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Vector Generalized Finite Element Method

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Research Summary:

Vector generalized finite element method (VGFEM) has been recently introduced for the numerical solution of electromagnetic problems. VGFEM is based on the partition of unity method, which enriches the capabilities of finite element methods. The partition of unity in VGFEM enables flexibility in the choice of basis functions, use of different types of basis functions or mixed polynomial orders within a simulation, and inclusion of physics in solution space. These features can be exploited to yield more accurate, convergent, and, perhaps, efficient solutions to electromagnetic problems.

We are continuously developing the technique using the treasure of the finite element method (FEM). Our current research topics in VGFEM are as follows:

  1. Development of methodologies to use VGFEM with arbitrarily shaped PU domains
  2. Applications of the featires of VGFEM in EM simulations
      
    Scattering in a WR-90 waveguide.

  3. Formulations of absorbing boundary conditions and perfectly matched layers in the VGFEM framework for domain truncation
  4. Hybridization of the method with boundary integrals
      
    Scattering from cavity backed aparture using VGFEM-Boundary Integral: a=16.26 lambda, b=0.2 lambda, c=0.85 lambda at 12 GHz

  5. Hybridization of the method with other differential methods such as FEM, domain decomposition methods and Discontinuous Galerkin Methods
      
    Scattering in a WR-90 waveguide using Discontinuous Galerkin-VGFEM

      
    Wave propagation in a rectangular waveguide using hybrid FEM-VGFEM with p=0. (VGFEM region: z > 0.4, FEM region: z < 0.4)

  6. Development of time domain VGFEM (TD-VGFEM) for the analysis of transient EM problems
      
    Spatial, temporal, and p convergences of TD-VGFEM (Impedance BC). The problem is z polarized, y directed and modulated Gaussian pulse propagation in 1.0m x 0.5m x 0.75m domain. Time step is 1.7e-10 for the figure on the left and 0.43e-10 for the figure on the right.

Recent Related Publications:

[1] Tuncer, O., et al. "Further development of vector generalized finite element method and its hybridization with boundary integrals." IEEE Transactions on Antennas and Propagation 58.3 (2010): 887-899.

[2] Tuncer, O., B. Shanker, and L. C. Kempel. "A hybrid finite element–Vector generalized finite element method for electromagnetics." 2010 IEEE Antennas and Propagation Society International Symposium. IEEE, 2010.

[3] Tuncer, O., B. Shanker, and L. C. Kempel. "A vector generalized finite element-Boundary Integral formulation for scattering from cavity-backed apertures." 2009 IEEE Antennas and Propagation Society International Symposium. IEEE, 2009.

[4] Tuncer, O., B. Shanker, and L. C. Kempel. "A hybrid discontinuous Galerkin-vector generalized finite element method for electromagnetics." Antennas and Propagation (APSURSI), 2011 IEEE International Symposium on. IEEE, 2011.

[5] Tuncer, O., B. Shanker, and L. C. Kempel. "Tetrahedral-based vector generalized finite element method and its applications." IEEE Antennas and Wireless Propagation Letters 11 (2012): 945-948.