基于非结构网格有限差分法的扎染算法

应用有限差分法时遵循贴体坐标系下离散等价方程的离散准则，格式中仅用到当地网格点的坐标变换系数，据此可以构建新的非结构网格有限差分法，数值算例表明在空间二维情况下用连接离散点的3条网格线构造出来的一阶迎风格式可以稳定收敛。把这种非结构网格有限差分法推广到三维，在4条网格线基础上进行离散计算，然后提出一种局部区域多重网格、重复计算的新算法。首先在包含物体的整个区域采用直角坐标系下的均匀网格进行计算，然后消除物体内部点对物体外部流场区域的影响，最后在物面和直角网格之间很小的局部区域填补非结构网格进行计算，计算过程类似于中国传统的扎染工艺。处理流场内存在曲面物体边界问题时，相较于常规结构网格差分算法，扎染算法尽管在计算中包含需要特殊处理的无用网格点，也存在重复计算的局部区域，但是以这些计算能力的浪费为代价，换取了编程简便、内存量小、网格生成快捷、易于扩展网格规模等优势。定性分析表明，这种扎染算法非常适合研制大规模并行计算，建立的计算流体力学应用软件可以充分发挥数十万至百万处理器核心的超级计算机的效能。

Tie-dye algorithm based on finite difference method for unstructured grid

The application of the finite difference method follows the discretization criterion of the discrete equivalent equation in the body fitted coordinate system, and the coordinate transformation coefficient of the local grid points is used in the scheme, enabling the creation of a new finite difference method for unstructured grids. The numerical results show that the first-order upwind scheme constructed by the three-grid lines connecting discrete points can converge stably in two dimensional space. In this paper, we extend this unstructured grid finite difference method to three dimensions. The discrete calculation is performed on four lines, and a new algorithm based on multiple grids in the local area and repetitive calculation is then proposed. Firstly, the calculation is applied to the whole region, which contains the object, using the cartesian-coordinated uniform grid. The second step is to eliminate the influence on the external flow field from the internal points of the object. Finally, the unstructured grid is filled in the small local area between the object surface and the rectangle grid to conduct the calculation. This calculation process is similar to Chinese traditional tie-dye process. Compared with the structured grid finite difference method, the tie-dye algorithm contains some useless grid points which require special treatment, and some local areas are repeatedly calculated. However, its advantages include simple programming, small memory occupation, fast grid-generation, and easy grid-expansion, at the cost of computing capacity waste. Qualitative analysis shows that this tie-dye algorithm is suitable for the development of the parallel computing in a large scale. The computational fluid dynamics application based on this algorithm can fully utilize the efficiency of the supercomputers with hundred thousand to millions of processor cells.