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Modeling of the Perfectly Matched Layer Absorbing Boundaries and Intrinsic Attenuation in Explicit Finite-Element Methods

¹ Department of Earth Science and Institute for Crustal Studies
University of California, Santa Barbara
Santa Barbara, California 93106
 (S.M.)
² Institute for Crustal Studies
University of California, Santa Barbara
Santa Barbara, California 93106
 (P.L.)

^* Present address: Department of Geophysics, Stanford University, Panama Mall 397, Stanford, California 94305-2215.

We present an implementation of the perfectly matched layer (PML) absorbing boundary conditions and modeling of intrinsic attenuation (Q) in explicit finite-element simulations of wave propagation. The finite-element method uses one integration point and an hourglass control scheme, which leads to an easy extension of the velocity-stress implementation of PML to the finite-element method. Numerical examples using both regular and irregular elements in the PML region show excellent results: very few reflections are observed from the boundary for both body waves and surface waves—far superior to the classic first-order absorbing boundaries. The one-point integration also gives rise to an easy incorporation of the coarse-grain approach for modeling Q (Day, 1998). We implement the coarse-grain method in a structured finite-element mesh straightforwardly. We also apply the coarse-grain method to a widely used, slightly unstructured finite-element mesh, where unstructured finite elements are only used in the vertical velocity transition zones. A linear combination of eight relaxation mechanisms is used to simulate the target attenuation model over a wide frequency range. The relaxation time and weight of each relaxation mechanism are distributed in a spatially periodic manner to the center of each element. Stress relaxations caused by anelastic material response are calculated from elastic strains in the element and redistributed to the nodal forces of the element. Numerical simulation of anelastic wave propagation in a layered velocity structure with very small Qs using both the structured mesh and the unstructured mesh show excellent agreement with the analytical solutions when the viscoelastic modulus is calculated by a harmonic average over the coarse-grain unit. Our scheme greatly expands the use of PML and the coarse-grain method for modeling Q, so that these methods can be used in a versatile and efficient finite-element formulation.

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