混凝土细观力学模拟的代数多重网格预条件及其并行算法研究

Dynamic performance of concrete material is one of the key techniques of the seism-induced damage mechanics study of concrete dam, and meso-scale numerical simulation is one of the effective methods. However, the related sparse linear systems during the simulation is very large, and is the most time-consuming part. At the same time, to simulate more accurately with relatively less degree of freedoms, unstructured grids are often used. To solve the large-scale linear systems derived from this type of grids efficiently, the algebraic multigrid is potentially the most powerful preconditioner. In this project, for large scale sparse linear systems from meso-scale simulation of concrete, the method to construct highly efficient smoothers based on the known information such as symmetric positive definity and elemental stiffness matrix will be studied. And the technique to construct coarse grids with both the matrix entries and the nodal coordinates will be focused on. Then, Based on the analyses of the influence of harmonization to convegence,there will propose interpolation operators which are both consistent optimally with the soomther and economic in computation complexity. The interpolation is the most important part in algebraic multigrid and will attract most of the attention in this research. Furthermore, the diagonal dominance of the coefficient matrices on each level will be enhanced with reordering techniques and simple transforms, to improve the convergence of the derived algebraic multigrid algorithm. Finally, the parallel algorithm for the derived algebraic multigrid will be designed and be implemented , and the methods and algorithms will be incorporated into the meso-scale simulation of concrete, to improve the whole efficiency of the actual numerical simulation.

混凝土动态性能是混凝土大坝地震破坏机理的关键技术之一，基于有限元离散的细观数值模拟是重要研究手段之一，但其中涉及的稀疏线性方程组非常大，求解特别耗时。同时，为利用相对更少的自由度进行更准确模拟，经常采用非结构网格。为高效求解这种网格上的大规模线性方程组，代数多重网格是最有潜力的预条件技术。 本项目针对混凝土细观数值模拟程序中的大型稀疏线性方程组，结合系数矩阵对称正定与单元刚度矩阵已知等特点研究高效光滑算子的构造方法，并采用矩阵信息与节点坐标相结合的方法进行粗网格构造方法研究，再以协调性对收敛性的影响分析为基础，研究既与光滑算子具有最优协调性又能有效控制计算量的插值算子，之后，利用重排与简单变换强化各层上系数矩阵对角优势等方法，进一步改进代数多重网格算法的有效性。最后，进行代数多重网格算法的并行算法设计与程序实现，并将研究成果集成到混凝土细观数值模拟过程中，提高实际数值模拟的效率。

本项目主要从光滑算子构造方法、系数矩阵对角优势强化、粗网格构造与校正方法等三方面进行了研究。1)在光滑算子构造方法方面，提出了多行不完全LDU分解即MRILDU技术。该技术在ILUT技术上进行了两方面的改进，其一是用不完全LDU分解取代不完全LU分解；其二是一次计算多行，之后对这些行应用舍弃策略，提取出幅度相对较大的元素。2)在系数矩阵对角优势强化技术方面，进行了多项研究。其一是针对聚集型代数多重网格，提出了一种基于ILU(0)的算子平滑方法，以改进各层上系数矩阵的对角占有性。其二是在提出的自顶向下图分割聚集算法中，引入了边权的影响，因而在进行图分割时，尽量减少子图之间的连接边之和。其三是提出了一种基于反幂法和收缩技术计算Fidler向量的新算法，以在将子图间边权之和作为限制条件的情况下，通过谱方法准确、快速而有效地进行图的分割。3)在与光滑算子协调的粗网格校正方法方面，先提出了一种基于完全子图的聚集算法，并对多种两点聚集算法、现有的知名聚集算法、以及LIS软件包进行了实验研究。在此基础上，提出了基于坐标分割的聚集算法，并通过图分割技术将其推广到一般的对称正定稀疏线性方程组。之后，对该方法的参数敏感性与有效性进行了全面的分析与研究，并给出了一种较高效的并行算法。