Matthys M. Botha

Solving the volume integral equations of electromagnetic scattering

Time-harmonic electromagnetic scattering by inhomogeneous, three-dimensional structures within a free space environment can be described by electric- and magnetic field, volume integral equations involving the free space Green function. A comprehensive set of Galerkin projection formulations (also known as moment methods) for the numerical solution of these equations is presented, together with comparative numerical results. Such formulations are widely used for particle scattering analysis, optical near field calculation, etc. Results are obtained with higher-order, divergence-and curl-conforming basis functions on iso-parametric, tetrahedral meshes. The results demonstrate that all formulations converge with similar accuracy in the case of an analytically-solvable test problem. When modeling flux densities as solution variables, it is argued that solenoidal function spaces should be used, rather than the standard divergence-conforming function spaces; this assertion is supported by the results. Some of the formulations involve solving for curl-conforming fields; such fields can be discretized with fewer unknowns than divergence-conforming ones, implying lower computational costs. Additionally, some formulations yield system matrices which are approximately halfway sparse, meaning that computational costs will be down by a factor of 2 when iterative solvers are employed, which is the case for the widely-used fast methods.

On the variational formulation of hybrid finite element-boundary integral techniques for electromagnetic analysis

This paper addresses the time-harmonic, electromagnetic analysis of a three-dimensional inhomogeneous radiator/scatterer in free-space. Such analysis can be carried out by combining the finite element method (FEM) with the method of moments (MoM), which yields finite element-boundary integral (FE-BI) formulations. A general framework is presented, within which stationary FE-BI formulations can be established (variational boundary-value problems), which relate to equivalent, underlying variational principles (stationary functionals). The formulations are shown to be accurate, robust and computationally efficient. They avoid the problem of interior resonances without resorting to the combined field integral equation and they result in symmetric system matrices, which preserve reciprocity explicitly. Thus, the stationary FE-BI framework combines the FEM and MoM on the continuous level, as opposed to the usual approach of hybridization after discretization, which generally leads to asymmetric matrices. The stationary FE-BI framework allows one to solve either for the electric and magnetic fields on the volume of the problem domain, or for one volume and one exterior surface field quantity, with only marginal differences in computational cost. The volume-surface formulations have the same storage requirements as previous FE-BI formulations and can be more efficiently solved. The volume-volume formulations provide simultaneous solutions of the electric and magnetic fields, which could for instance be used to construct error estimators directly based on Maxwells equations.

A Family of Augmented Duffy Transformations for Near-Singularity Cancellation Quadrature

A new family of systematically constructed near-singularity cancellation transformations is presented, yielding quadrature rules for integrating near-singular kernels over triangular surfaces. This family results from a structured augmentation of the well-known Duffy transformation. The benefits of near-singularity cancellation quadrature are that no analytical integral evaluations are required and applicability in higher-order basis function and curvilinear settings. Six specific transformations are constructed for near-singularities of orders one, two and three. Two of these transformations are found to be equivalent to existing ones. The performance of the new schemes is thoroughly assessed and compared with that of existing schemes. Results for the gradient of the scalar Green function are also presented. For simplicity, static kernel results are shown. The new schemes are competitive with and in some cases superior to the existing schemes considered.

Rigorous, Auxiliary Variable-Based Implementation of a Second-Order ABC for the Vector FEM

The finite element method (FEM) is commonly used for electromagnetic radiation and scattering analysis. When an infinite, free space exterior domain needs to be incorporated into the method, a radiation boundary condition must be enforced. An approach which has received considerable attention, is to employ approximate conditions, known as absorbing boundary conditions (ABCs), that preserve the sparsity of the original FEM system upon discretization. In the case of time-harmonic analysis based on the vector wave equation in three dimensions, the symmetric, spherical Bayliss-Turkel-type ABCs of first- and second-orders are well-established. The second-order version is expected to be more accurate, however when using the standard curl-conforming approach to FEM discretization, an implementation difficulty is encountered, relating to successive derivatives being required of the nonconforming field components. This issue is addressed here by introducing a scheme where the nonconforming first-order derivatives are projected onto a suitably conforming auxiliary field, of which another derivative can then be taken instead. Additional computational costs are minimal and the scheme retains the symmetry of the standard formulation. Numerical results demonstrate the superior performance of the rigorously implemented second-order ABC over its first-order counterpart

Adaptive finite element-boundary integral analysis for electromagnetic fields in 3-D

This paper presents a complete adaptive finite element-boundary integral (FE-BI) analysis scheme for the time-harmonic, electromagnetic analysis of three-dimensional inhomogeneous scatterers/radiators in free-space. The adaptive scheme is based on an FE-BI formulation which yields electric and magnetic field solutions simultaneously. It employs a posteriori error estimates which exploit the availability of both field solutions and estimates error distributions and global solution quality for the electric and magnetic fields separately. It automatically determines which elements should be refined in order to equi-distribute the estimated error, based on the type of refinement requested (h,p or hp). This automatic determination is based on extrapolating the elemental error estimates. The algorithm terminates when specified tolerance levels are reached by the electric and/or magnetic field global solution quality estimates. The only required user specifications within the algorithm are the termination tolerances and the types of refinements to effect. Results are presented which show that within the scope of the presented error measures significant reductions in computational cost may be achieved. The proposed scheme could be used with other types of error estimates and it could be adapted to other FE or FE-BI formulations.

