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Journal articles on the topic 'Perfectly Matched Layers (PML)'

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Published: 26 January 2024

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1

Zhang, Jianfeng, and Hongwei Gao. "Irregular perfectly matched layers for 3D elastic wave modeling." GEOPHYSICS 76, no. 2 (March 2011): T27—T36. http://dx.doi.org/10.1190/1.3533999.

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We have developed a perfectly matched layer (PML) absorbing boundary condition that can be imposed along an arbitrary geometric boundary in 3D elastic wave modeling. The scheme is developed by using the local coordinate system-based PML splitting equations and integral approach of the PML equations under a discretization of tetrahedral grids. However, no explicit coordinate transformations arise. The local coordinate system-based PML splitting equations make it possible to decay incident waves around the direction normal to the irregular geometric boundaries, instead of a coordinate axis direction. Based on the resulting 3D irregular PML model, we can flexibly construct the computational domain with smaller nodes by cutting uninterested zones. This results in significant reductions in computational cost and memory requirements for 3D simulations. By building a smooth artificial boundary, the irregular PML model can avoid the respective treatments to the edges and corners of the artificial boundaries. Also, the irregular PML model may reduce the grazing incidence that makes the PML model less efficient by changing the shapes of the artificial boundaries. Numerical testing was used to demonstrate the performance of the irregular PML model.
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2

Ge, Ju, Liping Gao, and Rengang Shi. "Well-Designed Termination Wall of Perfectly Matched Layers for ATS-FDTD Method." International Journal of Antennas and Propagation 2019 (June 2, 2019): 1–6. http://dx.doi.org/10.1155/2019/6343641.

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This paper presents a well-designed termination wall for the perfectly matched layers (PML). This termination wall is derived from Mur’s absorbing boundary condition (ABC) with special difference schemes. Numerical experiments illustrate that PML and the termination wall works well with ATS-FDTD(Shi et al. 2015). With the help of termination wall, perfectly matched layers can be decreased to two layers only; meanwhile, the reflection error still reaches -60[dB] when complex waveguide is simulated by ATS-FDTD.
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3

Bunting, Gregory, Arun Prakash, Timothy Walsh, and Clark Dohrmann. "Parallel Ellipsoidal Perfectly Matched Layers for Acoustic Helmholtz Problems on Exterior Domains." Journal of Theoretical and Computational Acoustics 26, no. 02 (June 2018): 1850015. http://dx.doi.org/10.1142/s2591728518500159.

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Exterior acoustic problems occur in a wide range of applications, making the finite element analysis of such problems a common practice in the engineering community. Various methods for truncating infinite exterior domains have been developed, including absorbing boundary conditions, infinite elements, and more recently, perfectly matched layers (PML). PML are gaining popularity due to their generality, ease of implementation, and effectiveness as an absorbing boundary condition. PML formulations have been developed in Cartesian, cylindrical, and spherical geometries, but not ellipsoidal. In addition, the parallel solution of PML formulations with iterative solvers for the solution of the Helmholtz equation, and how this compares with more traditional strategies such as infinite elements, has not been adequately investigated. In this paper, we present a parallel, ellipsoidal PML formulation for acoustic Helmholtz problems. To faciliate the meshing process, the ellipsoidal PML layer is generated with an on-the-fly mesh extrusion. Though the complex stretching is defined along ellipsoidal contours, we modify the Jacobian to include an additional mapping back to Cartesian coordinates in the weak formulation of the finite element equations. This allows the equations to be solved in Cartesian coordinates, which is more compatible with existing finite element software, but without the necessity of dealing with corners in the PML formulation. Herein we also compare the conditioning and performance of the PML Helmholtz problem with infinite element approach that is based on high order basis functions. On a set of representative exterior acoustic examples, we show that high order infinite element basis functions lead to an increasing number of Helmholtz solver iterations, whereas for PML the number of iterations remains constant for the same level of accuracy. This provides an additional advantage of PML over the infinite element approach.
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Chen, Yong H., Weng Cho Chew, and Michael L. Oristaglio. "Application of perfectly matched layers to the transient modeling of subsurface EM problems." GEOPHYSICS 62, no. 6 (November 1997): 1730–36. http://dx.doi.org/10.1190/1.1444273.

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Berenger's perfectly matched layers (PML) have been found to be very efficient as a material absorbing boundary condition (ABC) for finite‐difference time‐domain (FDTD) modeling of lossless media. In this paper, we apply the PML technique to truncate the simulation region of conductive media. Examples are given to show some possible applications of the PML technique to subsurface problems with lossy media. To apply the PML ABC for lossy media, we first modify the original 3-D Maxwell's equations to achieve PML at the boundaries of the simulation region. The modified equations are then solved by using a staggered grid with a central‐differencing scheme. A 3-D FDTD code has been written on the basis of our PML formulation to simulate the electromagnetic field responses of a dipole source in both lossless and lossy media. The code is first tested against analytical solutions for homogeneous media of different losses and then applied to some subsurface problems, such as a geological fault and a buried gas tank. Very interesting propagation and scattering phenomena are observed from the simulation results. Some analyses are also given to explain the physical phenomena of the calculated waveforms.
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5

He, Yanbin, Tianning Chen, Jinghuai Gao, and Zhaoqi Gao. "Superior performance of optimal perfectly matched layers for modeling wave propagation in elastic and poroelastic media." Journal of Geophysics and Engineering 19, no. 1 (February 2022): 106–19. http://dx.doi.org/10.1093/jge/gxac002.

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Abstract The perfectly matched layer (PML) technique is a popular truncation method to model wave propagation in unbounded elastic media. Both numerical efficiency and high stability are important improvement areas in the field. In this study, we extend the optimal PML, previously proposed for acoustic media, to elastic and poroelastic media, which turns out to be more efficient and flexible than the classical PML. We investigate the accuracy and stability of the optimal PML by comparing it with the classical PML in several scenarios. First, the effectiveness of the optimal PML is studied using frequency-domain and time-domain simulations for isotropic and homogeneous elastic solids. The efficiency and accuracy of the optimal PML and classical PML are then compared across a wide range of Poisson's ratios of elastic media. The stability of the optimal PML and the classical PML is also compared taking into account the effect of the outer boundary conditions of the PML as well as the heterogeneity of the geological model. Moreover, the optimal PML is applied to poroelastic media to address the instability problem of the classical PML. Comprehensive analyses of numerical results show that the optimal PML can absorb the outgoing waves. This can be done using thinner layers and with higher accuracy than for classical PMLs. In addition, the long-time stability of optimal PML increases.
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6

Lei, Da, Liangyong Yang, Changmin Fu, Ruo Wang, and Zhongxing Wang. "The application of a novel perfectly matched layer in magnetotelluric simulations." GEOPHYSICS 87, no. 3 (March 29, 2022): E163—E175. http://dx.doi.org/10.1190/geo2020-0393.1.

