IgA-BEM for 3D Helmholtz problems using conforming and non-conforming multi-patch discretizations and B-spline tailored numerical integration

An Isogeometric Boundary Element Method (IgA-BEM) is considered for the numerical solution of Helmholtz problems on 3D bounded or unbounded domains, admitting a smooth multi-patch representation of their finite boundary surface. The discretization spaces are formed by C0 inter-patch continuous functional spaces whose restriction to a patch simplifies to the span of tensor product B-splines composed with the given patch NURBS parameterization. Both conforming and non-conforming spaces are allowed, so that local refinement is possible at the patch level. For regular and singular integration, the proposed model utilizes a numerical procedure defined on the support of each trial B-spline function, which makes possible a function-by-function implementation of the matrix assembly phase. Spline quasi-interpolation is the common ingredient of all the considered quadrature rules; in the singular case it is combined with a B-spline recursion over the spline degree and with a singularity extraction technique, extended to the multi-patch setting for the first time. A threshold selection strategy is proposed to automatically distinguish between nearly singular and regular integrals. The non-conforming C0 joints between spline spaces on different patches are implemented as linear constraints based on knot removal conditions, and do not require a hierarchical master-slave relation between neighbouring patches. Numerical examples on relevant benchmarks show that the expected convergence orders are achieved with uniform discretization and a small number of uniformly spaced quadrature nodes

On the polarizability and capacitance of the cube

An efficient integral equation based solver is constructed for the electrostatic problem on domains with cuboidal inclusions. It can be used to compute the polarizability of a dielectric cube in a dielectric background medium at virtually every permittivity ratio for which it exists. For example, polarizabilities accurate to between five and ten digits are obtained (as complex limits) for negative permittivity ratios in minutes on a standard workstation. In passing, the capacitance of the unit cube is determined with unprecedented accuracy. With full rigor, we develop a natural mathematical framework suited for the study of the polarizability of Lipschitz domains. Several aspects of polarizabilities and their representing measures are clarified, including limiting behavior both when approaching the support of the measure and when deforming smooth domains into a non-smooth domain. The success of the mathematical theory is achieved through symmetrization arguments for layer potentials.Comment: 33 pages, 7 figure

Quadrature methods for 2D and 3D problems

AbstractIn this paper we give an overview on well-known stability and convergence results for simple quadrature methods based on low-order composite quadrature rules and applied to the numerical solution of integral equations over smooth manifolds. First, we explain the methods for the case of second-kind equations. Then we discuss what is known for the analysis of pseudodifferential equations. We explain why these simple methods are not recommended for integral equations over domains with dimension higher than one. Finally, for the solution of a two-dimensional singular integral equation, we prove a new result on a quadrature method based on product rules

Accurate 2D MoM technique for arbitrary dielectric, magnetic and conducting media applied to shielding problems

Calculating interaction integrals in a Method of Moments technique is highly challenging in a conductive medium. The specific form of its wave number leads to a strongly oscillating and exponentially damped Green's function, making standard numerical evaluation schemes inapt to accurately evaluate the interaction integrals. In this paper, we present an accurate 2D Method of Moments technique for arbitrary dielectric, magnetic and conducting media and apply the method to solve shielding problems

Square lattice Ising model susceptibility: Series expansion method and differential equation for χ(3)\chi^{(3)}

In a previous paper (J. Phys. A {\bf 37} (2004) 9651-9668) we have given the Fuchsian linear differential equation satisfied by $\chi^{(3)}$, the ``three-particle'' contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) allowing one to obtain long series in polynomial time. The method is based on series expansion in the variables that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the isotropic square lattice Ising model susceptibility $\chi$. The integration rules are straightforward due to remarkable formulas we derived for these variables. We obtain without any numerical approximation $\chi^{(3)}$ as a fully integrated series in the variable $w=s/2/(1+s^{2})$, where $s =sh (2K)$, with $K=J/kT$ the conventional Ising model coupling constant. We also give some perspectives and comments on these results.Comment: 28 pages, no figur

Gravitational Radiation from Two-Body Systems

Thanks to the new generation of gravitational wave detectors LIGO and VIRGO, the theory of general relativity will face new and important confrontations to observational data with unprecedented precision. Indeed the detection and analysis of the gravitational waves from compact binary star systems requires beforehand a very precise solution of the two-body problem within general relativity. The approximation currently used to solve this problem is the post-Newtonian one, and must be pushed to high order in order to describe with sufficient accuracy (given the sensitivity of the detectors) the inspiral phase of compact bodies, which immediately precedes their final merger. The resulting post-Newtonian ``templates'' are currently known to 3.5PN order, and are used for searching and deciphering the gravitational wave signals in VIRGO and LIGO.Comment: 19 pages, to appear in the Proceedings of the Spanish Relativity Meeting ``A Century of Relativity Physics'' (ERE05), Edited by Lysiane Mornas and Joaquin Diaz-Alons