矩的显式积分算法研究与应用

矩的求解通常被用于求解有限元、体积、惯性矩等问题中.基于矩的叠加性,首先给出了在三维空间中计算域的离散方式,并推导了矩的显式积分公式,随后将其推广到n维空间中,该表达式易于在计算机上实现;设计了矩的并行计算算法,并通过Fortran和Python混编的方式,实现了矩的并行计算;对多重精度下的样例数据给出了一个算例,实现了零阶矩和二阶矩的计算,并和串行算法、逐次降维算法作出比较,进行了效率分析和误差分析.结果显示,矩的显式积分并行计算算法易于程序实现,并且在效率上高于串行算法,能够很容易推广到高维空间,该算法具有高度可并行性,误差主要来自计算域离散.

Explicit moment integration algorithm and its application

The calculation of moment is often used in finite element method, volume calculation, moment of inertia calculation, etc. A discrete method of the computational domain in three-dimensional space was proposed firstly based on the superposition of moment. An explicit formula was derived in three-dimensional space and then extended to n-dimensional space, which can be easily implemented on the computer. Secondly, a parallel algorithm of moment calculation was designed and implemented with mixed Fortran and Python. Thirdly, a zero-order and second-order moment was calculated in a multi-fidelity example. The efficiency of the algorithm was compared with a serial algorithm and a successive dimensionality reduction algorithm. Meanwhile, efficiency analysis and error analysis were presented. The result shows that the explicit moment integration algorithm can be easily implemented with programs and runs faster than the serial algorithm. It is highly parallel and can also be easily extended to a higher dimensional space. The algorithm is highly parallel, whose error mainly comes from the discrete process of the computational domain.