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Interior Penalty Discontinuous Galerkin Finite Element Method for the Time-Domain Maxwell's Equations
Dosopoulos, Stylianos

2012, Doctor of Philosophy, Ohio State University, Electrical and Computer Engineering.
This dissertation, investigates a discontinuous Galerkin (DG) methodology to solve Maxwell's equations in the time-domain.
More specifically, we focus on a Interior Penalty (IP) approach to derive a DG formulation. In general, discontinuous Galerkin methods decompose the computational domain into a number of disjoint polyhedral (elements). For each polyhedron, we define local basis functions and approximate the fields as a linear combination of these basis functions. To ensure equivalence to the original
problem the tangentially continuity of the electric and magnetic fields need to be enforced between polyhedra interfaces. This condition is applied in the weak sense by proper penalty terms on
the variational formulation also known as numerical fluxes.
Due to this way of coupling between adjacent polyhedra DG methods offer great flexibility and a nice set of properties such as, explicit time-marching, support for non-conformal meshes,
freedom in the choice of basis functions and high efficiency in parallelization. Here, we first introduce an Interior Penalty (IP) approach to derive a DG formulation and a physical interpretation of such an approach. This physical interpretation will provide a physical insight into the IP method and link important concepts like the duality pairing principle to a physical meaning. Furthermore, we discuss the time discretization and stability condition aspects of our scheme. Moreover, to address the issue of very small time steps in multi-scale applications we employ a local time-stepping (LTS) strategy which can greatly reduce the solution time. Secondly, we present an approach to incorporate a conformal Perfectly Matched Layer (PML) in our interior penalty discontinuous Galerkin time-domain (IPDGTD) framework. From a practical point of view, a conformal PML is easier to model compared to a Cartesian PML
and can reduce the buffer space between the structure and the
truncation boundary, thus potentially reducing the number of unknowns. Next, we discuss our approach to combine EM and circuit simulation into a single framework. We show how we incorporate passive lumped elements such as resistors, capacitors and inductors
in the IPDGTD framework. Practically, such a capability is useful since EM applications may often include lumped elements.Following, we present our design of a scalable parallel implementation of IPDGTD in order to exploit the inherit DG parallelism and significantly speed up computations. Our parallelization, is aimed to multi-core CPUs and/or graphics processor units (GPUs), for
shared and/or distributed memory systems. In this way all of MPI/CPU, MPI/GPU and MPI/OpenMP configurations can be used. Finally, we extend our IPDGTD to further include the case of non-conformal meshes. Since, in DG methods the tangentially continuity of the fields is enforced in a weak sense, DG methods naturally support non-conformal meshes. In cases of complicated geometries where a conformal mesh is nearly impossible to get, the ability to handle non-conformal meshes is important. The original geometry in divided into smaller pieces and each piece is meshed independently. Thus, meshing requirements can be greatly relaxed and a final mesh can be obtained for computation.
Jin-Fa Lee (Advisor)
Teixeira Fernando (Committee Member)
Krishnamurthy Ashok (Committee Member)

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Dosopoulos, S. (2012). Interior Penalty Discontinuous Galerkin Finite Element Method for the Time-Domain Maxwell's Equations. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Dosopoulos, Stylianos. "Interior Penalty Discontinuous Galerkin Finite Element Method for the Time-Domain Maxwell's Equations." Electronic Thesis or Dissertation. Ohio State University, 2012. OhioLINK Electronic Theses and Dissertations Center. 25 Nov 2017.

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Dosopoulos, Stylianos "Interior Penalty Discontinuous Galerkin Finite Element Method for the Time-Domain Maxwell's Equations." Electronic Thesis or Dissertation. Ohio State University, 2012. https://etd.ohiolink.edu/

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