论文标题
快速动态平滑的自适应网格划分方案，并具有可压缩流的应用
A fast dynamic smooth adaptive meshing scheme with applications to compressible flow
论文作者
论文摘要
我们开发了一种用于动态曲线网格生成的快速运行的平滑自适应网格链（SAM）算法，该算法基于时间相关的Monge-ampère（MA）方程的快速解决方案策略，$ \ det \ det \ nablaψ（x，x，x，x，x，x，x，x，x，x，x，t）= \ mathsf {g} \ circinsf {g} \ Circin（x，x，x，x，T）$。我们方法的新颖性是MA的一种新的所谓扰动公式，该公式通过组成了参考网格的一系列近乎身份变形的序列来构建解决方案映射$ψ$。然后，我们制定了一种新版本的变形方法，该方法可导致简单，快速，高阶精确的数值方案和动态SAM算法，当应用于时间依赖性的网格生成溶液中，该算法具有最佳的复杂性，以供诸如Euler气体动力学的EULER方程（例如气体动力学的Euler方程）。我们对具有较大变形的网格进行了一系列具有挑战性的2 $ d $和3 $ d $网状生成实验，并证明SAM能够生成与最先进的求解器相当的光滑网格，而运行速度约为200倍。然后将SAM算法与2 $ d $气体动力学的简单任意拉格朗日欧拉（ALE）方案耦合。具体来说，我们实施了$ c $ - 方法，并为接触不连续性开发了新的啤酒接口跟踪算法。我们针对NOH内爆问题以及经典的雷利 - 泰勒不稳定性问题执行数值实验。结果证实，使用我们的SAM-ALE算法的低分辨率模拟与高分辨率均匀网格运行相比。
We develop a fast-running smooth adaptive meshing (SAM) algorithm for dynamic curvilinear mesh generation, which is based on a fast solution strategy of the time-dependent Monge-Ampère (MA) equation, $\det \nabla ψ(x,t) = \mathsf{G} \circψ(x,t)$. The novelty of our approach is a new so-called perturbation formulation of MA, which constructs the solution map $ψ$ via composition of a sequence of near-identity deformations of a reference mesh. Then, we formulate a new version of the deformation method that results in a simple, fast, and high-order accurate numerical scheme and a dynamic SAM algorithm that is of optimal complexity when applied to time-dependent mesh generation for solutions to hyperbolic systems such as the Euler equations of gas dynamics. We perform a series of challenging 2$D$ and 3$D$ mesh generation experiments for grids with large deformations, and demonstrate that SAM is able to produce smooth meshes comparable to state-of-the-art solvers, while running approximately 200 times faster. The SAM algorithm is then coupled to a simple Arbitrary Lagrangian Eulerian (ALE) scheme for 2$D$ gas dynamics. Specifically, we implement the $C$-method and develop a new ALE interface tracking algorithm for contact discontinuities. We perform numerical experiments for both the Noh implosion problem as well as a classical Rayleigh-Taylor instability problem. Results confirm that low-resolution simulations using our SAM-ALE algorithm compare favorably with high-resolution uniform mesh runs.
