Discontinuous Galerkin Time-domain Research Articles

Numerical study of light backscattering from layers of absorbing irregular particles larger than the wavelength

We simulate light backscattering from monolayers of absorbing irregular particles larger than the wavelength using a numerically exact discontinuous Galerkin time domain (DGTD) method. Varying the real and imaginary parts of the complex refractive index m, particle size and the structure of a monolayer we study the evolution of the negative polarization (NP) feature that is observed for many particulate surfaces near backscattering. Simulations show that the NP is enhanced when increasing absorption from Im(m)=0.06 to 0.3. The real part Re(m) plays little role in this case. We confirm correlation of NP with the particle size predicted by approximate models and observed in laboratory measurements. We also demonstrate that the interplay between single- and multiple scattering can be controlled by the topography of a monolayer resulting in enhanced or damped NP.

A Hybrid 3-D-ADI-TDPE/DGTD Method for Multiscale Target Echo Simulation in Large-Scale Complex Environments

In this article, we propose a novel hybrid method for multiscale target echo simulation in large-scale complex environments. The three-dimensional-alternating direction implicit-time domain parabolic equation (3-D-ADI-TDPE) method with refractive index terms and the discontinuous Galerkin time-domain (DGTD) method are combined to solve the propagation of transient electromagnetic waves in large-scale regions and backscattering in multiscale target regions, respectively. At the same time, to realize the connection of two regions and the prediction of target echo, the transfer functions of wave propagation and target backscattering are obtained by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> transform (ZT). In addition, the target echo process is transformed into a cascade of transfer functions to realize simulation modularization and parallel processing. We also discuss the performance of the proposed method in terms of accuracy, memory consumption, and CPU processing time. Results indicate that the processing time required by the 3-D-ADI-TDPE/DGTD hybrid method is more than 11 times less than that of DGTD alone. Moreover, the ZT improves the flexibility of the hybrid method and provides a new concept for the simulation technology of radio wave propagation.

A Minimal Round-Trip Strategy Based on Graph Matching for Parallel DGTD Method With Local Time-Stepping

The load distribution in the local time stepping (LTS) method significantly impacts its computing efficiency. This letter proposes a minimal round-trip (MRT) strategy of the LTS method to balance the communication load of the discontinuous Galerkin time-domain (DGTD) method. By discovering the matching of the connected graph of computing nodes, independent communication with a similar load can be done in the same round trip to minimize waiting between nodes in nonblocking communication, thereby decreasing the communication time of the DGTD-LTS technique. The numerical results indicate that the MRT strategy reduces the communication time between processors by 50% and improves the parallel performance when the LTS method is implemented in the DGTD method. The parallel scale of the MRT approach may be increased to 16 000 nodes (1 040 000 cores) on the supercomputer, and the parallel efficiency is greater than 73.8%.

A Locally-Implicit Discontinuous Galerkin Time-Domain Method to Simulate Metasurfaces Using Generalized Sheet Transition Conditions

The generalized sheet transition conditions (GSTCs) are incorporated into a discontinuous Galerkin time-domain (DGTD) method to efficiently simulate metasurfaces. The numerical flux for GSTCs is derived for the first time using the Rankine–Hugoniot jump conditions. This numerical flux includes the time derivatives of fields, and therefore, explicit time integration schemes, which are traditionally used with DGTD, do not yield a stable time marching method. To alleviate this bottleneck, a new time marching scheme, which solves a local matrix system for the unknowns of the elements touching the same GSTC face, is developed. This locally implicit method maintains its high-parallel efficiency just like the traditional explicit DGTD schemes. Numerical results, which validate the accuracy of the proposed method against analytical solutions and demonstrate its applicability to the simulation of curved and space/time-varying metasurfaces, are presented.

