A fast and well-conditioned spectral method for singular integral equations

We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in ${\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is the bandwidth and $n$ is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to ${\cal O}(m n)$ operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface

Eliminating artificial boundary conditions in time-dependent density functional theory using Fourier contour deformation

We present an efficient method for propagating the time-dependent Kohn-Sham equations in free space, based on the recently introduced Fourier contour deformation (FCD) approach. For potentials which are constant outside a bounded domain, FCD yields a high-order accurate numerical solution of the time-dependent Schrodinger equation directly in free space, without the need for artificial boundary conditions. Of the many existing artificial boundary condition schemes, FCD is most similar to an exact nonlocal transparent boundary condition, but it works directly on Cartesian grids in any dimension, and runs on top of the fast Fourier transform rather than fast algorithms for the application of nonlocal history integral operators. We adapt FCD to time-dependent density functional theory (TDDFT), and describe a simple algorithm to smoothly and automatically truncate long-range Coulomb-like potentials to a time-dependent constant outside of a bounded domain of interest, so that FCD can be used. This approach eliminates errors originating from the use of artificial boundary conditions, leaving only the error of the potential truncation, which is controlled and can be systematically reduced. The method enables accurate simulations of ultrastrong nonlinear electronic processes in molecular complexes in which the inteference between bound and continuum states is of paramount importance. We demonstrate results for many-electron TDDFT calculations of absorption and strong field photoelectron spectra for one and two-dimensional models, and observe a significant reduction in the size of the computational domain required to achieve high quality results, as compared with the popular method of complex absorbing potentials

Eliminating artificial boundary conditions in time-dependent density functional theory using Fourier contour deformation

We present an efficient method for propagating the time-dependent Kohn-Sham equations in free space, based on the recently introduced Fourier contour deformation (FCD) approach. For potentials which are constant outside a bounded domain, FCD yields a high-order accurate numerical solution of the time-dependent Schrodinger equation directly in free space, without the need for artificial boundary conditions. Of the many existing artificial boundary condition schemes, FCD is most similar to an exact nonlocal transparent boundary condition, but it works directly on Cartesian grids in any dimension, and runs on top of the fast Fourier transform rather than fast algorithms for the application of nonlocal history integral operators. We adapt FCD to time-dependent density functional theory (TDDFT), and describe a simple algorithm to smoothly and automatically truncate long-range Coulomb-like potentials to a time-dependent constant outside of a bounded domain of interest, so that FCD can be used. This approach eliminates errors originating from the use of artificial boundary conditions, leaving only the error of the potential truncation, which is controlled and can be systematically reduced. The method enables accurate simulations of ultrastrong nonlinear electronic processes in molecular complexes in which the inteference between bound and continuum states is of paramount importance. We demonstrate results for many-electron TDDFT calculations of absorption and strong field photoelectron spectra for one and two-dimensional models, and observe a significant reduction in the size of the computational domain required to achieve high quality results, as compared with the popular method of complex absorbing potentials

3D Capacitance Extraction With the Method of Moments

In this thesis, the Method of Moments has been applied to calculate capacitance between two arbitrary 3D metal conductors or a capacitance matrix for a 3D multi-conductor system. Capacitance extraction has found extensive use for systems involving sets of long par- allel transmission lines in multi-dielectric environment as well as integrated circuit package including three-dimensional conductors located on parallel planes. This paper starts by reviewing fundamental aspects of transient electro-magnetics followed by the governing dif- ferential and integral equations to motivate the application of numerical methods as Method of Moments(MoM), Finite Element Method(FEM), etc. Among these numerical tools, the surface-based integral-equation methodology - MoM is ideally suited to address the prob- lem. It leads to a well-conditioned system with reduced size, as compared to volumetric methods. In this dissertation, the MoM Surface Integral Equation (SIE)-based modeling approach is developed to realize electrostatic capacitance extraction for 3D geometry. MAT- LAB is employed to validate its e?ciency and e?ectiveness along with design of a friendly GUI. As a base example, a parallel-plate capacitor is considered. We evaluate the accu- racy of the method by comparison with FEM simulations as well as the corresponding quasi-analytical solution. We apply this method to the parallel-plate square capacitor and demonstrate how far could the undergraduate result 0C = A ? =d\u27 be from reality. For the completion of the solver, the same method is applied to the calculation of line capacitance for two- and multi-conductor 2D transmission lines

A framework for mathematics curricula in engineering education: a report of the mathematics working group.

This document adapts the competence concept to the mathematical education of engineers and explains and illustrates it by giving examples. It also provides information for specifying the extent to which a competency should be acquired. It does not prescribe a particular level of progress for competence acquisition in engineering education. There are many different engineering branches and many different job profiles with various needs for mathematical competencies; consequently it is not appropriate to specify a fixed profile. The competence framework serves as an analytical framework for thinking about the current state in one’s own institution and also as a design framework for specifying the intended profile. A sketch of an example profile for a practice-oriented study course in mechanical engineering is given in the document. This document retains the list of content-related learning outcomes (slightly modified) that formed the ‘kernel’ of the previous curriculum document. These are still important because lecturers teaching application subjects want to be sure that students have at least an ‘initial familiarity’ with certain mathematical concepts and procedures which they need in their application modelling. In order to offer helpful orientation for designing teaching processes, teaching and learning environments and approaches are outlined which help students to obtain the competencies to an adequate degree. It is clear that such competencies cannot be obtained by simply listening to lectures, so adequate forms of active involvement of students need to be included. Moreover, in a competence-based approach the mathematical education must be integrated in the surrounding engineering study course to really achieve the ability to use mathematics in engineering contexts. The document presents several forms of how this integration can be realized. This integration is essential to the development of competencies and will require close co-operation between mathematics academics and their engineering counterparts. Finally, since assessment procedures determine to a great extent the behaviour of students, it is extremely important to address competency acquisition in assessment schemes. Ideas for doing this are also outlined in the document. The main purpose of this document is to provide orientation for those who set up concrete mathematics curricula for their specific engineering programme, and for lecturers who think about learning and assessment arrangements for achieving the intended level of competence acquisition. It also serves as a framework for the group’s future work and discussions

Differential/Difference Equations

The study of oscillatory phenomena is an important part of the theory of differential equations. Oscillations naturally occur in virtually every area of applied science including, e.g., mechanics, electrical, radio engineering, and vibrotechnics. This Special Issue includes 19 high-quality papers with original research results in theoretical research, and recent progress in the study of applied problems in science and technology. This Special Issue brought together mathematicians with physicists, engineers, as well as other scientists. Topics covered in this issue: Oscillation theory; Differential/difference equations; Partial differential equations; Dynamical systems; Fractional calculus; Delays; Mathematical modeling and oscillations