This work concerns the implementation of the hybridizable discontinuous Galerkin (HDG) method to solve the linear anisotropic elastic equation in the frequency domain. First-order formulation with the compliance tensor and Voigt notation are employed to provide a compact description of the discretized problem and flexibility with highly heterogeneous media. We further focus on the question of optimal choice of stabilization in the definition of HDG numerical traces. For this purpose, we construct a hybridized Godunov-upwind flux for anisotropic elasticity possessing three distinct wavespeeds. This stabilization removes the need to choose scaling factors, contrary to identity and Kelvin-Christoffel based stabilizations which are popular choices in literature. We carry out comparisons among these families for isotropic and anisotropic material, with constant background and highly heterogeneous ones, in two and three dimensions. They establish the optimality of the Godunov stabilization which can be used as a reference choice for generic material and different types of waves.

翻译：本文致力于研究混合可间断伽辽金（HDG）方法在频域内求解线性各向异性弹性方程的数值实现。采用包含柔度张量和Voigt记号的一阶公式，以简洁描述离散问题并适应高度非均匀介质。我们进一步聚焦于HDG数值通量定义中最优稳定化参数的选择问题。为此，针对具有三种不同波速的各向异性弹性介质，构建了混合Godunov迎风通量。与文献中常用的基于单位矩阵和Kelvin-Christoffel矩阵的稳定化方法不同，这种稳定化方案无需选择缩放因子。我们在二维和三维空间中，对各项同性与各向异性材料、恒定背景介质与高度非均匀介质进行了系统比较。结果表明，Godunov稳定化方法具有最优性，可将其作为通用材料及不同类型波动的参考选择。