Complex Frequency Shifted PML Research Articles

Penalty Sponge Layers (PSL) for hyperbolic systems. General formulation, well-posedness and stability

In this work we design and analyze an alternative to the Absorbing Boundary Conditions (ABC) or Perfectly Matched Layers (PML) technique to solve symmetric first order hyperbolic systems in unbounded domains. We propose a general formulation of absorbing layers that we call Penalty Sponge Layer (PSL). Well-posedness as well as stability properties are quite simply inherited from the original problem. Such a formulation is written in terms of a set of auxiliary variables similar to those encountered in the Complex Frequency Shifted PML (CFS-PML) technique, but this time they have the role of a penalty term, hence the name PSL. This second role of penalty, ensured by stretching an artificial frequency to infinity in the vicinity of the interface, makes the layer asymptotically non-reflecting. Numerical tests are performed to compare them with Standard Sponge Layers (SSL) and PMLs, and to validate their success, even in severe situations where the latter fail (e.g., the Euler system at oblique subsonic/critical or supersonic flow and anisotropic waves).

Simulating 3D periodic structures at oblique incidences with discontinuous Galerkin time-domain methods: theoretical and practical considerations

In this work, we focus on the development of the use of Periodic Boundary Conditions (PBC) with sources at oblique incidence in a Discontinuous Galerkin Time Domain (DGTD) framework. Whereas in the context of the Finite Difference Time Domain (FDTD) methods, an abundant literature can be found, for DGTD, the amount of contributions reporting on such methods is remarkably low. In this paper, we supplement the existing references using the field transform technique with an analysis of the continuous system using the method of characteristics and provide an energy estimate. Furthermore, we also study the discrete stability of the resulting DGTD scheme. Additional details about sources, observables (reflectance, transmittance and diffraction efficiency), and the use of Complex Frequency-Shifted Perfectly-Matched Layers (CFS-PMLs) in this framework are also given. After numerical validations, two realistic test-cases are considered in the context of nanophotonics with the Diogenes DGTD solver (http://diogenes.inria.fr).

Efficient WLP-FDTD With Complex Frequency-Shifted PML for Super-Resolution Analysis

In this letter, an improved domain decomposition (DD) finite-difference time-domain (FDTD) method based on the weighted Laguerre polynomials (WLPs) is proposed to simulate far-field super-resolution focusing of time reversal (TR) waves. The unconditionally stable WLP-FDTD method is very suitable for solving this multiscale problem involving special microstructures in a subwavelength array. The derived higher order perfectly matched layer with auxiliary differential equations can be placed very close to walls, and thus the simulation area can be reduced accordingly. Moreover, the reverse Cuthill–Mckee scheme for matrix bandwidth compression is employed to improve the efficiency of DD-WLP-FDTD. A numerical example of TR wave propagation is included to validate the accuracy and efficiency of the improved method.

Spectral-difference parallel algorithm for the seismic forward modeling in the presence of complex topography

A spectral-difference parallel algorithm for modeling acoustic and elastic wave fields for the 2.5D geometry in the presence of irregular surface topography is considered. The initial boundary-value problem is transformed to a series of boundary-value problems for elliptic equations via the integral Laguerre transform with respect to time. For solving difference equations, it is proposed to use efficient parallel procedures based on the fast Fourier transform and the dichotomy algorithm, the latter was designed for solving systems of linear algebraic equations (SLAEs) with tridiagonal and block-tridiagonal matrices. A modification of the dichotomy algorithm for diagonally dominant matrices, which makes it possible to reduce the time of preparatory computations and increase scalability of the method relative to the number of processors, is considered. The influence of different methods of curved boundary approximation on the quality of solution is investigated; practical evaluation of accuracy is performed. Calculations of the wave field with the use of high-resolution meshes for the Canadian Foothills medium model are presented. Implementation of the complex frequency-shifted PML boundary conditions for a dynamic elasticity problem is considered in the context of the spectral-difference approach.

Novel and efficient FDTD implementation of higher-order perfectly matched layer based on ADE method

Based on the stretched coordinate perfectly matched layer (SC-PML) formulations and the auxiliary differential equation (ADE) method, an efficient and unsplit-field implementation of the higher-order PML scheme with more than one pole is proposed to truncate the finite-difference time-domain (FDTD) lattices. The higher-order PML has the advantages of both the conventional PML and the complex frequency shifted PML (CFS-PML) in terms of absorbing performances. The proposed algorithm is validated through two numerical tests carried out in three dimensional and two dimensional domains. It is shown in the numerical simulations that the proposed PML formulations with the higher-order scheme are efficient in terms of attenuating both the low-frequency propagating waves and evanescent waves and reducing late-time reflections, and also hold much better absorbing performances than the conventional SC-PML and the convolutional PML (CPML) with the CFS scheme.

Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling

The complex-frequency-shifted perfectly matched layer (CFS-PML) technique can efficiently absorb near-grazing incident waves. In seismic wave modeling, CFS-PML has been implemented by the first-order-accuracy convolutional PML technique or second-order-accuracy recursive convolution PML technique. Both use different algorithms than the numerical scheme for the interior domain to update auxiliary memory variables in the PML and thus cannot be used directly with higher-order time-marching schemes. We work with an unsplit-field CFS-PML implementation using auxiliary differential equations (ADEs) to update the auxiliary memory variables. This ADE CFS-PML results in complete first-order differential equations. Thus, the numerical scheme for the interior domain can be used to solve ADE CFS-PML equations. We have implemented ADE CFS-PML in the finite-difference time-domain method and in anonstaggered-grid finite-difference method with the fourth-order Runge-Kutta scheme, demonstrating its straightforward implementation in different numerical time-marching schemes. We have also theoretically analyzed the role of the scalingfactor of CFS-PML; it transforms the PML to a transversely isotropic material, reducing the effective wave speed normal to the PML layer and bending the wavefront toward the normal direction of the PML layer. Our numerical tests indicate that the optimal value reduces the points per dominant wavelength at the outermost boundary to three, about half the value required by the numerical scheme. We also have found that the PML equations should be derived taking the free-surface boundary condition into account in finite-difference methods. Otherwise, the free surface in the PML layer causes instability or ineffective absorption of surface waves. Tests show that we can use a narrow-slice mesh with ADE CFS-PML to simulate full wave propagation efficiently in models with complex structure.

A simple and efficient implementation of CFS-PML in the FDTD analysis of periodic structures

The use of perfectly matched layers (PMLs) is one of the most popular techniques employed for mesh truncation in finite-difference time-domain (FDTD) simulations. Its original form, however, has a poor absorption of evanescent waves, which are highly present in simulations of periodic structures. A better absorption has been achieved using a complex frequency shifted (CFS) PML, which has been implemented via a convolution scheme. In this letter, we present a simple and efficient implementation of the CFS PML and apply it to periodic structures.