曲线网格上拉格朗日流体动力学的基于分层正交基的间断Galerkin方法
A discontinuous Galerkin method based on a hierarchical orthogonal basis for Lagrangian hydrodynamics on curvilinear grids
摘要：
提出了一种新的高阶精度拉格朗日间断伽辽金（DG）流体动力学方法来模拟材料动力学（例如，气体、流体和固体），在立方体网格上具有高达四阶精度。变量，如特定的体积，速度，特定的总能量，和变形梯度场内的细胞，表示从一个新的层次正交基构造的多项式的质心，这是由于质量矩阵是对角的解决方案的时刻。在细胞边界的多项式的不连续性是通过解决一个多方向的黎曼问题在顶点的细胞和一维黎曼问题在额外的非顶点正交点沿边缘，使表面积分是准确的多项式阶。其独特之处在于曲线网格的顶点作为DG方法曲面积分的求积点。为了确保鲁棒的网格运动，黎曼问题的压力考虑了单元上的密度变化与来自子单元网格稳定（SMS）的密度场之间的差异。新的高阶精确拉格朗日DG流体动力学方法的准确性和鲁棒性证明了通过模拟一套具有挑战性的测试问题，包括气体和固体动力学问题的弯曲网格。
英文摘要：
We present a new high-order accurate Lagrangian discontinuous Galerkin (DG) hydrodynamic method to simulate material dynamics (for e.g., gasses, fluids, and solids) with up to fourth-order accuracy on cubic meshes. The variables, such as specific volume, velocity, specific total energy, and deformation gradient fields within a cell, are represented with a polynomial constructed from a novel hierarchical orthogonal basis about the center of mass, which decouples the moments of the solution because the mass matrix is diagonal. The discontinuity in the polynomials at the cell boundary is addressed by solving a multi-directional Riemann problem at the vertices of the cell and a 1D Riemann problem at additional non-vertex quadrature points along the edges so that the surface integral is exact for the polynomial order. The uniqueness lies in that the vertices of the curvilinear grid work as the quadrature points for the surface integral of DG methods. To ensure robust mesh motion, the pressure for the Riemann problem accounts for the difference between the density variation over the cell and a density field from subcell mesh stabilization (SMS). The accuracy and robustness of the new high-order accurate Lagrangian DG hydrodynamic method is demonstrated by simulating a diverse suite of challenging test problems covering gas and solid dynamic problems on curved grids.
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