Perfectly Matched Discrete Layers Research Articles

Perfectly matched discrete layers for modeling unbounded domains

The method of perfectly matched layers (PML) is based on the elegant idea of introducing attenuation while keeping impedance unaltered, thus resulting in effective absorption of waves without spurious reflections. Due to its simplicity and effectiveness, PML has been widely used over the past 25 years to model wave propagation in unbounded domains. However, an acknowledged challenge is that PML no longer preserves impedance once the domain and the PML region are discretized, leading to spurious reflection at the interface. This issue has been fixed with the help of so-called impedance preserving discretization, leading to the method of perfectly matched discrete layers (PMDL), developed more than a decade ago. The idea is based on the observation that midpoint integration completely eliminates the discretization error in impedance that arises from linear finite element discretization. Beyond eliminating reflections at the interface, impedance preserving discretization is shown to be crucial to achieving stability of absorbing boundary conditions in situations with differing signs of phase and group velocities, e.g., anisotropic media. This talk will be an exposition of the development of the PMDL by the author and his collaborators, staring with fundamental ideas impedance preserving discretization followed by more advanced ABCs for anisotropic and periodic media.

Dynamic soil-structure interaction analysis in time domain based on a modified version of perfectly matched discrete layers

Abstract Analysis of soil-structure interaction is commonly conducted by dividing the infinite domain of the soil into two domains: interior and exterior domains. The interior domain is bounded in a small region, while the exterior domain is replaced by artificial boundary conditions. The choice of artificial boundary conditions is a critical issue in the analysis of soil-structure interaction problems. Perfectly matched discrete layer (PMDL) has been proved as a good approach for modeling the exterior domain. In this study, a modified version of the PMDLs, i.e. PMDLs with analytical wavelengths (AW-PMDLs), is used in the soil-structure interaction analysis in time domain, which essentially can be regarded as an extension of the analysis in frequency domain, being previously proven to be effective. Numerical verifications are implemented. The results demonstrate that the proposed method performs well in the analysis of soil-structure interaction problems in time domain.

Seismic Soil–Structure Interaction Analysis of Concrete Gravity Dam Using Perfectly Matched Discrete Layers with Analytical Wavelengths

ABSTRACT For heavy structures like concrete gravity dams, soil–structure interaction has a great effect on the structural behavior. One of the important features of soil–structure interaction problems is absorbing boundary conditions for the unbounded soil domain. In this study, perfectly matched discrete layers with analytical wavelengths (AW-PMDLs), which was developed recently by the authors, is applied to concrete gravity dam analyses. Free-field motions during earthquakes with the presence of absorbing boundary conditions need to be included. A Free-field element is presented to take into account the free-field motions. The hydrodynamic pressure from the reservoir to the dam is introduced by using Westergaard added mass method. A procedure with the combination of the AW-PMDL elements, Free-field elements, and Westergaard added masses is proposed to analyze concrete gravity dams considering soil-–structure interaction. For the sake of numerical analyses, AW-PMDLs, Free-field elements, and Westergaard added masses are implemented by using user-defined elements (UEL) in ABAQUS. The validation of the proposed procedure is made by comparing the results with a previous study and the results obtained from SASSI, a program used to analyze the soil–structure interaction problems in the frequency domain. Then, the linear and nonlinear analyses of the dam system are observed for various soil stiffness. It can be seen that the nonlinear analyses need to be considered for a rigorous dam structure analysis.

Perfectly Matched Discrete Layers with Analytical Wavelengths for Soil–Structure Interaction Analysis

The importance of soil–structure interaction analysis has been proven by many researchers. It is obvious that soil media should be considered as an infinite domain to represent the radiation of waves into infinity. Perfectly matched discrete layer (PMDL) is one of the most promising methods to describe properly the infinite domain in soil media in frequency and time domains. In this research, a modified version of PMDLs that has a different strategy to determine their parameters is proposed. The method is named perfectly matched discrete layers with analytical wavelengths (AW-PMDLs). For verification of the proposed method, the dynamic compliances of strip foundations are analyzed and validated in the frequency domain. In the analyses, frequency-dependent system properties and hysteretic (material) damping are considered. The results show that the proposed procedure, AW-PMDL method, is effective for soil–structure interaction analysis in the frequency domain.

