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IEEE TRANSACTIONS ON MAGNETICS, VOL. XX, NO. Y, MONTH, 20031Development of Hybrid BoundaryIntegral-Generalized (Partition of Unity)Finite Element Solvers for the ScalarHelmholtz EquationC. Lu and B. Shanker2120 Engr. Bldg., Dept. Elec. & Comp. Engr.,Michigan State University, East Lansing, MI 48824AbstractFinite element based techniques have been one of the most popular methods used to model electromagnetic field behavior.Typically, the methods invoked involve representing the function of interest in terms of basis functions whose support is definedover a simplicial mesh. Since its introduction, the technique has seen steady improvement. The method now boasts of adaptiveh-p refinement. Recently, however, Babuska and his colleagues introduced the notion of “meshless finite elements” or theso-called Generalized Finite Element method (GFEM). This method does not rely on an underlying tesselation and admitsa larger class of basis functions. Application of this technique to analyze practical problems has largely been restricted tosystems that require the solution to the Poisson equation. Investigation of the applicability of this technique to wave scatteringproblems has been limited. The principal difficulty in analyzing scattering and radiation problems using this technique liesin developing appropriate boundary conditions to truncate the computational domain. Our method of choice is to imposeexact boundary conditions using boundary integrals–this is largely governed by the fact that these may be conformal to thesurface, and can be readily accelerated using existing fast solvers. We will thoroughly explore the applicability of this techniqueto two-dimensional electromagnetic systems. In doing so, we will explore the methods necessary to impose various boundaryconditions. This analysis will give us the background necessary to develop the framework for hybridizing boundary integraltechniques with GFEM, thus imposing an exact radiation boundary condition. The results obtained using this hybrid code willbe first validated against analytical data for a range of scenarios. To further validate the proposed approach for more complexscatterers, we integrate GFEM with perfectly matched layers, and compare results obtained for a complex scatterer. Severalresults that demonstrate the accuracy of the proposed method will also be presented.KeywordsGeneralized Finite Elements, Meshless, hp-adaptive, Boundary IntegralI. IntroductionThe state of art of FEM tools for electromagnetic analysis has grown by leaps and bounds over the past fewdecades [1]. Classical methods require an underlying tesselation on which basis functions are defined. Thesebasis functions are based on a span of polynomials, have finite support, and obey conditions at inter-elementboundaries. For instance, Whitney elements that are typically used in computational electromagnetics

IEEE TRANSACTIONS ON MAGNETICS, VOL. XX, NO. Y, MONTH, 20032satisfy either tangential or normal continuity across inter-element boundaries. A genre of higher order basisfunctions have also been presented and have been continually refined [2, 3]. Likewise, basis functions thatcan be used in a h, p-convergence setting have been presented and applied to several engineering problems.Indeed, in a series of excellent papers, it has been shown that it is possible to have true h, p–convergence [4,5].Classical FEM schemes require a simplicial structure that satisfies certain aspect ratios to define thespan of polynomials. In many situations, creating such a mesh can be laborious and time consuming. Forinstance, developing h, p-convergence meshes or analyzing time varying phenomena that requires re-meshingat every time instant can be laborious. Another handicap of classical FEM is that the ansatz space usedto approximate the local behavior is a span of polynomials. If analytic local behavior is known, then itmight be possible to use functions other than polynomials to approximate local behavior. The developmentof “meshless FEM” was motivated by the need to address these possible improvements. Intuitively, thesemethods work as follows: the domain being considered is partitioned into a union of patches or a “partitionof unity,” and on these patches, the local approximation is constructed using a span of functions [6]. Thus,the representation of the function is achieved via two functions; one that is defined on the partitions ofunity and the other on each of the patches. The basis functions describing the unknowns inherits the higherorder nature of approximation from the local basis functions and the smoothness of the functions definedon the partition of unity. As with classical FEM, using a span of different local approximations in differentregions is also possible. Thus, the meshless methods retain several features of classical FEM and provideadditional flexibility in terms of functions that are used and obfuscating the need for a simplical partitionof the domain.Several flavors of