A simplified impedance boundary algorithm in discontinuous Galerkin time-domain

&lt;sec&gt;Large-size conductive targets or coated targets are difficult problems in computational electromagnetics. In general, these problems can be classified as multi-scale problems. Multi-scale problems usually consume a large quantity of computational resources. A lot of efforts have been devoted to seeking for fast methods for these problems. When the skin depth is less than the size of a conductive target, the tangential component of the electric field and magnetic field over the surface of the target can be correlated by the surface impedance &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ \tilde Z $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M9.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M9.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. The &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ \tilde Z $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M10.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M10.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is usually a complex function of the frequency, and it can be used to formulate an impedance boundary condition (IBC) to describe iterative equations in time domain methods, avoiding the volumetric discretization of the target and improving computational efficiency. This condition is commonly known as the surface impedance boundary condition (SIBC). Similarly, for a conductor whose thickness is in the order of skin depth or less, it also has high resource requirements, if the target is of direct volume discretization. The transmission impedance boundary condition (TIBC) can be utilized instead of a coated object to reduce resource requirements. Therefore, there is no need to discretize volume.&lt;/sec&gt;&lt;sec&gt;There are few studies on the IBC scheme by using the discontinuous Galerkin time-domain (DGTD) method. Li et al. (Li P, Shi Y, Jiang L J, Bağcι H &lt;ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1109/TAP.2015.2491969"&gt;2015 &lt;i&gt;IEEE Trans. Antennas Propag.&lt;/i&gt; &lt;b&gt;63&lt;/b&gt; 5686&lt;/ext-link&gt;; Li P, Jiang L J, Bağcι H &lt;ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1109/TAP.2015.2426198"&gt;2015 &lt;i&gt;IEEE Trans. Antennas Propag.&lt;/i&gt; &lt;b&gt;63&lt;/b&gt; 3065 &lt;/ext-link&gt;; Li P, Jiang L J, Bağcι H &lt;ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1109/TAP.2018.2826567"&gt;2018 &lt;i&gt;IEEE Trans. Antennas Propag.&lt;/i&gt; &lt;b&gt;66&lt;/b&gt; 3590 &lt;/ext-link&gt;) discussed the IBC scheme by using the DGTD, which involves complex matrix operations in the processing of IBC. In the DGTD method, numerical flux is used to transmit data between neighboring elements, and the key to the IBC scheme in DGTD is how to handle numerical flux. We propose a DGTD method with a simple form and matrix-free IBC scheme. The key to dealing with IBC in DGTD is numerical flux. Unlike the way in the literature, the impedance &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$ \tilde Z $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M11.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M11.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is not approximated by rational functions in our study. A specfic function &lt;inline-formula&gt;&lt;tex-math id="M12"&gt;\begin{document}$ {\tilde Z_R} $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M12.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M12.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; obtained after the derivation in this work is approximated by rational functions in the Laplace domain through using the vector-fitting (VF) method, and its time-domain iteration scheme is given. This approach avoids matrix operations. The TIBC and SIBC processing schemes are also given. The advantage of the proposed method are that the upwind flux’s standard coefficients are retained and the complex frequency-time conversion problem is implemented by the vector-fitting method. The one-dimensional and three-dimensional examples also show the accuracy and effectiveness of our proposed method in this work.&lt;/sec&gt;

A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations

&lt;p style='text-indent:20px;'&gt;We present a non-intrusive model order reduction (NIMOR) approach with an offline-online decoupling for the solution of parameterized time-domain Maxwell's equations. During the offline stage, the training parameters are chosen by using Smolyak sparse grid method with an approximation level &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ L $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; (&lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ L\geq1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;) over a target parameterized space. For each selected parameter, the snapshot vectors are first produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. In order to minimize the overall computational cost in the offline stage and to improve the accuracy of the NIMOR method, a radial basis function (RBF) interpolation method is then used to construct more snapshot vectors at the sparse grid with approximation level &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ L+1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, which includes the sparse grids from approximation level &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ L $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. A nested proper orthogonal decomposition (POD) method is employed to extract time- and parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the reduced coefficient matrices of the high-fidelity solutions onto the reduced-order subspace spaned by the POD basis functions are extracted. Moreover, a Gaussian process regression (GPR) method is proposed to approximate the dominating time- and parameter-modes of the reduced coefficient matrices. During the online stage, the reduced-order solutions for new time and parameter values can be rapidly recovered via outputs from the regression models without using the DGTD method. Numerical experiments for the scattering of plane wave by a 2-D dielectric cylinder and a multi-layer heterogeneous medium nicely illustrate the performance of the NIMOR method.&lt;/p&gt;

