An integral equation method for the electromagnetic scattering from cavities

Consider a time‐harmonic electromagnetic plane wave incident on a cavity in a ground plane. The physical process is modelled by Maxwell's equations. In this paper, integral representations of the solutions to the model problem in both fundamental polarizations are derived and studied. Existence and uniqueness of the solutions for the integral equations are established. The integral equations approach forms a basis for numerical solution of the model problem. In particular, for each fundamental polarization, an integral formulation with Gårding‐type estimates is derived. These formulations provide a basis for variational boundary element methods for solving the cavity problem. The Gårding‐type estimates imply convergence results for conforming boundary element methods. Copyright © 2000 John Wiley & Sons, Ltd     

Combined boundary integral equations for acoustic scattering‐resonance problems

In this paper, boundary integral formulations for a time‐harmonic acoustic scattering‐resonance problem are analyzed. The eigenvalues of eigenvalue problems resulting from boundary integral formulations for scattering‐resonance problems split in general into two parts. One part consists of scattering‐resonances, and the other one corresponds to eigenvalues of some Laplacian eigenvalue problem for the interior of the scatterer. The proposed combined boundary integral formulations enable a better separation of the unwanted spectrum from the scattering‐resonances, which allows in practical computations a reliable and simple identification of the scattering‐resonances in particular for non‐convex domains. The convergence of conforming Galerkin boundary element approximations for the combined boundary integral formulations of the resonance problem is shown in canonical trace spaces. Numerical experiments confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Modified boundary integral formulations for the Helmholtz equation

In this paper we describe some modified regularized boundary integral equations to solve the exterior boundary value problem for the Helmholtz equation with either Dirichlet or Neumann boundary conditions. We formulate combined boundary integral equations which are uniquely solvable for all wave numbers even for Lipschitz boundaries Γ=∂Ω. This approach extends and unifies existing regularized combined boundary integral formulations.     

Mixed FEM and BEM coupling for the three‐dimensional magnetostatic problem

We present two new mixed finite element methods coupled with a boundary method for the three dimensional magnetostatic problem. Such formulations are obtained by coupling a finite element method inside a bounded domain with a boundary integral method involving either the Calderon equations or the inverse of Dirichlet Neumann operator to treat the exterior domain. First, we present the formulations and then prove that our mixed formulations are well posed and that they lead to a convergent Galerkin method. Finally, we give numerical results for a sphere immersed in a homogeneous (source) field in the two formulations. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 443–462, 2003     

Boundary element methods—An overview

Variational methods for boundary integral equations deal with the weak formulations of boundary integral equations. Their numerical discretizations are known as the boundary element methods. This paper gives an overview of the method from both theoretical and numerical point of view. It summarizes the main results obtained by the author and his collaborators over the last 30 years. Fundamental theory and various applications will be illustrated through simple examples. Some numerical experiments in elasticity as well as in fluid mechanics will be included to demonstrate the efficiency of the methods.     

边界元法中奇异积分计算的极坐标变换法

在边界元法中,奇异积分的处理是一个极为引人注目的问题.本文提出了一种在单元状态作极坐标变换的新的处理方法,它能显式地消除奇异积分的奇异性,使之成为常规积分,因而易于在边界元法中使用高次单元.计算实例表明,本文所提出的方法是有效的、方便的.     

BEM with linear complexity for the classical boundary integral operators

Alternative representations of boundary integral operators corresponding to elliptic boundary value problems are developed as a starting point for numerical approximations as, e.g., Galerkin boundary elements including numerical quadrature and panel-clustering. These representations have the advantage that the integrands of the integral operators have a reduced singular behaviour allowing one to choose the order of the numerical approximations much lower than for the classical formulations. Low-order discretisations for the single layer integral equations as well as for the classical double layer potential and the hypersingular integral equation are considered. We will present fully discrete Galerkin boundary element methods where the storage amount and the CPU time grow only linearly with respect to the number of unknowns.

     

A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator

The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some appropriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.     

Weak Solutions of Fluid–Solid Interaction Problems

This paper is concerned with various variational formulations for the fluid–solid interaction problems. The basic approach here is a coupling of field and boundary integral equation methods. In particular, Gårding's inequalities are established in appropriate Sobolev spaces for all the formulations. Existence and uniqueness results of the corresponding weak solutions are given under suitable assumptions.     