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Truncation boundaries are needed when simulating a region in magnetotelluric (MT) modeling. As an efficient alternative truncation boundary, perfectly matched layers (PMLs) have been widely applied in many high-frequency wavefield simulations. However, the governing equation of most electromagnetic exploration methods is for the diffusion field, in which the conduction current is significantly greater than the displacement current. Because the wave and diffusion fields have completely different relations for the frequency and constitutive parameters, conventional PMLs, which are mainly designed for the wavefield, are not a good choice for the diffusion field. For this reason, we propose a formula for a PML that covers the entire frequency band, including the wave and the diffusion fields. It is based on a uniaxial PML. Moreover, we derive a simplified form for the diffusion field. To check the feasibility and application potential of the proposed formula for a PML in MT simulations, we have implemented PMLs using a finite-element method and compared our results with those for a conventional long-distance extended grid. The results of the 1D simulation demonstrate that PMLs can achieve high accuracy and stable performance and have a broad application range. In 2D and 3D models, the air layer is an obstacle in the application of a PML. By selecting appropriate parameters for the PML, 2D and 3D models can achieve satisfactory performance. Therefore, the proposed PML is useful for MT simulations and can achieve satisfactory truncation performance.
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7

CHEW, W. C., and Q. H. LIU. "PERFECTLY MATCHED LAYERS FOR ELASTODYNAMICS: A NEW ABSORBING BOUNDARY CONDITION." Journal of Computational Acoustics 04, no. 04 (December 1996): 341–59. http://dx.doi.org/10.1142/s0218396x96000118.

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The use of perfectly matched layers (PML) has recently been introduced by Berenger as a material absorbing boundary condition (ABC) for electromagnetic waves. In this paper, we will first prove that a fictitious elastodynamic material half-space exists that will absorb an incident wave for all angles and all frequencies. Moreover, the wave is attenuative in the second half-space. As a consequence, layers of such material could be designed at the edge of a computer simulation region to absorb outgoing waves. Since this is a material ABC, only one set of computer codes is needed to simulate an open region. Hence, it is easy to parallelize such codes on multiprocessor computers. For instance, it is easy to program massively parallel computers on the SIMD (single instruction multiple data) mode for such codes. We will show two- and three-dimensional computer simulations of the PML for the linearized equations of elastodynamics. Comparison with Liao’s ABC will be given.
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8

Bérenger, Jean-Pierre. "Perfectly Matched Layer (PML) for Computational Electromagnetics." Synthesis Lectures on Computational Electromagnetics 2, no. 1 (January 2007): 1–117. http://dx.doi.org/10.2200/s00030ed1v01y200605cem008.

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9

Hervella-Nieto, Luis M., Andrés Prieto, and Sara Recondo. "Computation of Resonance Modes in Open Cavities with Perfectly Matched Layers." Proceedings 54, no. 1 (August 18, 2020): 2. http://dx.doi.org/10.3390/proceedings2020054002.

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During the last decade, several authors have addressed that the Perfectly Matched Layers (PML) technique can be used not only for the computation of the near-field in time-dependent and time-harmonic scattering problems, but also to compute numerically the resonances in open cavities. Despite such complex resonances are not natural eigen-frequencies of the physical system, the numerical determination of this kind of eigenvalues provides information about the model, what can be used in further applications. The present work will be focused on two main specific goals—firstly, the mathematical analysis of the frequency-dependent highly non-linear eigenvalue problem associated to the computation of resonances with the standard PML technique. Second, the implementation of a robust numerical method to approximate resonances in open cavities.
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10

Cao, Da, Naohisa Inoue, and Tetsuya Sakuma. "Finite element analysis of bending waves in Mindlin plates with Perfectly Matched Layers." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 265, no. 5 (February 1, 2023): 2527–34. http://dx.doi.org/10.3397/in_2022_0355.

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It is important to determine the boundary conditions of walls and floors precisely when simulating the building acoustics. For a certain room, the extension of the spans can be considered as infinite edges. The Perfect Matched Layer(PML) is an artificial absorbing domain for the wave propagations and is widely used in finite element analysis to simulate the acoustical free field conditions right now. In this paper, an effective PML technique for the plate structure will be presented. The PML formulation will be derived based on the Mindlin plate theory and the implementation method will be introduced. This technique will be validated through the numerical experiments. The accuracy and limits of the presented technique will be discussed based on the numerical results compared with the analytical results. The results show that the presented PML technique is effective and accurate to simulate the plate as an infinite large plate. It is expected to implement the technique in the further research of the structure borne sound, such as floor impact sounds.
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11

Nissen, Anna, and Gunilla Kreiss. "An Optimized Perfectly Matched Layer for the Schrödinger Equation." Communications in Computational Physics 9, no. 1 (January 2011): 147–79. http://dx.doi.org/10.4208/cicp.010909.010410a.

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AbstractWe derive a perfectly matched layer (PML) for the Schrödinger equation using a modal ansatz. We derive approximate error formulas for the modeling error from the outer boundary of the PML and the error from the discretization in the layer and show how to choose layer parameters so that these errors are matched and optimal performance of the PML is obtained. Numerical computations in 1D and 2D demonstrate that the optimized PML works efficiently at a prescribed accuracy for the zero potential case, with a layer of width less than a third of the de Broglie wavelength corresponding to the dominating frequency.
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12

Wang, Enjiang, José M. Carcione, Jing Ba, Mamdoh Alajmi, and Ayman N. Qadrouh. "Nearly perfectly matched layer absorber for viscoelastic wave equations." GEOPHYSICS 84, no. 5 (September 1, 2019): T335—T345. http://dx.doi.org/10.1190/geo2018-0732.1.

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We have applied the nearly perfectly matched layer (N-PML) absorber to the viscoelastic wave equation based on the Kelvin-Voigt and Zener constitutive equations. In the first case, the stress-strain relation has the advantage of not requiring additional physical field (memory) variables, whereas the Zener model is more adapted to describe the behavior of rocks subject to wave propagation in the whole frequency range. In both cases, eight N-PML artificial memory variables are required in the absorbing strips. The modeling simulates 2D waves by using two different approaches to compute the spatial derivatives, generating different artifacts from the boundaries, namely, 16th-order finite differences, where reflections from the boundaries are expected, and the staggered Fourier pseudospectral method, where wraparound occurs. The time stepping in both cases is a staggered second-order finite-difference scheme. Numerical experiments demonstrate that the N-PML has a similar performance as in the lossless case. Comparisons with other approaches (S-PML and C-PML) are carried out for several models, which indicate the advantages and drawbacks of the N-PML absorber in the anelastic case.
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13

Cho, Jeahoon, Min-Seok Park, and Kyung-Young Jung. "Perfectly Matched Layer for Accurate FDTD for Anisotropic Magnetized Plasma." Journal of Electromagnetic Engineering and Science 20, no. 4 (October 31, 2020): 277–84. http://dx.doi.org/10.26866/jees.2020.20.4.277.

