Low dispersive modeling of Rayleigh waves on partly staggered grids
Low dispersive modeling of Rayleigh waves on partly staggered grids
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部分交错网格上瑞利波的低色散建模
作者：
O. Rojas;B. Otero;J. Castillo;S. Day
在弹性介质中，部分交错网格 (PSG) 上自由表面 (FS) 边界条件的有限差分 (FD) 实现使用高色散真空公式 (VPSG)。 FS 边界嵌入到“真空”网格层（拉梅常数为零且密度值可忽略不计）中，其中离散运动方程允许计算表面位移。我们沿着平面 FS 放置一组新的复合（应力-位移）节点，并使用零牵引条件的单边模拟 FD 离散化进行位移计算 (MPSG)。在内部节点，MPSG 简化为标准 VPSG 方法，并沿单元对角线应用四阶中心 FD 进行交错微分，并及时与节点二阶 FD 相结合。我们对这些方法进行了兰姆问题的色散分析，并根据两个 FS 接收器处的加窗数值瑞利脉冲的相位差来估计色散曲线。对于给定的网格采样标准（例如，每个参考 S 波长 λS 有 6 个或 10 个节点），MPSG 色散误差仅为 VPSG 方法的四分之一。我们还量化了数值时间序列相对于分析波形的均方根 (RMS) 失配。当九个节点对传输中的最小 S 波长 λMINS$lambda _{ ext {MIN}}^{mathrm {S}}$ 进行采样时（沿距离 ∼$sim $145λMINS$lambda _{ ext {MIN}}^{mathrm {S}}$），MPSG RMS 失配几乎不超过 10%。在相同的测试中，VPSG RMS 失配超过 70%。我们还将 MPSG 与在标准交错网格上设计的一致的四阶模拟方法进行了比较。后者相当于前者在两倍密集的网格上的色散误差，并且仅在每个 λMINS$lambda _{ ext {MIN}}^{mathrm {S}}$ 具有 6 个或更少节点的网格上显示出更高的 RMS 精度。
In elastic media, finite-difference (FD) implementations of free-surface (FS) boundary conditions on partly staggered grid (PSG) use the highly dispersive vacuum formulation (VPSG). The FS boundary is embedded into a “vacuum” grid layer (null Lame’s constants and negligible density values) where the discretized equations of motion allow computing surface displacements. We place a new set of compound (stress-displacement) nodes along a planar FS and use unilateral mimetic FD discretization of the zero-traction conditions for displacement computation (MPSG). At interior nodes, MPSG reduces to standard VPSG methods and applies fourth-order centered FD along cell diagonals for staggered differentiation combined with nodal second-order FD in time. We perform a dispersion analysis of these methods on a Lamb’s problem and estimate dispersion curves from the phase difference of windowed numerical Rayleigh pulses at two FS receivers. For a given grid sampling criterion (e.g., six or ten nodes per reference S wavelength λS), MPSG dispersion errors are only a quarter of the VPSG method. We also quantify root-mean-square (RMS) misfits of numerical time series relative to analytical waveforms. MPSG RMS misfits barely exceed 10 % when nine nodes sample the minimum S wavelength λMINS$lambda _{ ext {MIN}}^{mathrm {S}}$ in transit (along distances ∼$sim $145λMINS$lambda _{ ext {MIN}}^{mathrm {S}}$). In same tests, VPSG RMS misfits exceed 70 %. We additionally compare MPSG to a consistently fourth-order mimetic method designed on a standard staggered grid. The latter equates the former’s dispersion errors on grids twice denser and shows higher RMS precision only on grids with six or less nodes per λMINS$lambda _{ ext {MIN}}^{mathrm {S}}$.