Numerical solutions of 3D isotropic elastodynamics form the key computational kernel for many isotropic elastic reverse time migration and full-waveform inversion applications. However, real-life scenarios often require computing solutions for computational domains characterized by non-Cartesian geometry (e.g., free-surface topography). One solution strategy is to compute the elastodynamic response on vertically deformed meshes designed to incorporate irregular topology. Using a tensorial formulation, we have developed and validated a novel system of semianalytic equations governing 3D elastodynamics in a stress-velocity formulation for a family of vertically deformed meshes defined by Bézier interpolation functions between two (or more) nonintersecting surfaces. The analytic coordinate definition also leads to a corresponding analytic free-surface boundary condition (FSBC) as well as expressions for wavefield injection and extraction. Theoretical examples illustrate the utility of the tensorial approach in generating analytic equations of 3D elastodynamics and the corresponding FSBCs for scenarios involving free-surface topography. Numerical examples developed using a fully staggered grid with a mimetic finite-difference formulation demonstrate the ability to model the expected full-wavefield behavior, including complex free-surface interactions.

中文翻译：

---

各向同性介质的张量弹性动力学

3D各向同性弹性动力学的数值解形成了许多各向同性弹性逆向时间偏移和全波形反演应用程序的关键计算核心。然而，现实生活中的场景通常需要针对以非笛卡尔几何（例如，自由表面地形）为特征的计算域的计算解决方案。一种解决方案策略是在设计为包含不规则拓扑的垂直变形网格上计算弹性动力响应。我们使用张量公式开发并验证了一种应力分析公式中控制3D弹性动力学的新型半解析方程组，该公式用于两个（或多个）非相交曲面之间的贝塞尔曲线插值函数定义的一系列垂直变形网格。解析坐标定义还导致相应的解析自由表面边界条件（FSBC）以及波场注入和提取的表达式。理论示例说明了张量方法在生成3D弹性动力学解析方程和相应的FSBC时在涉及自由表面地形的场景中的实用性。使用具有模拟有限差分公式的完全交错网格开发的数值示例证明了能够对预期的全波场行为（包括复杂的自由表面相互作用）进行建模的能力。理论示例说明了张量方法在生成3D弹性动力学解析方程和相应的FSBC的实用性中的实用性，其中涉及自由表面形貌。使用具有模拟有限差分公式的完全交错网格开发的数值示例证明了能够对预期的全波场行为（包括复杂的自由表面相互作用）进行建模的能力。理论示例说明了张量方法在生成3D弹性动力学解析方程和相应的FSBC时在涉及自由表面地形的场景中的实用性。使用具有模拟有限差分公式的完全交错网格开发的数值示例证明了能够对预期的全波场行为（包括复杂的自由表面相互作用）进行建模的能力。