function and the piecewise-linear trial function. At the discontinuous point of the coefficient, a special technique is used. The method is shown to be first-order accurate and singular perturbation parameter uniform convergence. Finally, numerical results are presented, which are in agreement with theoretical results.

Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions

In this paper, a space fractional di®usion equation (SFDE) with non- homogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix raised to the same fractional power. Analytic solutions of the SFDE are derived. Finally, some numerical results are given to demonstrate that the MTT is a computationally e±cient and accurate method for solving SFDE.