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In this work, we propose a stable perfectly matched layer (PML) for accurate finite-difference time-domain (FDTD) methods for analyzing electromagnetic wave propagation in the anisotropic magnetized plasma region. Toward this purpose, we apply the complex frequency-shifted PML systematically to the E-J collocated FDTD method. In specific, auxiliary PML variables are included in the matrix calculation involved in the final update equations of the E-J collocated FDTD method. Numerical examples are used to validate the proposed PML-FDTD for anisotropic magnetized plasma.
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14

Gao, Kai, and Lianjie Huang. "Optimal damping profile ratios for stabilization of perfectly matched layers in general anisotropic media." GEOPHYSICS 83, no. 1 (January 1, 2018): T15—T30. http://dx.doi.org/10.1190/geo2017-0430.1.

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Conventional perfectly matched layers (PMLs) can be unstable for certain kinds of anisotropic media. The multiaxial PML removes such instability using nonzero damping coefficients in the directions tangential with the PML interface. Although using nonzero damping profile ratios can stabilize PMLs, it is important to obtain the smallest possible damping profile ratios to minimize artificial reflections caused by these nonzero ratios, particularly for 3D general anisotropic media. Using the eigenvectors of the PML system matrix, we have developed a straightforward and efficient numerical algorithm to determine the optimal damping profile ratios to stabilize PMLs in 2D and 3D general anisotropic media. Numerical examples indicate that our algorithm provides optimal damping profile ratios to ensure the stability of PMLs and complex-frequency-shifted PMLs for elastic-wave modeling in 2D and 3D general anisotropic media.
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15

Hu, Wenyi, Aria Abubakar, and Tarek M. Habashy. "Application of the nearly perfectly matched layer in acoustic wave modeling." GEOPHYSICS 72, no. 5 (September 2007): SM169—SM175. http://dx.doi.org/10.1190/1.2738553.

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In this work, we successfully applied an alternative formulation of the perfectly matched layer (PML), the so-called nearly PML (NPML), to acoustic wave propagation modeling. The NPML formulation shows great advantages over the standard complex stretched coordinate PML. The NPML formulation deviates from the standard PML through an inexact variable change, but this fact only affects the wave behavior in the NPML layer, which is outside the region of interest. The equivalence of the wave-absorbing performance between these two PML formulations (the standard complex stretched coordinate PML formulation and the NPML formulation) in 3D Cartesian coordinates for acoustic wave propagation modeling is proved mathematically in this work. In time-domain methods, the advantages of the NPML over the standard PML were explained by both the analytical analysis and the numerical simulations in terms of implementation simplicity and computational efficiency. The computation time saving is up to 17% for the 2D example used in this work. For 3D problems, this computational saving is more significant. After theoretically analyzing the numerical reflections from the NPML and the standard PML, we concluded that these two PML formulations have exactly the same performance, even after spatial discretization. This conclusion is validated by numerical experiment. Finally, we tested the NPML in the Marmousi velocity model and found its wave-absorbing rate is high enough, even for this realistic structure.
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16

Shankar Badry, Ravi, Maruthi Kotti, and Pradeep Kumar Ramancharla. "A Comparative Study of Absorbing Layer Methods to Model Radiating Boundary Conditions for the Wave Propagation in Infinite Medium." International Journal of Engineering & Technology 7, no. 3.35 (September 2, 2018): 25. http://dx.doi.org/10.14419/ijet.v7i3.35.29141.

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Radiating boundary condition is an important consideration in the finite element modelling of unbounded media. Absorbing layer techniquessuch as Perfectly Matched Layers (PML) and Absorbing Layers by Increasing Damping (ALID) becoming popular as they are efficient in absorbing outward propagating waves energy. In this study, a comparative analysis has been carried out between PML and ALID+VABC (Absorbing Boundary conditions for Viscoelastic materials) methods. The methods are analyzedusing LS-DYNAexplicit solver and the efficiency is compared with standard solutions.The study concluded that PML requires less number of elements to model the boundary conditions when compared with ALID+VABC. But PMLrequires a smaller element length which increases overall computational time. Both the methods are efficient in absorbing the wave energy. However, PML requires additional implementation cost to solve the complex equations.
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17

Langlois, Christophe, Jean-Daniel Chazot, Emmanuel Perrey-Debain, and Benoit Nennig. "Partition of Unity Finite Element Method applied to exterior problems with Perfectly Matched Layers." Acta Acustica 4, no. 4 (2020): 16. http://dx.doi.org/10.1051/aacus/2020011.

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The Partition of Unity Finite Element Method (PUFEM) is now a well established and efficient method used in computational acoustics to tackle short-wave problems. This method is an extension of the classical finite element method whereby enrichment functions are used in the approximation basis in order to enhance the convergence of the method whilst maintaining a relatively low number of degrees of freedom. For exterior problems, the computational domain must be artificially truncated and special treatments must be followed in order to avoid or reduce spurious reflections. In recent papers, different Non-Reflecting Boundary Conditions (NRBCs) have been used in conjunction with the PUFEM. An alternative is to use the Perfectly Match Layer (PML) concept which consists in adding a computational sponge layer which prevents reflections from the boundary. In contrast with other NRBCs, the PML is not case specific and can be applied to a variety of configurations. The aim of this work is to show the applicability of PML combined with PUFEM for solving the propagation of acoustic waves in unbounded media. Performances of the PUFEM-PML are shown for different configurations ranging from guided waves in ducts, radiation in free space and half-space problems. In all cases, the method is shown to provide acceptable results for most applications, similar to that of local approximation of NRBCs.
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18

Zhang, Yongou, Zhongjian Ling, Hao Du, and Qifan Zhang. "Finite-Difference Frequency-Domain Scheme for Sound Scattering by a Vortex with Perfectly Matched Layers." Mathematics 11, no. 18 (September 18, 2023): 3959. http://dx.doi.org/10.3390/math11183959.

