An Efficient Unsplit Perfectly Matched Layer for Finite-Element Time-Domain Modeling of Elastodynamics in Cylindrical Co
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Pure and Applied Geophysics
An Efficient Unsplit Perfectly Matched Layer for Finite-Element Time-Domain Modeling of Elastodynamics in Cylindrical Coordinates FENG-XI ZHOU1 Abstract—The truncation of an unbounded medium into a finite domain and the elimination of reflections from artificial medium boundaries is important in numerical simulations of elastodynamic equations. The perfectly matched layer (PML) technique as a novel absorbing boundary has demonstrated very high efficiency for elastic wave equation models. Based on the stretched coordinate concept, an efficient and unsplit-field perfectly matched layer equation for elastic waves is formulated in a cylindrical coordinate system. By introducing integrated complex variables for radial direction and auxiliary functions, the PML formulation is extended in cylindrical coordinates based on the second-order elastic wave equation with displacements as basic unknowns. Finite-element time-domain modeling of the secondorder unsplit PML for the elastodynamic equation is presented, which is a standard displacement-based formulation for the computational domain including a PML region. The formulas for the special cases with 2-D axisymmetric coordinates and polar coordinates are also given. The efficiency of the proposed formulations is illustrated by a numerical example using the finite-element method. Keywords: Unsplit-field perfectly matched layer (PML), cylindrical coordinates, elastic wave equation, finite-element method.
1. Introduction In elastodynamic fields such as those involving the analysis of earthquake response or soil–structure interactions, it is necessary to solve the governing differential equations in infinite or semi-infinite regions. For many practical cases, with the exception of special cases involving simple configurations, it is difficult to solve these physical problems using
1 School of Civil Engineering, Lanzhou University of Technology, Lanzhou 730050, Gansu, China. E-mail: [email protected]
and LI-YE WANG1 analytical tools. Instead, numerical tools are generally applied. However, in the case of unbounded regions, domain-based numerical tools such as finitedifference or finite-element methods become impractical to use directly. One has to truncate the finite computational domain of interest in an infinite or semi-infinite region because of computer memory limitations and simulation time. As a result, some artificial medium boundaries need to be introduced, which can produce false reflections that influence the subsequent wave field analysis. In order to eliminate such false reflections while accurately simulating elastic wave propagation in unbounded media, two solutions have been proposed: absorbing boundary conditions and absorbing layers (Collino and Tsogka 2001). Since Engquist and Majda (1977) proposed absorbing boundary conditions for the acoustic wave equation, several absorbing boundary condition techniques have been developed; see the review papers (Givoli 1991; Tsynkov 1998; Hagstrom 1999; Givoli 2008; Higdon 1991). Absorbing layer
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