Reciprocal absorbing boundary condition with perfectly matched discrete layers for transient analysis of SV-P waves in a layered half-space

We present an extension of the combination of the Reciprocal Absorbing Boundary Condition (RABC) with Perfectly Matched Discrete Layers (PMDLs) for transient analysis of in-plane wave motion in a layered half-space. Due to the accuracy of RABC for multilayered systems, it is employed for domain truncation beyond vertical boundaries. PMDLs are particularly appropriate for handling the homogenous half-space and, herein, they are applied as excellent absorbers of waves into the half-space underlying the layered domain. Such a combination leads to solutions of problems of plane-strain wave-propagation in a layered half-space, directly in the time domain. Numerical examples are presented illustrating the accuracy, stability and effectiveness of the proposed combination. By combining two absorbing boundaries, the use of corner elements becomes unnecessary. Moreover, the long-term solution by RABCs-PMDLs is observed to be stable in the time-window of numerical examples.

Nonlinear soil-structure interaction analysis in poroelastic soil using mid-point integrated finite elements and perfectly matched discrete layers

A numerical approach for a nonlinear analysis of soil-structure interaction in poroelastic soil is proposed. Nonlinear behavior in the near-field region of soil is considered by conventional finite elements. The far-field region of soil is represented by mid-point integrated finite elements and perfectly matched discrete layers (PMDLs) in order to consider the energy radiation into infinity. The mid-point integrated finite elements can be formulated identically to conventional finite elements. Thus, PMDLs for poroelastic media are formulated in this study. A means by which to represent a layered poroelastic half-space with conventional finite elements, mid-point integrated finite elements, and the developed PMDLs is proposed. The proposed numerical approach is verified from various perspectives. The approach is applied to a nonlinear analysis of the earthquake responses of a structural system on poroelastic soil. It is demonstrated via the application that the proposed approach can be applied successfully to nonlinear dynamic soil-structure interaction problems.

Dispersion analysis of composite acousto-elastic waveguides

Abstract Propagation characteristics of guided waves are widely utilized for nondestructive characterization of laminated composites which are often immersed in fluid. These propagation characteristics are quantified through dispersion and attenuation curves, and need to be computed for a large number of estimated structure and material property combinations, over a wide range of frequencies. To solve this central problem, an efficient approach is proposed for dispersion analysis of three types of immersed waveguides: laminated plates; laminated rods and pipes; composite waveguides with generic cross-section. The approach is based on Semi-Analytical Finite Element (SAFE) method enhanced with two novel discretization methods: Complex-length Finite Element Method (CFEM) for the solid domain and Perfectly Matched Discrete Layers (PMDL) for the surrounding fluid. The resulting approach is computationally more efficient than the existing methods in that it keeps the underlying eigenvalue problem linear and substantially smaller. The paper presents the basic ideas of the proposed approach, specific algorithms for determining the discretization parameters, and open-source implementation of the resulting waveguide models. Several numerical examples are presented to illustrate the method's efficiency. Finally, the theoretical predictions from the method are validated using experimental observations for several structural members.

Reciprocal absorbing boundary condition with perfectly matched discrete layers for the time-domain propagation of SH waves in a layered half-space

In this paper, a reciprocal absorbing boundary condition (RABC) is combined with perfectly matched discrete layers (PMDLs) for the time-domain numerical analysis of SH-wave propagation in a layered half-space. The RABC is employed for representation of the layered stratum beyond the vertical consistent transmitting boundary while the PMDLs are applied in order to absorb waves into the underlying homogeneous half-space. The combination of RABC and PMDLs circumvents concerns with the treatment of layered media. The RABC handles layers without any approximation, other than discretization, while PMDLs are used exclusively for the simulation of the homogeneous medium underlying the layers.This combination leads to solution of the problem of anti-plane shear wave-propagation in a layered half-space directly in the time domain. Examples are presented demonstrating the accuracy and effectiveness of the proposed combination.