meshless methods exist; the principal difference between these lie in the manner in whichthe local approximations and functions on patches are specified. However, it has been shown that most ofthe these methods fit into the framework of the Partition of Unity FEM (PUFEM) or Generalized FEM(GFEM) introduced by Babuska and his colleagues [7, 8]. In what follows, we will use the term GFEM todescribe methods to be developed herein. The mathematical foundations of this algorithm have been laidout in great detail [7–9], and it has been shown that h-, p- and hp-adaptivity is easily achieved. Likewise,the efficacy of using a space of harmonic functions as local approximants have been demonstrated [9].The application of this technique to solving problems in electromagnetics has not been extensive. Principally (this is not a complete list), research has been conducted by [10–21]. To a large extent, our work in thisarea revolved around developing meshless methods for solving the diffusion equation in both the frequencyand time domain as applied to non-destructive evaluation [11–13]. The basis functions used in this analysisrelied on the element free Galerkin method (EFG) [22]. The method, particularly its scalar implementation, has been gaining a foothold in the research landscape insofar as application to magnetic field analysis.Recently, we have explored the viability of suitably modifying the EFG method to enable the analysis of

IEEE TRANSACTIONS ON MAGNETICS, VOL. XX, NO. Y, MONTH, 20033vector fields [15] and have developed meshless-PMLs to enable the analysis of open region problems [23]within the context of the EFG method. However, while the EFG method can be thought of as a subset ofGFEM, it does not lend itself readily to hp–adaptivity, whereas this is inherent in other GFEM methods.Having a proper understanding of the sources of error and the means though which one may control themis important. We have found that in most of the implementation, reason that the convergence is not of thesame order of the underlying scheme is largely due to the improper imposition of boundary conditions.Proper imposition of boundary conditions within a meshless scheme is challenging. Unlike classical FEM,the space of approximating functions are not interpolatory. This poses severe challenges in imposing Dirichletboundary conditions. More specifically, one cannot use the Lagrange multiplier technique as the approximation spaces have to obey the inf-sup condition, and it is not always possible to construct such spaces formeshless methods. This deficiency is directly linked to the difficulty in truncating the computational domainusing boundary integrals. This is due to two facts; (i) to solve the hybrid problem, one typically defines anauxiliary set of basis functions and unknowns to represent the tangential components of the fields [1]. Itimplies from the above discussion [24] that these basis functions, together with those used in the interior,should satisfy the inf-sup (Babuska-Brezzi) condition, and (ii) there are practical situations wherein oneuses a first kind Fredholm integral equation as the boundary is not closed. In these cases, the BI enforces aDirichlet type condition. Thus, the principal contribution of this paper is four fold:1. We present a scheme for implementing GFEM for the Helmholtz equation.2. We develop an adaptation of Nitsche’s method for implementing the Dirichlet boundary condition.3. We develop the hybrid GFEM-BI technique for domain truncation for both open and closed domains.4. We develop the methodology wherein local boundary conditions can be integrated with GFEM–morespecifically, the perfectly matched layer (PML). The development of this technique is a by-product of theneed to have an additional modality of validating the results obtained by the GFEM-BI scheme.Our ultimate aim is to develop a three-dimension vector solver for analyzing electromagnetics. The rationalefor embarking upon this specific problem is as follows: (a) This approach presented in this paper (basisfunctions/means to impose boundary conditions, etc) can readily used for solution of quasi-static electromagnetic phenomena and scalar wave equations; (b) It permits us to work out several mathematical andnumerical hurdles–the principal being the application of Dirichlet boundary condition and the accurate evaluation of integrals; (c) it is equally important to understand the manner in which boundary integral basedtechniques can be hybridized with this scheme. The advantage of hybridizing GFEM with BI is readily apparent; it imposes an exact boundary condition for open domain problems, and the computational cost canbe amortized using recent advances in the integral equation technology namely the fast multipole methodor a host of FFT-based schemes. This work has provided the mathematical basis for our subsequent workon three-dimensional vector solvers for electromagnetics problems. Our preliminary work in this area waspresented recently [?], and a paper is being prepared for submission.

