Novel Strategies for Efficient Computational Electromagnetic (CEM) Simulation of Microstrip Circuits, Antennas, Arrays, and Metamaterials Part-II: Characteristic Basis Function Method, Perfectly Matched Layer, GPU Acceleration

Rapid-prototyping plays a critical role in the design of antennas and related planar circuits for wireless communications, especially as we embrace the 5G/6G protocols going forward into the future. While there are a number of software modules commercially available for such rapid prototyping, often they are found to be not as reliable as desired, especially when they are based on approximate equivalent circuit models for various circuit components comprising the antenna system. Consequently, it becomes necessary to resort to the use of more sophisticated simulation techniques, based on full-wave solvers that are numerically rigorous, albeit computer-intensive. Furthermore, optimizing the dimensions of antennas and circuits to enhance the performance of the system is frequently desired, and this often exacerbates the problem since the simulation must be run a large number of times to achieve the performance goal—an optimized design. Consequently, it is highly desirable to develop accurate yet efficient techniques, both in terms of memory requirements and runtimes, to expedite the design process as much as possible. This is especially true when the antenna utilizes metamaterials and metasurfaces for their performance enhancement, as is often the case in modern designs. The purpose of this paper is to present strategies that address the bottlenecks encountered in the generation of Green’s Functions for layered media, especially in the millimeter wave frequency range where the dimensions of the antennas and the platforms upon which they are mounted can be several wavelengths in size. The paper is divided into two parts. Part-I covers the topics of construction of layered medium Green’s Function for millimeter wavelengths; the Equivalent Medium Approach (EMA) which obviates the need to construct Green’s Function for certain geometries; and the T-matrix approach for hybridizing the finite methods with the Method of Moments(MoM). In Part-II of this paper, we go on to discuss three other strategies for performance enhancement of CEM techniques: the Characteristic Basis Function Method (CBFM); mesh truncation for finite methods by using a new form of the Perfectly Matched Layer (PML); and GPU acceleration of MoM as well as FDTD (Finite Difference Time Domain) algorithms. The common theme between the two parts is the “performance enhancement” of CEM (Computational Electromagnetics) techniques, which provides the synergistic link between the two parts.

We present a detailed theoretical and numerical investigation of the perfectly matched layer (PML) concept as applied to the problem of mesh truncation in the finite-element method (FEM). We show that it is possible to extend the Cartesian PML concepts involving half-spaces to cylindrical and spherical geometries appropriate for closed boundaries in two and three dimensions by defining lossy anisotropic layers in the relevant coordinate systems. By using the method of separation of variables, it is possible to solve the boundary value problems in these geometries. The analytical solutions demonstrate that under certain conditions, outgoing waves are absorbed with negligible reflection, and the transmitted wave is attenuated within the PML. To reduce the white space in radiation or scattering problems, conformal PMLs are constructed via parametric mappings. It is also verified that the PML concept, which was originally introduced for problems governed by Maxwell's equations, can be extended to cases governed by the scalar Helmholtz equation. Finally, numerical results are presented to demonstrate the use of the PML in FEM mesh truncation.

The conformal perfectly matched layer (PML), i.e., an efficient absorbing boundary condition, is commonly employed to address the open-field scattering problem of electromagnetic wave. To develope a conformal PML exhibiting a significant absorption effect and small reflection error, the present study proposes the constitutive parameter optimization method of obliquely incident reflectivity in terms of the conformal PML. First, the recurrence formula of obliquely incident reflectivity is desired. Subsequently, by the sensitivity analysis of constitutive parameters, the major optimal design variables are determined for the conformal PML. Lastly, with the reflectivity of the conformal PML as the optimization target, this study adopts the genetic algorithm (GA), simulated annealing algorithm (SA) and particle swarm optimization algorithm (PSO) to optimize the constitutive parameters of the conformal PML. As revealed from the results, the optimization method is capable of significantly reducing the reflection error and applying to the parameter design of conformal PML.

The perfectly matched layer (PML) is very popular and efficient for grid truncation of open-region problems. The concept of PML was introduced by J. P. Berenger (see J. Comput. Phys., vol.114, p.185-200, 1994) together with a numerical implementation based on the finite-difference time-domain (FDTD) scheme. However, the basic FDTD scheme is formulated on Cartesian grids and in many cases an unnecessarily large free-space region must be discretized between the PML and the object under investigation. F.L. Teixeira et al. (see IEEE Trans. Antennas Propag., vol.49, p.902-7, 2001) developed a conformal PML for the FDTD scheme. We continue this effort by presenting a new conformal PML formulation for the finite element method (FEM) in the time domain. It is based on the anisotropic material which has been used for the time domain FEM by D. Jiao et al. (see IEEE Antennas and Propag. Soc. Int. Symp., vol.2, p.158-61, 2002). We present the 2D-version of the conformal PML, which is a special case of our 3D-formulation. Furthermore, we consider only radar cross section (RCS) computations based on the scattered field formulation and emphasize that our PML is applicable to the total field formulation as well as radiating structures.

