Stable and High-Order Accurate Boundary Treatments for the Elastic Wave Equation on Second-Order Form

Self Cite

We study the problem of two elastic half-planes in contact and the Stoneley interface wave that may exist at the interface between two different elastic materials, emphasis is put on the case when the half-planes are almost incompressible. We show that numerical simulations involving interface waves requires an unexpectedly high number of grid points per wavelength as the materials become more incompressible. Let λ, µ, ρ and λ , µ , ρ be the Lamé parameters and densities of the first and second half-plane, respectively. A theoretical study shows that if K is a real constant, λ = Kλ, µ = Kµ, ρ = Kρ and µ → 0, then for an accurate solution the required number of grid points per wavelength scales as (µ/λ) −1/p , where p is the order of accuracy of the numerical method. This requirement becomes very restrictive close to the incompressible limit µ λ, especially for lower order methods i.e., a small p. The theoretical findings are supported by numerical experiments that illustrate the demanding resolution requirement as well as the superiority of higher order methods. The scaling is also seen to hold for a more general choice of Lamé parameters. Numerical experiments when one of the half-planes is a vacuum are also presented, where the higher resolution requirement is illustrated in a numerical solution of Lamb's problem.

Section: The Numerical Methodssupporting

confidence: 79%

“…The elastic wave equation on the second order form (1) was discretized in [3]. To approximate spatial operators high order SBP operators were used.…”

Section: The Numerical Methodsmentioning

confidence: 99%

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Interface waves in almost incompressible elastic materials

Virta

Kreiss

2015

Self Cite

“…An energy estimate can be derived in the same way as in the Cartesian case, because the factor J in the scalar product cancels the 1/J on the right hand side of (36). Partial integration gives a spatial decomposition of the form (15). The only difference is that the matrices M jk , which describe the material properties in the Cartesian case, are replaced by the matrices N jk , which describe the corresponding material properties in parameter space r ∈ [0, 1] 3 .…”

Section: Boundary Conditionsmentioning

confidence: 99%

Wave propagation in anisotropic elastic materials and curvilinear coordinates using a summation-by-parts finite difference method

Petersson

Sjögreen

2015

We develop a fourth order accurate finite difference method for solving the three-dimensional elastic wave equation in general heterogeneous anisotropic materials on curvilinear grids. The proposed method is an extension of the method for isotropic materials, previously described in the paper by Sjögreen and Petersson [J. Sci. Comput. 52 (2012)]. The proposed method discretizes the anisotropic elastic wave equation in second order formulation, using a node centered finite difference method that satisfies the principle of summation by parts. The summation by parts technique results in a provably stable numerical method that is energy conserving. We also generalize and evaluate the super-grid far-field technique for truncating unbounded domains. Unlike the commonly used perfectly matched layers (PML), the super-grid technique is stable for general anisotropic material, because it is based on a coordinate stretching combined with an artificial dissipation. As a result, the discretization satisfies an energy estimate, proving that the numerical approximation is stable. We demonstrate by numerical experiments that sufficiently wide super-grid layers result in very small artificial reflections. Applications of the proposed method are demonstrated by three-dimensional simulations of anisotropic wave propagation in crystals.

“…Furthermore, the new discretization is presented for d spatial dimensions and applies to the generalized Laplace operator ∇ · b( x)∇. This paper only considers weak enforcement of boundary and interface conditions using SATs, but it is possible to combine SBP operators with strong enforcement by, for example, injection [4] or ghost points [20,31]. In fact, for certain classes of boundary and interface conditions, the ghost point technique yields a significantly smaller spectral radius than SATs [31].…”

Section: Introductionmentioning

confidence: 99%

Non-stiff boundary and interface penalties for narrow-stencil finite difference approximations of the Laplacian on curvilinear multiblock grids

Almquist

Dunham

2020

The Laplacian appears in several partial differential equations used to model wave propagation. Summation-by-parts-simultaneous approximation term (SBP-SAT) finite difference methods are often used for such equations, as they combine computational efficiency with provable stability on curvilinear multiblock grids. However, the existing SBP-SAT discretization of the Laplacian quickly becomes prohibitively stiff as grid skewness increases. The stiffness stems from the SATs that impose inter-block couplings and Dirichlet boundary conditions. We resolve this issue by deriving stable SATs whose stiffness is almost insensitive to grid skewness. The new discretization thus allows for large time steps in explicit time integrators, even on very skewed grids. It also applies to the variable-coefficient generalization of the Laplacian. We demonstrate the efficacy and versatility of the new method by applying it to acoustic wave propagation problems inspired by marine seismic exploration and infrasound monitoring of volcanoes.