Discontinuous Galerkin Time-domain Algorithm Research Articles

A robust discontinuous Galerkin time domain algorithm for penetrable thin layers

<sec>The thin layer problem is common in some engineering cases, such as a coating target or dielectric radome. If the skin depth of the medium behind the thin layer is much smaller than the medium scale, the transmission line impedance boundary can be used to solve the thin layer problem . When both sides of the thin layer are free space or the skin depth of the medium behind the thin layer is greater than or comparable to the scale of the medium, the transmission line impedance boundary condition is not applicable. Such problems need incident and reflected wave data and are also known as penetrable thin-layer problems.</sec><sec>When dealing with penetrable thin-layer problems, we need to give the electromagnetic wave data on both sides. In addition, the thin layer problem can belong with multi-scale problems. The EM simulation is difficult to implement because of the high resource consumption of accurate modeling.</sec><sec>The accurate modeling is a very expensive algorithm, and the mesh-size difference between the thin layer region and the surrounding ordinary region is too large, thus seriously affecting the quality of the elements (tetrahedrons). So, it is difficult to solve the problem of thin layer quickly by using the time-domain discontinuous Galerkin method or even by using local time steps or other optimization methods. To acquire a fast solution, the approximate boundary condition can be used to describe the thin layer. The DGTD has also the advantage of the geometric fitting property. If a thin layer model is curved, we can build a conformal geometric model to deal with the thin layer, thus avoiding the complicated interpolation correction scheme.</sec><sec>In this work, A robust DGTD method for penetrable thin layers is proposed. In our study, we find that the current excitation scheme is prone to divergence when large dielectric coefficients are dealt with and its robustness is poor. To solve this problem, we use the electromagnetic field correction scheme in DGTD to deal with the thin layer problems. In addition, engineering problems may encounter multiple thin layers. In this work, a simulation scheme for multiple thin layers is presented and relevant numerical examples are given to verify it. The results illustrate the accuracy and effectiveness of the method in this work, and the method presented in this work can be used for the fast calculation of complex thin layer problems.</sec>

Modeling of Metasurfaces Using Discontinuous Galerkin Time-Domain Method Based on Generalized Sheet Transition Conditions

A new discontinuous Galerkin time-domain (DGTD) method based on generalized sheet transition conditions (GSTCs) for analyzing the transient response of electromagnetic metasurfaces is presented. The discontinuities in electromagnetic fields around the vicinity of the metasurface are modeled by placing virtual edges on both sides of the metasurface. The GSTCs are incorporated in the updating equations of the standard DGTD algorithm by integrating the time-domain GSTCs formula with Galerkin’ s-weighted residual method. The DGTD-GSTC formulation can also simulate a bf piecewise linear approximation of a curved metasurfaces due to the unstructured mesh. This is the first time that DGTD is applied to the simulation of GSTCs. The implementation of this new method is verified by numerical cases.

A Fast, Hybrid, Time-Domain Discontinuous Galerkin-Physical Optics Method for Composite Electromagnetic Scattering Analysis

To accelerate the solution of transient electromagnetic scattering from composite scatters, a novel hybrid discontinuous Galerkin time domain (DGTD) and time-domain physical optics (TDPO) method is proposed. The DGTD method is used to solve the accurate scattering field of the multi-scale objects region, and a hybrid explicit-implicit time integration method is also used to improve the efficiency of multi-scale problems in the time domain. Meanwhile, the TDPO method is used to accelerate the speed of surface current integration in an electrically large region. In addition, the DGTDPO method considers the mutual coupling between two regions, and effectively reduces the number of numerical calculations for the other space of the composite target, thereby significantly reducing the computer memory consumption. Numerical results certified the high efficiency and accuracy of the hybrid DGTDPO. According to the results, in comparison with the DGTD algorithm in the entire computational domain, the DGTDPO method can reduce computing time and memory by 90% and 70% respectively. Meanwhile, the normalized root mean square deviation (NRMSD) of the time-domain, high-frequency approximation method is over 0.2, and that of the DGTDPO method is only 0.0971. That is, compared with the approximation methods, the hybrid method improves the accuracy by more than 64%.

Discontinuous Galerkin Time Domain Method with Dispersive Modified Debye Model and its Application to the Analysis of Optical Frequency Selective Surfaces

We develop a discontinuous Galerkin time domain (DGTD) algorithm with an experimentally validated modified Debye model (MDM) to take metal dispersion into consideration. The MDM equation is coupled with Maxwell’s equations and solved together through the auxiliary differential equation (ADE) method. A Runge-Kutta time-stepping scheme is proposed to update the semi-discrete transformed Maxwell’s equations and ADEs with high order accuracy. Then we employ the proposed algorithm to analyze an infinite doubly periodic frequency selective surface (FSS) operating in the optical regime that exhibits transmission enhancement due to the surface plasmatic effect. The accuracy and the efficiency enhancements are validated through a comparison with commercial simulation software. This work represents the first integration of MDM with DGTD, which enables the DGTD algorithm to efficiently analyze metallic structures in the optical regime.