HILBERT — a MATLAB implementation of adaptive 2D-BEM

We report on the Matlab program package HILBERT. It provides an easily-accessible implementation of lowest order adaptive Galerkin boundary element methods for the numerical solution of the Poisson equation in 2D. The library was designed to serve several purposes: The stable implementation of the integral operators may be used in research code. The framework of Matlab ensures usability in lectures on boundary element methods or scientific computing. Finally, we emphasize the use of adaptivity as general concept and for boundary element methods in particular. In this work, we summarize recent analytical results on adaptivity in the context of BEM and illustrate the use of HILBERT. Various benchmarks are performed to empirically analyze the performance of the proposed adaptive algorithms and to compare adaptive and uniform mesh-refinements. In particular, we do not only focus on mathematical convergence behavior but also on the usage of critical system resources such as memory consumption and computational time. In any case, the superiority of the proposed adaptive approach is empirically supported.     

Solving boundary value problems,integral, and integro-differential equations using Gegenbauer integration matrices

We introduce a hybrid Gegenbauer (ultraspherical) integration method (HGIM) for solving boundary value problems (BVPs), integral and integro-differential equations. The proposed approach recasts the original problems into their integral formulations, which are then discretized into linear systems of algebraic equations using Gegenbauer integration matrices (GIMs). The resulting linear systems are well-conditioned and can be easily solved using standard linear system solvers. A study on the error bounds of the proposed method is presented, and the spectral convergence is proven for two-point BVPs (TPBVPs). Comparisons with other competitive methods in the recent literature are included. The proposed method results in an efficient algorithm, and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems. The proposed method can be applied on a broad range of mathematical problems while producing highly accurate results. The developed numerical scheme provides a viable alternative to other solution methods when high-order approximations are required using only a relatively small number of solution nodes.     

Primal mixed formulations for the coupling of FEM and BEM. Part I: linear problems

We apply the boundary integral equation method and a primal mixed finite element approach to study the weak solvability and Galerkin approximations of linear interior transmission problems arising in potential theory and elastostatics. The existence and uniqueness of solution of the resulting weak formulations and of the associated discrete schemes are derived by using the classical theory for variational problems with constraints. Suitable finite element subspaces of Lagrange type satisfying the compatibility conditions are utilized for defining the Galerkin scheme. The error analysis and corresponding rates of convergence are also provided.     

Mixed boundary element solution for three-dimensional convection-diffusion problem with a velocity profile

A mixed boundary element formulation is presented for convection-diffusion problems with a velocity profile. In this formulation the convection-diffusion equation is considered as a nonlinear diffusion equation with inhomogeneous terms in which the convective term is involved additionally, because the spatial distribution of the drift velocity cannot be straightforwardly expressed in boundary integral form. Accordingly, a corresponding boundary integral equation may be described usually in the form of a so-called hybrid-type boundary integral equation.

In the present paper, mixed boundary elements are employed in a discrete model of the original convection-diffusion system. In the mixed element, potentials are approximated linearly, and their normal derivatives to boundaries are assumed constant. A simple iterative scheme is adopted in order to solve hybrid-type mixed boundary element equations. Simple three-dimensional models are dealt with in numerical experiments. The proposed approach gives more accurate and stable solutions compared with constant boundary elements which have been reported.     

Discrete finite element characterizations of source fields for volume and boundary constraints in electromagnetic problems

The consideration of electromagnetic field sources in potential formulations necessitates the definition of source fields. Such source fields are first defined for both volume and boundary constraints in static electromagnetic models. Then, automatic procedures are proposed to conveniently and efficiently characterize discrete source fields, with regard to their use in finite element formulations, their supports, their direct expression requiring no pre-computation, and their associated constraints. Two application examples are proposed to illustrate the approach.     

Iterative solvers and stabilisation for mixed electrostatic and magnetostatic formulations

Mixed electrostatic and magnetostatic finite element formulations are considered. Solution methods for the resulting indefinite algebraic systems are investigated. Methods developed for the mixed formulations of the Stokes equations are modified in order to apply to the Maxwell equations: an efficient block preconditioner is proposed and a stabilised formulation is described. The different methods are applied to 2D and 3D examples.     