A robust discontinuous Galerkin time domain algorithm for penetrable thin layers

&lt;sec&gt;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.&lt;/sec&gt;&lt;sec&gt;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.&lt;/sec&gt;&lt;sec&gt;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.&lt;/sec&gt;&lt;sec&gt;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.&lt;/sec&gt;

Efficient field-circuit co-simulation method for GaN-based high power microwave devices

Due to the development of the third-generation semiconductors representative of gallium nitride (GaN), the microwave power devices are developing towards higher power, higher efficiency and high integration. However, the electromagnetic field effects are more significant inside the device. As a result, circuit-level based simulation techniques can no longer satisfy the accuracy requirements of device design. Therefore it is necessary to urgently establish the field-circuit co-simulation techniques to couple the active GaN devices with passive electromagnetic structures. In this work, we propose a high-precision discontinuous Galerkin time-domain method to analyze the performances of GAN-based high-power microwave devices. The extracted large-signal compact model of the GaN HEMT is incorporated into the electromagnetic field equations. A local time-stepping technique is adopted to remove the constraints of nonlinear compact models and multiscale elements on the stability conditions of the global algorithm. The comparisons among numerical simulations, experimental results, and software calculations demonstrate the excellent accuracy and efficiency of the proposed method, which can provide a theoretical analysis and design tool for the high reliability design of advanced high-power microwave devices.

Novel Parallelization of Discontinuous Galerkin Method for Transient Electromagnetics Simulation Based on Sunway Supercomputers

A novel parallelization of discontinuous Galerkin time-domain (DGTD) method hybrid with the local time step (LTS) method on Sunway supercomputers for electromagnetic simulation is proposed. The proposed method includes a minimum number of roundtrip (MNR) strategy for processor-level parallelism and a double buffer strategy based on the remote memory access (RMA) of the Sunway processor. The MNR strategy optimizes the communication topology between nodes by recursively establishing the minimum spanning tree and the double buffer strategy is designed to make communication overlapped computation when RMA transmission. Combining the two methods, the proposed method achieves an unprecedented massively parallelism of the DGTD method. Several examples of radiation and scattering are used as cases to study the accuracy and validity of the proposed method. The numerical results show that the proposed method can effectively support 16,000 nodes (1,040,000 cores) parallelism on the Sunway supercomputer, which enables the DGTD method to solve the transient electromagnetic field in a very short time.

Simulation of the nonlinear Kerr and Raman effect with a parallel local time-stepping DGTD solver.

In this paper, an efficient discontinuous Galerkin time-domain (DGTD) method is proposed to solve Maxwell's equations for nonlinear Kerr or Raman media. Based on our previous work, an arbitrary high-order derivatives DGTD method with a local time-stepping scheme is introduced for simulating dynamic optical responses in nonlinear dispersive media such that the nonlinear effects do not impose constraints on the stability conditions for linear subdomains. Therefore, the scheme enables the simulations in the nonlinear and linear media regions with independent time-stepping increments, which greatly improves the efficiency of the time-domain analysis. Moreover, by applying an iteration solution scheme, the proposed method preserves the intrinsic local features, which is favorable for the realization of highly parallelized algorithms. Numerical examples demonstrate the accuracy and the efficiency of our proposed method. We believe the proposed method provides an effective tool for numerical analysis of nonlinear optical phenomena.

Novel adaptive locally implicit scheme of the discontinuous Galerkin time-domain method for distributed transient simulation of antennas and resonance models

Novel adaptive locally implicit scheme of the discontinuous Galerkin time-domain method for distributed transient simulation of antennas and resonance models

Space–Time Adaptive Modeling and Shape Optimization of Microwave Structures With Applications to Metasurface Design

This article presents a time-domain modeling and shape optimization framework for microwave structures, including metasurfaces, based on a nodal discontinuous Galerkin time-domain (DGTD) method. In particular, we employ an unstructured mesh and the inherent mesh refinement ability of DGTD to model multiscale geometries via adaptive mesh refinement. More importantly, we integrate a multitier local time-stepping technique into the time integration of DGTD, which significantly alleviates the cost introduced by mesh refinement. We further present a flexible parameterization approach by defining the contours of metasurface unit cells by B-spline curves, which allows us to explore a wide range of smooth shapes by only a few design variables. Besides, we adopt a polynomial chaos-Kriging (PCK) surrogate method to approximate the full-wave DGTD model, reducing the computational cost for optimization problems that require time-consuming simulations by three orders of magnitude. The B-spline parameterization and the PCK surrogate model are combined with a pattern search-based Pareto front algorithm to optimize metasurfaces for multiple design objectives. The proposed optimization framework is also applicable to other microwave structures. We demonstrate the effectiveness of our approach through the optimization of an omega-bianisotropic Huygens’ metasurface unit cell and an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula> -plane microwave T-junction.

