Discontinuous Galerkin modeling of 3D arbitrary anisotropic Q

Abstract

For wave propagation problems, conventional time-domain anelastic attenuation modeling involves either Caputo fractional time derivatives for an exactly constant-[Formula: see text] model, thus leading to globally temporal memory effects; or auxiliary partial differential equations (PDEs) for a nearly constant-[Formula: see text] model, thus resulting in globally spatial operators. Therefore, memory and time consumptions increase tremendously, compared with the purely elastic counterpart. Moreover, the numerical models are usually limited to isotropic or transversely isotropic attenuation, due to the ambiguity of anisotropic attenuation parameterization. Therefore, it is indispensable to investigate an efficient method, to easily incorporate the general anisotropic attenuation effects in the time domain. To tackle these problems, we have first developed a [Formula: see text]-transformation rule, via the correspondence principle, revealing the validity range for a large enough [Formula: see text] value. Then, we construct a new constitutive equation, by extending the generalized Maxwell body, from the isotropic viscoelastic media to fully anisotropic scenario, i.e., as complex as triclinic attenuation. As a result, global memory effects are effectively localized, with several anelastic functions subject to ordinary differential equations, while preserving the original governing equations. An efficient hp-adaptive discontinuous Galerkin (DG) time-domain algorithm is implemented, where the Riemann problem is exactly solved. Consequently, the extra computation cost to incorporate [Formula: see text] effects is nearly negligible. Furthermore, we derive an analytical solution for the general anisotropic attenuation to verify this DG implementation.