Algorithm of Incompressible Flow Simulation for Massively Parallel Computer in Small Granularity

当进行大规模的数值模拟时，由数百个处理器组成的分布式存储器的大规模并行计算机变得流行起来。不可压缩流的标准算法，如SIMPLE和MAC算法，即所谓的解耦算法，随着处理器数量的增加和粒度的变小，不能保持较大的并行效率值。研究表明，这种同时求解动量方程和连续性方程的耦合方法，即使在小粒度的情况下也能非常有效地进行大规模并行计算，而这种耦合方法之所以适用于大规模并行计算，是因为这种耦合方法占用的处理器间通信的开销时间很小。另一个原因是算法简单，因为主要是在控制体的每个单元中求解一个小矩阵。耦合算法的结果如下：在二维圆柱体…的计算中在网格数目为256×128的情况下，当处理器数目分别增加到256、512和1024个时，加速系数分别为235、374和491，这意味着并行算法的效率下降到92、73和48%。结果表明，即使网格点数很小，如128、64和32个，耦合方法也能获得足够的计算性能。例如，对于很小粒度的32个网格点，效率为50%，而标准SIMPLE算法的效率约为10%，在二维和三维矩形柱体的其他流动中也得到了类似的结果。最后利用该程序对三维圆柱体进行了数值模拟，证实了卡门旋涡出现在障碍物的后面，其结构与实际情况非常相似。这段代码将被开发成一个巨大的LES模拟，这是未来设计的解算器，并进一步发展成拥有超过一百万个处理器的“实时模拟器”。较少

Massively parallel computers of distributed memories which consist of more than several hundreds processors become popular when a huge scale of numerical simulations are carried out. The standard algorithm for incompressible flow such as SIMPLE and MAC,so-called "de-coupled algorithm", cannot keep the large value of parallel efficiency as the number of processors increases and the granularity becomes small. This research proposes that the coupled method, which solves momentum equations and continuity equation simultaneously, is very effective for massive parallel computing even in small granularity.The reason why this coupled method is suitable for massively parallel computing is the small occupation of the overhead time for communications between processors. The another reason is the simplicity of algorithm because the main part is to solve a small matrix in each cell of control volume.The results of coupled algorithm are as follows. In the calculations of two-dimensional circular cyl … More inder whose grid number is 256x128, the speed-up factor becomes 235,374 and 491 as the number of processors increases to 256,512 and 1024 respectively, which means that the efficiency of parallel algorithm decreases to 92,73 and 48%. It is found that the adequate performance is derived in the coupled method even if the number of grid points, that is, granularity, is very small such as 128,64 and 32. For example, the efficiency of 50% for the very small granularity of 32 grid points is surprisingly large comparing with around 10% in the standard algorithm of SIMPLE.The similar results are confirmed in the other flows of two- and three-dimensional rectangular cylinder. The final simulation of three-dimensional circular cylinder is carried out with this developed code, in which it is confirmed that the Karman vortex appears behind a obstacle and its structure is very similar to the exact one. This code will be developed to a huge simulation of LES which is a future solver for the design, and furthermore to "a real time simulator" with more than one million processors. Less