Analysis of the Shielding Effectiveness of Thin Wire Cages by a Hybrid DGTD-FDTD Method

Thin wire simulation is a typical multiscale problem in time-domain computational electromagnetic methods. This article addresses the shielding effectiveness of thin wire cages, and such models can be attributed to complex thin wire structures with multiwire junctions. The discontinuous Galerkin time-domain method for Maxwell's equations is used in our study, and the one-dimensional finite-difference time-domain method is used to construct the iterative scheme for the thin wire. The thin wire is expressed by the modified telegrapher's equations proposed by Holland. In this article, the mesh size is determined by the wavelength of the electromagnetic wave to be analyzed without considering the radius of the thin wire. There is no physical presence of thin wires in our model, and thin wires are expressed by current elements. The proposed method supports the simulation of arbitrarily oriented thin wires, and the proposed method can easily handle complex wire structures. The calculation efficiency can be greatly improved after avoiding multiscale modeling.

Efficient Full-Wave Modeling of Microwave Components With Interval-Valued Electromagnetic Parameters by Polynomial Chaos Expansion-Based DGTD Method

An efficient polynomial chaos expansion (PCE)-based discontinuous Galerkin time-domain (DGTD) method is proposed to analyze microwave components with interval-valued electromagnetic (EM) parameters. The proposed method retains the flexibility of the conventional DGTD method in the element-level domain decomposition and parallel computing. Numerical examples with the interval-valued electron density of plasma and permittivity of a dielectric block are presented to demonstrate the accuracy and capability of the proposed method. The scattering from a dielectric sphere with an interval-valued permittivity is evaluated to verify the applicability of the proposed method in unbounded scattering analysis. Numerical results illustrate that the proposed PCE-based DGTD method has great superiority in computational accuracy and efficiency over the Gaussian process (GP) technique and the Monte Carlo (MC) method in both deterministic and stochastic analyses with the interval-valued parameters, and it requires only a single simulation to obtain all results in the high-dimensional interval domains.

Simulation of Electromagnetic Waves in Plasma by Subdomain-Level Nonconformal DGTD Method

The subdomain-level discontinuous Galerkin time-domain (SL-DGTD) method in a 3-D case is proposed in this article to simulate electromagnetic waves in plasma. An auxiliary differential equation (ADE) technique is employed to manipulate the dispersive characteristic of plasma. A hybrid nonconformal domain decomposition method is used to improve computational efficiency and modeling accuracy. The fourth-order explicit Runge–Kutta integration scheme is implemented to update the fields in each time step. Finally, the proposed algorithm is validated by studying the cutoff frequency of plasma, the transmission coefficients of a plasma slab, and the scattering characteristics of a plasma-coated sphere.

A New Explicit–Implicit Hybridizable Discontinuous Galerkin Time-Domain Method for Electromagnetics

For time-domain electromagnetics, the choice of time scheme is very pivotal. Usually, an explicit time scheme has the stability constraint on small grid size. Though the implicit time scheme is unconditionally stable, it has a necessary expense of computing the global system at each time iteration. To alleviate the expensive computational cost, a new explicit-implicit hybridizable discontinuous Galerkin time-domain method that combines the explicit DGTD (exDGTD) method and our former imHDGTD method will be proposed for the first time. Here let’s call it exDGTD-imHDGTD (ex-imHDGTD). This letter gives its implementation, including a new transmission condition. Unlike the traditional explicit–implicit methods, imHDGTD is used to replace traditional implicit algorithms for the refined region, which leads to a remarkable reduction of degrees of freedom (DOFs) and a better matrix solving ability. Numerical results show that the proposed ex–imHDGTD is very effective both on accuracy and performance.