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Understanding the effect of vortexes on sound propagation is of great significance in the field of target detection and acoustic imaging. A prediction algorithm of the two-dimensional vortex scattering is realized based on a finite-difference frequency-domain (FDFD) numerical scheme with perfectly matched layers (PML). Firstly, the governing equation for flow–sound interaction is given based on the perturbation theory, and the FDFD program is built. Subsequently, the mesh independence is verified, and the result has a good convergence when the mesh corresponds to over 15 nodes per wavelength. Then, computational parameters of the PML are discussed to achieve better absorbing boundary conditions. Finally, the results of this algorithm are compared with previous literature data. Results show that for different cortex scattering cases, the absorption coefficient should vary linearly with the density of the medium and the incident wave frequency. When the thickness of the PML boundary is greater than 2.5 times the wavelength, the PML boundary can absorb the scattering sound effectively. This provides a reliable algorithm for the numerical study of the effect of vortexes on sound propagation.
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19

Zeng, Y. Q., J. Q. He, and Q. H. Liu. "The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media." GEOPHYSICS 66, no. 4 (July 2001): 1258–66. http://dx.doi.org/10.1190/1.1487073.

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The perfectly matched layer (PML) was first introduced by Berenger as a material absorbing boundary condition (ABC) for electromagnetic waves. In this paper, a method is developed to extend the perfectly matched layer to simulating seismic wave propagation in poroelastic media. This nonphysical material is used at the computational edge of a finite‐difference algorithm as an ABC to truncate unbounded media. The incorporation of PML in Biot’s equations is different from other PML applications in that an additional term involving convolution between displacement and a loss coefficient in the PML region is required. Numerical results show that the PML ABC attenuates the outgoing waves effectively.
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20

Liang, Chao, and Xueshuang Xiang. "Convergence of an Anisotropic Perfectly Matched Layer Method for Helmholtz Scattering Problems." Numerical Mathematics: Theory, Methods and Applications 9, no. 3 (July 20, 2016): 358–82. http://dx.doi.org/10.4208/nmtma.2016.m1505.

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AbstractThe anisotropic perfectly matched layer (APML) defines a continuous vector field outside a rectangle domain and performs the complex coordinate stretching along the vector field. Inspired by [Z. Chen et al., Inverse Probl. Imag., 7, (2013):663–678] and based on the idea of the shortest distance, we propose a new approach to construct the vector field which still allows us to prove the exponential decay of the stretched Green function without the constraint on the thickness of the PML layer. Moreover, by using the reflection argument, we prove the stability of the PML problem in the PML layer and the convergence of the PML method. Numerical experiments are also included.
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21

Jung, Kyung-Young, Hyeongdong Kim, and Kwang-Cheol Ko. "A modified perfectly matched layer (PML) for waveguide problems." Microwave and Optical Technology Letters 18, no. 5 (August 5, 1998): 360–62. http://dx.doi.org/10.1002/(sici)1098-2760(19980805)18:5<360::aid-mop16>3.0.co;2-6.

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22

Chen, Hanming, Hui Zhou, He Lin, and Shangxu Wang. "Application of perfectly matched layer for scalar arbitrarily wide-angle wave equations." GEOPHYSICS 78, no. 1 (January 1, 2013): T29—T39. http://dx.doi.org/10.1190/geo2012-0062.1.

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Arbitrarily wide-angle wave equation (AWWE) is a space domain, high-order one-way wave equation (OWWE). Its accuracy can be arbitrarily increased, and it is amenable to easy numerical implementation. Those properties make it outstanding among the existing OWWEs and further enable it to be a desirable tool for migration. We extend the perfectly matched layer (PML) to 3D scalar AWWE to provide a good approach to suppress artifacts arising at truncation boundaries. We follow the concept of complex coordinate stretching, and the derivation procedure of PML for AWWE is straightforward. An existing finite-difference scheme is adopted to fit the split PML formulation and its stability is observed through numerical examples. The performance of the developed PML condition is compared with two different wave-equation based absorbing boundary conditions. Numerical results illustrate that the PML condition used in AWWE propagator can effectively absorb the propagating waves and evanescent waves at a price of limited additional computation cost.
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23

Wang, Cong, and Jian Xun Zhang. "Convolution Approximating Perfect Matched Layer Absorbing Boundary Condition of 3D Scalar Acoustic Wave Equation." Advanced Materials Research 971-973 (June 2014): 1095–98. http://dx.doi.org/10.4028/www.scientific.net/amr.971-973.1095.

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Because traditional displacement form of 3D acoustic perfectly matched layer (PML) absorbing condition needs to split the displacement into three parts, which requires solving a third-order differential equation in time and occupies a large amount of memory. In order to solve the above problems, this paper puts forward an Convolution approximating PML absorbing boundary condition based on the previous works, and discusses the basic construction of the traditional perfectly matched layer absorbing boundary condition and the new arithmetic in detail, then the new method is compared with absorbing condition of low order paraxial approximation and traditional PML, investigating the absorbing effects of 3D acoustic wave’s numerical records.
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24

Chen, Wei, and Song Ping Wu. "Perfectly Matched Layer as an Absorbing Boundary Condition for Computational Aero-Acoustic." Advanced Materials Research 726-731 (August 2013): 3153–58. http://dx.doi.org/10.4028/www.scientific.net/amr.726-731.3153.

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Advances in Computational Aeroacoustics (CAA) depend critically on the availability of accurate, nondispersive, least dissipative computation algorithm as well as high quality numerical boundary treatments. This paper focuses on the Perfectly Matched Layer (PML) for external boundaries in CAA. To achieve low dissipation and dispersion errors, Dispersion-Relation-Preserving (DRP) Schemes are used for spatial discretization of the acoustic equations. The classical fourth-order Runge-Kutta time scheme is applied to the acoustic equations for time discretization. Four cases are given to illustrate the 2D PML equations for the linearized/nonlinear Euler equations in Cartesian coordinates and Cylindrical coordinates. The results show that the PML is effective as absorbing boundary condition. Those are basis for PML in actual computations of acoustic problems.
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Mi, Yongzhen, and Xiang Yu. "An isogeometric formulation of locally-conformal perfectly matched layer for acoustic scattering problems." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 263, no. 6 (August 1, 2021): 829–33. http://dx.doi.org/10.3397/in-2021-1660.

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This paper presents an isogeometric formulation of the locally-conformal perfectly matched layer (PML) for time-harmonic acoustic scattering problems. The new formulation is a generalization of the conventional locally-conformal PML, in which the NURBS patch supporting the PML domain is transformed from real space to complex space. This is achieved by complex coordinate stretching, based on a stretching vector field indicating the directions in which incident sound waves are absorbed. The performance of the isogeometric PML formulation is discussed through several acoustic scattering problems, spanning from one to three dimensions. It is found that the proposed method presents superior computational accuracy, high geometric adaptivity, and good robustness against challenging geometric features. The geometry-preserving ability inherent in the isogeometric framework could bring extra benefits by eliminating geometric errors that are unavoidable in the conventional PML. Meanwhile, these properties are not sensitive to the location of the sound source or the depth of the PML domain.
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26

Wu, Xinming, and Weiying Zheng. "An Adaptive Perfectly Matched Layer Method for Multiple Cavity Scattering Problems." Communications in Computational Physics 19, no. 2 (February 2016): 534–58. http://dx.doi.org/10.4208/cicp.040215.280815a.

