Conformal Perfectly Matched Layer Research Articles

High-Order Conformal Perfectly Matched Layer for the DGTD Method

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.

Parameter Design of Conformal PML Based on 2D Monostatic RCS Optimization

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.

Constitutive Parameter Optimization Method of Obliquely Incident Reflectivity for Conformal PML

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.

Analytical Derivation of Ellipsoid Conformal Perfectly Matched Layers for Electromagnetic Simulations

The 3-D ellipsoid conformal perfectly matched layer (PML) is proposed and realized by the discontinuous Galerkin time-domain (DGTD) method. The interface between free space and the PML region is an ellipsoid in this scheme. The analytical formulas are derived from differential geometry. The numerical results show that compared with planar-PML, the ellipsoid-PML (EPML) avoids the corner problem and provides an efficient absorbing boundary condition for the DGTD simulation. Meanwhile, this new EPML scheme has higher accuracy than that of the old EPML scheme that we proposed.

Anisotropic Locally Conformal Perfectly Matched Layer for Higher Order Curvilinear Finite-Element Modeling

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.

Layer-oriented Element Integration Algorithm of Conformal Perfectly Matched Layer

As an efficient artificial truncating boundary condition, conformal perfectly matched layer (CPML) is a kind of multilayer anisotropic absorbing media. To reduce computing effort of CPML, this article proposes a layer-oriented element integration algorithm. In this algorithm, the relative dielectric constant and permeability are considered as constants for each the very thin monolayer of CPML, and the element integration of multilayer along the normal direction is substituted by the element integration of monolayer. The CPML is divided into multilayer elements, whose element matrixes are evaluated by monolayer integration. Numerical examples show that the layer-oriented element integration algorithm is reliable and the CPML formed with this algorithm possesses high-efficient absorbing performance.

Conformal Perfectly Matched Layer for the Mixed Finite Element Time-Domain Method

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).

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

In the numerical solution of some boundary value problems by the finite element method (FEM), the unbounded domain must be truncated by an artificial absorbing boundary or layer to have a bounded computational domain. The perfectly matched layer (PML) approach is based on the truncation of the computational domain by a reflectionless artificial layer which absorbs outgoing waves regardless of their frequency and angle of incidence. In this paper, we present the near-field numerical performance analysis of our new PML approach, which we call as locally-conformal PML, using Monte Carlo simulations. The locally-conformal PML method is an easily implementable conformal PML implementation, to the problem of mesh truncation in the FEM. The most distinguished feature of the method is its simplicity and flexibility to design conformal PMLs over challenging geometries, especially those with curvature discontinuities, in a straightforward way without using artificial absorbers. The method is based on a special complex coordinate transformation which is ‘locally-defined’ for each point inside the PML region. The method can be implemented in an existing FEM software by just replacing the nodal coordinates inside the PML region by their complex counterparts obtained via complex coordinate transformation. We first introduce the analytical derivation of the locally-conformal PML method for the FEM solution of the two-dimensional scalar Helmholtz equation arising in the mathematical modeling of various steady-state (or, time-harmonic) wave phenomena. Then, we carry out its numerical performance analysis by means of some Monte Carlo simulations which consider both the problem of constructing the two-dimensional Green’s function, and some specific cases of electromagnetic scattering.

Multicenter perfectly matched layer implementation for finite element mesh truncation

AbstractWe present the multicenter perfectly matched layer (PML) technique, which is an easy and practical conformal PML implementation, obtained by the complex coordinate stretching, to the problem of mesh truncation in the finite element method. After developing the analytical background of this method, we demonstrate its performance in electromagnetic radiation/scattering problems. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 827–832, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22258

Locally‐conformal perfectly matched layer implementation for finite element mesh truncation

AbstractIn this article, we introduce the locally conformal perfectly matched layer (PML) technique, which is an easily implementable conformal PML implementation, obtained via complex coordinate transformation, for the purpose of mesh truncation in the finite element method. After deriving the governing equations, we test this technique using electromagnetic scattering problems. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 1836–1839, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21788

A finite‐element realization of a 3‐D conformal PML

AbstractThe 3‐D conformal perfectly matched layer (PML) is realized by the finite‐element method (FEM). We outline the procedure, and demonstrate that it works well on highly curved scatterers. A significant challenge in this implementation is ensuring the continuity of the physical parameters in the PML region. We present explanation from the computer‐aided geometric design (CAGD) point of view, and a normal curvature‐based approximation is proposed to overcome this difficulty. © 2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 30: 170–173, 2001.

Conformal PML-FDTD schemes for electromagnetic field simulations: a dynamic stability study

We present a study on the dynamic stability of the perfectly matched layer (PML) absorbing boundary condition for finite-difference time-domain (FDTD) simulations of electromagnetic radiation and scattering problems in body-conformal orthogonal grids. This work extends a previous dynamic stability analysis of Cartesian, cylindrical and spherical PMLs to the case of a conformal PML. It is shown that the conformal PML defined over surface terminations with positive local radii of curvature (concave surfaces as viewed from inside the computational domain) is dynamically stable, while the conformal PML defined over surface terminations with a negative local radius (convex surfaces as viewed from inside the computational domain) is dynamically unstable. Numerical results illustrate the analysis.

Application of a hyperbolic grid generation technique to a conformal PML implementation

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.

Analytical derivation of a conformal perfectly matched absorber for electromagnetic waves

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.

Investigation of nonplanar perfectly matched absorbers for finite-element mesh truncation

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.