Accuracy of Near-Singularity Cancellation Quadrature Schemes for the Dynamic MoM Kernel

Near-singularity cancellation quadrature schemes are used for certain surface integrals in the method of moments (MoM). Schemes are typically derived for the static kernel, but then applied to the dynamic kernel in practice. In this letter, the effect upon quadrature error of adding the complex exponential factor to the integrand is evaluated for various schemes. Both theoretical and numerical results are presented. Integration of a weakly near-singular test kernel over a triangular surface is considered. It is found that a number of schemes can handle the dynamic factor well. The Radial-Angular-R1-Sqrt scheme is found to be the most efficient for the weak near-singularity. Results also show that for weak near-singularities, schemes tailored to strong near-singularities are less efficient.

An explicit a posteriori error indicator for electromagnetic, finite element-boundary integral analysis

Within the finite element method (FEM), there is always an error associated with the solution. Quantifying this error is a very important aspect of FE analysis, since it can increase confidence in a solution and can be used to drive adaptive refinements of the discretization, optimizing the quality of the solution relative to the number of degrees of freedom and in turn reducing computational cost. This paper presents an a posteriori error indicator for time-harmonic, electromagnetic analysis in three dimensions, when using a Galerkin FE approach with curl-conforming elements and including a nonlocal, boundary integral boundary condition as well as a dominant-mode port boundary condition. The error indicator is residual-based, of an explicit nature and constitutes a bound on an error energy measure. The error indicator is evaluated by presenting results relating to common microwave engineering applications, which shows that it can be used successfully to estimate the relative error distribution and that it can successfully serve within an adaptive algorithm.

The implicit, element residual method for a posteriori error estimation in FE-BI analysis

Hybrid finite element-boundary integral (FE-BI) formulations are widely used for electromagnetic analysis. In such analysis, estimation of the mesh-wide error distribution and global solution quality may be used to drive adaptive refinements. In this paper, an a posteriori error estimator based on the implicit, element residual method is constructed for the cavity FE-BI formulation, with dominant-mode coaxial port excitation. It is based on solving elemental variational boundary value problems of the error, using approximate, localized boundary conditions. Results show that the scheme can be applied to practical problems, yielding significant reductions in computational cost.

Bcl-2 confers survival in cisplatin treated cervical cancer cells: circumventing cisplatin dose-dependent toxicity and resistance

BackgroundCisplatin is the main chemotherapeutic drug for the treatment of cervical cancers, however resistance to cisplatin is increasingly common and therefore has limited the efficacy and use of this drug in the clinic. Dose-dependent toxicity poses an additional challenge since patients suffer long-term and often permanent side-effects after treatment. Bcl-2 up-regulation has been implicated in the resistance to cisplatin in a variety of cancer cell lines, however its role in cervical cancer is confounding.MethodsA low, non-cytotoxic concentration of cisplatin was used in the treatment of HeLa and CaSki cells. Bcl-2 expression was determined through Western blotting and immunocytochemistry before and after treatment with cisplatin. To assess the reliance of the cervical cancer cells on Bcl-2 in the presence of cisplatin, Bcl-2 knock-down was achieved through RNA interference, where after apoptosis was assessed through PARP cleavage (Western blotting), Caspase activity (Caspase-Glo©) and PI inclusion analysis (Flow cytometry). Finally, pre-malignant and malignant cervical tissue was analysed for the presence of Bcl-2 through Western blotting and immunofluorescence.ResultsCervical cancer cells upregulate Bcl-2 when treated with a non-cytotoxic concentration of cisplatin, which when silenced, effectively enhanced cisplatin sensitivity, and therefore significantly induced apoptosis. Analysis of the expression profile of Bcl-2 in cervical tissue revealed its up-regulation in cervical carcinoma, which agrees with results obtained from the in vitro data.ConclusionsOur data strongly suggest that utilising a lower dose of cisplatin is feasible when combined with Bcl-2 silencing as an adjuvant treatment, thereby improving both the dose-dependent toxicity, as well as cervical cancer resistance.

A nontoxic concentration of cisplatin induces autophagy in cervical cancer: selective cancer cell death with autophagy inhibition as an adjuvant treatment.

Background Increasing resistance to cisplatin as well as the severity of the adverse effects limit the use of this drug, particularly at high doses. Evidence has implicated the importance of autophagy in cancer resistance as well as the fact that various chemotherapy agents induce autophagy in cancer cells. We therefore aimed to first assess the role of autophagy in cisplatin treatment and second to assess whether a nontoxic concentration of cisplatin, together with autophagy inhibition, is able to maintain its cancer-specific cytotoxic action. Methods Three human cervical cell lines were used: a noncancerous ectocervical epithelial cell line (Ect1/E6E7) and 2 cancerous cervical cell lines (HeLa and CaSki). Autophagy was monitored through the presence of the classical protein markers LC-3 II and p62 under basal and treatment conditions, and inhibited using bafilomycin and autophagy protein 5 (ATG5) siRNA under treatment conditions. Cell death was analyzed through examination of the apoptotic markers PARP and caspase-3 through Western blotting, as well as the Caspase-Glo assay to confirm caspase-3/7 activity. Cervical biopsies were analyzed for the presence of LC-3 using Western blotting and immunofluorescence to determine if a correlation between autophagic levels and the progression of the disease exists. Results Cervical cancer cells exhibit increased basal autophagic levels in comparison to the noncancerous counterparts. Cisplatin treatment enhanced autophagic activity in all 3 cell lines. Inhibition of this autophagic response together with cisplatin treatment leads to significant increases in cancer cell death. Expression profiles of LC-3 in normal, premalignant (low-grade and high-grade squamous intraepithelial lesion), and cancerous cervical tissue revealed that autophagy is significantly up-regulated in HSILs and carcinoma cervical tissue, which emphasized the role of autophagy in the progression of the disease. Conclusions The inhibition of autophagy improves the cytotoxicity of a nontoxic concentration of cisplatin and provides a promising new avenue for the future treatment of cervical cancer.