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AbstractA uniaxial perfectly matched layer (PML) method is proposed for solving the scattering problem with multiple cavities. By virtue of the integral representation of the scattering field, we decompose the problem into a system of single-cavity scattering problems which are coupled with Dirichlet-to-Neumann maps. A PML is introduced to truncate the exterior domain of each cavity such that the computational domain does not intersect those for other cavities. Based on the a posteriori error estimates, an adaptive finite element algorithm is proposed to solve the coupled system. The novelty of the proposed method is that its computational complexity is comparable to that for solving uncoupled single-cavity problems. Numerical experiments are presented to demonstrate the efficiency of the adaptive PML method.
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Deng, Chunhua, Ma Luo, Mengqing Yuan, Bo Zhao, Mingwei Zhuang, and Qing Huo Liu. "The Auxiliary Differential Equations Perfectly Matched Layers Based on the Hybrid SETD and PSTD Algorithms for Acoustic Waves." Journal of Theoretical and Computational Acoustics 26, no. 01 (March 2018): 1750031. http://dx.doi.org/10.1142/s2591728517500311.

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The perfectly matched layer (PML) absorbing boundary condition has been proven to absorb body waves and surface waves very efficiently at non-grazing incidence. However, the traditional PML would generate large spurious reflections at grazing incidence, for example, when the sources are located near the truncating boundary and the receivers are at a large offset. In this paper, a new PML implementation is presented for the boundary truncation in three-dimensional spectral element time domain (SETD) for solving acoustic wave equations. This method utilizes pseudospectral time-domain (PSTD) method to solve first-order auxiliary differential equations (ADEs), which is more straightforward than that in the classical FEM framework.
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Zhao, Li. "The generalized theory of perfectly matched layers (GT-PML) in curvilinear co-ordinates." International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 13, no. 5 (2000): 457–69. http://dx.doi.org/10.1002/1099-1204(200009/10)13:5<457::aid-jnm377>3.0.co;2-y.

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Guo, Hong Wei, Shang Xu Wang, Nai Chuan Guo, and Wei Chen. "Wave Equation Simulation by Finite-Element Method with Perfectly Matched Layer." Advanced Materials Research 524-527 (May 2012): 96–100. http://dx.doi.org/10.4028/www.scientific.net/amr.524-527.96.

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In numerical simulation, the treatment of boundary conditions is of great significance. In this paper, we have deduced the one order governing equations of the acoustic wave finite-element method with perfectly matched layer (PML) for the first time. The one order equations are easier to realize than the two order form and have a good absorption effect. Then, we have analyzed the absorption effect of the absorbing boundary conditions (ABCs) and the PML. Finally, we get some useful conclusions.
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Kim, Dojin. "A Modified PML Acoustic Wave Equation." Symmetry 11, no. 2 (February 2, 2019): 177. http://dx.doi.org/10.3390/sym11020177.

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In this paper, we consider a two-dimensional acoustic wave equation in an unbounded domain and introduce a modified model of the classical un-split perfectly matched layer (PML). We apply a regularization technique to a lower order regularity term employed in the auxiliary variable in the classical PML model. In addition, we propose a staggered finite difference method for discretizing the regularized system. The regularized system and numerical solution are analyzed in terms of the well-posedness and stability with the standard Galerkin method and von Neumann stability analysis, respectively. In particular, the existence and uniqueness of the solution for the regularized system are proved and the Courant-Friedrichs-Lewy (CFL) condition of the staggered finite difference method is determined. To support the theoretical results, we demonstrate a non-reflection property of acoustic waves in the layers.
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Komatitsch, Dimitri, and Roland Martin. "An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation." GEOPHYSICS 72, no. 5 (September 2007): SM155—SM167. http://dx.doi.org/10.1190/1.2757586.

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The perfectly matched layer (PML) absorbing boundary condition has proven to be very efficient from a numerical point of view for the elastic wave equation to absorb both body waves with nongrazing incidence and surface waves. However, at grazing incidence the classical discrete PML method suffers from large spurious reflections that make it less efficient for instance in the case of very thin mesh slices, in the case of sources located close to the edge of the mesh, and/or in the case of receivers located at very large offset. We demonstrate how to improve the PML at grazing incidence for the differential seismic wave equation based on an unsplit convolution technique. The improved PML has a cost that is similar in terms of memory storage to that of the classical PML. We illustrate the efficiency of this improved convolutional PML based on numerical benchmarks using a finite-difference method on a thin mesh slice for an isotropic material and show that results are significantly improved compared with the classical PML technique. We also show that, as the classical PML, the convolutional technique is intrinsically unstable in the case of some anisotropic materials.
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Liu, Yikuo. "Application of perfectly matched layers in 3D transient controlled-source electromagnetic modeling by the rapid expansion method." GEOPHYSICS 85, no. 1 (November 22, 2019): E15—E26. http://dx.doi.org/10.1190/geo2018-0753.1.

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I have developed an extension of the rapid expansion method (REM) for 3D time-domain controlled-source electromagnetic modeling that includes perfectly matched layers (PMLs) as the absorbing boundary. The REM solves the time-domain electric field by a weighted summation of the Chebyshev polynomials. The results are free of temporal dispersion and accurate to the Nyquist frequency, yet the domain of Chebyshev polynomials lacks an accurate absorbing boundary. I find that by introducing a fictitious magnetic field in the Chebyshev domain, the recursion of the Chebyshev polynomials obeys a discrete coupled wave equation, which shares a similarity with the propagation of EM waves in a lossless medium. The time and frequency components in the Chebyshev domain are derived based on the eigenvalues of the propagation matrix, and the PML theory designed for EM waves can be extended to the Chebyshev domain in a straightforward way. Numerical tests against analytical solution and spectral methods show an excellent agreement after PML solves the boundary problem in the Chebyshev domain, which demonstrates the accuracy of the REM algorithm and the usefulness of the PML absorbing boundary.
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Ji, Jinzu, and Feiliang Liu. "Study on Influence of Thickness and Electromagnetic Parameter of Perfectly Matched Layer (PML)." Open Electrical & Electronic Engineering Journal 8, no. 1 (September 16, 2014): 50–55. http://dx.doi.org/10.2174/1874129001408010050.

