求解一类网格结构模型的代数多重网格法

An algebraic multigrid method for a class of discrete models of lattice structures

针对网格结构离散模型的特点,设计了一类适用于求解大规模二维网格结构数值计算的代数多重网格方法,详细描述了代数多重网格方法中粗化策略与插值算子的构造,并在此基础上得到了一类以该代数多重网格为预条件子的预处理方法。数值试验表明,本文建立的代数多重网格方法及相应的预处理方法是健壮的,具有较好的数值效率,非常适合于大规模网格结构材料的数值计算。近似连续模型的建立为代数多重网格方法的可靠性和计算的准确性提供了有效的理论基础。

A new type of algebraic multigrid (AMG) methods are designed to find the solution of the discrete models possessing some special properties corresponding to lattice structures. The selection of coarser grids and the construction of interpolation operators based on element agglomeration and energy optimization are discussed in detail. The methods are suitable to the large-scale numerical computations for lattice structure of two dimension. At the same time, a preconditioned conjugate gradient method with AMG as a preconditioner (AMG-CG) is obtained. Numerical results have shown that the constructed AMG and AMG-CG methods are highly efficient and robust. In addition, the approximately continuous models of lattice structure provide some theoretical basis for the reliability of the AMG algorithm and the veracity of computational results.