IEEE TRANSACTIONS ON MAGNETICS, VOL. XX, NO. Y, MONTH, 20034This paper proceeds along the following lines; in the next section, we formulate the problem in detail. Herewe introduce the concepts of GFEM, discretization of the domain, basis functions, methods for integration,and methods for implementing various boundary condition. The last includes different types of boundaryintegral techniques and a local absorbing boundary condition. Next, we demonstrate the accuracy andconvergence of the GFEM and GFEM-BI via a series of analytical comparisons. We shall also demonstratethe accuracy of this scheme by comparing the results obtained against those obtained by truncating thedomain using a PML as an absorbing boundary condition. Finally, the paper will conclude with directionson our future research.II. FormulationConsider a multiply connected domain Ω whose interior boundaries are denoted by Ω : Γ SiΓi . It isassumed that this domain is embedded in a domain Ωe and its exterior boundary Γe is defined as Γe : Ωe Ω̄.Interior to the domain Ω, the function u(x) satisfies¡ · [α(x) ] ω 2 γ(x) u(x) f (x)Bi {u(x)} gi (x) for x Γi(1)Be {u(x)} ge (x) for x ΓeIn the above equations it is assumed that x Rd , Be and Bi are differential operators, and gi (x) is thefunction that is imposed on Γi . Here, d 2, 3, α(x) and γ(x) are material parameters. The function ofinterest u(x) is used to denote the ẑ component of either the electric or magnetic field. The rationale ofdefining Γe explicitly is to impose appropriate boundary conditions that enable the analysis of scatteringproblems. The parameters α(x) and β(x) can stand for either the permittivity or permeability, dependingon the variable that u(x) represents. Solution to this problem using the standard finite element methodrequires an underlying tesselation on which basis functions are defined. These basis functions are based ona span of polynomials, have finite support, and obey conditions at inter-element boundaries. For instance,Whitney elements that are typically used in computational electromagnetics satisfy either tangential ornormal continuity across inter-element boundaries. Higher order basis functions based on these elementshave also been presented and have been continually refined [2, 3]. Likewise, basis functions that can be usedin a h, p-convergence setting have been presented and applied to several engineering problems. Indeed, in aseries of excellent papers, it has been shown that it is possible to have true hp–adaptivity [4, 5]. On the otherhand, meshless methods attach a patch or volumina to each point whose union forms an open covering ofthe domain. The local shape functions are constructed within each domain. Several different flavors of thesemethods exists [8]. In this paper we will base our development on the generalized finite element method(GFEM) [8].

IEEE TRANSACTIONS ON MAGNETICS, VOL. XX, NO. Y, MONTH, 20035A. Generalized Finite Element MethodA.1 Basis functionsThe presentation of the fundamentals of basis functions is a repetition of those in [9, 25]. Inclusion of thisdescription is purely for the sake of completeness. GFEM is based on a set of N nodes located at xi in the ªvicinity of the domain Ω such that xi Rd : xi Ω, i 1, · · · , N . Associated with each of these nodes is a ªSpatch or volumina denoted by Ωi of size hi such that Ω CΩ : i Ωi and Ωi x Rd : x xi hi Nd(k)(k)(k)(k)Rd . Specifically, a patch Ωi is defined as Ωi k 1 Ωi , Ωi {x(k) R, xi x(k) hi }. Figure1 describes such a construction. Typically, there are no restrictions on the shape of the domain. To a largeextent, these are chosen depending on the underlying basis functions. Associated with each patch are basisfunctions that will be used for Galerkin testing and source. The basis function is a product of two functions,ψi (x) and vi (x): functions ψi (x) form a partition of unity subordina