Perfectly Matched Layer Layers Research Articles

Numerical Solution of a One-Dimensional Nonlocal Helmholtz Equation by Perfectly Matched Layers

We consider the computation of a nonlocal Helmholtz equation by using Perfectly Matched Layer (PML). We first derive the nonlocal PML equation by extending PML modifications from the local operator to the nonlocal operator of integral form. We then give stability estimates of some weighted average value of the nonlocal Helmholtz solution and prove that (i) the weighted average value of the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the weighted average value of the nonlocal Helmholtz solution itself decays exponentially outside some domain. Particularly for a typical kernel function $\gamma_1(s)=\frac12 e^{-| s|}$, we obtain the Green's function of the nonlocal Helmholtz equation, and use the Green's function to further prove that (i) the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the nonlocal Helmholtz solution itself decays exponentially outside some domain. Based on our theoretical analysis, the truncated nonlocal problems are discussed and an asymptotic compatibility scheme is also introduced to solve the resulting truncated problems. Finally, numerical examples are provided to verify the effectiveness and validation of our nonlocal PML strategy and theoretical findings.

Time-domain PML formulation for modeling viscoelastic waves with Rayleigh-type damping in an unbounded domain: Theory and application in ABAQUS

The Perfectly Matched Layer (PML) method is a powerful approach to absorb outgoing waves from a finite computational domain in various media, such as elastic, poroelastic, anisotropic, and viscoelastic, by means of layers of artificial material placed at the finite domain boundaries. However, its use in most finite-element method (FEM) codes, which take into account viscous damping by employing only the Rayleigh and Lindsday damping formulation, is inadequate. This paper introduces a viscoelastic Perfectly Matched Layer (PML) formulation with Rayleigh-type damping for finite-element time-domain analyses. The PML formulations in the frequency- and time-domain are derived using a two-parameter complex coordinate stretching function. The displacement field is the only unknown variable in the formulation, so that the approach can be readily implemented in general finite-element codes. The weak form formulation in the time-domain is implemented in ABAQUS/Standard by merging it with a user-defined element (UEL) subroutine in Fortran90. The UEL is developed for 2D plane-strain (PE4ML) and plane-stress (PS4ML) four-node, linear, isoparametric, quadrilateral elements. The Hilber-Hughes-Taylor implicit time integration scheme is used to evaluate the unknown displacements, velocities, and accelerations. The validity and efficiency of the viscoelastic PML formulation and the UEL subroutine are examined with two numerical examples, namely a single-layer and a multi-layered soil profile with different damping ratios and properties. The results also highlight the poor performance of elastic PML layers, when they are placed at the boundaries of a viscoelastic interior domain, because, as the damping ratio increases, the reflection at the interface of the interior domain and the elastic PML domain increases. Long-time stability of the model is also examined, and no instabilities are observed.

Ellipsoidal perfectly matched layers for acoustics

In this talk, we present a novel ellipsoidal formulation and massively parallel implementation of a perfectly matched layer (PML) for acoustics and structural acoustics. Perfectly matched layers and infinite elements are commonly used for finite element simulations of acoustic waves on unbounded domains. PML is a more recent technology that is gaining popularity due to ease of implementation and effectiveness as an absorbing boundary condition. In this study, we extend well-known curvilinear formulations for PML to ellipsoidal domains, thus allowing for a minimal volume encapsulation of the structure of interest. We discuss the issues involved in parallelization and compare the performance of PML against infinite elements on a set of representative acoustic problems on exterior domains. We examine the conditioning of the linear systems generated by the two techniques by examining the number of Krylov-iterations needed for convergence to a fixed solver tolerance. We also examine the scalability of the methods in terms of the number of Helmholtz solver iterations as the number of PML layers and infinite element order are increased. [Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL850000.]

Frequency-domain acoustic-wave modeling with hybrid absorbing boundary conditions

Acoustic waves are modeled using a 2D frequency-domain finite-difference scheme with three different artificial damping boundaries: the absorbing boundary condition (ABC), perfectly matched layer (PML), and a hybrid strategy combining PML and ABC. We evaluated the features of each simulation for a given artificial boundary condition for homogeneous and heterogeneous media. For shorter PML layers, the hybrid absorbing scheme gave better results when compared with the traditional PML technique, with a significant reduction of the spurious reflections and a shorter computational run time.