Perfectly matched discrete layers for three-dimensional nonlinear soil–structure interaction analysis

The dynamic behavior of structures established in flexible ground can be greatly influenced by soil–structure interaction. The challenge with regard to soil–structure interaction is to model the nonlinear behavior of soil in the vicinity of the foundation, including the boundary nonlinearity at the interface and the radiation of energy into infinity simultaneously. In this study, a three-dimensional time-domain formulation of perfectly matched discrete layers (PMDLs) is developed. It can be combined with a detailed finite-element model of the near-field region surrounding the foundation. The developed PMDL formulation can minimize the modeling region. A procedure to determine the parameters of the three-dimensional PMDLs to model a layered half-space effectively and accurately is proposed. Green’s functions are calculated in half-spaces with the developed method and found to agree well with existing solutions. The developed PMDL formulation is applied to a nonlinear three-dimensional soil–structure interaction analysis, confirming its capability.

A two‐level domain decomposition method with accurate interface conditions for the Helmholtz problem

SummaryA new and efficient two‐level, non‐overlapping domain decomposition (DD) method is developed for the Helmholtz equation in the two Lagrange multiplier framework. The transmission conditions are designed by utilizing perfectly matched discrete layers (PMDLs), which are a more accurate representation of the exterior Dirichlet‐to‐Neumann map than the polynomial approximations used in the optimized Schwarz method. Another important ingredient affecting the convergence of a DD method, namely, the coarse space augmentation, is also revisited. Specifically, the widely successful approach based on plane waves is modified to that based on interface waves, defined directly on the subdomain boundaries, hence ensuring linear independence and facilitating the estimation of the optimal size for the coarse problem. The effectiveness of both PMDL‐based transmission conditions and interface‐wave‐based coarse space augmentation is illustrated with an array of numerical experiments that include comprehensive scalability studies with respect to frequency, mesh size and the number of subdomains. Copyright © 2015 John Wiley & Sons, Ltd.

Efficient computation of dispersion curves for multilayered waveguides and half-spaces

Motivated by the need to compute dispersion curves for layered media in the contexts of geophysical inversion and nondestructive testing, a novel discretization approach, termed complex-length finite element method (CFEM), is developed and shown to be more efficient than the existing finite element approaches. The new approach is exponentially convergent based on two key features: unconventional stretching of the mesh into complex space and midpoint integration for evaluating the contribution matrices. For modeling the layered half-spaces of infinite depth, we couple CFEM with the method of perfectly matched discrete layers (PMDL) to minimize the errors due to mesh truncation. A number of numerical examples are used to investigate the efficiency of the proposed methods. It is shown that the suggested combination of CFEM and PMDL drastically reduces the number of elements, while requiring minor modifications to the existing finite element codes. It is concluded that the methods’ exponential convergence and sparse computation associated with linear finite elements, result in significant reduction in the overall computational cost.

Nonlinear analysis of soil–structure interaction using perfectly matched discrete layers

When nonlinear behaviors of soil are important in a soil–structure interaction system, radiation of energy into the infinity of the soil as well as the nonlinearity must be considered rigorously. In this study, perfectly matched discrete layers (PMDLs) are employed to represent the radiation of energy rigorously. A time-domain formulation for a soil–structure interaction is given using the layers. To represent a layered half-space effectively and accurately, a method to determine PMDL parameters for the half-space is proposed. It is demonstrated that the proposed PMDL system can be applied successfully to problems of nonlinear soil–structure interaction.