A novel time‐domain numerical methodology for the electromagnetic analysis of an H‐plane tee power divider

A time-domain numerical methodology for the transient analysis of multisection waveguide power dividers is proposed. Waveguide power dividers are essential components of microwave systems for high-power applications. A numerical strategy based on the 3D discontinuous Galerkin time-domain finite element method is used to estimate the electromagnetic propagation inside a power divider with three ports. The proposed approach leads to simulations that are fast, robust, and accurate, this being the result of a parallel local discretisation of the full-wave time-domain Maxwell's equations on unstructured tetrahedral meshes. The stability of the overall numerical scheme is obtained using an upwind numerical flux, an implicit second-order accurate time integration scheme, and Whitney's edge elements. Good agreements are obtained when the computed results are compared with other HFSS simulation results, and with measurement results found in the literature.

Simulation of the interaction of light with 3‐D metallic nanostructures using a proper orthogonal decomposition‐Galerkin reduced‐order discontinuous Galerkin time‐domain method

AbstractIn this artice, we report on a reduced‐order model (ROM) based on the proper orthogonal decomposition (POD) technique for the system of 3‐D time‐domain Maxwell's equations coupled to a Drude dispersion model, which is employed to describe the interaction of light with nanometer scale metallic structures. By using the singular value decomposition (SVD) method, the POD basis vectors are extracted offline from the snapshots produced by a high order discontinuous Galerkin time‐domain (DGTD) solver. With a Galerkin projection and a second order leap‐frog (LF2) time discretization, a discrete ROM is constructed. The stability condition of the ROM is then analyzed. In particular, when the boundary is a perfect electric conductor condition, the global energy of the ROM is bounded, which is consistent with the characteristics of global energy in the DGTD method. It is shown that the ROM based on Galerkin projection can maintain the stability characteristics of the original high dimensional model. Numerical experiments are presented to verify the accuracy, demonstrate the capabilities of the POD‐based ROM and assess its efficiency for 3‐D nanophotonic problems.

A Hybridizable Discontinuous Galerkin Time-Domain Method With Robin Transmission Condition for Transient Thermal Analysis of 3-D Integrated Circuits

In this article, a discontinuous Galerkin time-domain (DGTD) method hybridized with Robin transmission condition (RTC-DGTD) is proposed for the transient thermal analysis of 3-D integrated circuits (ICs). As a kind of domain decomposition method (DDM), the DGTD method employs a term named numerical flux for the information exchange of solutions between neighboring subdomains. The numerical flux is a function of both temperature <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> and heat flux q, and thus for the heat equation, apart from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> , the variable q has to be solved as well. To reach this, the traditional DGTD method decomposes the second-order partial-differential thermal equation into two first-order ones, which results that both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> and q need to be solved in the whole computational domain, thereby the computational cost is extremely expensive. In this work, to get this trouble-treated, the proposed DGTD method is directly applied to discretize the second-order heat equation. An RTC is established at the interfaces of subdomains, which serves as the second governing equation and meanwhile bridges another connection of solutions in adjacent subdomains. In this way, the new variable q is now only present at the interfaces of subdomains. Consequently, the total number of unknowns is significantly reduced compared with that in the traditional DGTD method. To further speed up the time-marching scheme, an adaptive time-stepping method in terms of the proportional–integral–derivative (PID) control algorithm is employed. Besides, by introducing a restriction operator, the globally coupled matrix equation is torn into a number of small matrix equations pertinent to each subdomain, which is later solved via a finite-element tearing and interconnecting (FETI)-like method. To validate and verify the accuracy and efficiency of the proposed algorithm, several representative examples are studied and compared with COMSOL simulations.

Electrostatic Discharge Simulation Using a GPU-Accelerated DGTD Solver Targeting Modern Graphics Processors

The presence of graphics processors (GPUs) in supercomputers constantly increased in the past decade. Finite-difference time-domain (FDTD) and discontinuous Galerkin time-domain (DGTD) are traditionally used on GPUs for their scalability; however, the limitations of past hardware required particular care in the implementation to obtain good performance. In this work, we discuss an implementation of DGTD for Maxwell’s equations on modern GPUs, and we assess its performance on the simulation of an electrostatic discharge.