Comparative evaluation of PML technique for hydrodynamic impact loading on spar-type floating platform

The structural dynamic stability of spar-type floating platform for offshore wind turbine subject to hydrodynamic impact is numerically investigated by the dynamic response analysis. The unbounded flow domain of sea water is truncated to a bounded finite domain and the reflection of out-going hydrodynamic impact wave at the artificial boundary of truncated domain is minimized by the perfectly matched layer (PML) technique. The generalized transport equations governing the non-viscous compressible water flow is split into three PML equations by introducing the direction-wise absorption coefficients and the state variables. The coupled fluid–structure interaction problem is approximated by the iterative Eulerian FVM–Lagrangian FEM with the Euler–Lagrange coupling scheme. The impact-induced hydrodynamic pressure is calculated by the JWL equation of state and the mixture of un-reacted explosive and reaction products is defined by the reacted volume fraction. It is confirmed from the numerical experiments that the wave reflection phenomenon at the artificial boundary is substantially reduced when compared with the case without using the PML technique. And, the remarkable amplitude difference and the fluctuation with several subsequent peaks in the time responses of rigid body motion and effective stress of the floating platform are successfully suppressed when PML layers are used.

Stability of finite difference numerical simulations of acoustic logging-while-drilling with different perfectly matched layer schemes

In acoustic logging-while-drilling (ALWD) finite difference in time domain (FDTD) simulations, large drill collar occupies, most of the fluid-filled borehole and divides the borehole fluid into two thin fluid columns (radius ∼27 mm). Fine grids and large computational models are required to model the thin fluid region between the tool and the formation. As a result, small time step and more iterations are needed, which increases the cumulative numerical error. Furthermore, due to high impedance contrast between the drill collar and fluid in the borehole (the difference is >30 times), the stability and efficiency of the perfectly matched layer (PML) scheme is critical to simulate complicated wave modes accurately. In this paper, we compared four different PML implementations in a staggered grid finite difference in time domain (FDTD) in the ALWD simulation, including field-splitting PML (SPML), multiaxial PML(MPML), non-splitting PML (NPML), and complex frequency-shifted PML (CFS-PML). The comparison indicated that NPML and CFS-PML can absorb the guided wave reflection from the computational boundaries more efficiently than SPML and M-PML. For large simulation time, SPML, M-PML, and NPML are numerically unstable. However, the stability of M-PML can be improved further to some extent. Based on the analysis, we proposed that the CFS-PML method is used in FDTD to eliminate the numerical instability and to improve the efficiency of absorption in the PML layers for LWD modeling. The optimal values of CFS-PML parameters in the LWD simulation were investigated based on thousands of 3D simulations. For typical LWD cases, the best maximum value of the quadratic damping profile was obtained using one d 0. The optimal parameter space for the maximum value of the linear frequency-shifted factor (α 0) and the scaling factor (β 0) depended on the thickness of the PML layer. For typical formations, if the PML thickness is 10 grid points, the global error can be reduced to <1% using the optimal PML parameters, and the error will decrease as the PML thickness increases.

Optimal configurations for perfectly matched layers in FDTD simulations

To be effective a perfectly matched layer (PML) must be reflectionless to all impinging waves over a broad band of frequencies whilst not overly increasing the computational burden. This can be difficult to achieve as one must find a balance between the discretization error, the number of PML layers and the reflection coefficient. This work presents an optimized solution for simulating PMLs for the finite-difference time-domain (FDTD) technique using both the mono-objective genetic algorithm (GA) and multiobjective genetic algorithm (MGA). The main goal is to implement a tool capable of finding the best choice of parameters for the conductivity profile to achieve a minimum for both the reflection and computation time

On Optimal Finite-Difference Approximation of PML

A technique derived from two related methods suggested earlier by some of the authors for optimization of finite-difference grids and absorbing boundary conditions is applied to discretization of perfectly matched layer (PML) absorbing boundary conditions for wave equations in Cartesian coordinates. We formulate simple sufficient conditions for optimality and implement them. It is found that the minimal error can be achieved using pure imaginary coordinate stretching. As such, the PML discretization is algebraically equivalent to the rational approximation of the square root on [0,1] conventionally used for approximate absorbing boundary conditions. We present optimal solutions for two cost functions, with exponential (and exponential of the square root) rates of convergence with respect to the number of the discrete PML layers using a second order finite-difference scheme with optimal grids. Results of numerical calculations are presented.