Radiation and Scattering by Complex Conformal Antennas on a Circular Cylinder

A technique for characterizing and designing complex conformal antennas flush-mounted on a singly-curved surface is presented. This approach is based on the hybrid finite element–boundary integral (FE–BI) method. A related method was proposed in the past utilizing cylindrical-shell finite element and roof-top rectangular basis functions for the boundary integral. Although that method proved very powerful for analyzing cylindrical–rectangular patch arrays flush-mounted to a circular cylinder, the requirement for uniform meshing in the aperture ultimately limited its usefulness. In this present formulation, tetrahedral elements are used to expand the volumetric electric fields while similar basis functions are used for the boundary integral. The curvature of the aperture is explicitly included via the use of the circular cylinder dyadic Green's function. After presentation of the formulation and validation using several well-understood examples, an example is presented that illustrates the capabilities of this method for modeling complex conformal antennas heretofore not examined by rigorous methods due to inherent limitations of the various published methods.     

A Symmetric Galerkin Boundary Element Method for Linear Poroelasticity

In linear poroelasticity so far only collocation boundary element methods have been available. However, in some applications, e.g., when coupling with finite elements is desired, a symmetric formulation is preferable. Choosing a Galerkin approach which involves the second boundary integral equation, such a formulation is possible. Here, a previously presented integration by part technique for the regularization of the first boundary integral equation is extended to the second boundary integral equation as well. While the weakly singular representation of the double layer operator has been presented before, the emphasis lies here on the so called hyper-singular boundary integral operator. Due to the regularization, this operator can be evaluated numerically and, hence, be used within a numerical scheme for the first time. Different numerical studies will be presented to show the behavior of the established symmetric Galerkin boundary element method, also comparing it with collocation boundary element methods. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Indirect boundary integral formulation for the solution of shear‐thinning flow inside single rotor mixers

An efficient indirect boundary integral formulation for the evaluation of inelastic non‐Newtonian shear‐thinning flows at low Reynolds number is presented in this article. The formulation is based on the solution of a homogeneous Stokes flow field and the use of a particular solution for the nonlinear non‐Newtonian terms that yields the complete solution to the problem. Matrix multiplications are reduced in comparison to other means of handling nonlinear terms in boundary integral formulations such as the dual reciprocity method. The iterative solution of the nonlinear system of equations has been performed with a modified Newton‐Raphson method obtaining accurate results for values of the power law index as low as 0.4 without domain partitioning. Geometries such as Couette flow and a typical industrial polymer mixer have been analyzed with the proposed method obtaining good results with a reduction in computational cost compared with other equivalent formulations. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27:1610–1627, 2011     

Potential-based numerical solution of Dirichlet problems for the Helmholtz equation

Three-dimensional Dirichlet problems for the Helmholtz equation are considered in generalized formulations. By applying single-layer potentials, they are reduced to Fredholm boundary integral equations of the first kind. The equations are discretized using a special averaging method for integral operators with weak singularities in the kernels. As a result, the integral equations are approximated by systems of linear algebraic equations with easy-to-compute coefficients, which are solved numerically by applying the generalized minimal residual method. A modification of the method is proposed that yields solutions in the spectra of interior Dirichlet problems and integral operators when the integral equations are not equivalent to the original differential problems and are not well-posed. Numerical results are presented for assessing the capabilities of the approach.     

A symmetric boundary element method for the Stokes problem in multiple connected domains

We consider a symmetric Galerkin boundary element method for the Stokes problem with general boundary conditions including slip conditions. The boundary value problem is reformulated as Steklov–Poincaré boundary integral equation which is then solved by a standard approximation scheme. An essential tool in our approach is the invertibility of the single layer potential which requires the definition of appropriate factor spaces due to the topology of the domain. Here we describe a modified boundary element approach to solve Dirichlet boundary value problems in multiple connected domains. A suitable extension of the standard single layer potential leads to an operator which is elliptic on the original function space. Copyright © 2003 John Wiley & Sons, Ltd.