非线性动力方程精细积分级数解的并行算法

The Parallel Algorithms of Series Solution under Precise Integration for Nonlinear Dynamic Equations

非线性动力方程通过变量变换可以转化为一阶微分方程,该方程的解由表示初值影响的齐次方程解和反映荷载作用的积分之和组成.其中:第一项用指数矩阵计算;第二项在文中采用级数解计算(设计了3种相应的并行算法),算法1对级数解的每一项先做若干个向量的线性组合,再做矩阵向量乘1次;算法2与算法1原理相同,只是将矩阵的幂运算转换成乘积;算法3先做若干个矩阵向量乘,再做若干个向量的线性组合.算法1的并行效率最好,但存储空间需求大,不利于大型结构的求解.算法2、3利用动力方程的稀疏变换改善了算法1的不足,算法3中级数解每一项计算均在其前一项基础上进行,一般能比算法2节省时间.最后,给出了算例验证,三种算法都获得了较好的加速比.

Nonlinear dynamic equations can be changed into one order differential equations.The solution of the equations is composed of two parts: the homogeneous solution caused by the initial value and the particular solution caused by the loading item.The first part of computation is based on the precise solution of exponential matrix,and the second part is solved by series solution.Three parallel algorithms were presented.For every item of the series solution, the first algorithm calculates one linear combination of several vectors,and then one matrix-vector product.The second algorithm has the principle of the first algorithm,but the second algorithm changes power of matrix into product of matrix.The third algorithm calculates several matrix-vector products,and then one linear combination of several vectors.The first algorithm has the best parallel efficiency.But it needs a great deal of memory and is not benefit to large-scale question.Based on the sparse transform of dynamic equations,the second and third algorithms improve the deficiency of the first algorithm.In general,the third algorithm spends less time than the second algorithm,because the third algorithm calculates every item of series solution on the basis of its preceding(item.) Finally,these algorithms were demonstrated by a numerical example and have higher speedup.