Comment on “a PML‐FDTD formulation for the simulation of optical structures”

AbstractThe focus of this comment is to discuss an important point regarding the derivation of the wave equation in the perfectly matched layer (PML). It is demonstrated that the distance dependency of the PML conductivity profile should be taken into consideration when deriving the wave equation in the PML region. Significant improvements in the absorbing performance are achieved with a small number of PML layers. A numerical test is carried out in one dimension to validate the proposed formulations. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 36: 74–76, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10677

Split-field and anisotropic-medium PML-FDTD implementations for inhomogeneous media

In this paper, we present three-dimensional finite-difference time-domain (FDTD) algorithms for transient simulation of electromagnetic-wave propagation in arbitrary inhomogeneous media, which incorporate the perfectly matched layer (PML) absorbing boundary condition. We discuss the choice of constitutive parameters inside the PML layers to match the interior inhomogeneous media for planar interfaces and corner regions. We illustrate the method using both a split-fleld PML-FDTD formulation in Cartesian coordinates and an anisotropic medium (unsplit) PML-FDTD formulation in cylindrical coordinates.

Application of PML to open boundary problem using TLM method

Numerical results obtained by directly applying the perfectly-matched layer (PML) to an open boundary problem, i.e. the calculation of PCB crosstalk using the TLM, are presented. While evanescent waves cannot be absorbed even by a large number of PML layers, a single layer is sufficient provided it is placed sufficiently far away.

Use of PML absorbing layers for the truncation of the head model in cellular telephone simulations

An efficient implementation of the perfectly matched layer (PML) boundary has been used to truncate a 3-mm resolution head model used for cellular telephone simulations. An extensive analysis of the model truncation effects along all three axes has been performed. A basic observation is that a considerable fraction of the power radiated by a cellular telephone is absorbed in the proximal ear region, and there is no interest for safety certification and antenna design in retaining electromagnetic-field information in the weakly exposed regions. The authors have progressively reduced the finite difference time-domain space in the ear-to-ear, back-to-front, and bottom-to-top directions by embedding the weakly exposed sides of the head in the PML layers. Results show that, at the lower frequency of 835 MHz, only truncations in the ear-to-ear direction is appropriate for specific-absorption-rate (SAR) accuracy. However, at the personal communication system frequency of 1900 MHz, 1- and 10-g SARs within 1% of accuracy can be obtained by retaining just 4% of the original volume of the head model. This method indicates that high-resolution cellular telephone simulations can be performed with tremendous savings in execution times and memory requirements. All of the SAR results presented in this paper have been obtained with a laptop computer, and execution times as low as 1 min have been obtained for the fully optimized simulations at 1900 MHz. Furthermore, it is shown that by using a truncated half-model, it is possible to obtain accurate radiation patterns at both frequencies of 835 and 1900 MHz. Since both the SAR evaluation and radiation pattern calculation are needed for new antenna design, this should result in a highly efficient algorithm for electromagnetic design of new personal wireless devices.

Extending PML absorbing boundary condition to truncate microstrip line in nonuniform 3D FDTD grid

The Berenger's perfectly matched layer (PML) absorbing boundary condition is applied to terminate a microstrip line with metal strip and multilayer dielectric media extending into the PML regions. In order to handle the inhomogeneous properties of the PML layers better, the integral form of the Maxwell's equations, approximated by central-difference expressions instead of exponential-difference expressions, are utilized to deduce the finite-difference time-domain (FDTD) equations for updating field components inside PML media. The derived formulas are proven to be flexible and convenient for generating and handling the PML layers in a nonuniform FDTD grid. Simulation results show that the proposed multilayer PML boundary condition can effectively absorb the outgoing wave with at least -80-dB reflection coefficient under a complicated boundary situation.

On the optimal design of the PML absorbing boundary condition for the FDTD code

An analytical method to predict and optimize the performance of Berenger's (see J. Computat. Phys., vol.114, p.185-200, 1994) perfectly matched layer (PML) absorbing boundary condition (ABC) for finite-difference time-domain (FDTD) simulations is described. The shape of the conductivity in the PML layers has to be chosen carefully to obtain the best performance for a given number of layers. The relative error is shown to be the composite of two distinct effects: the theoretical reflection coefficient given by the PML layers backed by a metal plane and the second-order error in the differential intrinsic in the FDTD formulation. A theoretical expression to evaluate this error as a function of the number of PML layers and the shape of the conductivity is given, and the result is compared to that obtained for several FDTD test cases. The good agreement of the shapes of the theoretical and numerically derived curves allows the use of the theoretical formulation to optimize the PML region as a function of the shape of the conductivity, resolution, and number of layers.