有限元分析快速解法

基于结构分析有限元方程组的特征，提出了在刚度矩阵及其因子的超方程概念下的细胞稀疏索引存贮方案。与传统的稀疏索引存贮方案相比，它可以减少磁盘空间和内存的占用量约30％。同时，这一存贮方案也可以减少关于索引的操作.结合双向循环展开技术，发展了一种适合于多维有限元分析的快速稀疏直接静力求解方法。工程算例表明，所建议的方案在存贮量和速度方面显著地改进了直接求解法的效率。

FAST SOLUTION ALGORITHM IN FINITE ELEMENT ANALYSIS

In early nineteen seventy's, the skyline active column algorithm is considered as an efficient one for LDLT or Cholesky factorization of the positive definite sparse linear equations Ax = f. But beneath the skyline there are usually many zeros in almost each column. All these zeros occupied the storage allocations and participated operations. These zeros caused also extra I/O between the core and the secondary memory. As the size of linear equations increasing, the skyline approach encounters difficulties of storage shortage and lower efficiency. Comparing to conventional skyline and half-bandwidth storage schemes, the sparse storage scheme needs only allocations for non-zero entries of the stiffness matrix and the whole matrix is manipulated in a very smart way. Therefore sparse solver requires less disk space and is faster than conventional direct solvers with skyline or half-bandwidth storage scheme for large-scale problems. In reference 3] the super-equation was introduced to sparse matrix computation with the conventional sparse storage scheme for the factor Ll of K in order to perform one-way unrolling. In this paper, we extent the super-equation concept to the global stiffness matrix A of three dimensional FEA and proposes cell sparse storage scheme. The proposed cell sparse storage scheme provides an excellent data structure for two-way loop unrolling, to which we unrolled two outer loops of the triply nested loop in so-called JKI LDLr factorization. This essential changes led to a big improvement in efficiency and the disk space requirement. The cell sparse storage reduced about 70% of core memory as well as disk space requirement for engineering FEA in comparison with the skyline storage scheme. Even compared to conventional sparse storage scheme, 30% of memory requirement was reduced. The combination of the cell sparse technology and loop unrolling in the new solution strategy achieved very high efficiency in sense of elapsed time and disk space requirement. Eight examples showed that the proposed algorithm was 7-57 times faster than the skyline solver and the disk space requirement reduced to 1/3 for engineering FEA. The higher speed came from the following aspects: 1) the number of operations of the LDL1 factorization in the sparse storage scheme is much less than that in the skyline scheme; 2) the loop unrolling reduces the data traffic between RAM and cache, and even registers; 3) there are less indices in the cell sparse storage scheme, thus the index manipulations decreased; and 4) smaller disk and core space requirement reduces the I/O.