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It is well known that graphene demonstrates spatial dispersion properties, i.e., its conductivity is nonlocal and a function of spectral wavenumber (momentum operator) $\mathbf {q}$. In this paper, to account for effects of spatial dispersion on transmission of high-speed signals along graphene nanoribbon (GNR) interconnects, a discontinuous Galerkin time-domain (DGTD) algorithm is proposed. The atomically thick GNR is modeled using a nonlocal transparent surface impedance boundary condition (SIBC) incorporated into the DGTD scheme. Since the conductivity is a complicated function of $\mathbf {q}$ (and one cannot find an analytical Fourier transform pair between $\mathbf {q}$ and spatial differential operators), an exact time-domain SIBC model cannot be derived. To overcome this problem, the conductivity is approximated by its Taylor series in spectral domain under low- $\mathbf {q}$ assumption. This approach permits expressing the time-domain SIBC in the form of a second-order partial differential equation (PDE) in current density and electric field intensity. To permit easy incorporation of this PDE with the DGTD algorithm, three auxiliary variables, which degenerate the second-order (temporal and spatial) differential operators to first-order ones, are introduced. Regarding to the temporal dispersion effects, the auxiliary differential equation method is utilized to eliminate the expensive temporal convolutions. To demonstrate the applicability of the proposed scheme, numerical results, which involve characterization of spatial dispersion effects on the transfer-impedance matrix of GNR interconnects, are presented. [ABSTRACT FROM AUTHOR]

In this paper, a discontinuous Galerkin time-domain (DGTD) method is developed to analyze the power-ground planes taking into account the decoupling capacitors. In the presence of decoupling capacitors, the whole physical system can be split into two subsystems: 1) the field subsystem that is governed by Maxwell’s equations that will be solved by the DGTD method, and 2) the circuit subsystem including the capacitor and its parasitic inductor and resistor, which is going to be characterized by the modified nodal analysis algorithm constructed circuit equations. With the aim to couple the two subsystems together, a lumped port is defined over a coaxial surface between the via barrel and the ground plane. To reach the coupling from the field to the circuit subsystem, a lumped voltage source calculated by the integration of electric field along the radial direction is introduced. On the other hand, to facilitate the coupling from the circuit to field subsystem, a lumped port current source calculated from the circuit equation is introduced, which serves as an impressed current source for the field subsystem. With these two auxiliary terms, a hybrid field-circuit matrix equation is established, which enables the field and circuit subsystems are solved in a synchronous scheme. Furthermore, the arbitrarily shaped antipads are considered by enforcing the proper wave port excitation using the magnetic surface current source derived from the antipads supported electric eigenmodes. In this way, the S-parameters corresponding to different modes can be conveniently extracted. To further improve the efficiency of the proposed algorithm in handling multiscale meshes, the local time-stepping marching scheme is applied. The proposed algorithm is verified by several representative examples. [ABSTRACT FROM PUBLISHER]

A discontinuous Galerkin time-domain (DGTD) method analyzing signal/power integrity on multilayered power-ground parallel plate pairs is proposed. The excitation is realized by introducing wave ports on the antipads where electric/magnetic current sources are represented in terms of the eigenmodes of the antipads. Since closed-forms solutions do not exist for the eigenmodes of the arbitrarily shaped antipads, they have to be calculated using numerical schemes. Spatial orthogonality of the eigenmodes permits determination of each mode's temporal expansion coefficient by integrating the product of the electric field and the mode over the wave port. The temporal mode coefficients are then Fourier transformed to accurately calculate the S-parameters corresponding to different modes. Additionally, to generalize the DGTD to manipulate dispersive media, the auxiliary differential equation method is employed. This is done by introducing a time-dependent polarization volume current as an auxiliary unknown and the constitutive relation between this current and the electric field as an auxiliary equation. Consequently, computationally expensive temporal convolution is avoided. Various numerical examples, which demonstrate the applicability, robustness, and accuracy of the proposed method, are presented. [ABSTRACT FROM PUBLISHER]

In this paper, the electromagnetic (EM) features of graphene are characterized by a discontinuous Galerkin time-domain (DGTD) algorithm with a resistive boundary condition (RBC). The atomically thick graphene is equivalently modeled using an RBC by regarding the graphene as an infinitesimally thin conductive sheet. To incorporate RBC into the DGTD analysis, the surface conductivity of the graphene composed of contributions from both intraband and interband terms is first approximated by rational basis functions using the fast-relaxation vector-fitting (FRVF) method in the Laplace domain. Next, through the inverse Laplace transform, the corresponding time-domain matrix equations in integral can be obtained. Finally, these matrix equations are solved by time-domain finite integral technique (FIT). For elements not touching the graphene sheet, however, the well-known Runge-Kutta (RK) method is employed to solve the two first-order time-derivative Maxwell’s equations. The application of the surface boundary condition significantly alleviates the memory consuming and the limitation of time step size required by Courant–Friedrichs–Lewy (CFL) condition. To validate the proposed algorithm, various numerical examples are presented and compared with available references. [ABSTRACT FROM PUBLISHER]

