流动与传热计算中的高效稳健并行代数多重网格方法研究

Parallel algebraic multigrid (AMG)is one of the most important issues in modern computational fluid dynamic and numerical heat transfer, with which it is possible to achieve the sophisticated computation of various complicated flow and heat transfer problems. Though certain developments and progresses have been made in this field, a series of problems must be solved first to realize large-scale engineering applications, for instance the urgent demand for efficient grid operator, interpolation operator and smoothing operator along with the superior grid coarsening strategy which hold scalable property. The present project aims to propose the novel grid operator, interpolation operator and smoothing operator desirable for flow and heat transfer computation based on the fundamental principles, like conservativeness, mobility and boundedness, and study their influences on the efficiency of parallel computing. For the multivariable coupling problem, such as coupling of velocity and pressure, the relationship between error attenuation and coupling smoothing operator will be studied. In addition, the present project plans to reveal the essential reason that the grid coarsening strategy affects the efficiency of parallel computing by theoretical analysis, and to put forward new grid coarsening strategy based on local information. Through comprehensive and systematic research and analysis mentioned above, the project could make desirable progress for solution of complex flow and heat transfer problems using parallel AMG, which would promote broader applications of this method in thermal science and engineering.

并行代数多重网格是当今计算流体力学和计算传热学的一个重要课题，通过采用该方法使得精细求解各种复杂流动与传热问题成为可能。尽管代数多重网格的并行计算已经取得了一定的进展，要实现其大规模工程实际应用，尚需解决一系列的难题，包括亟须提出有效的网格算子、插值算子、光顺算子和并行扩展性能优良的网格粗化策略。本项目针对并行代数多重网格中存在的上述问题开展研究，基于守恒性、迁移性和有界性等原则，提出适用于流动和传热数值计算的新的网格算子、插值算子和光顺算子，研究这些算子对并行计算效率的影响；对速度、压力等多变量耦合问题，研究误差衰减与耦合光顺算子的关系；采用理论分析方法揭示网格粗化策略影响代数多重网格并行计算效率的本质原因，在此基础上提出基于局部信息的新网格粗化原则。通过以上系统研究，在运用并行代数多重网格方法求解复杂流动和传热问题上取得重要进展，促进该方法在热流科学及工程实际中的应用。

多重网格方法通过不同疏密的网格来消除不同谐波频率的误差，克服了固定网格不能有效衰减各频率误差的缺点。由于影响多重网格方法收敛速度的因素众多，其实际加速效果并没有达到理论水平，严重限制了多重网格方法的大规模工程实际应用。. 本项目针对目前流动与传热数值计算中多重网格方法应用存在的诸多问题，开展了高效稳健的多重网格方法研究。发展了贴体坐标系下基于同位网格SIMPLE算法的几何多重网格算法，数值计算效率约提高10~100倍，该算法健壮性较高、通用性较强，可用于求解不规则计算区域上复杂流动与传热耦合问题，进一步拓宽了几何多重网格方法的适用范围；通过该方法深入研究了不规则计算区域上顶盖驱动流、自然对流及混合对流的基准解。基于有限容积法能量守恒原则，提出了几何多重网格中最优余量限定算子的确定方法，进一步提高了几何多重网格方法的加速效果。针对代数多重网格的网格粗化问题，提出了基于局部信息的整体强弱连接系数判别法，通过整体强连接系数对网格点进行二次准确粗化判断，选取的粗网格点能较好地反映问题的奇异性，保证了数值计算的物理性质不失真；发展了基于局部信息的二次粗化阈值判别法，使得网格粗化更加精确；发展了基于局部信息优先原则的网格粗化策略来精确表征网格点之间的整体强弱连接强度，弥补了经典网格粗化策略中集合势不能精确表征网格点的整体强弱连接强度进而导致网格粗化路径不唯一和计算效率不高的不足，有效优化了网格粗化路径并在一定程度上提高了求解阶段的计算效率。针对多个流动与传热问题实现了多重网格的GPU并行计算，进一步提高了多重网格方法的效率。. 在本项目的支持下，已发表国内外期刊论文和会议论文24篇，其中SCI收录12篇，EI收录3篇，国际/国内会议论文8篇；授权发明专利15项；培养研究生5名。