Efficient and accurate domain-truncation techniques for seismic soil-structure interaction

We modify the formulation of a recently developed absorbing boundary condition (ABC), the perfectly matched discrete layers (PMDL), to incorporate the excitation coming from the exterior such as earthquake waves. The modified formulation indicates that the effect of the exterior excitation can be incorporated into PMDL ABCs (traditionally designed to treat only interior excitation) simply by applying appropriate forces on the nodes connected to the first PMDL layer. Numerical results are presented to clearly illustrate the effectiveness of the proposed method.

Absorbing boundary conditions for scalar waves in anisotropic media. Part 1: Time harmonic modeling

With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general anisotropic media, we investigate the accuracy aspects of local ABCs designed for the scalar anisotropic wave equation in the frequency domain (time harmonic case). The ABC analyzed in this paper is the perfectly matched discrete layers (PMDL). PMDL is a simple variant of perfectly matched layers (PML) and is equivalent to rational approximation-based local ABCs. Specifically, we derive a sufficient condition for PMDL to accurately absorb wave modes with outgoing group velocities and this condition turns out to be a simple bound on the PMDL parameters. The reflection coefficient derived in this paper clearly reveals that the PMDL absorption is based on group velocities, and not phase velocities, and hence a PMDL can be designed to correctly identify and accurately absorb all outgoing wave modes (even those with opposing signs of phase and group velocities). The validity of the sufficient condition is demonstrated through a series of frequency domain simulations. In part 2 of this paper [S. Savadatti, M.N. Guddati, Absorbing boundary conditions for scalar waves in anisotropic media. Part 2: Time-dependent modeling, J. Comput. Phys. (2010), doi:10.1016/j.jcp.2010.05.017], the accuracy condition presented here is shown to govern both the well-posedness and accuracy aspects of PMDL designed for transient (time-dependent) modeling of scalar waves in anisotropic media.

Absorbing boundary conditions for scalar waves in anisotropic media. Part 2: Time-dependent modeling

With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general anisotropic media, we investigate the well-posedness and accuracy aspects of local ABCs designed for the transient modeling of the scalar anisotropic wave equation. The ABC analyzed in this paper is the perfectly matched discrete layers (PMDL), a simple variant of perfectly matched layers (PML) that is also equivalent to rational approximation based ABCs. Specifically, we derive the necessary and sufficient condition for the well-posedness of the initial boundary value problem (IBVP) obtained by coupling an interior and a PMDL ABC. The derivation of the reflection coefficient presented in a companion paper (S. Savadatti, M.N. Guddati, J. Comput. Phys., 2010, doi:10.1016/j.jcp.2010.05.018) has shown that PMDL can correctly identify and accurately absorb outgoing waves with opposing signs of group and phase velocities provided the PMDL layer lengths satisfy a certain bound. Utilizing the well-posedness theory developed by Kreiss for general hyperbolic IBVPs, and the well-posedness conditions for ABCs derived by Trefethen and Halpern for isotropic acoustics, we show that this bound on layer lengths also ensures well-posedness. The time discretized form of PMDL is also shown to be theoretically stable and some instability related to finite precision arithmetic is discussed.

Phonon absorbing boundary conditions for molecular dynamics

With the goal of minimizing the domain size for molecular dynamics (MD) simulations, we develop a new class of absorbing boundary conditions (ABCs) that mimic the phonon absorption properties of an unbounded exterior. The proposed MD-ABCs are extensions of perfectly matched discrete layers (PMDLs), originally developed as an absorbing boundary condition for continuous wave propagation problems. Called MD-PMDL, this extension carefully targets the absorption of phonons, the high frequency waves, whose propagation properties are completely different from continuous waves. This paper presents the derivation of MD-PMDL for general lattice systems, followed by explicit application to one-dimensional and two-dimensional square lattice systems. The accuracy of MD-PMDL for phonon absorption is proven by analyzing reflection coefficients, and demonstrated through numerical experiments. Unlike existing MD-ABCs, MD-PMDL is local in both space and time and thus more efficient. Based on their favorable properties, it is concluded that MD-PMDL could provide a more effective alternative to existing MD-ABCs.