A hybrid electromagnetics (EM)-circuit simulator exploiting the discontinuous Galerkin time domain (DGTD) method and the modified nodal analysis (MNA) algorithm is developed for analyzing hybrid distributive and nonlinear multiport lumped circuit systems. The computational domain is split into two subsystems. One is the EM subsystem that is analyzed by DGTD, while the other is the circuit subsystem that is solved by the MNA method. The coupling between the EM and circuit subsystems is enforced at the lumped port where related field and circuit unknowns are coupled via the use of numerical flux, port voltages, and current sources. Since the spatial operations of DGTD are localized, thanks to the use of numerical flux, coupling matrices between EM and circuit subsystems are small and are directly inverted. To handle nonlinear devices within the circuit subsystem, the standard Newton-Raphson method is applied to the nonlinear coupling matrix system. In addition, a local time-stepping scheme is applied to improve the efficiency of the hybrid solver. Numerical examples including single and multiport linear/nonlinear circuit networks are presented to validate the proposed solver. [ABSTRACT FROM PUBLISHER]

In an electrostatic simulation, an equipotential condition with an undefined/floating potential value has to be enforced on the surface of an isolated conductor. If this conductor is charged, a nonzero charge condition is also required. While implementation of these conditions using a traditional finite element method (FEM) is not straightforward, they can be easily discretized and incorporated within a discontinuous Galerkin (DG) method. However, DG discretization results in a larger number of unknowns as compared to FEM. In this work, a hybridizable DG (HDG) method is proposed to alleviate this problem. Floating potential boundary conditions, possibly with different charge values, are introduced on surfaces of each isolated conductor and are weakly enforced in the global problem of HDG. The unknowns of the global HDG problem are those only associated with the nodes on the mesh skeleton and their number is much smaller than the total number of unknowns required by DG. Numerical examples show that the proposed method is as accurate as DG while it improves the computational efficiency significantly. [ABSTRACT FROM AUTHOR]

The time domain combined field integral equation (TD-CFIE), which is constructed from a weighted sum of the time domain electric and magnetic field integral equations (TD-EFIE and TD-MFIE) for analyzing transient scattering from closed perfect electrically conducting bodies, is free from spurious resonances. The standard marching-on-in-time technique for discretizing the TD-CFIE uses Galerkin and collocation schemes in space and time, respectively. Unfortunately, the standard scheme is theoretically not well understood: stability and convergence have been proven for only one class of space-time Galerkin discretizations. Moreover, existing discretization schemes are nonconforming, i.e., the TD-MFIE contribution is tested with divergence conforming functions instead of curl conforming functions. We therefore introduce a novel space-time mixed Galerkin discretization for the TD-CFIE. A family of temporal basis and testing functions with arbitrary order is introduced. It is explained how the corresponding interactions can be computed efficiently by existing collocation-in-time codes. The spatial mixed discretization is made fully conforming and consistent by leveraging both Rao–Wilton–Glisson and Buffa–Christiansen basis functions and by applying the appropriate bi-orthogonalization procedures. The combination of both techniques is essential when high accuracy over a broad frequency band is required. [ABSTRACT FROM PUBLISHER]

A discontinuous Galerkin finite element method (DG-FEM) with a highly accurate time integration scheme for solving Maxwell equations is presented. The new time integration scheme is in the form of traditional predictor-corrector algorithms, PE(CE)^m, but it uses coefficients that are obtained using a numerical scheme with fully controllable accuracy. Numerical results demonstrate that the proposed DG-FEM uses larger time steps than DG-FEM with classical PE(CE)^m schemes when high accuracy, which could be obtained using high-order spatial discretization, is required. [ABSTRACT FROM PUBLISHER]

Wrapping a computation domain with a perfectly matched layer (PML) is one of the most effective methods of imitating/approximating the radiation boundary condition in the Maxwell and wave equation solvers. Many PML implementations often use a smoothly increasing attenuation coefficient to increase the absorption for a given layer thickness, and, at the same time, to reduce the numerical reflection from the interface between the computation domain and the PML. In discontinuous Galerkin time-domain (DGTD) methods, using a PML coefficient that varies within a mesh element requires a different mass matrix to be stored for every element and therefore significantly increases the memory footprint. In this work, this bottleneck is addressed by applying a weight-adjusted approximation to these mass matrices. The resulting DGTD scheme has the same advantages as the scheme that stores individual mass matrices, namely, higher accuracy (due to reduced numerical reflection) and increased meshing flexibility (since the PML does not have to be defined layer by layer), but it requires significantly less memory. [ABSTRACT FROM AUTHOR]