We introduce a conformal perfectly matched layer (PML) for the finite-element time-domain (FETD) solution of transient Maxwell equations in open domains. The conformal PML is implemented in a mixed FETD setting based on a direct discretization of the first-order coupled Maxwell curl equations (as opposed to the second-order vector wave equation) that employs edge elements (Whitney 1-form) to expand the electric field and face elements (Whitney 2-form) to expand the magnetic field. We show that the conformal PML can be easily incorporated into the mixed FETD algorithm by utilizing PML constitutive tensors whose discretization is naturally decoupled from that of Maxwell curl equations (spatial derivatives). Compared to the conventional (rectangular) PML, a conformal PML allows for a considerable reduction on the amount of buffer space in the computational domain around the scatterer(s).

In this study, 2D finite element (FE) solving process with the conformal perfectly matched layer (PML) is elucidated to perform the electromagnetic scattering computation. With the 2D monostatic RCS as the optimization objective, a sensitivity analysis of the basic design parameters of conformal PML (e.g., layer thickness, loss factor, extension order and layer number) is conducted to identify the major parameters of conformal PML that exerts more significant influence on 2D RCS. Lastly, the major design parameters of conformal PML are optimized by the simulated annealing algorithm (SA). As revealed from the numerical examples, the parameter design and optimization method of conformal PML based on SA is capable of enhancing the absorption effect exerted by the conformal PML and decreasing the error of the RCS calculation. It is anticipated that the parameter design method of conformal PML based on RCS optimization can be applied to the cognate absorbing boundary and 3D electromagnetic computation.

Near-field performance analysis of locally-conformal perfectly matched absorbers via Monte Carlo simulations

Based on the discontinuous Galerkin time-domain (DGTD) method, a new conformal perfectly matched layer (PML) is introduced to truncate the 3-D open domain. Compared with the traditional (rectangular or cubic) PML, the conformal PML is a smooth convex shell, which significantly reduces the buffer space in the computational domain. In this article, we construct the conformal PML in an orthogonal curvilinear coordinate system. Conductivities are defined to absorb the outgoing waves, depending on the distance from the sampling point to the non-PML region and the principal curvature radii of the sampling point, which are calculated utilizing the Weingarten transformation. The analytical expression of 3-D conformal PML is derived with the complex coordinate stretching technique. Furthermore, to reduce the total degrees of freedom (DoFs) while maintaining accuracy, the hierarchical vector basis functions are chosen to discretize the conformal PML and physical region. Numerical results validate its good absorption performance.

A perfectly matched layer (PML) method is proposed for electrically large curvilinear meshes based on a higher order finite-element modeling paradigm and the concept of transformation electromagnetics. The method maps the non-Maxwellian formulation of the locally conformal PML to a purely Maxwellian implementation using continuously varying anisotropic and inhomogeneous material parameters. An approach to the implementation of a conformal PML for higher order meshes is also presented, based on a method of normal projection for PML mesh generation around an already existing convex volume mesh of a dielectric scatterer, with automatically generated constitutive material parameters. Once the initial mesh is generated, a PML optimization method based on gradient descent is implemented to most accurately match the PML material parameters to the geometrical interface. The numerical results show that the implementation of a conformal PML in the higher order finite-element modeling paradigm dramatically reduces the reflection error when compared to traditional PMLs with piecewise constant material parameters. The ability of the new PML to accurately and efficiently model scatterers with a large variation in geometrical shape and those with complex material compositions is demonstrated in examples of a dielectric almond and a continuously inhomogeneous and anisotropic transformation-optics cloaking structure, respectively.

