Anisotropic Modeling with Geometric Multigrid Preconditioned Finite Element Method

Formation anisotropy in complicated geophysical environments can have a significant impact on data interpretation of electromagnetic surveys. To facilitate full 3D modeling of arbitrary anisotropy, we have adopted an h-version geometric multigrid preconditioned finite-element method based on vector basis functions. By using a structured mesh, instead of an unstructured one, our method can conveniently construct the restriction and prolongation operators for multigrid implementation, and then recursively coarsen the grid with the F-cycle coarsening scheme. The geometric multigrid method is used as a preconditioner for the biconjugate-gradient stabilized method to efficiently solve the linear system resulting from the finite-element method. Our method avoids the need of interpolation for arbitrary anisotropy modeling as in Yee’s grid-based finite-difference method, and it is also more capable of large-scale modeling with respect to the p-version geometric multigrid preconditioned finite-element method. A numerical example in geophysical well logging is included to demonstrate its numerical performance. Our h-version geometric multigrid preconditioned finite-element method is expected to help formation anisotropy characterization with electromagnetic surveys in complicated geophysical environments.

On the Q operator and the spectrum of the XXZ model at root of unity

The spin-\frac{1}{2}12 Heisenberg XXZ chain is a paradigmatic quantum integrable model. Although it can be solved exactly via Bethe ansatz techniques, there are still open issues regarding the spectrum at root of unity values of the anisotropy. We construct Baxter’s Q operator at arbitrary anisotropy from a two-parameter transfer matrix associated to a complex-spin auxiliary space. A decomposition of this transfer matrix provides a simple proof of the transfer matrix fusion and Wronskian relations. At root of unity a truncation allows us to construct the Q operator explicitly in terms of finite-dimensional matrices. From its decomposition we derive truncated fusion and Wronskian relations as well as an interpolation-type formula that has been conjectured previously. We elucidate the Fabricius–McCoy (FM) strings and exponential degeneracies in the spectrum of the six-vertex transfer matrix at root of unity. Using a semicyclic auxiliary representation we give a conjecture for creation and annihilation operators of FM strings for all roots of unity. We connect our findings with the `string-charge duality’ in the thermodynamic limit, leading to a conjecture for the imaginary part of the FM string centres with potential applications to out-of-equilibrium physics.

Stress and Displacement Intensity Factors of Cracks in Anisotropic Media

A relation connecting stress intensity factors (SIF) with displacement intensity factors (DIF) at the crack front is derived by solving a pseudodifferential equation connecting stress and displacement discontinuity fields for a plane crack in an elastic anisotropic medium with arbitrary anisotropy. It is found that at a particular point on the crack front, the vector valued SIF is uniquely determined by the corresponding DIF evaluated at the same point.

Full wave sensitivity of SK(K)S phases to arbitrary anisotropy in the upper and lower mantle

SUMMARY Core-refracted phases such as SKS and SKKS are commonly used to probe seismic anisotropy in the upper and lowermost portions of the Earth’s mantle. Measurements of SK(K)S splitting are often interpreted in the context of ray theory, and their frequency dependent sensitivity to anisotropy remains imperfectly understood, particularly for anisotropy in the lowermost mantle. The goal of this work is to obtain constraints on the frequency dependent sensitivity of SK(K)S phases to mantle anisotropy, particularly at the base of the mantle, through global wavefield simulations. We present results from a new numerical approach to modelling the effects of seismic anisotropy of arbitrary geometry on seismic wave propagation in global 3-D earth models using the spectral element solver AxiSEM3D. While previous versions of AxiSEM3D were capable of handling radially anisotropic input models, here we take advantage of the ability of the solver to handle the full fourth-order elasticity tensor, with 21 independent coefficients. We take advantage of the computational efficiency of the method to compute wavefields at the relatively short periods (5 s) that are needed to simulate SK(K)S phases. We benchmark the code for simple, single-layer anisotropic models by measuring the splitting (via both the splitting intensity and the traditional splitting parameters ϕ and δt) of synthetic waveforms and comparing them to well-understood analytical solutions. We then carry out a series of numerical experiments for laterally homogeneous upper mantle anisotropic models with different symmetry classes, and compare the splitting of synthetic waveforms to predictions from ray theory. We next investigate the full wave sensitivity of SK(K)S phases to lowermost mantle anisotropy, using elasticity models based on crystallographic preferred orientation of bridgmanite and post-perovskite. We find that SK(K)S phases have significant sensitivity to anisotropy at the base of the mantle, and while ray theoretical approximations capture the first-order aspects of the splitting behaviour, full wavefield simulations will allow for more accurate modelling of SK(K)S splitting data, particularly in the presence of lateral heterogeneity. Lastly, we present a cross-verification test of AxiSEM3D against the SPECFEM3D_GLOBE spectral element solver for global seismic waves in an anisotropic earth model that includes both radial and azimuthal anisotropy. A nearly perfect agreement is achieved, with a significantly lower computational cost for AxiSEM3D. Our results highlight the capability of AxiSEM3D to handle arbitrary anisotropy geometries and its potential for future studies aimed at unraveling the details of anisotropy at the base of the mantle.

Interfacial acoustic waves in one-dimensional anisotropic phononic bicrystals with a symmetric unit cell

The paper is concerned with the interfacial acoustic waves localized at the internal boundary of two different perfectly bonded semi-infinite one-dimensional phononic crystals represented by periodically layered or functionally graded elastic structures. The unit cell is assumed symmetric relative to its midplane, whereas the constituent materials may be of arbitrary anisotropy. The issue of the maximum possible number of interfacial waves per full stop band of a phononic bicrystal is investigated. It is proved that, given a fixed tangential wavenumber, the lowest stop band admits at most one interfacial wave, while an upper stop band admits up to three interfacial waves. The results obtained for the case of generally anisotropic bicrystals are specialized for the case of a symmetric sagittal plane.