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Perfectly matched layer (PML) absorbing boundary condition (ABC) is an important technique in finitedifference time-domain (FDTD) when simulating infinite area's electromagnetic behaviour in finite area. A big enough computational area was designed in which there is no influence of reflective wave. The radiation of sinusoidal line source in this area was the benchmark for assessment of reflectivity. The total wave with boundary condition subtracted by benchmark was reflective wave. Influences of the PML absorbing layer's thickness, electromagnetic parameter distribution and loss tangent were studied via this method. The results show that 15-mesh-layer may elaborate the PML's ability and the reflectivity may be only 0.003. Better absorbing effect can be achieved when loss tangent is distributed in second power and the corresponding loss tangent in cut off boundary is 2.5.
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Yang, Jixin, Xiao He, and Hao Chen. "Processing the Artificial Edge-Effects for Finite-Difference Frequency-Domain in Viscoelastic Anisotropic Formations." Applied Sciences 12, no. 9 (May 7, 2022): 4719. http://dx.doi.org/10.3390/app12094719.

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Real sedimentary media can usually be characterized as transverse isotropy. To reveal wave propagation in the true models and improve the accuracy of migrations and evaluations, we investigated the algorithm of wavefield simulations in an anisotropic viscoelastic medium. The finite difference in the frequency domain (FDFD) has several advantages compared with that in the time domain, e.g., implementing multiple sources, multi-scaled inversion, and introducing attenuation. However, medium anisotropy will lead to the complexity of the wavefield in the calculation. The damping profile of the conventional absorption boundary is only defined in one single direction, which produces instability when the wavefields of strong anisotropy are reflected on that truncated boundary. We applied the multi-axis perfectly matched layer (M-PML) to the wavefield simulations in anisotropic viscoelastic media to overcome this issue, which defines the damping profiles along different axes. In the numerical examples, we simulated seismic wave propagation in three viscous anisotropic media and focused on the wave attenuation in the absorbing layers using time domain snapshots. The M-PML was more effective for wave absorption compared to the conventional perfectly matched layer (PML). In strongly anisotropic media, the PML became unstable, and prominent reflections appeared at truncated boundaries. In contrast, the M-PML remained stable and efficient in the same model. Finally, the modeling of the stratified cross-well model showed the applicability of this proposed algorithm to heterogeneous viscous anisotropic media. The numerical algorithm can analyze wave propagation in viscoelastic anisotropic media. It also provides a reliable forward operator for waveform inversion, wave equation travel-time inversion, and seismic migration in anisotropic viscoelastic media.
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Chen, Hanming, Hui Zhou, and Yanqi Li. "Application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide-angle wave equation." GEOPHYSICS 79, no. 6 (November 1, 2014): T313—T321. http://dx.doi.org/10.1190/geo2014-0103.1.

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A classical split perfectly matched layer (PML) method has recently been applied to the scalar arbitrarily wide-angle wave equation (AWWE) in terms of displacement. However, the classical split PML obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence. Our goal was to improve the computational efficiency of AWWE and to enhance the suppression of edge reflections by applying a convolutional PML (CPML). We reformulated the original AWWE as a first-order formulation and incorporated the CPML with a general complex frequency shifted stretching operator into the renewed formulation. A staggered-grid finite-difference (FD) method was adopted to discretize the first-order equation system. For wavefield depth continuation, the first-order AWWE with the CPML saved memory compared with the original second-order AWWE with the conventional split PML. With the help of numerical examples, we verified the correctness of the staggered-grid FD method and concluded that the CPML can efficiently absorb evanescent and propagating waves.
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Zeng, Chong, Jianghai Xia, Richard D. Miller, and Georgios P. Tsoflias. "Application of the multiaxial perfectly matched layer (M-PML) to near-surface seismic modeling with Rayleigh waves." GEOPHYSICS 76, no. 3 (May 2011): T43—T52. http://dx.doi.org/10.1190/1.3560019.

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Perfectly matched layer (PML) absorbing boundaries are widely used to suppress spurious edge reflections in seismic modeling. When modeling Rayleigh waves with the existence of the free surface, the classical PML algorithm becomes unstable when the Poisson’s ratio of the medium is high. Numerical errors can accumulate exponentially and terminate the simulation due to computational overflows. Numerical tests show that the divergence speed of the classical PML has a nonlinear relationship with the Poisson’s ratio. Generally, the higher the Poisson’s ratio, the faster the classical PML diverges. The multiaxial PML (M-PML) attenuates the waves in PMLs using different damping profiles that are proportional to each other in orthogonal directions. The proportion coefficients of the damping profiles usually vary with the specific model settings. If they are set appropriately, the M-PML algorithm is stable for high Poisson’s ratio earth models. Through numerical tests of 40 models with Poisson’s ratios that varied from 0.10 to 0.49, we found that a constant proportion coefficient of 1.0 for the x- and z-directional damping profiles is sufficient to stabilize the M-PML for all 2D isotropic elastic cases. Wavefield simulations indicate that the instability of the classical PML is strongly related to the wave phenomena near the free surface. When applying the multiaxial technique only in the corners of the PML near the free surface, the original M-PML technique can be simplified without losing its stability. The simplified M-PML works efficiently for homogeneous and heterogeneous earth models with high Poisson’s ratios. The analysis in this paper is based on 2D finite difference modeling in the time domain that can easily be extended into the 3D domain with other numerical methods.
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Connolly, David P., Antonios Giannopoulos, and Michael C. Forde. "A higher order perfectly matched layer formulation for finite-difference time-domain seismic wave modeling." GEOPHYSICS 80, no. 1 (January 1, 2015): T1—T16. http://dx.doi.org/10.1190/geo2014-0157.1.

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We have developed a higher order perfectly matched layer (PML) formulation to improve the absorption performance for finite-difference time-domain seismic modeling. First, we outlined a new unsplit “correction” approach, which allowed for traditional, first-order PMLs to be added directly to existing codes in a straightforward manner. Then, using this framework, we constructed a PML formulation that can be used to construct higher order PMLs of arbitrary order. The greater number of degrees of freedom associated with the higher order PML allow for enhanced flexibility of the PML stretching functions, thus potentially facilitating enhanced absorption performance. We found that the new approach can offer increased elastodynamic absorption, particularly for evanescent waves. We also discovered that the extra degrees of freedom associated with the higher order PML required careful optimization if enhanced absorption was to be achieved. Furthermore, these extra degrees of freedom increased the computational requirements in comparison with first-order schemes. We reached our formulations using one compact equation thus increasing the ease of implementation. Additionally, the formulations are based on a recursive integration approach that reduce PML memory requirements, and do not require special consideration for corner regions. We tested the new formulations to determine their ability to absorb body waves and surface waves. We also tested standard staggered grid stencils and rotated staggered grid stencils.
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38

Zhang, Bo. "PML’s Optimal Choice in Solving Helmholtz Equation." Applied Mechanics and Materials 39 (November 2010): 312–16. http://dx.doi.org/10.4028/www.scientific.net/amm.39.312.

