Error estimates for numerical solutions of one- and two-dimensional integral equations

This thesis is concerned with the improvement of numerical methods, specifically boundaryelement methods (BEMs), for solving Fredholm integral equations in both one- and two dimensions. The improvements are based on novel (computer-algebra-based) error analyses that yield explicit forms of correction terms for a priori incorporation into BEM methods employing piecewise-polynomial interpolation in the numerical approximation. The work is motivated by the aim of reducing errors of BEM methods for low-degree interpolating polynomials, without significantly increasing the computational cost associated with higher-degree interpolation. The present thesis develops, implements and assesses improved BEMs on two fronts. First, a modified Nystr¨om method is developed for the solution of one-dimensional Fredholm integral equations of the second kind (FIE2s). The method is based upon optimal approximation and inclusion of an explicit form of orthogonal-polynomial integration error, and it can be extended to systems of integral equations. It is validated, in both the single and system cases, on challenging FIE2s that contain a finite number of (integrable) singularities, or points of limited differentiability, within the integral kernels. Second, BEMs are developed for solving two-dimensional FIE2s in the widely applicable context of harmonic boundary value problems in which the boundary conditions may be either continuous or discontinuous. In the latter case, modifying the BEM to conquer the adverse effect (on convergence with decreasing mesh size) caused by boundary singularities requires considerable additional theory and implementation; the motivation for doing so is that such singularities arise naturally in the modelling of, e.g., stress fractures in solid mechanics and dielectrics in electrostatics. For both non-singular and singular BVPs, standard BEMs are improved herein by optimal approximation and inclusion of explicit forms of Lagrange-interpolation integration errors. The modified BEMs are validated against pseudo-analytic results obtained by a conformal transformation method, for which a novel implementation of the inverse transformation (needed to recover the physical solution) is included explicitly by use of an algebraic manipulator. Through a set of test problems with known (or otherwise computable) solutions, both the one- and two-dimensional modified methods, for both regular and singular BVPs, are demonstrated to show marked improvements in performance over their unmodified counterparts

The nature of ordered structures in melt spun iron-silicon alloys

Increasing the silicon content is known to improve the magnetic properties of iron-silicon alloys (I). However, the associated loss of ductility has prevented the use of high silicon alloys in practical applications. The melt spinning of high silicon alloys can,produce wide tapes with reasonable ductility that are now regarded as one of the promising materials for magnetic applications (2-8). Investigations of the magnetic and mechanical properties consistently indicate a variation with silicon content (2- , 8-II). The best property combinations seem to exist for 6.5 wt % silicon alloy. Narita and Enokizono (10) have shown that in bulk Fe-Si alloys with high silicon contents, the B2 and DO3 ordering lead to different property changes and that cooling rate has major influence on magnetic properties. The magnetic and mechanical properties of rapidly solidified melt spun alloys are shown to be significantly different (3-5,7- 10-12). For example, the permeability of melt spun tapes is different for different alloys. It has a maximum value for a 6.5 wt % Si alloy which is different behavior from that observed in a well annealed sample (12). In order to understand the reasons for such behavior, a clear understanding of the structure is necessary. There exists considerable confusion in the literature regarding the structure of the melt spun alloys (3-5,7-8,13). For example, DO3 ordering is assumed to be predominant in melt spun 6.5 wt %'SJ alloy by many investigators (4-7). Chang et al. (5) have presented TEM evidence of it Jn the shape of a 010 superlattice spot observable Jn a [100] zone selected area diffraction pattern. However, this is not adequate to differentiate between the B2 and DO3 ordering. A different conclusion that the structure is a predominantly disordered phase (A2) with a very small amount of B2, was reported by Enokizono et al. (3). They have also reported the absence of domain structure. Guntherodt (7) and Warlimont (8) highlighted the suppression of the ordering reaction by melt spinning as the main reason for the improved ductility of the melt spun tapes

The Nature of Ordered Structures in Melt Spun Iron-Silicon Alloys

Increasing the silicon content is known to improve the magnetic properties of iron-silicon alloys (I). However, the associated loss of ductility has prevented the use of high silicon alloys in practical applications. The melt spinning of high silicon alloys can,produce wide tapes with reasonable ductility that are now regarded as one of the promising materials for magnetic applications (2-8). Investigations of the magnetic and mechanical properties consistently indicate a variation with silicon content (2- , 8-II). The best property combinations seem to exist for 6.5 wt % silicon alloy. Narita and Enokizono (10) have shown that in bulk Fe-Si alloys with high silicon contents, the B2 and $DO_3$ ordering lead to different property changes and that cooling rate has major influence on magnetic properties. The magnetic and mechanical properties of rapidly solidified melt spun alloys are shown to be significantly different (3-5,7- 10-12). For example, the permeability of melt spun tapes is different for different alloys. It has a maximum value for a 6.5 wt % Si alloy which is different behavior from that observed in a well annealed sample (12). In order to understand the reasons for such behavior, a clear understanding of the structure is necessary. There exists considerable confusion in the literature regarding the structure of the melt spun alloys (3-5,7-8,13). For example, $DO_3$ ordering is assumed to be predominant in melt spun 6.5 wt %'SJ alloy by many investigators (4-7). Chang et al. (5) have presented TEM evidence of it Jn the shape of a 010 superlattice spot observable Jn a [100] zone selected area diffraction pattern. However, this is not adequate to differentiate between the B2 and $DO_3$ ordering. A different conclusion that the structure is a predominantly disordered phase (A2) with a very small amount of B2, was reported by Enokizono et al. (3). They have also reported the absence of domain structure. Guntherodt (7) and Warlimont (8) highlighted the suppression of the ordering reaction by melt spinning as the main reason for the improved ductility of the melt spun tapes