Due to the temperature-dependent resistivity of power distribution network (PDN) interconnects, a wiser and necessary strategy is to proceed the electrical–thermal cosimulation in order to include the thermal effects caused by Joule heating. As a natural domain decomposition method (DDM), in this article, a discontinuous Galerkin (DG) method is proposed to facilitate the steady-state electrical and thermal coanalysis. With the intention to avoid solving a globally coupled steady-state matrix system equations resulted from the implicit numerical flux in DG, the block Thomas method is deployed to solve the entire domain in a subdomain-by-subdomain scheme. As a direct solver, the block Thomas method is free of convergence problem frequently occurring in iterative methods, such as block Gauss–Seidel method. The capability of the proposed DG method in handling multiscale and complex 3-D PDNs is validated by several representative examples. [ABSTRACT FROM AUTHOR]

With the purpose of avoiding very fine mesh cells in the proximity of a thin wire, the modified telegrapher’s equations (MTEs) are employed to describe the thin wire voltage and current distributions, which consequently results in reduced number of unknowns and augmented Courant–Friedrichs–Lewy (CFL) number. As hyperbolic systems, both the MTEs and the Maxwell’s equations are solved by the discontinuous Galerkin time-domain (DGTD) method. In realistic situations, the thin wires could be either driven or loaded by circuit networks. The thin wire–circuit interface performs as a boundary condition for the thin wire solver, where the thin wire voltage and current used for the incoming flux evaluation involved in the DGTD-analyzed MTEs are not available. To obtain this voltage and current, an auxiliary current flowing through the thin wire–circuit interface is introduced at each interface. Corresponding auxiliary equations derived from the invariable property of characteristic variable for hyperbolic systems are developed and solved together with the circuit equations established by the modified nodal analysis (MNA) modality. Furthermore, in order to characterize the field and thin wire interactions, a weighted electric field and a volume current density are added into the MTEs and Maxwell–Ampere’s law equation, respectively. To validate the proposed algorithm, three representative examples are presented. [ABSTRACT FROM PUBLISHER]

To avoid straightforward volumetric discretization, a discontinuous Galerkin time-domain (DGTD) method integrated with the impedance boundary condition (IBC) is presented in this paper to analyze the scattering from objects with finite conductivity. Two situations are considered. 1) The skin depth is smaller than the thickness of the conductive volume. 2) The skin depth is larger than the thickness of a thin conductive sheet. For the first situation, a surface impedance boundary condition (SIBC) is employed, wherein the surface impedance usually exhibits a complex relation with the frequency. To incorporate the SIBC into DGTD, the surface impedance is first approximated by rational functions in the Laplace domain using the fast relaxation vector-fitting (FRVF) technique. Via inverse Laplace transform, the time-domain DGTD matrix equations can be obtained conveniently in integral form with respect to time t. For the second situation, a transmission IBC (TIBC) is used to include the transparent effects of the fields. In the TIBC, the tangential magnetic field jump is related with the tangential electric field via the surface conductivity. In this work, a specifically designed DGTD algorithm with TIBC is developed to model the graphene up to the terahertz (THz) band. In order to incorporate the TIBC into DGTD without involving the time-domain convolution, an auxiliary surface polarization current governed by a first-order differential equation is introduced over the graphene. For open-region scattering problems, the DGTD algorithm is further hybridized with the time-domain boundary integral (TDBI) method to rigorously truncate the computational domain. To demonstrate the accuracy and applicability of the proposed algorithm, several representative examples are provided. [ABSTRACT FROM PUBLISHER]

Stability of time domain integral equation (TDIE) solvers has remained an elusive goal for many years. Advancement of this research has largely progressed on four fronts: 1) Exact integration, 2) Lubich quadrature, 3) smooth temporal basis functions, and 4) space-time separation of convolutions with the retarded potential. The latter method's efficacy in stabilizing solutions to the time domain electric field integral equation (TD-EFIE) was previously reported for first-order surface descriptions (flat elements) and zeroth-order functions as the temporal basis. In this work, we develop the methodology necessary to extend the scheme to higher order surface descriptions as well as to enable its use with higher order basis functions in both space and time. These basis functions are then used in a space-time Galerkin framework. A number of results are presented that demonstrate convergence in time. The viability of the space-time separation method in producing stable results is demonstrated experimentally for these examples. [ABSTRACT FROM PUBLISHER]

A scheme hybridizing discontinuous Galerkin time-domain (DGTD) and time-domain boundary integral (TDBI) methods for accurately analyzing transient electromagnetic scattering is proposed. Radiation condition is enforced using the numerical flux on the truncation boundary. The fields required by the flux are computed using the TDBI from equivalent currents introduced on a Huygens' surface enclosing the scatterer. The hybrid DGTDBI ensures that the radiation condition is mathematically exact and the resulting computation domain is as small as possible since the truncation boundary conforms to scatterer's shape and is located very close to its surface. Locally truncated domains can also be defined around each disconnected scatterer additionally reducing the size of the overall computation domain. Numerical examples demonstrating the accuracy and versatility of the proposed method are presented. [ABSTRACT FROM PUBLISHER]