Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems.

Isogeometric analysis is a topic of considerable interest in the field of numerical analysis. The boundary element method (BEM) requires only the bounding surface of geometries to be described; this makes non-uniform rational B-splines (NURBS), which commonly describe the bounding curve or surface of geometries in CAD software, appear to be a natural tool for the approach. This isogeometric analysis BEM (IGABEM) provides accuracy benefits over conventional BEM schemes due to the analytical geometry provided by NURBS. When applied to wave problems, it has been shown that enriching BEM approximations with a partition-of-unity basis, in what has become known as the PU-BEM, allows highly accurate solutions to be obtained with a much reduced number of degrees of freedom. This paper combines these approaches and presents an extended isogeometric BEM (XIBEM) which uses partition-of-unity enriched NURBS functions; this new approach provides benefits which surpass those of both the IGABEM and the PU-BEM. Two numerical examples are given: a single scattering cylinder and a multiple-scatterer made up of two capsules and a cylinder

The Extended Isogeometric Boundary Element Method (XIBEM): An Enriched Collocation BEM for Wave Scattering Analysis

Isogeometric analysis [1] is an increasingly popular research topic. By using the functions that describe geometries in computer-aided design (CAD) directly in numerical analysis, it oers the possibility of running engineering simulations without the need for meshing. As the functions in CAD only describe the bounding surface of geometries, the boundary element method (BEM) is the ideal tool for isogeometric analysis. This combination provides solutions of higher accuracy than the conventional BEM due the exact geometry being utilised [2]. In this work, we make use of an isogeometric BEM to nd solutions to frequency-domain wave problems governed by the Helmholtz equation. We extend this method by using the partition-of-unity approach [3], in which the approximating functions are multiplied by families of plane waves. We call this plane-wave enriched approach the eXtended Isogeometric Boundary Element Method (XIBEM). When compared to conventional BEM and isogeometric BEM schemes, it requires signicantly fewer equations to provide a prescribed accuracy of solution for a given frequency and geometry of problem. An overview of the method will be presented with supporting numerical results

Extended isogeometric boundary element method (XIBEM) for acoustic wave scattering problems

Isogeometric analysis is the concept of using the same functions that describe a geometry in computer-aided design to approximate unknown elds in numerical simulations. This has become a topic of considerable interest to the boundary integral methods community. This paper introduces an eXtended Isogeometric Boundary Element Method (XIBEM), in which isogeometric functions approximating wave potential are enriched using the partition-of-unity method. In this new method, the isogeometric basis is formed from a space of non-uniform rational B-spline (NURBS) functions multiplied by families of plane waves. Using numerical examples, it is shown that this reduces the total number of equations that need to be solved for a given frequency and geometry of problem; this improves the accuracy of and extends the supported frequency range of the boundary element method to include short wave diraction problems

Novel basis functions for the partition of unity boundary element method for 2D Helmholtz problems

The boundary element method (BEM) is a popular tool for wave scattering problems. To reduce the number of degrees of freedom required, the partition of unity BEM (PUBEM) was developed in which the approximation space is enriched with a linear combination of plane-waves. Recent work has shown that the element ends are more susceptible to errors in the approximation than the mid-element regions. In this paper we propose that this is due to the reduced order of continuity in the Lagrangian shape function component of the basis functions. It will demonstrated that choosing trigonometric shapes functions, rather than classical quadratic shape functions, provides accuracy benefits

The equal spacing of N points on a sphere with application to partition-of-unity wave diffraction problems

This paper addresses applications involving the selection of a set of points on a sphere, in which the uniformity of spacing can be of importance in enhancing the computational performance and/or the accuracy of some simulation. For the authors, the motivation for this work arises from the need to specify wave directions in a partition-of-unity approach for numerical analysis of wave diffraction problems. A new spacing method is presented, based on a physical analogy in which an arbitrary number of charged particles are held in static equilibrium on a spherical surface. The new method, referred to in this paper as the Coulomb force method, offers an improvement over simpler methods, e.g., latitude/longitude and discretised cube methods, in terms of both the uniformity of spacing and the arbitrary nature of the number of points N that can be considered. A simple extension to the algorithm allows points to be biased towards a direction of choice. Numerical results of a wave scattering problem solved with a partition-of-unity boundary element method demonstrate the benefits of the algorithm

Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems

A boundary element method (BEM), based on non-uniform rational B-splines (NURBS), is used to find solutions to three-dimensional wave scattering problems governed by the Helmholtz equation. The method is extended in a partition-of-unity sense, multiplying the NURBS functions by families of plane waves; this method is called the eXtended Isogeometric Boundary Element Method (XIBEM). In this paper, the collocation XIBEM formulation is described and numerical results are given. The numerical results are compared against closed-form or converged solutions. Comparisons are made against the conventional boundary element method and the non-enriched isogeometric BEM (IGABEM). When compared to non-enriched boundary element simulations, using XIBEM significantly reduces the number of degrees of freedom required to obtain a solution of a given error; thus, with a fixed computational resource, problems of a shorter wavelength can be solved

Isogeometric partition-of-unity boundary integral method for acoustic wave scattering problems

Isogeometric analysis is the concept of using the same functions that describe a geometry in computer-aided design to approximate unknown elds in numerical simulations. This has become a topic of considerable interest to the boundary integral methods community. This paper introduces an eXtended Isogeometric Boundary Element Method (XIBEM), in which isogeometric functions approximating wave potential are enriched using the partition-of-unity method. In this new method, the isogeometric basis is formed from a space of non-uniform rational B-spline (NURBS) functions multiplied by families of plane waves. Using numerical examples, it is shown that this reduces the total number of equations that need to be solved for a given frequency and geometry of problem; this improves the accuracy of and extends the supported frequency range of the boundary element method to include short wave diraction problems