Method of lines transpose: an efficient A-stable solver for wave propagation

Building upon recent results obtained in [7,8,9], we describe an efficient second order, A-stable scheme for solving the wave equation, based on the method of lines transpose (MOLT), and the resulting semi-discrete (i.e. continuous in space) boundary value problem. In [7], A-stable schemes of high order were derived, and in [9] a high order, fast O(N) spatial solver was derived, which is matrix-free and is based on dimensional-splitting. In this work, are interested in building a wave solver, and our main concern is the development of boundary conditions. We demonstrate all desired boundary conditions for a wave solver, including outflow boundary conditions, in 1D and 2D. The scheme works in a logically Cartesian fashion, and the boundary points are embedded into the regular mesh, without incurring stability restrictions, so that boundary conditions are imposed without any reduction in the order of accuracy. We demonstrate how the embedded boundary approach works in the cases of Dirichlet and Neumann boundary conditions. Further, we develop outflow and periodic boundary conditions for the MOLT formulation. Our solver is designed to couple with particle codes, and so special attention is also paid to the implementation of point sources, and soft sources which can be used to launch waves into waveguides

Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University

This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield

Fundamental solution based numerical methods for three dimensional problems: efficient treatments of inhomogeneous terms and hypersingular integrals

In recent years, fundamental solution based numerical methods including the meshless method of fundamental solutions (MFS), the boundary element method (BEM) and the hybrid fundamental solution based finite element method (HFS-FEM) have become popular for solving complex engineering problems. The application of such fundamental solutions is capable of reducing computation requirements by simplifying the domain integral to the boundary integral for the homogeneous partial differential equations. The resulting weak formulations, which are of lower dimensions, are often more computationally competitive than conventional domain-type numerical methods such as the finite element method (FEM) and the finite difference method (FDM). In the case of inhomogeneous partial differential equations arising from transient problems or problems involving body forces, the domain integral related to the inhomogeneous solutions term will need to be integrated over the interior domain, which risks losing the competitive edge over the FEM or FDM. To overcome this, a particular treatment to the inhomogeneous term is needed in the solution procedure so that the integral equation can be defined for the boundary. In practice, particular solutions in approximated form are usually applied rather than the closed form solutions, due to their robustness and readiness. Moreover, special numerical treatment may be required when evaluating stress directly on the domain surface which may give rise to hypersingular integral formulation. This thesis will discuss how the MFS and the BEM can be applied to the three-dimensional elastic problems subjected to body forces by introducing the compactly supported radial basis functions in addition to the efficient treatment of hypersingular surface integrals. The present meshless approach with the MFS and the compactly supported radial basis functions is later extended to solve transient and coupled problems for three-dimensional porous media simulation

Numerical methods for computing Casimir interactions

We review several different approaches for computing Casimir forces and related fluctuation-induced interactions between bodies of arbitrary shapes and materials. The relationships between this problem and well known computational techniques from classical electromagnetism are emphasized. We also review the basic principles of standard computational methods, categorizing them according to three criteria---choice of problem, basis, and solution technique---that can be used to classify proposals for the Casimir problem as well. In this way, mature classical methods can be exploited to model Casimir physics, with a few important modifications.Comment: 46 pages, 142 references, 5 figures. To appear in upcoming Lecture Notes in Physics book on Casimir Physic

International Workshop on Finite Elements for Microwave Engineering

When Courant prepared the text of his 1942 address to the American Mathematical Society for publication, he added a two-page Appendix to illustrate how the variational methods first described by Lord Rayleigh could be put to wider use in potential theory. Choosing piecewise-linear approximants on a set of triangles which he called elements, he dashed off a couple of two-dimensional examples and the finite element method was born. … Finite element activity in electrical engineering began in earnest about 1968-1969. A paper on waveguide analysis was published in Alta Frequenza in early 1969, giving the details of a finite element formulation of the classical hollow waveguide problem. It was followed by a rapid succession of papers on magnetic fields in saturable materials, dielectric loaded waveguides, and other well-known boundary value problems of electromagnetics. … In the decade of the eighties, finite element methods spread quickly. In several technical areas, they assumed a dominant role in field problems. P.P. Silvester, San Miniato (PI), Italy, 1992 Early in the nineties the International Workshop on Finite Elements for Microwave Engineering started. This volume contains the history of the Workshop and the Proceedings of the 13th edition, Florence (Italy), 2016 . The 14th Workshop will be in Cartagena (Colombia), 2018