O. Tuncer

Further Development of Vector Generalized Finite Element Method and Its Hybridization With Boundary Integrals

Recently, vector generalized finite element method (VGFEM) was introduced for the solution of the vector Helmholtz equation, and its applicability was validated for canonical problems. VGFEM uses a local Helmholtz decomposition to construct basis functions in overlapping local domains of some canonical shape. While using a canonical shape for local domains adds flexibility to the method, one needs to provide information regarding boundaries of domains/inhomogeneities. The need for surface information proves to be a bottleneck in using the method for a larger class of problems. This paper is targeted towards overcoming these deficiencies; here, we will introduce the modifications to this method that permit interfacing with arbitrarily shaped local domains (to facilitate interfacing with existing meshing software), integrate this method with boundary integrals and provide a framework for studying dispersion. As will be apparent, the hybridization of the method with boundary integrals is not a simple adaptation of existing methods onto the VGFEM framework. Likewise, dispersion analysis is nontrivial due to the overlapping nature of VGFEM basis functions. A range of practical problems has been analyzed within the presented framework and results are compared either against measurements or existing FEM data to validate the presented methodology.

Characterization of Biaxial Anisotropic Material Using a Reduced Aperture Waveguide

A technique is introduced for measuring the electromagnetic properties of a biaxial anisotropic material sample using a reduced-aperture waveguide sample holder designed to accommodate a cubical sample. All the tensor material parameters can be determined by measuring the reflection and transmission coefficients of a single sample placed into several orientations. The theoretical reflection and transmission coefficients necessary to perform the material parameter extraction are obtained using a modal analysis technique. An optimization method that seeks to minimize the difference between theoretically computed and measured reflection and transmission coefficients is used to perform the extraction. Measurements of a stacked dielectric medium is characterized to demonstrate feasibility of the reduced-aperture waveguide approach.

A hybrid finite element – Vector generalized finite element method for electromagnetics

Vector generalized finite element method (VGFEM) has been successfully applied to solve open and closed domain electromagnetic problems [1, 2]. Theoretically, VGFEM is a powerful technique as it permits the use of a wide range of function spaces, and provides a rigorous framework for including different approximation spaces seamlessly. Doing so can provide better approximation, and perhaps, more efficient simulations. The first step in VGFEM is to define a partition of unity (PU) domain, or the domain of approximation function. In a departure from using the canonical domains as the PU domains [1], a methodology that permits the use of union of polyhedra has been developed and validated [2], where polyhedra can be transformed into a brick element. In this paper, our goal is two fold: (i) we will develop a framework for defining the PU domains as union of tetrahedrals, and (ii) integrate VGFEM with FEM. The extension of the VGFEM framework on tetrahedral elements is not straightforward since (i) definition of the PU function on an arbitrary polyhedra is difficult, and (ii) one needs to carefully handle the coupling of FEM and VGFEM basis functions. This paper presents the means through which these difficulties can be overcome. The rest of the paper is organized as follows: In the next Section, we present the framework of VGFEM with tetrahedral elements, and then explain the hybrid FEM-VGFEM technique. The new VGFEM framework is validated via a a set of simulations. Finally, conclusions and the future research are presented in the last Section.

Tetrahedral-Based Vector Generalized Finite Element Method and Its Applications

Vector generalized finite element method was first introduced as a meshless method. Its application to practical problems was stymied by the fact that one needs information of intersection of the partition of unity domains with the boundaries of the problem domain. The framework presented in Tuncer s work (IEEE Trans. Antennas Propag., vol. 58, no. 3, 887-899, Mar. 2010) overcame this bottleneck. In this letter, the framework is extended to tetrahedral meshes and applied to a number of electromagnetic problems. These examples are chosen so as to highlight some features of the method such as higher-order convergence, flexibility in the choice of basis functions, and use of different types of basis functions or mixed polynomial orders within a simulation.

A vector generalized finite element-Boundary Integral formulation for scattering from cavity-backed apertures

In this paper, we have developed a hybrid VGFEM-BI formulation for the analysis of scattering from filled cavities. Since surface basis functions are not supported in the framework of VGFEM, volumetric vector basis functions are forced to impose boundary conditions at the aperture. This is achieved by constructing conformal PU domains from the geometry mesh. The method has been validated by comparing the simulated radar cross sections of filled cavities against measurement and FEM-BI data. Our current research is on the application of the method for large scattering and radiation problems using tailored basis functions, and these results will be presented at the conference.

Discontinuous Galerkin Inspired Framework for Vector Generalized Finite-Element Methods

Discontinuous Galerkin methods have been extensively used in simulations of various electromagnetic problems, in both time and frequency domains. In this paper, we exploit features of these methods to provide a framework that can be used to overcome a fundamental bottleneck in using as well as augmenting the capabilities of vector generalized finite element methods (VGFEM). This approach permits the use of VGFEM for analysis of piecewise inhomogeneous domains without defining additional constraints on basis functions and allows hybridization with other finite element formulations. Interior penalty discontinuous Galerkin method (IP-DGM) is used to enable communication among partitioned domains. We will present several results that demonstrate applications of these hybrid methods as well as their convergence characteristics.

A hybrid discontinuous Galerkin-vector generalized finite element method for electromagnetics

In this paper, we present development of a hybrid discontinuous Galerkin-vector generalized finite element method (DG-VGFEM) to exploit advantages of both methods for frequency domain electromagnetic (EM) simulations. The mathematical framework of VGFEM permits the use of different types of basis functions or mixed polynomial orders within a simulation. Likewise, discontinuous Galerkin (DG) method enables handling of multi-material problems, flexibility in the mesh, and parallelization. These appealing features are exploited in this hybrid method to get better approximation, and perhaps, more efficient simulations of EM fields. The results presented demonstrate -h and -p convergences of the method as well as applications to some realistic problems.

Development of time domain vector generalized finite element method

Vector generalized finite element method (VGFEM) has been successfully applied for solving many electromagnetic (EM) problems in the frequency domain [1] – [3]. VGFEM is a powerful technique as it permits flexibility in the choice of basis functions, use of mixed polynomial orders, use of mixed basis functions in EM simulations, and inclusion of physics in solution space. In this paper, we develop a time domain VGFEM (TD-VGFEM) technique in order to use all these appealing features of the method in the solution of transient EM problems. Detailed bilinear formulation of TD-VGFEM is given for different boundary conditions. Temporal convergence and spatial convergence of the method are shown.

Development of vector basis functions in vector generalized finite element method for inhomogeneous domains

In this paper, we have developed additional basis functions for inhomogenous domains. The basis functions satisfy all the boundary condition requirements. These basis functions are defined on each subdomain and tested using hexahedral elements but they can be readily generalized to any linear subdomains. This enables us to use available meshing structures such that any material discontinuity in PU domain can be handled, and in fact, is the first step in making this method applicable the analysis of practical problems. The performance of the developed basis functions have been tested and validated by simulating eigenmodes of a cavity. Our current research is on application of the VGFEM with the proposed basis functions for practical problems, and these results will be presented at the conference.

Dispersion analysis in scalar generalized finite element method

A semi-analytic technique for the dispersion analysis of scalar generalized finite element method (GFEM) that can be easily applied to higher dimensions and vector GFEM has been developed. The phase error simulation results validate the O([h/lambda]2p) convergence rate of the Legendre polynomials. GFEM compared to FEM significantly suppresses the error for the higher orders. The phase error depends on the incident angle and it shows different behavior for each order. The error in discrete representation of the differential equation is shown to be related to the error in function representation. Results using different local approximation functions as well as generalization of the methodology to vector basis functions are presented.