The author offer a number of observations on some of the results of research concerned with the problem of mesh truncation with perfectly matched layers (PML). The conformal PML design appears to be useful in reducing the white space in the computational domain of the FEM. Analytical solutions have shown that it is possible to locate the PML arbitrarily close to the surface of a convex, electrically-large scatterer. This leads to the conclusion that the conformal PML is very effective for mesh truncation, especially at high frequencies. The PML concept is applicable not only to electromagnetic propagation problems described by the vector wave equation, but also to applications governed by the scalar Helmholtz equation. This, in turn, implies that PML absorbers can be used for acoustic scattering or radiation problems as well. The tensor constitutive parameters of a perfectly-matched anisotropic medium must satisfy the Kramers-Kronig relationships, such that the dynamical system governed by the constitutive relations is causal. A causal PML design ensures a better performance over the entire frequency spectrum. In the past, the perfectly-matched absorbers have been employed in FDTD and FEM applications. However, it seems feasible to use them for FETD mesh truncation as well. A causal/conformal PML design seems to offer an optimal solution to effective mesh truncation over a wider band of frequencies.

The core of domain decomposition method (DDM) is that it decomposes the original problem domain into several non-overlapping sub-domains, and solves every sub-domain independently. Since every sub-domain is solved independently by use of direct methods, we make the DDM be a parallelized algorithm. The conformal perfectly matched layer (PML) is introduced to this method for the analysis of electromagnetic scattering problems. This PML is based on a locally-defined complex coordinate transformation which has no explicit dependence on the differential geometric characteristics of the PML-free space interface. The DDM formulation is modified when the interface locates in the PML. We illustrate the application of the proposed approach via several 3D electromagnetic scattering problems.

Recently proposed anisotropic material-based conformal perfectly matched layer (PML) formulations are based on general orthogonal curvilinear coordinates and, consequently, their practical application to a conformal finite-difference time-domain (FDTD) method requires body-conformal orthogonal grids. In this work, we present an application of a two-dimensional (2-D) hyperbolic grid generation scheme for a numerical implementation of a conformal PML in conjunction with Janaswamy and Liu's FDTD method. Numerical results show that our hyperbolic grid-based conformal PML implementation provides an efficient absorbing boundary condition (ABC) for a body-conformal FDTD simulation involving complex geometries.

The modern graphics processing unit (GPU) programming technique is applied to speed up the simulation of the performance of GPS antenna installed on a self-driving car in practical environments with the method of moments (MoM) and physical optics (PO) hybrid technique. To take good advantage of GPU acceleration, the MoM part, ray tracing, PO current calculation, and post processing are all executed on GPU by developing a set of GPU code with CUDA. Numerical results show excellent accuracy and efficiency improvement.

We present an analytical derivation of a 3-D conformal perfectly matched layer (PML) for mesh termination in general orthogonal curvilinear coordinates. The derivation is based on the analytic continuation to complex space of the normal coordinate to the mesh termination. The resultant fields in the complex space do not obey Maxwell's equations. However, it is demonstrated that, through simple field transformations, a new set of fields can be introduced so that they obey Maxwell's equations for an anisotropic medium with properly chosen constitutive parameters depending on the local radii of curvature. The formulation presented here recovers, as particular cases, the previously proposed Cartesian, cylindrical, and spherical PMLs. A previously employed anisotropic (quasi-) PML for conformal terminations is shown to be the large radius of curvature approximation of the anisotropic conformal PML derived herein. © 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 17: 231–236, 1998.

In the numerical modeling of electromagnetic radiation and/or scattering problems by finite methods, the computational domain is usually truncated by an artificial boundary on which suitable absorbing boundary conditions (ABCs) are imposed. An alternative approach to mesh truncation, which was introduced by Berenger for finite difference time domain (FDTD) implementation, employs a region which is designed to absorb plane waves whose frequency and incident angle are arbitrary. Since the plane waves are transmitted into the region without any reflection, the region is called a perfectly matched layer (PML). In this chapter, our main objective is to systematically derive the partial differential equations satisfied within the PML media. It is demonstrated that both Maxwellian PMLs (the anisotropic and the bianisotropic PML media) as well as non-Maxwellian PMLs (Berenger PML) can be realized by assuming a field decay behavior within the PML. Causality and reciprocity issues, and their implications in the proper design and operation of perfectly matched absorbers, are also discussed. It is shown that if the constitutive parameters of the PML medium satisfy the Kramers-Kronig relations, the medium is causal and does not exhibit a singular behavior at lower frequencies. This, in turn, enables us to extend the PML concept to the static case. The reciprocity concept is also important in the design of PMLs, since the PML medium occupies a bounded domain in mesh truncation applications. The medium must be reciprocal and, as a consequence, the decay behavior of the waves traveling in the longitudinal direction must be identical to that of the waves reflected from the terminating boundary and traveling in the opposite direction. Some examples of causal/non-causal and reciprocal/non-reciprocal PMLs are given in the work to illustrate these behaviors.