Michael J. Bluck

Adsorption kinetics, capacity and mechanism of arsenate and phosphate on a bifunctional TiO2–Fe2O3 bi-composite

Mixed oxide TiO(2)-Fe(2)O(3) bi-composites have been recognised as efficient and economical sorbents with great promise for arsenic removal from groundwater. In this study, we use a fast, simple and inexpensive synthesis method for this type of bi-composite and assess its adsorption performance. The kinetics of arsenate and phosphate adsorption onto the bi-composite are determined, demonstrating rapid and stable uptake of both oxy-anions over several days and with improved performance compared to the widely used TiO(2) sorbent. A modified pseudo-second order rate equation is introduced, which allows the adsorption kinetics to be modelled as two simultaneous, parallel reaction pathways with separate kinetic parameters. This equation reproduces the experimental observations accurately across a wide range of timescales from minutes to days. Our experimental data agrees with previous interpretations of the adsorption mechanism including the formation of mono-dentate and bi-dentate inner-sphere surface complexes. The arsenate and phosphate uptake capacities of the bi-composite are reported. Equilibrium studies were conducted between pH 5 and 9 and interpreted within the Langmuir, Freundlich and Dubinin-Radushkevich isotherm models.

Costs and cost scaling in time-domain integral-equation analysis of electromagnetic scattering

Computation of scattering from multi-wavelength bodies is expensive, and costs scale with up to the sixth power of the incident frequency. Conventional integral-equation time-domain methods have costs scaling with the fifth power. Here are described modifications to the IETD approach that offer the prospect of a reduction in cost scaling, to possibly the third power of frequency, and an associated large reduction in cost. The approach exploits the pulsed nature of the illumination, which results in surface fields that are small most of the time over most of the body, on bodies that are electrically large. Neglect of these produces some modest increase in error, but allows large reductions in cost and storage requirements. In the examples shown, cost reductions by amounts approaching two orders of magnitude are obtained, with the factor by which costs are reduced itself increasing with roughly the square of the body size; storage requirements are rendered essentially negligible.

ANALYSIS OF THREE-DIMENSIONAL TRANSIENT ACOUSTIC WAVE PROPAGATION USING THE BOUNDARY INTEGRAL EQUATION METHOD

This paper describes a boundary integral equation (boundary element) method for the solution of a variety of transient acoustic problems. The spatial and temporal discretization employs quadratic isoparametric elements with high-order Gauss quadrature, and the ensuing equations are implicit. The implicit formulation both eliminates the instabilities reported in explicit treatments, and permits a freedom of choice of timestep which can reduce costs dramatically. The accuracy of the approach is demonstrated by comparison with the analytical solution for a sphere. Results for more demanding sphere–cone–sphere geometries extending to seven wavelengths long are presented, and compared to those obtained from a related frequency domain approach.

Electromagnetic scattering from 3-D curved dielectric bodies using time-domain integral equations

A boundary integral equation (BIE) approach is developed to calculate transient scattering from dielectric bodies. The treatment is directly in terms of the E and H fields rather than magnetic and electric currents. It employs curvilinear (quadratic) modeling, which allows accurate representation of arbitrarily shaped curved bodies. The treatment is isoparametric with the same quadratic representation of the spatial field variation and with the temporal variation modeled by similar quadratic elements. Integration employs high-order Gaussian quadrature with special treatment of the singular and hypersingular integrals that arise. The treatment is implicit, requiring the solution of a sparse matrix equation at each timestep. This adds only trivially to the cost at each timestep and, by freeing the timestep from the constraint that it be smaller than the smallest nodal spacing, can greatly reduce the number of timesteps that must be employed. Additionally, it produces stable results without resort to the averaging processes proposed elsewhere. Example calculations of scattering from a sphere, a cube, and an almond are presented and compared with earlier published transient results and with results from a frequency domain treatment. Good agreement and improved accuracy is found.

An accurate method for the calculation of singular integrals arising in time-domain integral equation analysis of electromagnetic scattering

An accurate method for the evaluation of the Cauchy principal value integrals arising in time-domain electromagnetic wave scattering is presented. This is applied to a boundary integral equation (BIE) method employing quadratic curvilinear surface elements where such singularities do not vanish (as they generally do when simpler but less accurate discretizations are employed). The technique involves weakening the singularity in the original kernel to a degree where conventional integration methods may be employed and transforming the strong singularity to a line integral in a form which allows cancellation of its singular components. The effectiveness of the method is demonstrated by scattering calculations on a variety of targets and the computational cost of the approach is trivial.

Applications of differential forms to boundary integral equations

In this paper we discuss the application of differential forms to integral equations arising in the study of electromagnetic wave propagation. The usual Stratton-Chu integral equations are derived in terms of differential forms and corresponding Galerkin formulations are constructed. All numerical schemes require the specification of basis functions and the use of differential forms provides a very general method for the construction of arbitrary order basis functions on curvilinear geometries. It is noted that the lowest order approximations on flat geometries reduce to forms essential equivalent to the standard Rao-Wilton-Glisson functions. The effect on accuracy is investigated for electric field integral equation and magnetic field integral equation formulations for a range of bases. Hierarchical classes of functions are also developed, as are transition elements useful in p-adaptive schemes where variable orders of approximation are sought.

Parallelisation issues for high speed time domain integral equation analysis

Computation of scattering from multi-wavelength bodies presents large computational requirements. Here is discussed the parallelisation strategy required for effective exploitation of a novel fast algorithm for integral equation time domain (IETD) analyses. The development and examples are couched in terms of electromagnetic scattering and radar cross section evaluation.

High-Order Discrete Helmholtz Decompositions for the Electric Field Integral Equation

We develop a differential form based formalism to address the problem of low-frequency breakdown of the electric field integral equation (EFIE). Note, in this formalism we approximate the surface magnetic field, not the surface current as is conventionally the case. A discrete Helmholtz decomposition is achieved for both triangular and quadrilateral curvilinear meshes based on a star-cotree decomposition. These decompositions are based upon the construction of a canonical basis which ab-initio possess the required separation into irrotational and nonirrotational spaces. This makes the process of construction clear and generally applicable. The construction of appropriate bases is demonstrated for a range of interpolation orders. The effects of these constructions is demonstrated on a simple flat PEC plate problem

Extending integral equation time domain acoustic scattering analysis to larger problems

A method is presented to accelerate the execution of integral equation time domain analyses of exterior acoustic scattering problems. Conventionally, these have costs which scale with the fifth power of the frequency of the excitation, and practical limits to such computations are reached when bodies approach perhaps ∼5–10 wavelengths long. The fast approach presented is based on exploiting the pulsed nature of the illumination to omit much nugatory calculation. There is an associated slight accuracy loss; this is investigated. The method has costs which can scale with frequency to the power as low as ∼3, such that, for example, costs on a 18·5 wavelength body are reduced by a factor of about 28, with this factor itself increasing with roughly the square of the body size. Associated with the reduction in operations is a reduction in the scaling of storage required, from the third to the second power of frequency. Examples of analysis of large scatterers are presented, extending to a ∼22000 node ‘almond’. Copyright

A multilevel hierarchical preconditioner for the electric field integral equation

This paper considers the solution of the electric field integral equation (EFIE). Hierarchical conforming bases are developed which are subsequently used in the construction of a multilevel Schwarz type preconditioner. The effectiveness of this approach is demonstrated by the computation of scattering from a range of perfectly conducting objects including spheres, cubes, plates and dihedrals. The resulting schemes are shown to be faster than conventional schemes by approximately an order of magnitude.