A Hybrid Mesh DGTD Algorithm Based on Virtual Element for Tetrahedron and Hexahedron

In the discontinuous Galerkin time-domain (DGTD) method, tetrahedral meshes are typically used to discretize the computational domain. However, due to their different topological structures, when tetrahedrons or hexahedrons of the same discrete scale are used to divide the same computational domain, there are more tetrahedral mesh units, resulting in higher temporal and spatial complexity. In this article, a hybrid mesh DGTD algorithm based on virtual elements for tetrahedra and hexahedra is proposed to improve the computational efficiency compared with the traditional DGTD method. The fine-structured computational domain and the outer region are computed using tetrahedral and hexahedral meshes, respectively. During calculation, the numerical flux of each interface element requires data from the other subdomain. In this article, virtual elements are established at each subdomain interface to enable the exchange of field values for ease of processing. Edge assignment or field interpolation is applied to sample points to create corresponding virtual units to calculate fringe fields. Finally, the field value of each virtual element is assigned to the corresponding real element for the numerical flux calculation. Numerical results verify the correctness and efficiency of the proposed method.

GPU-Accelerated Hybrid Discontinuous Galerkin Time Domain Algorithm With Universal Matrices and Local Time Stepping Method

In this article, a graphic processing unit (GPU)-based acceleration implementation of a hybrid discontinuous Galerkin time domain method based on Maxwell’s equations and Helmholtz vector wave equation (HDGTD) has been developed. The computational domain is discretized by tetrahedrons and the resultant meshes are categorized into two regions solved by Maxwell’s equations and Helmholtz vector wave equation, respectively. The hierarchical vector basis functions are used to expand the unknowns in the HDGTD method, and a universal matrix technique is proposed to decompose the geometry-dependent matrices in each tetrahedron into the summation of universal matrices defined in barycentric coordinates, thus giving rise to a great decrease of the memory usage. A local time stepping (LTS) method based on a simple interpolation is introduced in the proposed HDGTD method to achieve highly efficient solution of multiscale problems. Two kinds of compute unified device architecture (CUDA)-based mapping techniques, i.e., 1-D and 2-D blocks, are implemented to achieve a tradeoff between the parallel speedup and the memory usage. With the 1-D block mapping, over 590 times speedup can be achieved, and in the case of the 2-D block mapping, over 150 times acceleration and 13 times memory reduction are obtained. Some practical complex examples are given to demonstrate a good performance of the proposed parallel method.

Discontinuous Galerkin Time-Domain Modeling of Graphene Nanoribbon Incorporating the Spatial Dispersion Effects

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.

Transient Heterogeneous Electromagnetic Simulation With DGTD and Behavioral Macromodel

A novel hybrid field-circuit simulation method is proposed for transient heterogeneous electromagnetic simulations. It adopts the discontinuous Galerkin time domain (DGTD) method and behavioral macromodel trained by artificial neural network to analyze the electromagnetic structure and circuit network, respectively. Due to the feature of DGTD method and behavioral macromodel, the proposed method can handle not only electromagnetic structures with complex geometries and materials but also manage circuits with unknown internal details. It provides a new approach that has the trainable accuracy to meet today's IP projection requestions. It is also the first time for DGTD algorithm to work with the behavior model. Numerical examples have been benchmarked to demonstrate the capability of the proposed method.

Transient Thermal Analysis of 3-D Integrated Circuits Packages by the DGTD Method

Since accurate thermal analysis plays a critical role in the thermal design and management of the 3-D system-level integration, in this paper, a discontinuous Galerkin time-domain (DGTD) algorithm is proposed to achieve this purpose. Such as the parabolic partial differential equation (PDE), the transient thermal equation cannot be directly solved by the DGTD method. To address this issue, the heat flux, as an auxiliary variable, is introduced to reduce the Laplace operator to a divergence operator. The resulting PDE is hyperbolic, which can be further written into a conservative form. By properly choosing the definition of the numerical flux used for the information exchange between neighboring elements, the hyperbolic thermal PDE can be solved by the DGTD together with the auxiliary differential equation. The proposed algorithm is a kind of element-level domain decomposition method, which is suitable to deal with multiscale geometries in 3-D integrated systems. To verify the accuracy and robustness of the developed DGTD algorithm, several representative examples are benchmarked.

DGTD Analysis of Electromagnetic Scattering From Penetrable Conductive Objects With IBC

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.

Field-based DGTD/PIC technique for general and stable simulation of interaction between light and electron bunches

We introduce a hybrid technique based on the discontinuous Galerkin time domain (DGTD) and the particle in cell (PIC) simulation methods for the analysis of interaction between light and charged particles. The DGTD algorithm is a three-dimensional, dual-field and fully explicit method for efficiently solving Maxwell equations in the time domain on unstructured grids. On the other hand, the PIC algorithm is a versatile technique for the simulation of charged particles in an electromagnetic field. This paper introduces a novel strategy for combining both methods to solve for the electron motion and field distribution when an optical beam interacts with an electron bunch in a very general geometry. The developed software offers a complete and stable numerical solution of the problem for arbitrary charge and field distributions in the time domain on unstructured grids. For this purpose, an advanced search algorithm is developed for fast calculation of field data at charge points and for later importing to the PIC simulations. In addition, we propose a field-based coupling between the two methods resulting in a stable and precise time marching scheme for both fields and charged particle motion. To benchmark the solver, some examples are numerically solved and compared with analytical solutions. Eventually, the developed software is utilized to simulate the field emission from a flat metal plate and a silicon nano-tip. In the future, we will use this technique for the simulation and design of ultrafast compact x-ray sources.