To Stagger or Not to Stagger : Lebedev Grids for Elasticity: Algebraic Structure, Stability Analysis, and Numerical Experiments

Sammanfattning

This thesis investigates staggered finite-difference discretisations for elastic wave propagation in anisotropic media. Wave propagation in anisotropic elastic media poses a challenge for conventional staggered-grid finite-difference methods when the material stiffness matrix contains coupling between normal and shear components. This work therefore derives, analyses, and implements the elastic Lebedev grid, which is constructed to naturally address this difficulty. A preliminary two-dimensional acoustic model problem is used to motivate the staggered-grid arrangement through comparison with a centred SBP-projection discretisation. For elasticity, a binary-grid notation is introduced to formulate the Virieux and Lebedev staggered-grid schemes in a unified block-matrix framework. The summation-by-parts structure of the discrete operators is used to establish discrete energy conservation for the source-free semi-discrete system, and a stability condition is derived for the corresponding leapfrog time integration. We present numerical experiments that illustrate the progression from isotropic and VTI propagation on a conventional staggered grid to TTI and layered VTI-TTI media on the Lebedev grid. The results demonstrate the role of the Lebedev arrangement in treating rotated anisotropic stiffness matrices with normal-shear constitutive coupling.