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To solve truncation questions of calculation area(unbounded) of Helmhotlz equation, Berenger first proposed concept of Perfectly Matched Layer(PML) in 1994, the method optimizes boundary conditions and reduces computation quantities greatly. By choosing constants p,d,e of PML parameters , we obtain an optimal PML parameter in this paper . The final numerical experiments show that the result obtained by the PML parameter is almost same as accurate result of references [4].
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39

Wang, Tsili, and Xiaoming Tang. "Finite‐difference modeling of elastic wave propagation: A nonsplitting perfectly matched layer approach." GEOPHYSICS 68, no. 5 (September 2003): 1749–55. http://dx.doi.org/10.1190/1.1620648.

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In this paper, we present a nonsplitting perfectly matched layer (NPML) method for the finite‐difference simulation of elastic wave propagation. Compared to the conventional split‐field approach, the new formulation solves the same set of equations for the boundary and interior regions. The nonsplitting formulation simplifies the perfectly matched layer (PML) algorithm without sacrificing the accuracy of the PML. In addition, the NPML requires nearly the same amount of computer storage as does the split‐field approach. Using the NPML, we calculate dipole and quadrupole waveforms in a logging‐while‐drilling environment. We show that a dipole source produces a strong pipe flexural wave that distorts the formation arrivals of interest. A quadrupole source, however, produces clean formation arrivals. This result indicates that a quadrupole source is more advantageous over a dipole source for shear velocity measurement while drilling.
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Martin, Roland, Dimitri Komatitsch, and Abdelâziz Ezziani. "An unsplit convolutional perfectly matched layer improved at grazing incidence for seismic wave propagation in poroelastic media." GEOPHYSICS 73, no. 4 (July 2008): T51—T61. http://dx.doi.org/10.1190/1.2939484.

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The perfectly matched layer (PML) absorbing technique has become popular in numerical modeling in elastic or poroelastic media because of its efficiency in absorbing waves at nongrazing incidence. However, after numerical discretization, at grazing incidence, large spurious oscillations are sent back from the PML into the main domain. The PML then becomes less efficient when sources are located close to the edge of the truncated physical domain under study, for thin slices or for receivers located at a large offset. We develop a PML improved at grazing incidence for the poroelastic wave equation based on an unsplit convolutional formulation of the equation as a first-order system in velocity and stress. We show its efficiency for both nondissipative and dissipative Biot porous models based on a fourth-order staggered finite-difference method used in a thin mesh slice. The results obtained are improved significantly compared with those obtained with the classical PML.
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Tang, Jinxuan, Hui Zhou, Chuntao Jiang, Muming Xia, Hanming Chen, and Jinxin Zheng. "A Perfectly Matched Layer Technique Applied to Lattice Spring Model in Seismic Wavefield Forward Modeling for Poisson’s Solids." Bulletin of the Seismological Society of America 112, no. 2 (November 23, 2021): 608–21. http://dx.doi.org/10.1785/0120210166.

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ABSTRACT As a complementary way to traditional wave-equation-based forward modeling methods, lattice spring model (LSM) is introduced into seismology for wavefield modeling owing to its remarkable stability, high-calculation accuracy, and flexibility in choosing simulation meshes, and so forth. The LSM simulates seismic-wave propagation from a micromechanics perspective, thus enjoying comprehensive characterization of elastic dynamics in complex media. Incorporating an absorbing boundary condition (ABC) is necessary for wavefield modeling to avoid the artificial reflections caused by truncated boundaries. To the best of our knowledge, the perfectly matched layer (PML) method has been a routine ABC in the wave-equation-based numerical modeling of wave physics. However, it has not been used in the nonwave-equation-based LSM simulations. In this work, we want to apply PML to LSM to attenuate the boundary reflections. We divide the whole simulation region into PML region and inner region, PML region surrounds the inner region. To incorporate PML to LSM, we establish elastic-wave equations corresponding to LSM. The simulation in the PML region is conducted using the established wave equations and the simulation in the inner region is conducted using LSM. Three simulation examples show that the PML scheme is effective and outperforms Gaussian ABC.
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42

Huang, W. P., C. L. Xu, W. Lui, and K. Yokoyama. "The perfectly matched layer (PML) boundary condition for the beam propagation method." IEEE Photonics Technology Letters 8, no. 5 (May 1996): 649–51. http://dx.doi.org/10.1109/68.491568.

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HÜPPE, ANDREAS, and MANFRED KALTENBACHER. "STABLE MATCHED LAYER FOR THE ACOUSTIC CONSERVATION EQUATIONS IN THE TIME DOMAIN." Journal of Computational Acoustics 20, no. 01 (March 2012): 1250004. http://dx.doi.org/10.1142/s0218396x11004511.

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In recent years the development of free field radiation conditions in the time domain has become a topic of intensive research. Perfectly matched layer (PML) approaches for the frequency domain are well known. In the time domain, on the other hand, they suffer in many cases from highly increased complexity and instabilities. In this paper, we introduce a PML for the conservation equations of linear acoustics. The used formulation requires three auxiliary variables in 3D and circumvents thereby convolution integrals and higher order time derivatives. Furthermore, we prove the weak stability of the proposed formulation and show their good absorption properties by means of numerical examples.
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Dietiker, J.-F., K. A. Hoffmann, M. Papadakis, and R. Agarwal. "Development of Three-Dimensional PML Boundary Conditions for Aeroacoustics Applications." International Journal of Aeroacoustics 1, no. 3 (September 2002): 307–27. http://dx.doi.org/10.1260/147547202320962600.

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Perfectly Matched Layer (PML) boundary conditions are derived in generalized curvilinear coordinates for three-dimensional aeroacoustic applications. The resulting governing equations are solved numerically by a four-stage Runge-Kutta scheme, with 4th/6th order compact finite difference formulation. The PML equations are programmed in a subroutine, which is easily incorporated to the main program LINEULER (Linearized Euler's equation solver). Two and three-dimensional benchmarks problems are solved to investigate the efficiency and accuracy of the PML boundary conditions. Investigations on the PML parameters have been conducted to determine the optimum combination of parameters used in the computations.
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Yue, Yong Qing, Chun Hui Zhu, and Nai Xing Feng. "Z-Transform Implementation of the CFS-PML for Truncating 3D Meta-Material FDTD Domains." Advanced Materials Research 986-987 (July 2014): 3–7. http://dx.doi.org/10.4028/www.scientific.net/amr.986-987.3.

