Application of r-grid adaptive for shock capturing in discontinuous Galerkin finite element method
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摘要:
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。
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关键词:
- 间断Galerkin (DG)有限元 /
- r型网格自适应 /
- 驱动力 /
- Venkatakrishnan 限制器 /
- 激波
Abstract: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.
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图 1 r型网格自适应网格点移动位移计算流程
Figure 1. Calculation process of grid point displacement in r-grid adaptive
图 4 不同方法压力分布比较
Figure 4. Comparison of pressure distribution between different methods
图 5 NACA0012翼型初始计算网格及流场
Figure 5. Initial computational grid and flowfield of NACA0012 airfoil
图 6 NACA0012翼型第1次自适应后计算网格及流场
Figure 6. Computational grid and flowfield of NACA0012 airfoil after first r-adaptation
图 7 NACA0012翼型第3次自适应后计算网格及流场
Figure 7. Computational grid and flowfield of NACA0012 airfoil after third r-adaptation
图 8 NACA0012翼型第5次自适应后计算网格及流场
Figure 8. Computational grid and flowfield of NACA0012 airfoil after fifth r-adaptation
图 9 NACA0012翼型第11次自适应后计算网格及流场
Figure 9. Computational grid and flowfield of NACA0012 airfoil after eleventh r-adaptation
图 12 z=0.25 m处压力分布及激波处局部放大图
Figure 12. Pressure distribution at section z=0.25 m and local enlarged view of shock wave
图 13 z=0.068 m处压力分布及激波处局部放大图
Figure 13. Pressure distribution at section z=0.068 m and local enlarged view of shock wave
图 14 自适应过程阻力系数随计算步数变化曲线
Figure 14. Curve of drag coefficient varying with iteration steps in adaptive process
图 15 阻力系数及限制器系数平均值随自适应步数变化
Figure 15. Resistance and average value of limitation coefficients with adaptive steps
图 16 三维双半球-圆柱初始计算网格及流场
Figure 16. Initial computational grid and flowfield of 3D double hemispherical cylinder
图 17 双半球-圆柱第2次自适应计算网格及流场
Figure 17. Computational grid and flowfield of double hemispherical cylinder after second r-adaptation
图 18 双半球-圆柱第4次自适应计算网格及流场
Figure 18. Computational grid and flowfield of double hemispherical cylinder after fourth r-adaptation
图 19 双半球-圆柱初始x=0.22 m处空间网格及流场
Figure 19. Spatial grid and flowfield at x=0.22 m section of double hemispherial cylinder
图 20 双半球-圆柱第2次自适应后x=0.22 m处空间网格及流场
Figure 20. Spatial grid and flowfield at x=0.22 m section of double hemispherial cylinder after second r-adaptation
图 21 双半球-圆柱第4次自适应后x=0.22 m处空间网格及流场
Figure 21. Spatial grid and flowfield at x=0.22 m section of double hemispherial cylinder after fourth r-adaptation
图 22 自适应前后对称面z=0.04 m截面压力分布
Figure 22. Pressure distribution at z=0.04 m section of symmetry plane before and after adaptation
图 23 自适应前后下部圆柱y=0截面表面压力分布
Figure 23. Surface pressure distribution of lower cylinder at y=0 section before and after adaptation
图 24 自适应前后上部圆柱y=0截面表面压力分布
Figure 24. Surface pressure distribution of upper cylinder at y=0 section before and after adaptation
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