
科学技术与工程

中国技术经济学会
考虑约束的CSC格式总体矩阵存储算法
CSC Format Global Matrix Storage Algorithm Considering Constraints
为了提高有限元分析效率,减少内存消耗,对总体矩阵的存储算法进行研究。将约束条件统一处理为单元层面的矩阵分块,后续以刚度矩阵为例,确定了约束条件下刚度矩阵非零元素分布规律,并提出了适用于二维、三维有限元刚度矩阵非零元素数量计算方法及相应计算公式,同时检验了正确性;推导了一维等带宽存储与原始方阵地址对应关系,并将其作为辅助数组应用于CSC格式总体矩阵集成过程中,提高了总体矩阵组集效率,避免了节点编号优化难题。最后,以北京市房山线长稻区间四跨单线梁桥验证所提出算法的正确性与实用性,结果表明:该算法分析时总体矩阵组集时间减少30%、存储空间节省68%以上、保证了计算精度的同时线性方程组计算效率提升超过71%。
In order to improve the efficiency of finite element analysis and reduce memory consumption, the storage algorithm of the overall matrix was studied. The constraints were uniformly processed as matrix partitions at the element level. Subsequently, taking the stiffness matrix as an example, the distribution pattern of non-zero elements under the constrained condition was determined. A calculation method and corresponding formulas suitable for the number of non-zero elements in the stiffness matrix of two-dimensional and three-dimensional finite elements were proposed, and the correctness was verified. The correspondence between one-dimensional equal bandwidth storage and the original square matrix address was deduced, and it was applied as an auxiliary array in the process of integrating the overall matrix in CSC format, which has improved the efficiency of the overall matrix assembly and avoided the difficulty of node numbering optimization. Finally, the correctness and practicability of the proposed algorithm were verified by using thefourspan single-line girder bridge in the Changdao section of the Fangshan Line in Beijing. The results show that the overall matrix assembly time is reduced by 30%, the storage space is saved by more than 68%, and the calculation efficiency of linear equations is improved by more than 71% while ensuring the calculation accuracy.
有限单元法 / 刚度矩阵 / 非零元素 / 压缩存储 / Cholesky分解
finite element method / stiffness matrix / non-zero elements / compressed storage / cholesky decomposition
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