r型网格自适应在间断Galerkin有限元激波捕捉中的应用

间断Galerkin（DG）有限元方法因计算精度高、适用于非结构网格等特点得到广泛研究和应用，其在数值模拟包含强间断流场时存在残差收敛性和计算鲁棒性差问题，均匀分布的网格加剧这一问题并影响激波分辨率。针对该问题，发展了r型网格自适应方法，实现间断Galerkin有限元数值模拟过程中网格自适应加密。基于网格点归一化的压力值作为r型网格自适应中网格点移动驱动力的重要权值，并将网格自适应后的网格点位移变化量与网格点之间的初始位移之比作为驱动力的另一重要权值，实现网格沿激波方向各向异性自适应加密，并且激波附近网格点的相邻网格点同步向激波方向移动。发展了适合间断Galerkin有限元方法的Venkatakrishnan限制器。并列NACA0012翼型超声速算例及三维并列圆柱相互干扰算例结果表明:基于r型网格自适应的间断Galerkin有限元方法能够清晰锐利捕捉激波，提高模拟精度，具有良好的收敛性和鲁棒性。 

Application of r-grid adaptive for shock capturing in discontinuous Galerkin finite element method

The discontinuous Galerkin (DG) method has been widely studied and applied because of its high-order accuracy and applicability to the unstructured grid. However, it still has problems such as poor convergence and limited robustness in numerical simulation flowfield with strong discontinuity. This problem is exacerbated by the uniformly distributed grid, which results in poor shock resolution. In order to solve the problem, an r-grid adaptive method was developed to aggregation and refinement grids in process of DG numerical simulation. The normalized pressure of grid points was taken as an important weight to calculate the driving force of grid points. At the same time, the ratio of the displacement variation of grid points to the initial distance between grid points was taken as another important weight. A Venkatakrishnan limiter suitable for DG was developed. Numerical results of interaction between two parallel NACA0012 airfoils and interaction between two parallel cylinders showed that the DG method based on r-grid adaptation can capture shock clearly and sharply, improve simulation accuracy, and has good convergence and robustness.