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Efficient Z-transform implementa-tion of the complex frequency-shifted perfectly matched layer (CFS-PML) based on the stretched coordinate PML (SC-PML) formulations and the D-B formulations is proposed for truncating meta-material finite-difference time-domain (FDTD) lattices. In the proposed PML formulations, the Z-transform method is incorporated into the CFS-PML FDTD implementation. The main advantage of the proposed formulations can allow direct FDTD implementation of the Maxwell’s equations in the PML regions. A numerical test has been carried out in a three dimensions (3-D) FDTD domain to validate the proposed formulations. It is shown that the proposed formulations with CFS scheme are efficient in holding good absorbing performances.
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Pan, Guangdong, and Aria Abubakar. "Iterative solution of 3D acoustic wave equation with perfectly matched layer boundary condition and multigrid preconditioner." GEOPHYSICS 78, no. 5 (September 1, 2013): T133—T140. http://dx.doi.org/10.1190/geo2012-0287.1.

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We tested a biconjugate gradient stabilized (BiCGSTAB) solver using a multigrid-based preconditioner for solving the acoustic wave (Helmholtz) equation in the frequency domain. The perfectly matched layer (PML) method was used as the radiation boundary condition (RBC). The equation was discretized using either a second- or fourth-order finite-difference (FD) scheme. The convergence of an iterative solver depended strongly on the RBC used because the spectrum of the discretized equation also depends on it. We used a geometric multigrid approach to construct a preconditioner for our FD frequency-domain (FDFD) forward solver equipped with the PML boundary condition. For efficiency, this preconditioner was only constructed using a second-order FD scheme with negligible attenuation inside the PML domain. The preconditioner was used for accelerating the convergence rate of the FDFD forward solver for cases when the discretization grids were oversampled (i.e., when the number of discretization points per minimum wavelength was greater than 10). The number of multigrid levels was also chosen adaptively depending on the number of discretization grids. We found that the multigrid preconditioner can speed up the total computational time of the BiCGSTAB solver for oversampled cases or at low frequencies. We also observed that the BiCGSTAB solver using an accurate PML boundary condition converged for realistic SEG benchmark models at high frequencies.
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Ma, Xiao, Yangjia Li, and Jiaxing Song. "A stable auxiliary differential equation perfectly matched layer condition combined with low-dispersive symplectic methods for solving second-order elastic wave equations." GEOPHYSICS 84, no. 4 (July 1, 2019): T193—T206. http://dx.doi.org/10.1190/geo2018-0572.1.

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The stable implementation of the perfectly matched layer (PML), one of the most effective and popular artificial boundary conditions, has attracted much attention these years. As a type of low-dispersive and symplectic method for solving seismic wave equations, the nearly-analytic symplectic partitioned Runge-Kutta (NSPRK) method has been combined with split-field PML (SPML) and convolutional complex-frequency shifted PML (C-CFS-PML) previously to model acoustic and short-time elastic wave modelings, not yet successfully applied to long-time elastic wave propagation. In order to broaden the application of NSPRK and more general symplectic methods for second-order seismic models, we formulate an auxiliary differential equation (ADE)-CFS-PML with a stabilizing grid compression parameter. This includes deriving the ADE-CFS-PML equations and formulating an adequate time integrator to properly embed their numerical discretizations in the main symplectic numerical methods. The resulting (N)SPRK+ADE-CFS-PML algorithm can help break through the constraint of at most second-order temporal accuracy that used to be imposed on SPML and C-CFS-PML. Especially for NSPRK, we implement the strategy of neglecting the treatment of third-order spatial derivatives in the PML domain and obtain an efficient absorption effect. Related acoustic and elastic wave simulations illustrate the enhanced numerical accuracy of our ADE-CFS-PML compared with SPML and C-CFS-PML. The elastic wave simulation in a homogeneous isotropic medium shows that compared to NSPRK+C-CFS-PML, the NSPRK+ADE-CFS-PML is numerically stable throughout a simulation time of 2 s. The synthetic seismograms of the 2D acoustic SEG salt model and the two-layer elastic model demonstrate the effectiveness of NSPRK+ADE-CFS-PML for complex elastic models. The stabilization effect of the grid compression parameter is verified in the final homogeneous isotropic elastic model with free-surface boundary.
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Duan, Xiaoqi, Xue Jiang, and Weiying Zheng. "Exponential convergence of Cartesian PML method for Maxwell’s equations in a two-layer medium." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 3 (April 9, 2020): 929–56. http://dx.doi.org/10.1051/m2an/2019082.

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The perfectly matched layer (PML) method is extensively studied for scattering problems in homogeneous background media. However, rigorous studies on the PML method in layered media are very rare in the literature, particularly, for three-dimensional electromagnetic scattering problems. Cartesian PML method is favorable in numerical solutions since it is apt to deal with anisotropic scatterers and to construct finite element meshes. Its theories are more difficult than circular PML method due to anisotropic wave-absorbing materials. This paper presents a systematic study on the Cartesian PML method for three-dimensional electromagnetic scattering problem in a two-layer medium. We prove the well-posedness of the PML truncated problem and that the PML solution converges exponentially to the exact solution as either the material parameter or the thickness of PML increases. To the best of the authors’ knowledge, this is the first theoretical work on Cartesian PML method for Maxwell’s equations in layered media.
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Ma, Xiao, Dinghui Yang, Xueyuan Huang, and Yanjie Zhou. "Nonsplit complex-frequency shifted perfectly matched layer combined with symplectic methods for solving second-order seismic wave equations — Part 1: Method." GEOPHYSICS 83, no. 6 (November 1, 2018): T301—T311. http://dx.doi.org/10.1190/geo2017-0603.1.

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The absorbing boundary condition plays an important role in seismic wave modeling. The perfectly matched layer (PML) boundary condition has been established as one of the most effective and prevalent absorbing boundary conditions. Among the existing PML-type conditions, the complex frequency shift (CFS) PML attracts considerable attention because it can handle the evanescent and grazing waves better. For solving the resultant CFS-PML equation in the time domain, one effective technique is to apply convolution operations, which forms the so-called convolutional PML (CPML). We have developed the corresponding CPML conditions with nonconstant grid compression parameter, and used its combination algorithms specifically with the symplectic partitioned Runge-Kutta and the nearly analytic SPRK methods for solving second-order seismic wave equations. This involves evaluating second-order spatial derivatives with respect to the complex stretching coordinates at the noninteger time layer. Meanwhile, two kinds of simplification algorithms are proposed to compute the composite convolutions terms contained therein.
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Zhang, Yongjie, and Xiaofeng Deng. "Parameter Design of Conformal PML Based on 2D Monostatic RCS Optimization." Applied Computational Electromagnetics Society 36, no. 6 (August 6, 2021): 726–33. http://dx.doi.org/10.47037/2020.aces.j.360614.

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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.
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