齐次扩容精细算法

钟万勰院士创立的线性定常系统的精细算法HPD具有非常重要的工程实用价值。对于非齐次线性定常系统，钟构造了在一个积分步长内将激励项线性化的处理方法LHPD，Lin^3]等通过Fourier级数展开和寻找有解析形式的特解的方法，构造了HPD－F算法，这两种算法有一个共同点，即算法的实现需要求解系统矩阵及相关长阵的逆矩阵，数学上，也即隐含要求系统的矩阵及其相关矩阵非奇异，这样，就产生以下两个问题：1．当系统矩阵及其相关矩阵奇异时，如何设计这类动力响应问题的精细格式？2．算法的实现，需要设计高精度的矩阵求逆算法，而矩阵求逆的工作量是奶大的.本文借助齐次扩容技巧，设计了求解非齐次线性定常系统的一类新的精细算法－齐次扩容精细算法HHPD。该算法不涉及矩阵求逆运算，有效地解决 上述两个问题，并且具有设计合理，易于实现等特点，本文最后就几个典型算例，应用齐次扩容精细算法求解，与文献相比，数值结果更为理想。

Homogenization high precision direct integration

The arithmetic of HPD (High Precise Direct integration) proposed by Wanxie Zhong is very valuable in engineering. For the dynamic response of structures, Zhong established the algorithm of LHPD, in which the load is linearized within one time-step. Through combining with the Fourier method and special solution, Lin put forward the HPD-F. However, both of the algorithms require computing the inverse of the systematic matrix and its relative matrixes - in mathematics, which means, it requires the matrixes to be non-singular. Then there exit two problems: 1.when the systematic matrix or its relative matrixes is singular, how to devise the HPD? 2.the implementation of the algorithms demands to the high-precise solution to inverse matrixes. In this article, using the technology of homogenization, we devise the method of the HHPD to compute the dynamic response of structures. The HHPD is not involved the computation of the inverse matrixes, and solve successfully the above problems. Moreover, This algorithm has several advantages, such as simpler in principle, easier to generalize and implement etc. At last, the results of two examples show that HHPD is more effective.